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https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | simp [fittingSphereT, fittingSphereConvex, optimal, feasible] at h_opt ⊢ | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt : (fittingSphereConvex n m x).optimal (c, t)
⊢ (fittingSphereT n m x).optimal (c, t) | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
⊢ 0 ≤ t + ‖c‖ ^ 2 ∧
∀ (a : Fin n → ℝ) (b : ℝ),
0 ≤ b + ‖a‖ ^ 2 →
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | constructor | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
⊢ 0 ≤ t + ‖c‖ ^ 2 ∧
∀ (a : Fin n → ℝ) (b : ℝ),
0 ≤ b + ‖a‖ ^ 2 →
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2) | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
⊢ 0 ≤ t + ‖c‖ ^ 2
case right
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
⊢ ∀ (a : Fin n → ℝ) (b : ℝ),
0 ≤ b + ‖a‖ ^ 2 →
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | let a := Vec.norm x ^ 2 - 2 * mulVec x c | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
⊢ 0 ≤ t + ‖c‖ ^ 2 | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
⊢ 0 ≤ t + ‖c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | have h_ls : optimal (leastSquaresVec a) t := by
refine ⟨trivial, ?_⟩
intros y _
simp [objFun, leastSquaresVec]
exact h_opt c y | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
⊢ 0 ≤ t + ‖c‖ ^ 2 | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
⊢ 0 ≤ t + ‖c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | have h_t_eq := leastSquaresVec_optimal_eq_mean hm a t h_ls | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
⊢ 0 ≤ t + ‖c‖ ^ 2 | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
⊢ 0 ≤ t + ‖c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | have h_c2_eq : ‖c‖ ^ 2 = (1 / m) * ∑ i : Fin m, ‖c‖ ^ 2 := by
simp [sum_const]
field_simp | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
⊢ 0 ≤ t + ‖c‖ ^ 2 | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ 0 ≤ t + ‖c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | have h_t_add_c2_eq : t + ‖c‖ ^ 2 = (1 / m) * ∑ i, ‖(x i) - c‖ ^ 2 := by
rw [h_t_eq]; dsimp [mean]
rw [h_c2_eq, mul_sum, mul_sum, mul_sum, ← sum_add_distrib]
congr; funext i; rw [← mul_add]
congr; simp [Vec.norm]
rw [norm_sub_sq (𝕜 := ℝ) (E := Fin n → ℝ)]
simp [a]; congr | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ 0 ≤ t + ‖c‖ ^ 2 | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
⊢ 0 ≤ t + ‖c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | rw [← rpow_two, h_t_add_c2_eq] | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
⊢ 0 ≤ t + ‖c‖ ^ 2 | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
⊢ 0 ≤ 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | apply mul_nonneg (by norm_num) | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
⊢ 0 ≤ 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2 | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
⊢ 0 ≤ ∑ i : Fin m, ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | apply sum_nonneg | case left
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
⊢ 0 ≤ ∑ i : Fin m, ‖x i - c‖ ^ 2 | case left.h
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
⊢ ∀ i ∈ univ, 0 ≤ ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | intros i _ | case left.h
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
⊢ ∀ i ∈ univ, 0 ≤ ‖x i - c‖ ^ 2 | case left.h
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
i : Fin m
a✝ : i ∈ univ
⊢ 0 ≤ ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | exact sq_nonneg _ | case left.h
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
i : Fin m
a✝ : i ∈ univ
⊢ 0 ≤ ‖x i - c‖ ^ 2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | refine ⟨trivial, ?_⟩ | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
⊢ (leastSquaresVec a).optimal t | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
⊢ ∀ (y : ℝ), (leastSquaresVec a).feasible y → (leastSquaresVec a).objFun t ≤ (leastSquaresVec a).objFun y |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | intros y _ | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
