Datasets:
internlm
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Lean-Github

Dataset card Data Studio Files Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
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stringlengths
1
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stringlengths
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2.09M
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/HypersonicShapeDesign.lean
HypersonicShapeDesign.one_sub_bβ‚š_nonneg
[58, 1]
[60, 22]
unfold bβ‚š
a b : ℝ ⊒ 0 ≀ 1 - bβ‚š
a b : ℝ ⊒ 0 ≀ 1 - 0.65
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/HypersonicShapeDesign.lean
HypersonicShapeDesign.one_sub_bβ‚š_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/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
calc _ = βˆ‘ i, ((a i) ^ 2 - 2 * (a i) * x + (x ^ 2)) := by congr; funext i; simp; ring _ = βˆ‘ i, ((a i) ^ 2) - 2 * x * βˆ‘ i, (a i) + n * (x ^ 2) := by rw [sum_add_distrib, sum_sub_distrib, ← sum_mul, ← mul_sum]; simp [sum_const]; ring _ = n * mean (a ^ 2) - 2 * x * n * mean a + n * (x ^ 2) := by simp [mean]; field_simp; ring _ = n * ((x - mean a) ^ 2 + (mean (a ^ 2) - (mean a) ^ 2)) := by simp [mean]; field_simp; ring
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2))
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
congr
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ i : Fin n, (a i - x) ^ 2 = βˆ‘ i : Fin n, (a i ^ 2 - 2 * a i * x + x ^ 2)
case e_f n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (fun i => (a i - x) ^ 2) = fun i => a i ^ 2 - 2 * a i * x + x ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
funext i
case e_f n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (fun i => (a i - x) ^ 2) = fun i => a i ^ 2 - 2 * a i * x + x ^ 2
case e_f.h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ i : Fin n ⊒ (a i - x) ^ 2 = a i ^ 2 - 2 * a i * x + x ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
ring
case e_f.h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ i : Fin n ⊒ (a i - x) ^ 2 = a i ^ 2 - 2 * a i * x + x ^ 2
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
rw [sum_add_distrib, sum_sub_distrib, ← sum_mul, ← mul_sum]
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ i : Fin n, (a i ^ 2 - 2 * a i * x + x ^ 2) = βˆ‘ i : Fin n, a i ^ 2 - 2 * x * βˆ‘ i : Fin n, a i + ↑n * x ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ x : Fin n, a x ^ 2 - (2 * βˆ‘ i : Fin n, a i) * x + βˆ‘ x_1 : Fin n, x ^ 2 = βˆ‘ i : Fin n, a i ^ 2 - 2 * x * βˆ‘ i : Fin n, a i + ↑n * x ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
simp [sum_const]
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ x : Fin n, a x ^ 2 - (2 * βˆ‘ i : Fin n, a i) * x + βˆ‘ x_1 : Fin n, x ^ 2 = βˆ‘ i : Fin n, a i ^ 2 - 2 * x * βˆ‘ i : Fin n, a i + ↑n * x ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (2 * βˆ‘ i : Fin n, a i) * x = 2 * x * βˆ‘ i : Fin n, a i
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
ring
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (2 * βˆ‘ i : Fin n, a i) * x = 2 * x * βˆ‘ i : Fin n, a i
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
simp [mean]
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ i : Fin n, a i ^ 2 - 2 * x * βˆ‘ i : Fin n, a i + ↑n * x ^ 2 = ↑n * mean (a ^ 2) - 2 * x * ↑n * mean a + ↑n * x ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ x : Fin n, a x ^ 2 - 2 * x * βˆ‘ i : Fin n, a i = ↑n * ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2) - 2 * x * ↑n * ((↑n)⁻¹ * βˆ‘ i : Fin n, a i)
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
field_simp
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ x : Fin n, a x ^ 2 - 2 * x * βˆ‘ i : Fin n, a i = ↑n * ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2) - 2 * x * ↑n * ((↑n)⁻¹ * βˆ‘ i : Fin n, a i)
