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stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
11.2k
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stringlengths 3
2.09M
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stringlengths 6
2.09M
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https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/Diagonalize.lean | QuadraticForm.anisotropic_of_Equivalent | [71, 1] | [81, 16] | simp only [Isometry.map_app, AddEquivClass.map_eq_zero_iff] at h | case intro
ι : Type ?u.84107
R✝ : Type ?u.84110
K : Type ?u.84113
M✝ : Type ?u.84116
M₁ : Type ?u.84119
M₂ : Type u_1
M₃ : Type ?u.84125
V : Type ?u.84128
R : Type u_2
M : Type u_3
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : FiniteDimensional R M
inst✝² : Invertible 2
Q✝ : QuadraticForm R M
inst✝¹ : AddCommGroup M₂
inst✝ : Module R M₂
Q : QuadraticForm R M
S : QuadraticForm R M₂
val : Isometry Q S
x : M₂
hx : ↑S x = 0
h : ↑Q (↑(Isometry.symm val) x) = 0 → ↑(Isometry.symm val) x = 0
⊢ x = 0 | case intro
ι : Type ?u.84107
R✝ : Type ?u.84110
K : Type ?u.84113
M✝ : Type ?u.84116
M₁ : Type ?u.84119
M₂ : Type u_1
M₃ : Type ?u.84125
V : Type ?u.84128
R : Type u_2
M : Type u_3
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : FiniteDimensional R M
inst✝² : Invertible 2
Q✝ : QuadraticForm R M
inst✝¹ : AddCommGroup M₂
inst✝ : Module R M₂
Q : QuadraticForm R M
S : QuadraticForm R M₂
val : Isometry Q S
x : M₂
hx : ↑S x = 0
h : ↑S x = 0 → x = 0
⊢ x = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/Diagonalize.lean | QuadraticForm.anisotropic_of_Equivalent | [71, 1] | [81, 16] | specialize h hx | case intro
ι : Type ?u.84107
R✝ : Type ?u.84110
K : Type ?u.84113
M✝ : Type ?u.84116
M₁ : Type ?u.84119
M₂ : Type u_1
M₃ : Type ?u.84125
V : Type ?u.84128
R : Type u_2
M : Type u_3
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : FiniteDimensional R M
inst✝² : Invertible 2
Q✝ : QuadraticForm R M
inst✝¹ : AddCommGroup M₂
inst✝ : Module R M₂
Q : QuadraticForm R M
S : QuadraticForm R M₂
val : Isometry Q S
x : M₂
hx : ↑S x = 0
h : ↑S x = 0 → x = 0
⊢ x = 0 | case intro
ι : Type ?u.84107
R✝ : Type ?u.84110
K : Type ?u.84113
M✝ : Type ?u.84116
M₁ : Type ?u.84119
M₂ : Type u_1
M₃ : Type ?u.84125
V : Type ?u.84128
R : Type u_2
M : Type u_3
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : FiniteDimensional R M
inst✝² : Invertible 2
Q✝ : QuadraticForm R M
inst✝¹ : AddCommGroup M₂
inst✝ : Module R M₂
Q : QuadraticForm R M
S : QuadraticForm R M₂
val : Isometry Q S
x : M₂
hx : ↑S x = 0
h : x = 0
⊢ x = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/Diagonalize.lean | QuadraticForm.anisotropic_of_Equivalent | [71, 1] | [81, 16] | simpa using h | case intro
ι : Type ?u.84107
R✝ : Type ?u.84110
K : Type ?u.84113
M✝ : Type ?u.84116
M₁ : Type ?u.84119
M₂ : Type u_1
M₃ : Type ?u.84125
V : Type ?u.84128
R : Type u_2
M : Type u_3
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : FiniteDimensional R M
inst✝² : Invertible 2
Q✝ : QuadraticForm R M
inst✝¹ : AddCommGroup M₂
inst✝ : Module R M₂
Q : QuadraticForm R M
S : QuadraticForm R M₂
val : Isometry Q S
x : M₂
hx : ↑S x = 0
h : x = 0
⊢ x = 0 | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/Diagonalize.lean | QuadraticForm.equivalent_weightedSumSquares_units_of_nondegenerate'' | [109, 1] | [116, 10] | sorry | ι : Type ?u.120511
R✝ : Type ?u.120514
K : Type ?u.120517
M✝ : Type ?u.120520
M₁ : Type ?u.120523
M₂ : Type ?u.120526
M₃ : Type ?u.120529
V : Type u_1
R : Type ?u.120535
M : Type ?u.120538
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : FiniteDimensional R M
inst✝² : Invertible 2
Q✝ : QuadraticForm R M
inst✝¹ : AddCommGroup V
inst✝ : Module ℚ V
a : ℕ
Q : QuadraticForm ℚ V
h : 0 < FiniteDimensional.finrank ℚ V
ha : a = FiniteDimensional.finrank ℚ V
hQ : BilinForm.Nondegenerate (↑associated Q)
⊢ ∃ w hw1 hw0, Equivalent Q (weightedSumSquares ℚ w) | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Isotropic_of_zero_quadForm_dim_ge1 | [169, 1] | [179, 8] | rw [QuadraticForm.Isotropic] | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
⊢ Isotropic Q | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
⊢ ¬Anisotropic Q |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Isotropic_of_zero_quadForm_dim_ge1 | [169, 1] | [179, 8] | rw [QuadraticForm.Anisotropic] | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
