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stringclasses 147
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stringclasses 147
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stringlengths 7
101
| full_name
stringlengths 1
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stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
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stringlengths 3
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2.09M
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https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | have hx : 0 ≤ x := by linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
⊢ 0 < (a * x) ^ 2 - x ^ 2 | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
⊢ 0 < (a * x) ^ 2 - x ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | have hx2 : 0 < x ^ (2 : ℝ) := by positivity | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
⊢ 0 < (a * x) ^ 2 - x ^ 2 | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
hx2 : 0 < x ^ 2
⊢ 0 < (a * x) ^ 2 - x ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | rw [sub_pos, Real.mul_rpow ha hx, ← div_lt_iff hx2, div_self (ne_of_gt hx2)] | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
hx2 : 0 < x ^ 2
⊢ 0 < (a * x) ^ 2 - x ^ 2 | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
hx2 : 0 < x ^ 2
⊢ 1 < a ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | simp [ha1, abs_eq_self.mpr ha] | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
hx2 : 0 < x ^ 2
⊢ 1 < a ^ 2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
⊢ 0 ≤ a | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
⊢ 1 < a | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
⊢ 0 ≤ x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | positivity | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
⊢ 0 < x ^ 2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_add | [17, 1] | [24, 73] | rintro x ⟨hf, hg⟩ | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
⊢ f.eigenspace a ⊓ g.eigenspace b ≤ (f + g).eigenspace (a + b) | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : x ∈ ↑(f.eigenspace a)
hg : x ∈ ↑(g.eigenspace b)
⊢ x ∈ (f + g).eigenspace (a + b) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_add | [17, 1] | [24, 73] | simp only [eigenspace, SetLike.mem_coe, LinearMap.mem_ker,
LinearMap.sub_apply, algebraMap_end_apply] at hf hg | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : x ∈ ↑(f.eigenspace a)
hg : x ∈ ↑(g.eigenspace b)
⊢ x ∈ (f + g).eigenspace (a + b) | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : f x - a • x = 0
hg : g x - b • x = 0
⊢ x ∈ (f + g).eigenspace (a + b) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_add | [17, 1] | [24, 73] | simp only [eigenspace, map_add, LinearMap.mem_ker, LinearMap.sub_apply,
LinearMap.add_apply, algebraMap_end_apply] | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : f x - a • x = 0
hg : g x - b • x = 0
⊢ x ∈ (f + g).eigenspace (a + b) | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : f x - a • x = 0
hg : g x - b • x = 0
⊢ f x + g x - (a • x + b • x) = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_add | [17, 1] | [24, 73] | rw [← add_sub, add_comm (a • x), ← sub_sub, hg, add_sub, add_zero, hf] | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : f x - a • x = 0
hg : g x - b • x = 0
⊢ f x + g x - (a • x + b • x) = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_one | [26, 1] | [29, 64] | apply eq_top_iff.2 | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
⊢ eigenspace 1 1 = ⊤ | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
⊢ ⊤ ≤ eigenspace 1 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_one | [26, 1] | [29, 64] | intros x _ | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
⊢ ⊤ ≤ eigenspace 1 1 | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
a✝ : x ∈ ⊤
⊢ x ∈ eigenspace 1 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_one | [26, 1] | [29, 64] | simp only [mem_eigenspace_iff, LinearMap.one_apply, one_smul] | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
a✝ : x ∈ ⊤
⊢ x ∈ eigenspace 1 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.has_eigenvector_one | [35, 1] | [36, 60] | rw [eigenspace_one] | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
hx : x ≠ 0
⊢ x ∈ eigenspace 1 1 | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
hx : x ≠ 0
⊢ x ∈ ⊤ |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.has_eigenvector_one | [35, 1] | [36, 60] | apply Submodule.mem_top | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
hx : x ≠ 0
⊢ x ∈ ⊤ | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.log_nonneg_of_ge_one | [17, 1] | [18, 41] | linarith | x : ℝ
h : 0 ≤ x - 1
⊢ 1 ≤ x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.log_pos_of_gt_one | [20, 1] | [21, 38] | linarith | x : ℝ
h : 0 < x - 1
⊢ 1 < x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one | [38, 1] | [45, 32] | have hx1 : x ≤ 1 := by linarith | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