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (βˆ‘ x : Fin n, a x ^ 2 - 2 * x * βˆ‘ i : Fin n, a i) * ↑n = (βˆ‘ x : Fin n, a x ^ 2) * ↑n - 2 * x * ↑n * βˆ‘ i : Fin n, a i
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
ring
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (βˆ‘ x : Fin n, a x ^ 2 - 2 * x * βˆ‘ i : Fin n, a i) * ↑n = (βˆ‘ x : Fin n, a x ^ 2) * ↑n - 2 * x * ↑n * βˆ‘ i : Fin n, a i
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
simp [mean]
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ↑n * mean (a ^ 2) - 2 * x * ↑n * mean a + ↑n * x ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2))
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ↑n * ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2) - 2 * x * ↑n * ((↑n)⁻¹ * βˆ‘ i : Fin n, a i) + ↑n * x ^ 2 = ↑n * ((x - (↑n)⁻¹ * βˆ‘ i : Fin n, a i) ^ 2 + ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2 - ((↑n)⁻¹ * βˆ‘ i : Fin n, a i) ^ 2))
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
field_simp
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ↑n * ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2) - 2 * x * ↑n * ((↑n)⁻¹ * βˆ‘ i : Fin n, a i) + ↑n * x ^ 2 = ↑n * ((x - (↑n)⁻¹ * βˆ‘ i : Fin n, a i) ^ 2 + ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2 - ((↑n)⁻¹ * βˆ‘ i : Fin n, a i) ^ 2))
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ((βˆ‘ x : Fin n, a x ^ 2) * ↑n - 2 * x * ↑n * βˆ‘ i : Fin n, a i + ↑n * x ^ 2 * ↑n) * (↑n ^ 2 * (↑n * ↑n ^ 2)) = ↑n * ((x * ↑n - βˆ‘ i : Fin n, a i) ^ 2 * (↑n * ↑n ^ 2) + ((βˆ‘ x : Fin n, a x ^ 2) * ↑n ^ 2 - ↑n * (βˆ‘ i : Fin n, a i) ^ 2) * ↑n ^ 2) * ↑n
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
ring
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ((βˆ‘ x : Fin n, a x ^ 2) * ↑n - 2 * x * ↑n * βˆ‘ i : Fin n, a i + ↑n * x ^ 2 * ↑n) * (↑n ^ 2 * (↑n * ↑n ^ 2)) = ↑n * ((x * ↑n - βˆ‘ i : Fin n, a i) ^ 2 * (↑n * ↑n ^ 2) + ((βˆ‘ x : Fin n, a x ^ 2) * ↑n ^ 2 - ↑n * (βˆ‘ i : Fin n, a i) ^ 2) * ↑n ^ 2) * ↑n
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
simp [optimal, feasible, leastSquares] at h
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : (leastSquares a).optimal x ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
replace h : βˆ€ y, (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 := by intros y have hy := h y have h_rw_x := leastSquares_alt_objFun hn a x have h_rw_y := leastSquares_alt_objFun hn a y simp only [rpow_two] at h_rw_x h_rw_y ⊒ rwa [h_rw_x, h_rw_y, mul_le_mul_left (by positivity), add_le_add_iff_right] at hy
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have hmean := h (mean a)
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ (mean a - mean a) ^ 2 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
simp at hmean
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ (mean a - mean a) ^ 2 ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ 0 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have h_sq_eq_zero := le_antisymm hmean (sq_nonneg _)
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ 0 ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ 0 h_sq_eq_zero : (x - mean a) ^ 2 = 0 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
rwa [sq_eq_zero_iff, sub_eq_zero] at h_sq_eq_zero
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ 0 h_sq_eq_zero : (x - mean a) ^ 2 = 0 ⊒ x = mean a
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
intros y
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have hy := h y
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have h_rw_x := leastSquares_alt_objFun hn a x
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have h_rw_y := leastSquares_alt_objFun hn a y