⊢ ¬Anisotropic Q | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
⊢ ¬∀ (x : W), ↑Q x = 0 → x = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Isotropic_of_zero_quadForm_dim_ge1 | [169, 1] | [179, 8] | have h: ∃ (w : W), w ≠ 0 := by
simpa only [ne_eq, rank_zero_iff_forall_zero, not_forall] using h₂ | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
⊢ ¬∀ (x : W), ↑Q x = 0 → x = 0 | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
h : ∃ w, w ≠ 0
⊢ ¬∀ (x : W), ↑Q x = 0 → x = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Isotropic_of_zero_quadForm_dim_ge1 | [169, 1] | [179, 8] | obtain ⟨w, hw⟩ := h | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
h : ∃ w, w ≠ 0
⊢ ¬∀ (x : W), ↑Q x = 0 → x = 0 | case intro
V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
w : W
hw : w ≠ 0
⊢ ¬∀ (x : W), ↑Q x = 0 → x = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Isotropic_of_zero_quadForm_dim_ge1 | [169, 1] | [179, 8] | have : Q w = 0 := by
rw [h₁]
simp only [zero_apply] | case intro
V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
w : W
hw : w ≠ 0
⊢ ¬∀ (x : W), ↑Q x = 0 → x = 0 | case intro
V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
w : W
hw : w ≠ 0
this : ↑Q w = 0
⊢ ¬∀ (x : W), ↑Q x = 0 → x = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Isotropic_of_zero_quadForm_dim_ge1 | [169, 1] | [179, 8] | tauto | case intro
V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
w : W
hw : w ≠ 0
this : ↑Q w = 0
⊢ ¬∀ (x : W), ↑Q x = 0 → x = 0 | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Isotropic_of_zero_quadForm_dim_ge1 | [169, 1] | [179, 8] | simpa only [ne_eq, rank_zero_iff_forall_zero, not_forall] using h₂ | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
⊢ ∃ w, w ≠ 0 | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Isotropic_of_zero_quadForm_dim_ge1 | [169, 1] | [179, 8] | rw [h₁] | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
w : W
hw : w ≠ 0
⊢ ↑Q w = 0 | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
w : W
hw : w ≠ 0
⊢ ↑0 w = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Isotropic_of_zero_quadForm_dim_ge1 | [169, 1] | [179, 8] | simp only [zero_apply] | V : Type ?u.14055
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h₁ : Q = 0
h₂ : Module.rank k W ≠ 0
w : W
hw : w ≠ 0
⊢ ↑0 w = 0 | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.anisotropic_of_quadForm_dim_zero | [184, 1] | [189, 13] | intro (w : W) | V : Type ?u.23522
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h : Module.rank k W = 0
⊢ Anisotropic Q | V : Type ?u.23522
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h : Module.rank k W = 0
w : W
⊢ ↑Q w = 0 → w = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.anisotropic_of_quadForm_dim_zero | [184, 1] | [189, 13] | intro | V : Type ?u.23522
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h : Module.rank k W = 0
w : W
⊢ ↑Q w = 0 → w = 0 | V : Type ?u.23522
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h : Module.rank k W = 0
w : W
a✝ : ↑Q w = 0
⊢ w = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.anisotropic_of_quadForm_dim_zero | [184, 1] | [189, 13] | rw [rank_zero_iff_forall_zero] at h | V : Type ?u.23522
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h : Module.rank k W = 0
w : W
a✝ : ↑Q w = 0
⊢ w = 0 | V : Type ?u.23522
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h : ∀ (x : W), x = 0
w : W
a✝ : ↑Q w = 0
⊢ w = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.anisotropic_of_quadForm_dim_zero | [184, 1] | [189, 13] | exact h w | V : Type ?u.23522
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℚ V
inst✝³ : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝² : Field k
inst✝¹ : AddCommGroup W
inst✝ : Module k W
Q : QuadraticForm k W
h : ∀ (x : W), x = 0
w : W
a✝ : ↑Q w = 0
⊢ w = 0 | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | intro F | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
⊢ ∀ (F : QuadraticForm ℚ V), Hasse_Minkowski F | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ Hasse_Minkowski F |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | rw [Hasse_Minkowski] | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ Hasse_Minkowski F | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ Isotropic F ↔ EverywhereLocallyIsotropic F |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | constructor | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ Isotropic F ↔ EverywhereLocallyIsotropic F | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ Isotropic F → EverywhereLocallyIsotropic F