⊢ 0 ≤ 1 / x ^ 2 - 1 | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ 1 / x ^ 2 - 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one | [38, 1] | [45, 32] | have h0x : 0 ≤ x := by positivity | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ 1 / x ^ 2 - 1 | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
⊢ 0 ≤ 1 / x ^ 2 - 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one | [38, 1] | [45, 32] | have h0x2 : 0 < x ^ (2 : ℝ) := by positivity | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
⊢ 0 ≤ 1 / x ^ 2 - 1 | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
h0x2 : 0 < x ^ 2
⊢ 0 ≤ 1 / x ^ 2 - 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one | [38, 1] | [45, 32] | rw [le_sub_iff_add_le, zero_add, le_div_iff h0x2, one_mul] | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
h0x2 : 0 < x ^ 2
⊢ 0 ≤ 1 / x ^ 2 - 1 | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
h0x2 : 0 < x ^ 2
⊢ x ^ 2 ≤ 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one | [38, 1] | [45, 32] | simpa [abs_eq_self.mpr h0x] | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
h0x2 : 0 < x ^ 2
⊢ x ^ 2 ≤ 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one | [38, 1] | [45, 32] | linarith | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
⊢ x ≤ 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one | [38, 1] | [45, 32] | positivity | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one | [38, 1] | [45, 32] | positivity | x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
⊢ 0 < x ^ 2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_sq_nonneg_of_le_one | [73, 1] | [77, 31] | have hx1 : x ≤ 1 := by linarith | x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
⊢ 0 ≤ 1 - x ^ 2 | x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ 1 - x ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_sq_nonneg_of_le_one | [73, 1] | [77, 31] | rw [le_sub_iff_add_le, zero_add] | x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ 1 - x ^ 2 | x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ x ^ 2 ≤ 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_sq_nonneg_of_le_one | [73, 1] | [77, 31] | simpa [abs_eq_self.mpr h1] | x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ x ^ 2 ≤ 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_sq_nonneg_of_le_one | [73, 1] | [77, 31] | linarith | x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
⊢ x ≤ 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.exp_sub_one_pos_of_pos | [107, 1] | [108, 23] | simp [h] | x : ℝ
h : 0 < x
⊢ 0 < x.exp - 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_div_exp_pos_of_pos | [121, 1] | [122, 37] | field_simp | x : ℝ
h : 0 < x
⊢ 0 < 1 - 1 / x.exp | x : ℝ
h : 0 < x
⊢ 0 < x.exp |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.one_sub_div_exp_pos_of_pos | [121, 1] | [122, 37] | positivity | x : ℝ
h : 0 < x
⊢ 0 < x.exp | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | have ha : 0 ≤ a := by linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
⊢ 0 < (a * x) ^ 2 - x ^ 2 | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
⊢ 0 < (a * x) ^ 2 - x ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | have ha1 : 1 < a := by linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
⊢ 0 < (a * x) ^ 2 - x ^ 2 | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
⊢ 0 < (a * x) ^ 2 - x ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | have hx : 0 ≤ x := by linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
⊢ 0 < (a * x) ^ 2 - x ^ 2 | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
⊢ 0 < (a * x) ^ 2 - x ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | have hx2 : 0 < x ^ (2 : ℝ) := by positivity | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
⊢ 0 < (a * x) ^ 2 - x ^ 2 | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
hx2 : 0 < x ^ 2
⊢ 0 < (a * x) ^ 2 - x ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | rw [sub_pos, Real.mul_rpow ha hx, ← div_lt_iff hx2, div_self (ne_of_gt hx2)] | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
hx2 : 0 < x ^ 2
⊢ 0 < (a * x) ^ 2 - x ^ 2 | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
hx2 : 0 < x ^ 2
⊢ 1 < a ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | simp [ha1, abs_eq_self.mpr ha] | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
hx2 : 0 < x ^ 2
⊢ 1 < a ^ 2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
⊢ 0 ≤ a | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
⊢ 1 < a | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | linarith | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
⊢ 0 ≤ x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/Arith/PositivityExt.lean | Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos | [135, 1] | [143, 35] | positivity | a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
hx : 0 ≤ x