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ i : Fin n, (a i - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
simp only [rpow_two] at h_rw_x h_rw_y ⊒
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ i : Fin n, (a i - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ x : Fin n, (a x - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
rwa [h_rw_x, h_rw_y, mul_le_mul_left (by positivity), add_le_add_iff_right] at hy
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ x : Fin n, (a x - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
positivity
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ≀ ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_x : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ x : Fin n, (a x - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ 0 < ↑n
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
apply leastSquares_optimal_eq_mean hn a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : (leastSquaresVec a).optimal x ⊒ x = mean a
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : (leastSquaresVec a).optimal x ⊒ (leastSquares a).optimal x
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
simp [leastSquaresVec, leastSquares, optimal, feasible] at h ⊒
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : (leastSquaresVec a).optimal x ⊒ (leastSquares a).optimal x
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), Vec.sum ((a - Vec.const n x) ^ 2) ≀ Vec.sum ((a - Vec.const n y) ^ 2) ⊒ βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
intros y
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), Vec.sum ((a - Vec.const n x) ^ 2) ≀ Vec.sum ((a - Vec.const n y) ^ 2) ⊒ βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), Vec.sum ((a - Vec.const n x) ^ 2) ≀ Vec.sum ((a - Vec.const n y) ^ 2) y : ℝ ⊒ βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
simp only [Vec.sum, Pi.pow_apply, Pi.sub_apply, Vec.const, rpow_two] at h
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), Vec.sum ((a - Vec.const n x) ^ 2) ≀ Vec.sum ((a - Vec.const n y) ^ 2) y : ℝ ⊒ βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ y : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
exact h y
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ y : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - 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 [fittingSphereT, fittingSphereConvex, optimal, feasible] at h_opt ⊒
n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : (fittingSphereConvex n m x).optimal (c, t) ⊒ (fittingSphereT n m x).optimal (c, t)
n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) ⊒ 0 ≀ t + β€–cβ€– ^ 2 ∧ βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), 0 ≀ b + β€–aβ€– ^ 2 β†’ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) ⊒ 0 ≀ t + β€–cβ€– ^ 2 ∧ βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), 0 ≀ b + β€–aβ€– ^ 2 β†’ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2)
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) ⊒ 0 ≀ t + β€–cβ€– ^ 2 case right n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) ⊒ βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), 0 ≀ b + β€–aβ€– ^ 2 β†’ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) ⊒ 0 ≀ t + β€–cβ€– ^ 2
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c ⊒ 0 ≀ t + β€–cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c ⊒ 0 ≀ t + β€–cβ€– ^ 2
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t ⊒ 0 ≀ t + β€–cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t ⊒ 0 ≀ t + β€–cβ€– ^ 2
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a ⊒ 0 ≀ t + β€–cβ€– ^ 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 : β€–cβ€– ^ 2 = (1 / m) * βˆ‘ i : Fin m, β€–cβ€– ^ 2 := by simp [sum_const] field_simp
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a ⊒ 0 ≀ t + β€–cβ€– ^ 2