case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ EverywhereLocallyIsotropic F → Isotropic F |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | contrapose | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ Isotropic F → EverywhereLocallyIsotropic F | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ ¬EverywhereLocallyIsotropic F → ¬Isotropic F |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | intro | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ ¬EverywhereLocallyIsotropic F → ¬Isotropic F | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬EverywhereLocallyIsotropic F
⊢ ¬Isotropic F |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | rw [QuadraticForm.Isotropic] | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬EverywhereLocallyIsotropic F
⊢ ¬Isotropic F | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬EverywhereLocallyIsotropic F
⊢ ¬¬Anisotropic F |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | simp only [not_not] | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬EverywhereLocallyIsotropic F
⊢ ¬¬Anisotropic F | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬EverywhereLocallyIsotropic F
⊢ Anisotropic F |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | apply anisotropic_of_quadForm_dim_zero _ _ F hV | case mp
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬EverywhereLocallyIsotropic F
⊢ Anisotropic F | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | contrapose | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ EverywhereLocallyIsotropic F → Isotropic F | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ ¬Isotropic F → ¬EverywhereLocallyIsotropic F |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | intro | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
⊢ ¬Isotropic F → ¬EverywhereLocallyIsotropic F | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
⊢ ¬EverywhereLocallyIsotropic F |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | rw [QuadraticForm.EverywhereLocallyIsotropic] | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
⊢ ¬EverywhereLocallyIsotropic F | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
⊢ ¬((∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)) ∧ Isotropic (baseChange ℝ F)) |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | push_neg | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
⊢ ¬((∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)) ∧ Isotropic (baseChange ℝ F)) | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
⊢ (∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)) → ¬Isotropic (baseChange ℝ F) |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | intro | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
⊢ (∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)) → ¬Isotropic (baseChange ℝ F) | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
_✝ : ∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)
⊢ ¬Isotropic (baseChange ℝ F) |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | simp only [not_not] | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
_✝ : ∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)
⊢ ¬Isotropic (baseChange ℝ F) | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
_✝ : ∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)
⊢ Anisotropic (baseChange ℝ F) |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | apply anisotropic_of_quadForm_dim_zero | case mpr
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
_✝ : ∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)
⊢ Anisotropic (baseChange ℝ F) | case mpr.h
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
_✝ : ∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)
⊢ Module.rank ℝ (TensorProduct ℚ ℝ V) = 0 |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski0 | [191, 1] | [207, 50] | rw [← base_change_module_rank_preserved, hV] | case mpr.h
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F✝ : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
hV : Module.rank ℚ V = 0
F : QuadraticForm ℚ V
a✝ : ¬Isotropic F
_✝ : ∀ (p : ℕ) [inst : Fact (Nat.Prime p)], Isotropic (baseChange ℚ_[p] F)