⊢ ∀ (y : ℝ), (leastSquaresVec a).feasible y → (leastSquaresVec a).objFun t ≤ (leastSquaresVec a).objFun y | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
y : ℝ
a✝ : (leastSquaresVec a).feasible y
⊢ (leastSquaresVec a).objFun t ≤ (leastSquaresVec a).objFun y |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | simp [objFun, leastSquaresVec] | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
y : ℝ
a✝ : (leastSquaresVec a).feasible y
⊢ (leastSquaresVec a).objFun t ≤ (leastSquaresVec a).objFun y | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
y : ℝ
a✝ : (leastSquaresVec a).feasible y
⊢ Vec.sum ((a - Vec.const m t) ^ 2) ≤ Vec.sum ((a - Vec.const m y) ^ 2) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | exact h_opt c y | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
y : ℝ
a✝ : (leastSquaresVec a).feasible y
⊢ Vec.sum ((a - Vec.const m t) ^ 2) ≤ Vec.sum ((a - Vec.const m y) ^ 2) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | simp [sum_const] | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
⊢ ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2 | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
⊢ ‖c‖ ^ 2 = (↑m)⁻¹ * (↑m * ‖c‖ ^ 2) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | field_simp | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
⊢ ‖c‖ ^ 2 = (↑m)⁻¹ * (↑m * ‖c‖ ^ 2) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | rw [h_t_eq] | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2 | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ mean a + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | dsimp [mean] | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ mean a + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2 | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ 1 / ↑m * ∑ i : Fin m, a i + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | rw [h_c2_eq, mul_sum, mul_sum, mul_sum, ← sum_add_distrib] | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ 1 / ↑m * ∑ i : Fin m, a i + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2 | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ ∑ x : Fin m, (1 / ↑m * a x + 1 / ↑m * ‖c‖ ^ 2) = ∑ i : Fin m, 1 / ↑m * ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | congr | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ ∑ x : Fin m, (1 / ↑m * a x + 1 / ↑m * ‖c‖ ^ 2) = ∑ i : Fin m, 1 / ↑m * ‖x i - c‖ ^ 2 | case e_f
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ (fun x => 1 / ↑m * a x + 1 / ↑m * ‖c‖ ^ 2) = fun i => 1 / ↑m * ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | funext i | case e_f
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
⊢ (fun x => 1 / ↑m * a x + 1 / ↑m * ‖c‖ ^ 2) = fun i => 1 / ↑m * ‖x i - c‖ ^ 2 | case e_f.h
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ 1 / ↑m * a i + 1 / ↑m * ‖c‖ ^ 2 = 1 / ↑m * ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | rw [← mul_add] | case e_f.h
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ 1 / ↑m * a i + 1 / ↑m * ‖c‖ ^ 2 = 1 / ↑m * ‖x i - c‖ ^ 2 | case e_f.h
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ 1 / ↑m * (a i + ‖c‖ ^ 2) = 1 / ↑m * ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | congr | case e_f.h
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ 1 / ↑m * (a i + ‖c‖ ^ 2) = 1 / ↑m * ‖x i - c‖ ^ 2 | case e_f.h.e_a
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ a i + ‖c‖ ^ 2 = ‖x i - c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | rw [norm_sub_sq (𝕜 := ℝ) (E := Fin n → ℝ)] | case e_f.h.e_a
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ a i + ‖c‖ ^ 2 = ‖x i - c‖ ^ 2 | case e_f.h.e_a
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ a i + ‖c‖ ^ 2 = ‖x i‖ ^ 2 - 2 * RCLike.re ⟪x i, c⟫_ℝ + ‖c‖ ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | simp [a] | case e_f.h.e_a
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ a i + ‖c‖ ^ 2 = ‖x i‖ ^ 2 - 2 * RCLike.re ⟪x i, c⟫_ℝ + ‖c‖ ^ 2 | case e_f.h.e_a
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ Vec.norm x i ^ 2 - 2 * (x *ᵥ c) i = ‖x i‖ ^ 2 - 2 * ⟪x i, c⟫_ℝ |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | congr | case e_f.h.e_a