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ 0 ≀ t + β€–cβ€– ^ 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 + β€–cβ€– ^ 2 = (1 / m) * βˆ‘ i, β€–(x i) - cβ€– ^ 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 (π•œ := ℝ) (E := Fin n β†’ ℝ)] simp [a]; congr
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ 0 ≀ t + β€–cβ€– ^ 2
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 ⊒ 0 ≀ t + β€–cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 ⊒ 0 ≀ t + β€–cβ€– ^ 2
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 ⊒ 0 ≀ 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 ⊒ 0 ≀ 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2
case left n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 ⊒ 0 ≀ βˆ‘ i : Fin m, β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 ⊒ 0 ≀ βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2
case left.h n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 ⊒ βˆ€ i ∈ univ, 0 ≀ β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 ⊒ βˆ€ i ∈ univ, 0 ≀ β€–x i - cβ€– ^ 2
case left.h n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 i : Fin m a✝ : i ∈ univ ⊒ 0 ≀ β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2 i : Fin m a✝ : i ∈ univ ⊒ 0 ≀ β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c ⊒ (leastSquaresVec a).optimal t
n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c ⊒ βˆ€ (y : ℝ), (leastSquaresVec a).feasible y β†’ (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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c ⊒ βˆ€ (y : ℝ), (leastSquaresVec a).feasible y β†’ (leastSquaresVec a).objFun t ≀ (leastSquaresVec a).objFun y
n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c y : ℝ a✝ : (leastSquaresVec a).feasible y ⊒ (leastSquaresVec a).objFun t ≀ (leastSquaresVec a).objFun y
n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a ⊒ β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2
n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a ⊒ β€–cβ€– ^ 2 = (↑m)⁻¹ * (↑m * β€–cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a ⊒ β€–cβ€– ^ 2 = (↑m)⁻¹ * (↑m * β€–cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2
n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ mean a + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ mean a + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2
n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ 1 / ↑m * βˆ‘ i : Fin m, a i + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ 1 / ↑m * βˆ‘ i : Fin m, a i + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 2
n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ βˆ‘ x : Fin m, (1 / ↑m * a x + 1 / ↑m * β€–cβ€– ^ 2) = βˆ‘ i : Fin m, 1 / ↑m * β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ βˆ‘ x : Fin m, (1 / ↑m * a x + 1 / ↑m * β€–cβ€– ^ 2) = βˆ‘ i : Fin m, 1 / ↑m * β€–x i - cβ€– ^ 2
case e_f n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ (fun x => 1 / ↑m * a x + 1 / ↑m * β€–cβ€– ^ 2) = fun i => 1 / ↑m * β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 ⊒ (fun x => 1 / ↑m * a x + 1 / ↑m * β€–cβ€– ^ 2) = fun i => 1 / ↑m * β€–x i - cβ€– ^ 2
case e_f.h n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ 1 / ↑m * a i + 1 / ↑m * β€–cβ€– ^ 2 = 1 / ↑m * β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ 1 / ↑m * a i + 1 / ↑m * β€–cβ€– ^ 2 = 1 / ↑m * β€–x i - cβ€– ^ 2
case e_f.h n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ 1 / ↑m * (a i + β€–cβ€– ^ 2) = 1 / ↑m * β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ 1 / ↑m * (a i + β€–cβ€– ^ 2) = 1 / ↑m * β€–x i - cβ€– ^ 2
case e_f.h.e_a n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ a i + β€–cβ€– ^ 2 = β€–x i - cβ€– ^ 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 (π•œ := ℝ) (E := Fin n β†’ ℝ)]