⊢ Module.rank ℝ (TensorProduct ℚ ℝ V) = 0 | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski_of_Equivalent | [220, 1] | [226, 74] | simp only [Hasse_Minkowski, Isotropic, EverywhereLocallyIsotropic] at * | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q S : QuadraticForm ℚ V
h : Equivalent Q S
⊢ Hasse_Minkowski Q ↔ Hasse_Minkowski S | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q S : QuadraticForm ℚ V
h : Equivalent Q S
⊢ (¬Anisotropic Q ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] Q)) ∧ ¬Anisotropic (baseChange ℝ Q)) ↔
(¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] S)) ∧ ¬Anisotropic (baseChange ℝ S)) |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski_of_Equivalent | [220, 1] | [226, 74] | simp only [anisotropic_iff _ _ h] | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q S : QuadraticForm ℚ V
h : Equivalent Q S
⊢ (¬Anisotropic Q ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] Q)) ∧ ¬Anisotropic (baseChange ℝ Q)) ↔
(¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] S)) ∧ ¬Anisotropic (baseChange ℝ S)) | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q S : QuadraticForm ℚ V
h : Equivalent Q S
⊢ (¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] Q)) ∧ ¬Anisotropic (baseChange ℝ Q)) ↔
(¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] S)) ∧ ¬Anisotropic (baseChange ℝ S)) |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski_of_Equivalent | [220, 1] | [226, 74] | rw [anisotropic_iff _ _ (baseChange.Equivalent ℝ _ _ h)] | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q S : QuadraticForm ℚ V
h : Equivalent Q S
⊢ (¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] Q)) ∧ ¬Anisotropic (baseChange ℝ Q)) ↔
(¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] S)) ∧ ¬Anisotropic (baseChange ℝ S)) | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q S : QuadraticForm ℚ V
h : Equivalent Q S
⊢ (¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] Q)) ∧ ¬Anisotropic (baseChange ℝ S)) ↔
(¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] S)) ∧ ¬Anisotropic (baseChange ℝ S)) |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.Hasse_Minkowski_of_Equivalent | [220, 1] | [226, 74] | conv in (Anisotropic (baseChange _ Q)) =>
rw [anisotropic_iff _ _ (baseChange.Equivalent (R := ℚ) ℚ_[p] _ _ h)] | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q S : QuadraticForm ℚ V
h : Equivalent Q S
⊢ (¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] Q)) ∧ ¬Anisotropic (baseChange ℝ S)) ↔
(¬Anisotropic S ↔
(∀ (p : ℕ) [inst : Fact (Nat.Prime p)], ¬Anisotropic (baseChange ℚ_[p] S)) ∧ ¬Anisotropic (baseChange ℝ S)) | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.HasseMinkowski_of_degenerate | [228, 1] | [230, 8] | sorry | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q : QuadraticForm ℚ V
hQ : ¬BilinForm.Nondegenerate (↑associated Q)
⊢ Hasse_Minkowski Q | no goals |
https://github.com/alexjbest/ant-lorentz.git | 1f97add294b2d50f99537c15583666d78b0d7e24 | AntLorentz/HasseMinkowski2.lean | QuadraticForm.ex | [235, 1] | [246, 13] | obtain ⟨w, hw1, hw0, hEQ⟩ := equivalent_weightedSumSquares_units_of_nondegenerate'' Q _ h.symm hQ | V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q : QuadraticForm ℚ V
h : FiniteDimensional.finrank ℚ V = 2
hQ : BilinForm.Nondegenerate (↑associated Q)
⊢ Hasse_Minkowski Q | case intro.intro.intro
V : Type
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℚ V
inst✝² : FiniteDimensional ℚ V
F : QuadraticForm ℚ V
k W : Type
inst✝¹ : Field k
inst✝ : AddCommGroup W
Q : QuadraticForm ℚ V
h : FiniteDimensional.finrank ℚ V = 2
hQ : BilinForm.Nondegenerate (↑associated Q)
w : Fin 2 → ℤ
hw1 : w { val := 0, isLt := (_ : 0 < 2) } = 1
hw0 : ∀ (i : Fin 2), Squarefree (w i)
hEQ : Equivalent Q (weightedSumSquares ℚ w)
⊢ Hasse_Minkowski Q |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [301, 1] | [307, 7] | rcases sosx with ⟨a, b, xeq⟩ | α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y) | case intro.intro
α : Type u_1
inst✝ : CommRing α
x y : α
sosy : SumOfSquares y
a b : α
xeq : x = a ^ 2 + b ^ 2
⊢ SumOfSquares (x * y) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [301, 1] | [307, 7] | rcases sosy with ⟨c, d, yeq⟩ | case intro.intro
α : Type u_1
inst✝ : CommRing α
x y : α
sosy : SumOfSquares y
a b : α
xeq : x = a ^ 2 + b ^ 2