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
i : Fin m
⊢ Vec.norm x i ^ 2 - 2 * (x *ᵥ c) i = ‖x i‖ ^ 2 - 2 * ⟪x i, c⟫_ℝ | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | norm_num | n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
a : Fin m → ℝ := Vec.norm x ^ 2 - 2 * x *ᵥ c
h_ls : (leastSquaresVec a).optimal t
h_t_eq : t = mean a
h_c2_eq : ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖c‖ ^ 2
h_t_add_c2_eq : t + ‖c‖ ^ 2 = 1 / ↑m * ∑ i : Fin m, ‖x i - c‖ ^ 2
⊢ 0 ≤ 1 / ↑m | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | intros c' x' _ | case right
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
⊢ ∀ (a : Fin n → ℝ) (b : ℝ),
0 ≤ b + ‖a‖ ^ 2 →
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2) | case right
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
c' : Fin n → ℝ
x' : ℝ
a✝ : 0 ≤ x' + ‖c'‖ ^ 2
⊢ Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c' - Vec.const m x') ^ 2) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/FittingSphere.lean | FittingSphere.optimal_convex_implies_optimal_t | [157, 1] | [191, 22] | exact h_opt c' x' | case right
n m : ℕ
x : Fin m → Fin n → ℝ
hm : 0 < m
c : Fin n → ℝ
t : ℝ
h_opt :
∀ (a : Fin n → ℝ) (b : ℝ),
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ a - Vec.const m b) ^ 2)
c' : Fin n → ℝ
x' : ℝ
a✝ : 0 ≤ x' + ‖c'‖ ^ 2
⊢ Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c - Vec.const m t) ^ 2) ≤
Vec.sum ((Vec.norm x ^ 2 - 2 * x *ᵥ c' - Vec.const m x') ^ 2) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.aₚ_nonneg | [45, 1] | [47, 22] | unfold aₚ | a b : ℝ
⊢ 0 ≤ aₚ | a b : ℝ
⊢ 0 ≤ 5e-2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.aₚ_nonneg | [45, 1] | [47, 22] | norm_num | a b : ℝ
⊢ 0 ≤ 5e-2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.bₚ_nonneg | [52, 1] | [53, 22] | unfold bₚ | a b : ℝ
⊢ 0 ≤ bₚ | a b : ℝ
⊢ 0 ≤ 0.65 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.bₚ_nonneg | [52, 1] | [53, 22] | norm_num | a b : ℝ
⊢ 0 ≤ 0.65 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.bₚ_lt_one | [55, 1] | [56, 22] | unfold bₚ | a b : ℝ
⊢ bₚ < 1 | a b : ℝ
⊢ 0.65 < 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.bₚ_lt_one | [55, 1] | [56, 22] | norm_num | a b : ℝ
⊢ 0.65 < 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.one_sub_bₚ_nonneg | [58, 1] | [60, 22] | unfold bₚ | a b : ℝ
⊢ 0 ≤ 1 - bₚ | a b : ℝ
⊢ 0 ≤ 1 - 0.65 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.one_sub_bₚ_nonneg | [58, 1] | [60, 22] | norm_num | a b : ℝ
⊢ 0 ≤ 1 - 0.65 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Finset.one_add_prod_le_prod_one_add | [19, 1] | [32, 37] | 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 }
_ ≤ ∑ t : Finset n, (∏ _a : n in t, 1 : ℝ) * ∏ a : n in univ \ t, f a := by
{ convert @Finset.sum_le_univ_sum_of_nonneg (Finset n) ℝ _ _ _ _ _
simp [hf, prod_nonneg] }
_ = ∏ i, (1 + f i) := by
{ rw [prod_add, powerset_univ] } | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ 1 + ∏ i : n, f i ≤ ∏ i : n, (1 + f i) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Finset.one_add_prod_le_prod_one_add | [19, 1] | [32, 37] | 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 }
_ ≤ ∑ t : Finset n, (∏ _a : n in t, 1 : ℝ) * ∏ a : n in univ \ t, f a := by
{ convert @Finset.sum_le_univ_sum_of_nonneg (Finset n) ℝ _ _ _ _ _
simp [hf, prod_nonneg] }
_ = ∏ i, (1 + f i) := by
{ rw [prod_add, powerset_univ] } | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ 1 + ∏ i : n, f i ≤ ∏ i : n, (1 + f i) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Finset.one_add_prod_le_prod_one_add | [19, 1] | [32, 37] | simp | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ 1 + ∏ i : n, f i = (∏ _a : n, 1) * ∏ a ∈ univ \ univ, f a + (∏ a ∈ ∅, 1) * ∏ a ∈ univ \ ∅, f a | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Finset.one_add_prod_le_prod_one_add | [19, 1] | [32, 37] | rw [Finset.sum_pair] | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ (∏ _a : n, 1) * ∏ a ∈ univ \ univ, f a + (∏ a ∈ ∅, 1) * ∏ a ∈ univ \ ∅, f a =