case e_f.h.e_a n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ a i + β€–cβ€– ^ 2 = β€–x i - cβ€– ^ 2
case e_f.h.e_a n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ a i + β€–cβ€– ^ 2 = β€–x iβ€– ^ 2 - 2 * RCLike.re βŸͺx i, c⟫_ℝ + β€–cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ a i + β€–cβ€– ^ 2 = β€–x iβ€– ^ 2 - 2 * RCLike.re βŸͺx i, c⟫_ℝ + β€–cβ€– ^ 2
case e_f.h.e_a n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ Vec.norm x i ^ 2 - 2 * (x *α΅₯ c) i = β€–x iβ€– ^ 2 - 2 * βŸͺx 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 i : Fin m ⊒ Vec.norm x i ^ 2 - 2 * (x *α΅₯ c) i = β€–x iβ€– ^ 2 - 2 * βŸͺx 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) a : Fin m β†’ ℝ := Vec.norm x ^ 2 - 2 * x *α΅₯ c h_ls : (leastSquaresVec a).optimal t h_t_eq : t = mean a h_c2_eq : β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–cβ€– ^ 2 h_t_add_c2_eq : t + β€–cβ€– ^ 2 = 1 / ↑m * βˆ‘ i : Fin m, β€–x i - cβ€– ^ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) ⊒ βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), 0 ≀ b + β€–aβ€– ^ 2 β†’ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2)
case right n m : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) c' : Fin n β†’ ℝ x' : ℝ a✝ : 0 ≀ x' + β€–c'β€– ^ 2 ⊒ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ 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 : β„• x : Fin m β†’ Fin n β†’ ℝ hm : 0 < m c : Fin n β†’ ℝ t : ℝ h_opt : βˆ€ (a : Fin n β†’ ℝ) (b : ℝ), Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ a - Vec.const m b) ^ 2) c' : Fin n β†’ ℝ x' : ℝ a✝ : 0 ≀ x' + β€–c'β€– ^ 2 ⊒ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c - Vec.const m t) ^ 2) ≀ Vec.sum ((Vec.norm x ^ 2 - 2 * x *α΅₯ c' - Vec.const m x') ^ 2)
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Lib/Math/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
repr_gramSchmidt_diagonal
[18, 1]
[24, 67]
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]
π•œ : Type u_2 E : Type u_1 inst✝⁡ : RCLike π•œ inst✝⁴ : NormedAddCommGroup E inst✝³ : InnerProductSpace π•œ E ΞΉ : Type u_3 inst✝² : LinearOrder ΞΉ inst✝¹ : LocallyFiniteOrderBot ΞΉ inst✝ : IsWellOrder ΞΉ fun x x_1 => x < x_1 i : ΞΉ b : Basis ΞΉ π•œ E ⊒ (b.repr (gramSchmidt π•œ (⇑b) i)) i = 1
π•œ : Type u_2 E : Type u_1 inst✝⁡ : RCLike π•œ inst✝⁴ : NormedAddCommGroup E inst✝³ : InnerProductSpace π•œ E ΞΉ : Type u_3 inst✝² : LinearOrder ΞΉ inst✝¹ : LocallyFiniteOrderBot ΞΉ inst✝ : IsWellOrder ΞΉ fun x x_1 => x < x_1 i : ΞΉ b : Basis ΞΉ π•œ E ⊒ βˆ€ x ∈ Finset.Iio i, (b.repr ↑((orthogonalProjection (Submodule.span π•œ {gramSchmidt π•œ (⇑b) x})) (b i))) i = 0
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Lib/Math/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
repr_gramSchmidt_diagonal
[18, 1]
[24, 67]
intros j hj
π•œ : Type u_2 E : Type u_1 inst✝⁡ : RCLike π•œ inst✝⁴ : NormedAddCommGroup E inst✝³ : InnerProductSpace π•œ E ΞΉ : Type u_3 inst✝² : LinearOrder ΞΉ inst✝¹ : LocallyFiniteOrderBot ΞΉ inst✝ : IsWellOrder ΞΉ fun x x_1 => x < x_1 i : ΞΉ b : Basis ΞΉ π•œ E ⊒ βˆ€ x ∈ Finset.Iio i, (b.repr ↑((orthogonalProjection (Submodule.span π•œ {gramSchmidt π•œ (⇑b) x})) (b i))) i = 0
π•œ : Type u_2 E : Type u_1 inst✝⁡ : RCLike π•œ inst✝⁴ : NormedAddCommGroup E inst✝³ : InnerProductSpace π•œ E ΞΉ : Type u_3 inst✝² : LinearOrder ΞΉ inst✝¹ : LocallyFiniteOrderBot ΞΉ inst✝ : IsWellOrder ΞΉ fun x x_1 => x < x_1 i : ΞΉ b : Basis ΞΉ π•œ E j : ΞΉ hj : j ∈ Finset.Iio i ⊒ (b.repr ↑((orthogonalProjection (Submodule.span π•œ {gramSchmidt π•œ (⇑b) j})) (b i))) i = 0
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Lib/Math/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
repr_gramSchmidt_diagonal
[18, 1]
[24, 67]
rw [Finset.mem_Iio] at hj