⊢ SumOfSquares (x * y) | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares (x * y) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [301, 1] | [307, 7] | rw [xeq, yeq] | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares (x * y) | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [301, 1] | [307, 7] | use a * c - b * d, a * d + b * c | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | case h
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul | [301, 1] | [307, 7] | ring | case h
α : Type u_1
inst✝ : CommRing α
x y a b : α
xeq : x = a ^ 2 + b ^ 2
c d : α
yeq : y = c ^ 2 + d ^ 2
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [341, 1] | [346, 7] | rcases sosx with ⟨a, b, rfl⟩ | α : Type u_1
inst✝ : CommRing α
x y : α
sosx : SumOfSquares x
sosy : SumOfSquares y
⊢ SumOfSquares (x * y) | case intro.intro
α : Type u_1
inst✝ : CommRing α
y : α
sosy : SumOfSquares y
a b : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * y) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [341, 1] | [346, 7] | rcases sosy with ⟨c, d, rfl⟩ | case intro.intro
α : Type u_1
inst✝ : CommRing α
y : α
sosy : SumOfSquares y
a b : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * y) | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [341, 1] | [346, 7] | use a * c - b * d, a * d + b * c | case intro.intro.intro.intro
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ SumOfSquares ((a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2)) | case h
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S02_The_Existential_Quantifier.lean | C03S02.sumOfSquares_mul' | [341, 1] | [346, 7] | ring | case h
α : Type u_1
inst✝ : CommRing α
a b c d : α
⊢ (a ^ 2 + b ^ 2) * (c ^ 2 + d ^ 2) = (a * c - b * d) ^ 2 + (a * d + b * c) ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma3 | [99, 1] | [102, 8] | intro x y ε epos ele1 xlt ylt | ⊢ ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma3 | [99, 1] | [102, 8] | sorry | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | intro x y ε epos ele1 xlt ylt | ⊢ ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | calc
|x * y| = |x| * |y| := by apply abs_mul
_ ≤ |x| * ε := by apply mul_le_mul; linarith; linarith; apply abs_nonneg; apply abs_nonneg;
_ < 1 * ε := by rw [mul_lt_mul_right epos]; linarith
_ = ε := by apply one_mul | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| < ε | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | apply abs_mul | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x * y| = |x| * |y| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | apply mul_le_mul | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| * |y| ≤ |x| * ε | case h₁
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| ≤ |x|
case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | linarith | case h₁
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| ≤ |x|
case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | linarith | case h₂
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |y| ≤ ε
case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | apply abs_nonneg | case c0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |y|
case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | apply abs_nonneg | case b0
x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 0 ≤ |x| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | rw [mul_lt_mul_right epos] | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| * ε < 1 * ε | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| < 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | linarith | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ |x| < 1 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [125, 1] | [140, 30] | apply one_mul | x y ε : ℝ
epos : 0 < ε
ele1 : ε ≤ 1
xlt : |x| < ε
ylt : |y| < ε
⊢ 1 * ε = ε | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.Subset.trans | [506, 1] | [507, 8] | sorry | α : Type u_1
r s t : Set α
⊢ r ⊆ s → s ⊆ t → r ⊆ t | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [196, 1] | [204, 21] | apply le_min | a b c d : ℝ
⊢ min a b + c ≤ min (a + c) (b + c) | case h₁
a b c d : ℝ
⊢ min a b + c ≤ a + c
case h₂
a b c d : ℝ
⊢ min a b + c ≤ b + c |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [196, 1] | [204, 21] | apply add_le_add_right | case h₂
a b c d : ℝ
⊢ min a b + c ≤ b + c | case h₂.bc
a b c d : ℝ
⊢ min a b ≤ b |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [196, 1] | [204, 21] | apply min_le_right | case h₂.bc
a b c d : ℝ