∑ x ∈ {univ, ∅}, (∏ _a ∈ x, 1) * ∏ a ∈ univ \ x, f a | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ univ ≠ ∅ |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Finset.one_add_prod_le_prod_one_add | [19, 1] | [32, 37] | simp | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ univ ≠ ∅ | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ¬univ = ∅ |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Finset.one_add_prod_le_prod_one_add | [19, 1] | [32, 37] | exact Finset.univ_nonempty.ne_empty | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ¬univ = ∅ | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Finset.one_add_prod_le_prod_one_add | [19, 1] | [32, 37] | convert @Finset.sum_le_univ_sum_of_nonneg (Finset n) ℝ _ _ _ _ _ | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∑ x ∈ {univ, ∅}, (∏ _a ∈ x, 1) * ∏ a ∈ univ \ x, f a ≤ ∑ t : Finset n, (∏ _a ∈ t, 1) * ∏ a ∈ univ \ t, f a | case convert_3
n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∀ (x : Finset n), 0 ≤ (∏ _a ∈ x, 1) * ∏ a ∈ univ \ x, f a |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Finset.one_add_prod_le_prod_one_add | [19, 1] | [32, 37] | simp [hf, prod_nonneg] | case convert_3
n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∀ (x : Finset n), 0 ≤ (∏ _a ∈ x, 1) * ∏ a ∈ univ \ x, f a | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Finset.one_add_prod_le_prod_one_add | [19, 1] | [32, 37] | rw [prod_add, powerset_univ] | n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∑ t : Finset n, (∏ _a ∈ t, 1) * ∏ a ∈ univ \ t, f a = ∏ i : n, (1 + f i) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.IsHermitian.eigenvectorMatrix_inv_mul | [46, 1] | [47, 41] | apply Basis.toMatrix_mul_toMatrix_flip | n : Type u_1
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : LinearOrder n
inst✝² : LocallyFiniteOrderBot n
𝕜 : Type u_2
inst✝¹ : DecidableEq 𝕜
inst✝ : RCLike 𝕜
A : Matrix n n 𝕜
hA : A.IsHermitian
⊢ hA.eigenvectorMatrixInv * hA.eigenvectorMatrix = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.IsHermitian.spectral_theorem'' | [50, 1] | [54, 67] | rw [conjTranspose_eigenvectorMatrix, Matrix.mul_assoc, ← spectral_theorem,
← Matrix.mul_assoc, eigenvectorMatrix_mul_inv, Matrix.one_mul] | n : Type u_1
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : LinearOrder n
inst✝² : LocallyFiniteOrderBot n
𝕜 : Type u_2
inst✝¹ : DecidableEq 𝕜
inst✝ : RCLike 𝕜
A : Matrix n n 𝕜
hA : A.IsHermitian
⊢ hA.eigenvectorMatrix * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * hA.eigenvectorMatrix.conjTranspose = A | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.sqrt_mul_sqrt | [63, 1] | [78, 11] | simp [IsHermitian.sqrt, Matrix.mul_assoc] | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
⊢ ⋯.sqrt * ⋯.sqrt =
⋯.eigenvectorMatrix *
((diagonal fun i => (⋯.eigenvalues i).sqrt) * (⋯.eigenvectorMatrix.transpose * ⋯.eigenvectorMatrix) *
diagonal fun i => (⋯.eigenvalues i).sqrt) *
⋯.eigenvectorMatrix.transpose | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.sqrt_mul_sqrt | [63, 1] | [78, 11] | rw [← conjTranspose_eq_transpose, hA.1.conjTranspose_eigenvectorMatrix,
hA.1.eigenvectorMatrix_inv_mul, Matrix.mul_one, diagonal_mul_diagonal,
← hA.1.conjTranspose_eigenvectorMatrix] | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
⊢ ⋯.eigenvectorMatrix *
((diagonal fun i => (⋯.eigenvalues i).sqrt) * (⋯.eigenvectorMatrix.transpose * ⋯.eigenvectorMatrix) *
diagonal fun i => (⋯.eigenvalues i).sqrt) *
⋯.eigenvectorMatrix.transpose =
A | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
⊢ (⋯.eigenvectorMatrix * diagonal fun i => (⋯.eigenvalues i).sqrt * (⋯.eigenvalues i).sqrt) *
⋯.eigenvectorMatrix.conjTranspose =
A |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.sqrt_mul_sqrt | [63, 1] | [78, 11] | convert hA.1.spectral_theorem'' | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
⊢ (⋯.eigenvectorMatrix * diagonal fun i => (⋯.eigenvalues i).sqrt * (⋯.eigenvalues i).sqrt) *
⋯.eigenvectorMatrix.conjTranspose =