π•œ : Type u_2 E : Type u_1 inst✝⁡ : RCLike π•œ inst✝⁴ : NormedAddCommGroup E inst✝³ : InnerProductSpace π•œ E ΞΉ : Type u_3 inst✝² : LinearOrder ΞΉ inst✝¹ : LocallyFiniteOrderBot ΞΉ inst✝ : IsWellOrder ΞΉ fun x x_1 => x < x_1 i : ΞΉ b : Basis ΞΉ π•œ E j : ΞΉ hj : j ∈ Finset.Iio i ⊒ (b.repr ↑((orthogonalProjection (Submodule.span π•œ {gramSchmidt π•œ (⇑b) j})) (b i))) i = 0
π•œ : Type u_2 E : Type u_1 inst✝⁡ : RCLike π•œ inst✝⁴ : NormedAddCommGroup E inst✝³ : InnerProductSpace π•œ E ΞΉ : Type u_3 inst✝² : LinearOrder ΞΉ inst✝¹ : LocallyFiniteOrderBot ΞΉ inst✝ : IsWellOrder ΞΉ fun x x_1 => x < x_1 i : ΞΉ b : Basis ΞΉ π•œ E j : ΞΉ hj : j < i ⊒ (b.repr ↑((orthogonalProjection (Submodule.span π•œ {gramSchmidt π•œ (⇑b) j})) (b i))) i = 0
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Lib/Math/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
repr_gramSchmidt_diagonal
[18, 1]
[24, 67]
simp [orthogonalProjection_singleton, gramSchmidt_triangular hj]
π•œ : Type u_2 E : Type u_1 inst✝⁡ : RCLike π•œ inst✝⁴ : NormedAddCommGroup E inst✝³ : InnerProductSpace π•œ E ΞΉ : Type u_3 inst✝² : LinearOrder ΞΉ inst✝¹ : LocallyFiniteOrderBot ΞΉ inst✝ : IsWellOrder ΞΉ fun x x_1 => x < x_1 i : ΞΉ b : Basis ΞΉ π•œ E j : ΞΉ hj : j < i ⊒ (b.repr ↑((orthogonalProjection (Submodule.span π•œ {gramSchmidt π•œ (⇑b) j})) (b i))) i = 0
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Tactic/DCP/AtomLibrary/Fns/Norm.lean
CvxLean.norm2β‚‚_eq_norm2
[57, 1]
[58, 36]
simp [Norm.norm, sqrt_eq_rpow]
x y : ℝ ⊒ β€–![x, y]β€– = (x ^ 2 + y ^ 2).sqrt
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
calc _ = βˆ‘ i, ((a i) ^ 2 - 2 * (a i) * x + (x ^ 2)) := by congr; funext i; simp; ring _ = βˆ‘ i, ((a i) ^ 2) - 2 * x * βˆ‘ i, (a i) + n * (x ^ 2) := by rw [sum_add_distrib, sum_sub_distrib, ← sum_mul, ← mul_sum]; simp [sum_const]; ring _ = n * mean (a ^ 2) - 2 * x * n * mean a + n * (x ^ 2) := by simp [mean]; field_simp; ring _ = n * ((x - mean a) ^ 2 + (mean (a ^ 2) - (mean a) ^ 2)) := by simp [mean]; field_simp; ring
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2))
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
congr
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ i : Fin n, (a i - x) ^ 2 = βˆ‘ i : Fin n, (a i ^ 2 - 2 * a i * x + x ^ 2)
case e_f n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (fun i => (a i - x) ^ 2) = fun i => a i ^ 2 - 2 * a i * x + x ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
funext i
case e_f n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (fun i => (a i - x) ^ 2) = fun i => a i ^ 2 - 2 * a i * x + x ^ 2
case e_f.h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ i : Fin n ⊒ (a i - x) ^ 2 = a i ^ 2 - 2 * a i * x + x ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
ring
case e_f.h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ i : Fin n ⊒ (a i - x) ^ 2 = a i ^ 2 - 2 * a i * x + x ^ 2
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
rw [sum_add_distrib, sum_sub_distrib, ← sum_mul, ← mul_sum]
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ i : Fin n, (a i ^ 2 - 2 * a i * x + x ^ 2) = βˆ‘ i : Fin n, a i ^ 2 - 2 * x * βˆ‘ i : Fin n, a i + ↑n * x ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ x : Fin n, a x ^ 2 - (2 * βˆ‘ i : Fin n, a i) * x + βˆ‘ x_1 : Fin n, x ^ 2 = βˆ‘ i : Fin n, a i ^ 2 - 2 * x * βˆ‘ i : Fin n, a i + ↑n * x ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
simp [sum_const]
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ x : Fin n, a x ^ 2 - (2 * βˆ‘ i : Fin n, a i) * x + βˆ‘ x_1 : Fin n, x ^ 2 = βˆ‘ i : Fin n, a i ^ 2 - 2 * x * βˆ‘ i : Fin n, a i + ↑n * x ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (2 * βˆ‘ i : Fin n, a i) * x = 2 * x * βˆ‘ i : Fin n, a i
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
ring
