url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
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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