A | case h.e'_2.h.e'_5.h.e'_6.h.e'_5.h
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
x✝ : n
⊢ (⋯.eigenvalues x✝).sqrt * (⋯.eigenvalues x✝).sqrt = (RCLike.ofReal ∘ ⋯.eigenvalues) x✝ |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.sqrt_mul_sqrt | [63, 1] | [78, 11] | rw [← Real.sqrt_mul (hA.eigenvalues_nonneg _), Real.sqrt_mul_self (hA.eigenvalues_nonneg _)] | case h.e'_2.h.e'_5.h.e'_6.h.e'_5.h
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
x✝ : n
⊢ (⋯.eigenvalues x✝).sqrt * (⋯.eigenvalues x✝).sqrt = (RCLike.ofReal ∘ ⋯.eigenvalues) x✝ | case h.e'_2.h.e'_5.h.e'_6.h.e'_5.h
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
x✝ : n
⊢ ⋯.eigenvalues x✝ = (RCLike.ofReal ∘ ⋯.eigenvalues) x✝ |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.sqrt_mul_sqrt | [63, 1] | [78, 11] | simp | case h.e'_2.h.e'_5.h.e'_6.h.e'_5.h
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
x✝ : n
⊢ ⋯.eigenvalues x✝ = (RCLike.ofReal ∘ ⋯.eigenvalues) x✝ | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.IsHermitian.one_add | [85, 1] | [86, 55] | dsimp [IsHermitian] | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ (1 + A).IsHermitian | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ (1 + A).conjTranspose = 1 + A |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.IsHermitian.one_add | [85, 1] | [86, 55] | rw [IsHermitian.add _ hA] | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ (1 + A).conjTranspose = 1 + A | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ IsHermitian 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.IsHermitian.one_add | [85, 1] | [86, 55] | simp | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ IsHermitian 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosDef.PosDef_sqrt | [95, 1] | [102, 68] | unfold IsHermitian.sqrt | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.sqrt.PosDef | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ((⋯.eigenvectorMatrix * diagonal fun i => (⋯.eigenvalues i).sqrt) * ⋯.eigenvectorMatrix.transpose).PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosDef.PosDef_sqrt | [95, 1] | [102, 68] | refine'
PosDef.conjTranspose_mul_mul _ (hA.1.eigenvectorMatrixᵀ)
(PosDef_diagonal (fun i => Real.sqrt_pos.2 (hA.eigenvalues_pos i))) _ | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ((⋯.eigenvectorMatrix * diagonal fun i => (⋯.eigenvalues i).sqrt) * ⋯.eigenvectorMatrix.transpose).PosDef | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.eigenvectorMatrix.transpose.det ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosDef.PosDef_sqrt | [95, 1] | [102, 68] | rw [det_transpose] | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.eigenvectorMatrix.transpose.det ≠ 0 | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.eigenvectorMatrix.det ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosDef.PosDef_sqrt | [95, 1] | [102, 68] | apply det_ne_zero_of_right_inverse hA.1.eigenvectorMatrix_mul_inv | n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.eigenvectorMatrix.det ≠ 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | refine' ⟨PosDef.det_ne_zero, _⟩ | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
⊢ M.PosDef ↔ M.det ≠ 0 | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
⊢ M.det ≠ 0 → M.PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | intro hdet | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
⊢ M.det ≠ 0 → M.PosDef | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
⊢ M.PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | refine' ⟨hM.1, _⟩ | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
⊢ M.PosDef | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
⊢ ∀ (x : n → ℝ), x ≠ 0 → 0 < star x ⬝ᵥ M.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | intros x hx | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