⊢ 0 < x ^ 2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular | [57, 1] | [65, 68] | letI : Unique {a // id a = k} := ⟨⟨⟨k, rfl⟩⟩, fun j => Subtype.ext j.property⟩ | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular | [57, 1] | [65, 68] | have h := congr_fun (congr_fun
(toSquareBlock_inv_mul_toSquareBlock_eq_one hM k) ⟨k, rfl⟩) ⟨k, rfl⟩ | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : (M⁻¹.toSquareBlock id k * M.toSquareBlock id k) ⟨k, ⋯⟩ ⟨k, ⋯⟩ = 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular | [57, 1] | [65, 68] | dsimp only [HMul.hMul, dotProduct] at h | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : (M⁻¹.toSquareBlock id k * M.toSquareBlock id k) ⟨k, ⋯⟩ ⟨k, ⋯⟩ = 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
∑ i : { a // id a = k }, Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ i) (M.toSquareBlock id k i ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular | [57, 1] | [65, 68] | rw [@Fintype.sum_unique _ _ _ _] at h | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
∑ i : { a // id a = k }, Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ i) (M.toSquareBlock id k i ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ default) (M.toSquareBlock id k default ⟨k, ⋯⟩) = OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular | [57, 1] | [65, 68] | simp at h | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ default) (M.toSquareBlock id k default ⟨k, ⋯⟩) = OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular | [57, 1] | [65, 68] | rw [← h] | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular | [57, 1] | [65, 68] | simp [toSquareBlock, toSquareBlockProp] | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹ k k) (M k k) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular | [57, 1] | [65, 68] | rfl | α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹ k k) (M k k) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular | [67, 1] | [75, 68] | letI : Unique {a // OrderDual.toDual a = k} :=
⟨⟨⟨k, rfl⟩⟩, fun j => Subtype.ext j.property⟩ | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular | [67, 1] | [75, 68] | have h := congr_fun (congr_fun
(toSquareBlock_inv_mul_toSquareBlock_eq_one hM k) ⟨k, rfl⟩) ⟨k, rfl⟩ | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k * M.toSquareBlock (⇑OrderDual.toDual) k) ⟨k, ⋯⟩ ⟨k, ⋯⟩ = 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular | [67, 1] | [75, 68] | dsimp [HMul.hMul, dotProduct] at h | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k * M.toSquareBlock (⇑OrderDual.toDual) k) ⟨k, ⋯⟩ ⟨k, ⋯⟩ = 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
∑ i : { a // OrderDual.toDual a = k },
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ i) (M.toSquareBlock (⇑OrderDual.toDual) k i ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular | [67, 1] | [75, 68] | rw [@Fintype.sum_unique _ _ _ this] at h | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
∑ i : { a // OrderDual.toDual a = k },
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ i) (M.toSquareBlock (⇑OrderDual.toDual) k i ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ default)
(M.toSquareBlock (⇑OrderDual.toDual) k default ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular | [67, 1] | [75, 68] | simp at h | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ default)
(M.toSquareBlock (⇑OrderDual.toDual) k default ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular | [67, 1] | [75, 68] | rw [← h] | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k = 1 | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k =
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular | [67, 1] | [75, 68] | simp [toSquareBlock, toSquareBlockProp] | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k =
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹ k k) (M k k) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean | Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular | [67, 1] | [75, 68] | rfl | α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹ k k) (M k k) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_zero | [20, 1] | [22, 39] | funext i j | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
⊢ toUpperTri 0 = 0 | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
i j : m
⊢ toUpperTri 0 i j = 0 i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_zero | [20, 1] | [22, 39] | simp [Matrix.toUpperTri] | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
i j : m
⊢ toUpperTri 0 i j = 0 i j | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_smul | [24, 1] | [28, 34] | funext i j | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
κ : ℝ
⊢ κ • A.toUpperTri = (κ • A).toUpperTri | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
κ : ℝ
i j : m
⊢ (κ • A.toUpperTri) i j = (κ • A).toUpperTri i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_smul | [24, 1] | [28, 34] | rw [Pi.smul_apply, Pi.smul_apply] | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
κ : ℝ
i j : m