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (2 * βˆ‘ i : Fin n, a i) * x = 2 * x * βˆ‘ i : Fin n, a i
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
simp [mean]
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ i : Fin n, a i ^ 2 - 2 * x * βˆ‘ i : Fin n, a i + ↑n * x ^ 2 = ↑n * mean (a ^ 2) - 2 * x * ↑n * mean a + ↑n * x ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ x : Fin n, a x ^ 2 - 2 * x * βˆ‘ i : Fin n, a i = ↑n * ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2) - 2 * x * ↑n * ((↑n)⁻¹ * βˆ‘ i : Fin n, a i)
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
field_simp
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ βˆ‘ x : Fin n, a x ^ 2 - 2 * x * βˆ‘ i : Fin n, a i = ↑n * ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2) - 2 * x * ↑n * ((↑n)⁻¹ * βˆ‘ i : Fin n, a i)
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (βˆ‘ x : Fin n, a x ^ 2 - 2 * x * βˆ‘ i : Fin n, a i) * ↑n = (βˆ‘ x : Fin n, a x ^ 2) * ↑n - 2 * x * ↑n * βˆ‘ i : Fin n, a i
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
ring
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ (βˆ‘ x : Fin n, a x ^ 2 - 2 * x * βˆ‘ i : Fin n, a i) * ↑n = (βˆ‘ x : Fin n, a x ^ 2) * ↑n - 2 * x * ↑n * βˆ‘ i : Fin n, a i
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
simp [mean]
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ↑n * mean (a ^ 2) - 2 * x * ↑n * mean a + ↑n * x ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2))
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ↑n * ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2) - 2 * x * ↑n * ((↑n)⁻¹ * βˆ‘ i : Fin n, a i) + ↑n * x ^ 2 = ↑n * ((x - (↑n)⁻¹ * βˆ‘ i : Fin n, a i) ^ 2 + ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2 - ((↑n)⁻¹ * βˆ‘ i : Fin n, a i) ^ 2))
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
field_simp
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ↑n * ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2) - 2 * x * ↑n * ((↑n)⁻¹ * βˆ‘ i : Fin n, a i) + ↑n * x ^ 2 = ↑n * ((x - (↑n)⁻¹ * βˆ‘ i : Fin n, a i) ^ 2 + ((↑n)⁻¹ * βˆ‘ x : Fin n, a x ^ 2 - ((↑n)⁻¹ * βˆ‘ i : Fin n, a i) ^ 2))
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ((βˆ‘ x : Fin n, a x ^ 2) * ↑n - 2 * x * ↑n * βˆ‘ i : Fin n, a i + ↑n * x ^ 2 * ↑n) * (↑n ^ 2 * (↑n * ↑n ^ 2)) = ↑n * ((x * ↑n - βˆ‘ i : Fin n, a i) ^ 2 * (↑n * ↑n ^ 2) + ((βˆ‘ x : Fin n, a x ^ 2) * ↑n ^ 2 - ↑n * (βˆ‘ i : Fin n, a i) ^ 2) * ↑n ^ 2) * ↑n
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_alt_objFun
[24, 1]
[41, 34]
ring
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ ⊒ ((βˆ‘ x : Fin n, a x ^ 2) * ↑n - 2 * x * ↑n * βˆ‘ i : Fin n, a i + ↑n * x ^ 2 * ↑n) * (↑n ^ 2 * (↑n * ↑n ^ 2)) = ↑n * ((x * ↑n - βˆ‘ i : Fin n, a i) ^ 2 * (↑n * ↑n ^ 2) + ((βˆ‘ x : Fin n, a x ^ 2) * ↑n ^ 2 - ↑n * (βˆ‘ i : Fin n, a i) ^ 2) * ↑n ^ 2) * ↑n
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
simp [optimal, feasible, leastSquares] at h
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : (leastSquares a).optimal x ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
replace h : βˆ€ y, (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 := by intros y have hy := h y have h_rw_x := leastSquares_alt_objFun hn a x have h_rw_y := leastSquares_alt_objFun hn a y simp only [rpow_two] at h_rw_x h_rw_y ⊒ rwa [h_rw_x, h_rw_y, mul_le_mul_left (by positivity), add_le_add_iff_right] at hy
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have hmean := h (mean a)
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ (mean a - mean a) ^ 2 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
simp at hmean
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ (mean a - mean a) ^ 2 ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ 0 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have h_sq_eq_zero := le_antisymm hmean (sq_nonneg _)