⊢ min a b ≤ b | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [196, 1] | [204, 21] | apply add_le_add_right | case h₁
a b c d : ℝ
⊢ min a b + c ≤ a + c | case h₁.bc
a b c d : ℝ
⊢ min a b ≤ a |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S04_More_on_Order_and_Divisibility.lean | C02S04.aux | [196, 1] | [204, 21] | apply min_le_left | case h₁.bc
a b c d : ℝ
⊢ min a b ≤ a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.div'_add_mod' | [382, 1] | [384, 45] | rw [div', mod'] | a b : ℤ
⊢ b * a.div' b + a.mod' b = a | a b : ℤ
⊢ b * ((a + b / 2) / b) + ((a + b / 2) % b - b / 2) = a |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.div'_add_mod' | [382, 1] | [384, 45] | linarith [Int.ediv_add_emod (a + b / 2) b] | a b : ℤ
⊢ b * ((a + b / 2) / b) + ((a + b / 2) % b - b / 2) = a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.abs_mod'_le | [386, 1] | [394, 11] | rw [mod', abs_le] | a b : ℤ
h : 0 < b
⊢ |a.mod' b| ≤ b / 2 | a b : ℤ
h : 0 < b
⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2 ∧ (a + b / 2) % b - b / 2 ≤ b / 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.abs_mod'_le | [386, 1] | [394, 11] | constructor | a b : ℤ
h : 0 < b
⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2 ∧ (a + b / 2) % b - b / 2 ≤ b / 2 | case left
a b : ℤ
h : 0 < b
⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2
case right
a b : ℤ
h : 0 < b
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.abs_mod'_le | [386, 1] | [394, 11] | have := Int.emod_lt_of_pos (a + b / 2) h | case right
a b : ℤ
h : 0 < b
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 | case right
a b : ℤ
h : 0 < b
this : (a + b / 2) % b < b
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.abs_mod'_le | [386, 1] | [394, 11] | have := Int.ediv_add_emod b 2 | case right
a b : ℤ
h : 0 < b
this : (a + b / 2) % b < b
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 | case right
a b : ℤ
h : 0 < b
this✝ : (a + b / 2) % b < b
this : 2 * (b / 2) + b % 2 = b
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.abs_mod'_le | [386, 1] | [394, 11] | have := Int.emod_lt_of_pos b zero_lt_two | case right
a b : ℤ
h : 0 < b
this✝ : (a + b / 2) % b < b
this : 2 * (b / 2) + b % 2 = b
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 | case right
a b : ℤ
h : 0 < b
this✝¹ : (a + b / 2) % b < b
this✝ : 2 * (b / 2) + b % 2 = b
this : b % 2 < 2
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.abs_mod'_le | [386, 1] | [394, 11] | revert this | case right
a b : ℤ
h : 0 < b
this✝¹ : (a + b / 2) % b < b
this✝ : 2 * (b / 2) + b % 2 = b
this : b % 2 < 2
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 | case right
a b : ℤ
h : 0 < b
this✝ : (a + b / 2) % b < b
this : 2 * (b / 2) + b % 2 = b
⊢ b % 2 < 2 → (a + b / 2) % b - b / 2 ≤ b / 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.abs_mod'_le | [386, 1] | [394, 11] | intro this | case right
a b : ℤ
h : 0 < b
this✝ : (a + b / 2) % b < b
this : 2 * (b / 2) + b % 2 = b
⊢ b % 2 < 2 → (a + b / 2) % b - b / 2 ≤ b / 2 | case right
a b : ℤ
h : 0 < b
this✝¹ : (a + b / 2) % b < b
this✝ : 2 * (b / 2) + b % 2 = b
this : b % 2 < 2
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.abs_mod'_le | [386, 1] | [394, 11] | linarith | case right
a b : ℤ
h : 0 < b
this✝¹ : (a + b / 2) % b < b
this✝ : 2 * (b / 2) + b % 2 = b
this : b % 2 < 2
⊢ (a + b / 2) % b - b / 2 ≤ b / 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.abs_mod'_le | [386, 1] | [394, 11] | linarith [Int.emod_nonneg (a + b / 2) h.ne'] | case left
a b : ℤ
h : 0 < b
⊢ -(b / 2) ≤ (a + b / 2) % b - b / 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | Int.mod'_eq | [402, 1] | [402, 91] | linarith [div'_add_mod' a b] | a b : ℤ
⊢ a.mod' b = a - b * a.div' b | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | aux | [412, 9] | [417, 17] | apply le_antisymm _ (sq_nonneg x) | α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ x ^ 2 = 0 | α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ x ^ 2 ≤ 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | aux | [412, 9] | [417, 17] | rw [← h] | α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ x ^ 2 ≤ 0 | α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ x ^ 2 ≤ x ^ 2 + y ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | aux | [412, 9] | [417, 17] | apply le_add_of_nonneg_right (sq_nonneg y) | α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ x ^ 2 ≤ x ^ 2 + y ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | sq_add_sq_eq_zero | [421, 1] | [433, 11] | constructor | α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