⊢ ∀ (x : n → ℝ), x ≠ 0 → 0 < star x ⬝ᵥ M.mulVec x | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | apply lt_of_le_of_ne' (hM.2 x) | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M.mulVec x | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ star x ⬝ᵥ M.mulVec x ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | 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] | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ star x ⬝ᵥ M.mulVec x ≠ 0 | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ ⋯.sqrt.mulVec (star x) ⬝ᵥ ⋯.sqrt.mulVec x ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | simp only [RCLike.re_to_real, star, id] | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ ⋯.sqrt.mulVec (star x) ⬝ᵥ ⋯.sqrt.mulVec x ≠ 0 | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ (⋯.sqrt.mulVec fun i => x i) ⬝ᵥ ⋯.sqrt.mulVec x ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | change @inner ℝ (EuclideanSpace ℝ _) _ (hM.1.sqrt.mulVec x) (hM.1.sqrt.mulVec x) ≠ 0 | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ (⋯.sqrt.mulVec fun i => x i) ⬝ᵥ ⋯.sqrt.mulVec x ≠ 0 | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | intro hinner | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ ≠ 0 | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ False |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | have sqrtMdet0 : hM.1.sqrt.det = 0 := by
refine' exists_mulVec_eq_zero_iff.1 ⟨x, hx, _⟩
rw [inner_self_eq_zero.1 hinner] | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ False | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
sqrtMdet0 : ⋯.sqrt.det = 0
⊢ False |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | rw [← hM.sqrt_mul_sqrt, det_mul, sqrtMdet0, mul_zero] at hdet | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
sqrtMdet0 : ⋯.sqrt.det = 0
⊢ False | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : 0 ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
sqrtMdet0 : ⋯.sqrt.det = 0
⊢ False |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | apply hdet rfl | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : 0 ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
sqrtMdet0 : ⋯.sqrt.det = 0
⊢ False | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | refine' exists_mulVec_eq_zero_iff.1 ⟨x, hx, _⟩ | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ ⋯.sqrt.det = 0 | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ ⋯.sqrt.mulVec x = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.PosSemidef.PosDef_iff_det_ne_zero | [104, 1] | [119, 17] | rw [inner_self_eq_zero.1 hinner] | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ ⋯.sqrt.mulVec x = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | let sqrtA := hA.1.sqrt | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
⊢ A.det + B.det ≤ (A + B).det | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
⊢ A.det + B.det ≤ (A + B).det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | have isUnit_det_sqrtA :=
isUnit_iff_ne_zero.2 hA.PosDef_sqrt.det_ne_zero | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
⊢ A.det + B.det ≤ (A + B).det | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
⊢ A.det + B.det ≤ (A + B).det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | have : IsUnit sqrtA :=
(isUnit_iff_isUnit_det _).2 isUnit_det_sqrtA | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
⊢ A.det + B.det ≤ (A + B).det | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
⊢ A.det + B.det ≤ (A + B).det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | have IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian := by
{ apply IsHermitian.nonsingular_inv (hA.posSemidef.PosSemidef_sqrt.1)
exact isUnit_det_sqrtA } | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
⊢ A.det + B.det ≤ (A + B).det | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
⊢ A.det + B.det ≤ (A + B).det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | have PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef :=
PosSemidef.mul_mul_of_IsHermitian hB IsHermitian_sqrtA | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
⊢ A.det + B.det ≤ (A + B).det | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
⊢ A.det + B.det ≤ (A + B).det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | let μ := PosSemidef_ABA.1.eigenvalues | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