⊢ (κ • A.toUpperTri) i j = (κ • A).toUpperTri i j | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
κ : ℝ
i j : m
⊢ κ • A.toUpperTri i j = (κ • A).toUpperTri i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_smul | [24, 1] | [28, 34] | simp only [Matrix.toUpperTri] | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
κ : ℝ
i j : m
⊢ κ • A.toUpperTri i j = (κ • A).toUpperTri i j | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
κ : ℝ
i j : m
⊢ (κ • if i ≤ j then A i j else 0) = if i ≤ j then (κ • A) i j else 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_smul | [24, 1] | [28, 34] | by_cases h : i ≤ j <;> simp [h] | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
κ : ℝ
i j : m
⊢ (κ • if i ≤ j then A i j else 0) = if i ≤ j then (κ • A) i j else 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_add | [30, 1] | [34, 34] | funext i j | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A B : Matrix m m ℝ
⊢ (A + B).toUpperTri = A.toUpperTri + B.toUpperTri | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A B : Matrix m m ℝ
i j : m
⊢ (A + B).toUpperTri i j = (A.toUpperTri + B.toUpperTri) i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_add | [30, 1] | [34, 34] | rw [Pi.add_apply, Pi.add_apply] | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A B : Matrix m m ℝ
i j : m
⊢ (A + B).toUpperTri i j = (A.toUpperTri + B.toUpperTri) i j | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A B : Matrix m m ℝ
i j : m
⊢ (A + B).toUpperTri i j = A.toUpperTri i j + B.toUpperTri i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_add | [30, 1] | [34, 34] | simp only [Matrix.toUpperTri] | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A B : Matrix m m ℝ
i j : m
⊢ (A + B).toUpperTri i j = A.toUpperTri i j + B.toUpperTri i j | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A B : Matrix m m ℝ
i j : m
⊢ (if i ≤ j then (A + B) i j else 0) = (if i ≤ j then A i j else 0) + if i ≤ j then B i j else 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.toUpperTri_add | [30, 1] | [34, 34] | by_cases h : i ≤ j <;> simp [h] | case h.h
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A B : Matrix m m ℝ
i j : m
⊢ (if i ≤ j then (A + B) i j else 0) = (if i ≤ j then A i j else 0) + if i ≤ j then B i j else 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.upperTriangular_toUpperTri | [36, 1] | [41, 18] | intros i j hij | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
⊢ A.toUpperTri.upperTriangular | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
i j : m
hij : id j < id i
⊢ A.toUpperTri i j = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.upperTriangular_toUpperTri | [36, 1] | [41, 18] | unfold toUpperTri | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
i j : m
hij : id j < id i
⊢ A.toUpperTri i j = 0 | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
i j : m
hij : id j < id i
⊢ (if i ≤ j then A i j else 0) = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.upperTriangular_toUpperTri | [36, 1] | [41, 18] | rw [if_neg] | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
i j : m
hij : id j < id i
⊢ (if i ≤ j then A i j else 0) = 0 | case hnc
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
i j : m
hij : id j < id i
⊢ ¬i ≤ j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.upperTriangular_toUpperTri | [36, 1] | [41, 18] | simpa using hij | case hnc
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
m : Type u_1
inst✝ : LinearOrder m
A : Matrix m m ℝ
i j : m
hij : id j < id i
⊢ ¬i ≤ j | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.upperTriangular.toUpperTri_eq | [43, 1] | [48, 44] | ext i j | n : Type
inst✝² : Fintype n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.upperTriangular
⊢ A.toUpperTri = A | case a
n : Type
inst✝² : Fintype n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.upperTriangular
i j : n
⊢ A.toUpperTri i j = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.upperTriangular.toUpperTri_eq | [43, 1] | [48, 44] | by_cases h : i ≤ j | case a
n : Type
inst✝² : Fintype n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.upperTriangular
i j : n
⊢ A.toUpperTri i j = A i j | case pos
n : Type
inst✝² : Fintype n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.upperTriangular
i j : n
h : i ≤ j
⊢ A.toUpperTri i j = A i j
case neg
n : Type
inst✝² : Fintype n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.upperTriangular
i j : n
h : ¬i ≤ j
⊢ A.toUpperTri i j = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.upperTriangular.toUpperTri_eq | [43, 1] | [48, 44] | simp [toUpperTri, h] | case pos
n : Type
inst✝² : Fintype n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.upperTriangular
i j : n
h : i ≤ j
⊢ A.toUpperTri i j = A i j
case neg
n : Type
inst✝² : Fintype n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.upperTriangular
i j : n
h : ¬i ≤ j
⊢ A.toUpperTri i j = A i j | case neg
n : Type
inst✝² : Fintype n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.upperTriangular