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ 0 ⊒ x = mean a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ 0 h_sq_eq_zero : (x - mean a) ^ 2 = 0 ⊒ x = mean a
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
rwa [sq_eq_zero_iff, sub_eq_zero] at h_sq_eq_zero
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2 hmean : (x - mean a) ^ 2 ≀ 0 h_sq_eq_zero : (x - mean a) ^ 2 = 0 ⊒ x = mean a
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
intros y
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ βˆ€ (y : ℝ), (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have hy := h y
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have h_rw_x := leastSquares_alt_objFun hn a x
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
have h_rw_y := leastSquares_alt_objFun hn a y
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ i : Fin n, (a i - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
simp only [rpow_two] at h_rw_x h_rw_y ⊒
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ i : Fin n, (a i - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ i : Fin n, (a i - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ x : Fin n, (a x - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
rwa [h_rw_x, h_rw_y, mul_le_mul_left (by positivity), add_le_add_iff_right] at hy
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 h_rw_x : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ x : Fin n, (a x - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ (x - mean a) ^ 2 ≀ (y - mean a) ^ 2
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquares_optimal_eq_mean
[43, 1]
[57, 52]
positivity
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 y : ℝ hy : ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ≀ ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_x : βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 = ↑n * ((x - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) h_rw_y : βˆ‘ x : Fin n, (a x - y) ^ 2 = ↑n * ((y - mean a) ^ 2 + (mean (a ^ 2) - mean a ^ 2)) ⊒ 0 < ↑n
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
apply leastSquares_optimal_eq_mean hn a
n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : (leastSquaresVec a).optimal x ⊒ x = mean a
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : (leastSquaresVec a).optimal x ⊒ (leastSquares a).optimal x
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
simp [leastSquaresVec, leastSquares, optimal, feasible] at h ⊒
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : (leastSquaresVec a).optimal x ⊒ (leastSquares a).optimal x
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), Vec.sum ((a - Vec.const n x) ^ 2) ≀ Vec.sum ((a - Vec.const n y) ^ 2) ⊒ βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
intros y
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), Vec.sum ((a - Vec.const n x) ^ 2) ≀ Vec.sum ((a - Vec.const n y) ^ 2) ⊒ βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), Vec.sum ((a - Vec.const n x) ^ 2) ≀ Vec.sum ((a - Vec.const n y) ^ 2) y : ℝ ⊒ βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
simp only [Vec.sum, Pi.pow_apply, Pi.sub_apply, Vec.const, rpow_two] at h
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ h : βˆ€ (y : ℝ), Vec.sum ((a - Vec.const n x) ^ 2) ≀ Vec.sum ((a - Vec.const n y) ^ 2) y : ℝ ⊒ βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ y : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/FittingSphere.lean
leastSquaresVec_optimal_eq_mean
[63, 1]
[70, 12]
exact h y
case h n : β„• hn : 0 < n a : Fin n β†’ ℝ x : ℝ y : ℝ h : βˆ€ (y : ℝ), βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2 ⊒ βˆ‘ x_1 : Fin n, (a x_1 - x) ^ 2 ≀ βˆ‘ x : Fin n, (a x - y) ^ 2
no goals