⊢ A.det + B.det ≤ (A + B).det | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A.det + B.det ≤ (A + B).det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | calc A.det + B.det
= A.det * (1 + (sqrtA⁻¹ * B * sqrtA⁻¹).det) := by
rw [det_mul, det_mul, mul_comm _ B.det, mul_assoc, ← det_mul, ← Matrix.mul_inv_rev,
hA.posSemidef.sqrt_mul_sqrt, mul_add, mul_one, mul_comm, mul_assoc, ← det_mul,
nonsing_inv_mul _ (isUnit_iff_ne_zero.2 hA.det_ne_zero), det_one, mul_one]
_ = A.det * (1 + ∏ i, μ i) := by
rw [PosSemidef_ABA.1.det_eq_prod_eigenvalues]
rfl
_ ≤ A.det * ∏ i, (1 + μ i) := by
apply (mul_le_mul_left hA.det_pos).2
apply Finset.one_add_prod_le_prod_one_add μ PosSemidef_ABA.eigenvalues_nonneg
_ = A.det * (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det := by
rw [mul_eq_mul_left_iff]; left; symm
rw [det_eq_prod_eigenvalues PosSemidef_ABA.1.eigenvectorBasis
(fun i => 1 + (PosSemidef_ABA.1.eigenvalues i)) _]
{ simp }
intro i
convert PosSemidef_ABA.1.has_eigenvector_one_add i using 1
simp only [map_add, toLin'_one, toLin'_mul, add_left_inj]
rfl
_ = (A + B).det := by
rw [← det_mul, ← det_conj this (A + B)]
apply congr_arg
rw [← hA.posSemidef.sqrt_mul_sqrt]
change sqrtA * sqrtA * (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = sqrtA * (sqrtA * sqrtA + B) * sqrtA⁻¹
rw [Matrix.mul_add, Matrix.mul_one, Matrix.mul_add, Matrix.add_mul,
Matrix.mul_assoc, Matrix.mul_assoc, Matrix.mul_assoc, Matrix.mul_assoc,
← Matrix.mul_assoc _ _ (B * _),
Matrix.mul_nonsing_inv _ isUnit_det_sqrtA, Matrix.one_mul, Matrix.mul_one,
hA.posSemidef.sqrt_mul_sqrt, Matrix.mul_assoc] | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A.det + B.det ≤ (A + B).det | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | apply IsHermitian.nonsingular_inv (hA.posSemidef.PosSemidef_sqrt.1) | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
⊢ sqrtA⁻¹.IsHermitian | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
⊢ IsUnit ⋯.sqrt.det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | exact isUnit_det_sqrtA | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
⊢ IsUnit ⋯.sqrt.det | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | rw [det_mul, det_mul, mul_comm _ B.det, mul_assoc, ← det_mul, ← Matrix.mul_inv_rev,
hA.posSemidef.sqrt_mul_sqrt, mul_add, mul_one, mul_comm, mul_assoc, ← det_mul,
nonsing_inv_mul _ (isUnit_iff_ne_zero.2 hA.det_ne_zero), det_one, mul_one] | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A.det + B.det = A.det * (1 + (sqrtA⁻¹ * B * sqrtA⁻¹).det) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | rw [PosSemidef_ABA.1.det_eq_prod_eigenvalues] | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A.det * (1 + (sqrtA⁻¹ * B * sqrtA⁻¹).det) = A.det * (1 + ∏ i : n, μ i) | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A.det * (1 + ∏ i : n, ↑(⋯.eigenvalues i)) = A.det * (1 + ∏ i : n, μ i) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | rfl | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A.det * (1 + ∏ i : n, ↑(⋯.eigenvalues i)) = A.det * (1 + ∏ i : n, μ i) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | apply (mul_le_mul_left hA.det_pos).2 | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A.det * (1 + ∏ i : n, μ i) ≤ A.det * ∏ i : n, (1 + μ i) | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ 1 + ∏ i : n, μ i ≤ ∏ i : n, (1 + μ i) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | apply Finset.one_add_prod_le_prod_one_add μ PosSemidef_ABA.eigenvalues_nonneg | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ 1 + ∏ i : n, μ i ≤ ∏ i : n, (1 + μ i) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | rw [mul_eq_mul_left_iff] | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A.det * ∏ i : n, (1 + μ i) = A.det * (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ∏ i : n, (1 + μ i) = (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det ∨ A.det = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | left | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ∏ i : n, (1 + μ i) = (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det ∨ A.det = 0 | case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ∏ i : n, (1 + μ i) = (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | symm | case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ∏ i : n, (1 + μ i) = (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det | case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det = ∏ i : n, (1 + μ i) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | rw [det_eq_prod_eigenvalues PosSemidef_ABA.1.eigenvectorBasis