⊢ x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 | case mp
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
⊢ x ^ 2 + y ^ 2 = 0 → x = 0 ∧ y = 0
case mpr
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
⊢ x = 0 ∧ y = 0 → x ^ 2 + y ^ 2 = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | sq_add_sq_eq_zero | [421, 1] | [433, 11] | rintro ⟨rfl, rfl⟩ | case mpr
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
⊢ x = 0 ∧ y = 0 → x ^ 2 + y ^ 2 = 0 | case mpr.intro
α : Type u_1
inst✝ : LinearOrderedRing α
⊢ 0 ^ 2 + 0 ^ 2 = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | sq_add_sq_eq_zero | [421, 1] | [433, 11] | norm_num | case mpr.intro
α : Type u_1
inst✝ : LinearOrderedRing α
⊢ 0 ^ 2 + 0 ^ 2 = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | sq_add_sq_eq_zero | [421, 1] | [433, 11] | intro h | case mp
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
⊢ x ^ 2 + y ^ 2 = 0 → x = 0 ∧ y = 0 | case mp
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ x = 0 ∧ y = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | sq_add_sq_eq_zero | [421, 1] | [433, 11] | constructor | case mp
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ x = 0 ∧ y = 0 | case mp.left
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ x = 0
case mp.right
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ y = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | sq_add_sq_eq_zero | [421, 1] | [433, 11] | rw [add_comm] at h | case mp.right
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ y = 0 | case mp.right
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : y ^ 2 + x ^ 2 = 0
⊢ y = 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | sq_add_sq_eq_zero | [421, 1] | [433, 11] | exact aux h | case mp.right
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : y ^ 2 + x ^ 2 = 0
⊢ y = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | sq_add_sq_eq_zero | [421, 1] | [433, 11] | exact aux h | case mp.left
α : Type u_1
inst✝ : LinearOrderedRing α
x y : α
h : x ^ 2 + y ^ 2 = 0
⊢ x = 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_nonneg | [451, 1] | [456, 18] | apply add_nonneg <;>
apply sq_nonneg | x : GaussInt
⊢ 0 ≤ x.norm | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_eq_zero | [459, 1] | [464, 6] | rw [norm, sq_add_sq_eq_zero, GaussInt.ext_iff] | x : GaussInt
⊢ x.norm = 0 ↔ x = 0 | x : GaussInt
⊢ x.re = 0 ∧ x.im = 0 ↔ x.re = re 0 ∧ x.im = im 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_eq_zero | [459, 1] | [464, 6] | rfl | x : GaussInt
⊢ x.re = 0 ∧ x.im = 0 ↔ x.re = re 0 ∧ x.im = im 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_pos | [467, 1] | [472, 21] | rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] | x : GaussInt
⊢ 0 < x.norm ↔ x ≠ 0 | x : GaussInt
⊢ 0 ≤ x.norm ∧ ¬x = 0 ↔ x ≠ 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_pos | [467, 1] | [472, 21] | simp [norm_nonneg] | x : GaussInt
⊢ 0 ≤ x.norm ∧ ¬x = 0 ↔ x ≠ 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mul | [475, 1] | [480, 7] | simp [norm] | x y : GaussInt
⊢ (x * y).norm = x.norm * y.norm | x y : GaussInt
⊢ (x.re * y.re - x.im * y.im) ^ 2 + (x.re * y.im + x.im * y.re) ^ 2 = (x.re ^ 2 + x.im ^ 2) * (y.re ^ 2 + y.im ^ 2) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mul | [475, 1] | [480, 7] | ring | x y : GaussInt
⊢ (x.re * y.re - x.im * y.im) ^ 2 + (x.re * y.im + x.im * y.re) ^ 2 = (x.re ^ 2 + x.im ^ 2) * (y.re ^ 2 + y.im ^ 2) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_conj | [499, 1] | [499, 76] | simp [norm] | x : GaussInt
⊢ x.conj.norm = x.norm | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | have norm_y_pos : 0 < norm y := by rwa [norm_pos] | x y : GaussInt
hy : y ≠ 0
⊢ (x % y).norm < y.norm | x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
⊢ (x % y).norm < y.norm |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | have H1 : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ | x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
⊢ (x % y).norm < y.norm | case H1
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
⊢ x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ (x % y).norm < y.norm |
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