i j : n
h : ¬i ≤ j
⊢ A.toUpperTri i j = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/ToUpperTri.lean | Matrix.upperTriangular.toUpperTri_eq | [43, 1] | [48, 44] | simp [toUpperTri, h, hA (lt_of_not_ge h)] | case neg
n : Type
inst✝² : Fintype n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.upperTriangular
i j : n
h : ¬i ≤ j
⊢ A.toUpperTri i j = A i j | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lowerInv_diagonal | [23, 1] | [29, 24] | simp only [lowerInv_eq_gramSchmidtBasis, gramSchmidtBasis] | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
⊢ lowerInv hS i i = 1 | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lowerInv_diagonal | [23, 1] | [29, 24] | letI := NormedAddCommGroup.ofMatrix hS.transpose | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1 | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lowerInv_diagonal | [23, 1] | [29, 24] | letI := InnerProductSpace.ofMatrix hS.transpose | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1 | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lowerInv_diagonal | [23, 1] | [29, 24] | rw [Basis.coe_mk, ← @repr_gramSchmidt_diagonal 𝕜 (n → 𝕜) _ _ _ n _ _ _ i (Pi.basisFun 𝕜 n)] | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1 | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))).transpose i i =
((Pi.basisFun 𝕜 n).repr (gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i)) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lowerInv_diagonal | [23, 1] | [29, 24] | simp [Basis.toMatrix] | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))).transpose i i =
((Pi.basisFun 𝕜 n).repr (gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i)) i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lower_eq_to_matrix | [31, 1] | [39, 72] | simp only [lower, lowerInv_eq_gramSchmidtBasis] | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ lower hS = ((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n))).transpose⁻¹ =
((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lower_eq_to_matrix | [31, 1] | [39, 72] | apply Matrix.inv_eq_left_inv | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n))).transpose⁻¹ =
((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose | case h
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ ((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose *
((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n))).transpose =
1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lower_eq_to_matrix | [31, 1] | [39, 72] | rw [← transpose_mul, Basis.toMatrix_mul_toMatrix_flip, transpose_one] | case h
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ ((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose *
((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n))).transpose =
1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lowerTriangular_lowerInv | [41, 1] | [42, 28] | apply lowerInv_triangular | 𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
n : Type u_1
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ (lowerInv hS).lowerTriangular | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.lower_diagonal | [54, 1] | [57, 77] | simpa using
diag_inv_mul_diag_eq_one_of_lowerTriangular (lowerTriangular_lower hS) i | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
⊢ lower hS i i = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.det_lowerInv | [59, 1] | [64, 25] | have h := det_of_lowerTriangular (lowerInv hS) (by apply lowerInv_triangular) | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ (lowerInv hS).det = 1 | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
⊢ (lowerInv hS).det = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.det_lowerInv | [59, 1] | [64, 25] | rw [h, Finset.prod_eq_one] | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
⊢ (lowerInv hS).det = 1 | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
⊢ ∀ x ∈ Finset.univ, lowerInv hS x x = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.det_lowerInv | [59, 1] | [64, 25] | intros | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
⊢ ∀ x ∈ Finset.univ, lowerInv hS x x = 1 | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
x✝ : n
a✝ : x✝ ∈ Finset.univ
⊢ lowerInv hS x✝ x✝ = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.det_lowerInv | [59, 1] | [64, 25] | rw [lowerInv_diagonal] | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
x✝ : n
a✝ : x✝ ∈ Finset.univ
⊢ lowerInv hS x✝ x✝ = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.det_lowerInv | [59, 1] | [64, 25] | apply lowerInv_triangular | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ (lowerInv hS).BlockTriangular ⇑OrderDual.toDual | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean | LDL.det_lower | [66, 1] | [68, 15] | simp [lower] | 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ (lower hS).det = 1 | 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 |
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