(fun i => 1 + (PosSemidef_ABA.1.eigenvalues i)) _] | case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det = ∏ i : n, (1 + μ i) | case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ↑(∏ i : n, (1 + ⋯.eigenvalues i)) = ∏ i : n, (1 + μ i)
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ∀ (j : n),
Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) j))
(⋯.eigenvectorBasis j) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | { simp } | case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ↑(∏ i : n, (1 + ⋯.eigenvalues i)) = ∏ i : n, (1 + μ i)
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ∀ (j : n),
Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) j))
(⋯.eigenvectorBasis j) | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ∀ (j : n),
Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) j))
(⋯.eigenvectorBasis j) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | intro i | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ∀ (j : n),
Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) j))
(⋯.eigenvectorBasis j) | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
i : n
⊢ Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) i))
(⋯.eigenvectorBasis i) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | convert PosSemidef_ABA.1.has_eigenvector_one_add i using 1 | n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
i : n
⊢ Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) i))
(⋯.eigenvectorBasis i) | case h.e'_6.h.h.h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
i : n
e_2✝ : (n → ℝ) = EuclideanSpace ℝ n
e_3✝ : EuclideanDomain.toCommRing = Real.commRing
he✝¹ : Pi.addCommGroup = WithLp.instAddCommGroup 2 ((i : n) → (fun x => ℝ) i)
he✝ : Pi.Function.module n ℝ ℝ = WithLp.instModule 2 ℝ ((i : n) → (fun x => ℝ) i)
⊢ toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = 1 + toLin' (sqrtA⁻¹ * B * sqrtA⁻¹) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | simp only [map_add, toLin'_one, toLin'_mul, add_left_inj] | case h.e'_6.h.h.h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
i : n
e_2✝ : (n → ℝ) = EuclideanSpace ℝ n
e_3✝ : EuclideanDomain.toCommRing = Real.commRing
he✝¹ : Pi.addCommGroup = WithLp.instAddCommGroup 2 ((i : n) → (fun x => ℝ) i)
he✝ : Pi.Function.module n ℝ ℝ = WithLp.instModule 2 ℝ ((i : n) → (fun x => ℝ) i)
⊢ toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = 1 + toLin' (sqrtA⁻¹ * B * sqrtA⁻¹) | case h.e'_6.h.h.h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
i : n
e_2✝ : (n → ℝ) = EuclideanSpace ℝ n
e_3✝ : EuclideanDomain.toCommRing = Real.commRing
he✝¹ : Pi.addCommGroup = WithLp.instAddCommGroup 2 ((i : n) → (fun x => ℝ) i)
he✝ : Pi.Function.module n ℝ ℝ = WithLp.instModule 2 ℝ ((i : n) → (fun x => ℝ) i)
⊢ LinearMap.id = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | rfl | case h.e'_6.h.h.h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
i : n
e_2✝ : (n → ℝ) = EuclideanSpace ℝ n
e_3✝ : EuclideanDomain.toCommRing = Real.commRing
he✝¹ : Pi.addCommGroup = WithLp.instAddCommGroup 2 ((i : n) → (fun x => ℝ) i)
he✝ : Pi.Function.module n ℝ ℝ = WithLp.instModule 2 ℝ ((i : n) → (fun x => ℝ) i)
⊢ LinearMap.id = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Subadditivity.lean | Matrix.det_add_det_le_det_add' | [121, 1] | [168, 57] | simp | case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ↑(∏ i : n, (1 + ⋯.eigenvalues i)) = ∏ i : n, (1 + μ i) | no goals |
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