Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 78 | 82 | theorem oangle_self (x : V) : o.oangle x x = 0 := by |
rw [oangle, kahler_apply_self, ← ofReal_pow]
convert QuotientAddGroup.mk_zero (AddSubgroup.zmultiples (2 * π))
apply arg_ofReal_of_nonneg
positivity
| 2,361 |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 86 | 87 | theorem left_ne_zero_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : x ≠ 0 := by |
rintro rfl; simp at h
| 2,361 |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 91 | 92 | theorem right_ne_zero_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : y ≠ 0 := by |
rintro rfl; simp at h
| 2,361 |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 96 | 97 | theorem ne_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : x ≠ y := by |
rintro rfl; simp at h
| 2,361 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 99 | 109 | theorem rotation_eq_matrix_toLin (θ : Real.Angle) {x : V} (hx : x ≠ 0) :
(o.rotation θ).toLinearMap =
Matrix.toLin (o.basisRightAngleRotation x hx) (o.basisRightAngleRotation x hx)
!![θ.cos, -θ.sin; θ.sin, θ.cos] := by |
apply (o.basisRightAngleRotation x hx).ext
intro i
fin_cases i
· rw [Matrix.toLin_self]
simp [rotation_apply, Fin.sum_univ_succ]
· rw [Matrix.toLin_self]
simp [rotation_apply, Fin.sum_univ_succ, add_comm]
| 2,362 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 114 | 119 | theorem det_rotation (θ : Real.Angle) : LinearMap.det (o.rotation θ).toLinearMap = 1 := by |
haveI : Nontrivial V :=
FiniteDimensional.nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
obtain ⟨x, hx⟩ : ∃ x, x ≠ (0 : V) := exists_ne (0 : V)
rw [o.rotation_eq_matrix_toLin θ hx]
simpa [sq] using θ.cos_sq_add_sin_sq
| 2,362 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 134 | 135 | theorem rotation_symm (θ : Real.Angle) : (o.rotation θ).symm = o.rotation (-θ) := by |
ext; simp [o.rotation_apply, o.rotation_symm_apply, sub_eq_add_neg]
| 2,362 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 140 | 140 | theorem rotation_zero : o.rotation 0 = LinearIsometryEquiv.refl ℝ V := by | ext; simp [rotation]
| 2,362 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 145 | 147 | theorem rotation_pi : o.rotation π = LinearIsometryEquiv.neg ℝ := by |
ext x
simp [rotation]
| 2,362 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 151 | 151 | theorem rotation_pi_apply (x : V) : o.rotation π x = -x := by | simp
| 2,362 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 155 | 157 | theorem rotation_pi_div_two : o.rotation (π / 2 : ℝ) = J := by |
ext x
simp [rotation]
| 2,362 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 49 | 55 | theorem continuousAt_oangle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∡ y.1 y.2.1 y.2.2) x := by |
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (o.continuousAt_oangle hf1 hf2).comp ((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd.fst)).continuousAt
| 2,363 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 60 | 60 | theorem oangle_self_left (p₁ p₂ : P) : ∡ p₁ p₁ p₂ = 0 := by | simp [oangle]
| 2,363 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 65 | 65 | theorem oangle_self_right (p₁ p₂ : P) : ∡ p₁ p₂ p₂ = 0 := by | simp [oangle]
| 2,363 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 75 | 76 | theorem left_ne_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ p₂ := by |
rw [← @vsub_ne_zero V]; exact o.left_ne_zero_of_oangle_ne_zero h
| 2,363 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 80 | 81 | theorem right_ne_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₃ ≠ p₂ := by |
rw [← @vsub_ne_zero V]; exact o.right_ne_zero_of_oangle_ne_zero h
| 2,363 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 85 | 86 | theorem left_ne_right_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ p₃ := by |
rw [← (vsub_left_injective p₂).ne_iff]; exact o.ne_of_oangle_ne_zero h
| 2,363 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#align_import geometry.euclidean.triangle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classica... | Mathlib/Geometry/Euclidean/Triangle.lean | 62 | 67 | theorem norm_sub_sq_eq_norm_sq_add_norm_sq_sub_two_mul_norm_mul_norm_mul_cos_angle (x y : V) :
‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - 2 * ‖x‖ * ‖y‖ * Real.cos (angle x y) := by |
rw [show 2 * ‖x‖ * ‖y‖ * Real.cos (angle x y) = 2 * (Real.cos (angle x y) * (‖x‖ * ‖y‖)) by ring,
cos_angle_mul_norm_mul_norm, ← real_inner_self_eq_norm_mul_norm, ←
real_inner_self_eq_norm_mul_norm, ← real_inner_self_eq_norm_mul_norm, real_inner_sub_sub_self,
sub_add_eq_add_sub]
| 2,364 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#align_import geometry.euclidean.triangle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classica... | Mathlib/Geometry/Euclidean/Triangle.lean | 71 | 75 | theorem angle_sub_eq_angle_sub_rev_of_norm_eq {x y : V} (h : ‖x‖ = ‖y‖) :
angle x (x - y) = angle y (y - x) := by |
refine Real.injOn_cos ⟨angle_nonneg _ _, angle_le_pi _ _⟩ ⟨angle_nonneg _ _, angle_le_pi _ _⟩ ?_
rw [cos_angle, cos_angle, h, ← neg_sub, norm_neg, neg_sub, inner_sub_right, inner_sub_right,
real_inner_self_eq_norm_mul_norm, real_inner_self_eq_norm_mul_norm, h, real_inner_comm x y]
| 2,364 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#align_import geometry.euclidean.triangle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classica... | Mathlib/Geometry/Euclidean/Triangle.lean | 79 | 104 | theorem norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi {x y : V}
(h : angle x (x - y) = angle y (y - x)) (hpi : angle x y ≠ π) : ‖x‖ = ‖y‖ := by |
replace h := Real.arccos_injOn (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x (x - y)))
(abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one y (y - x))) h
by_cases hxy : x = y
· rw [hxy]
· rw [← norm_neg (y - x), neg_sub, mul_comm, mul_comm ‖y‖, div_eq_mul_inv, div_eq_mul_inv,
mul_inv_rev, mul_... | 2,364 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#align_import geometry.euclidean.triangle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classica... | Mathlib/Geometry/Euclidean/Triangle.lean | 109 | 143 | theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by |
by_cases hxy : x = y
· rw [hxy, angle_self hy]
simp
· rw [Real.cos_add, cos_angle, cos_angle, cos_angle]
have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
app... | 2,364 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 36 | 42 | theorem oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 46 | 50 | theorem oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arccos (‖y‖ / ‖x + y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two h
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 54 | 61 | theorem oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_... | 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 65 | 69 | theorem oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arcsin (‖x‖ / ‖x + y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two h
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 73 | 79 | theorem oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arctan (‖y‖ / ‖x‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h) (o.left_ne_zero_of_oangle_eq_pi_div_two h)]
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 83 | 87 | theorem oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arctan (‖x‖ / ‖y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two h
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 91 | 96 | theorem cos_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.cos (o.oangle x (x + y)) = ‖x‖ / ‖x + y‖ := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
InnerProductGeometry.cos_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 584 | 588 | theorem oangle_right_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₂ p₃ p₁ = Real.arccos (dist p₃ p₂ / dist p₁ p₃) := by |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs,
angle_eq_arccos_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)]
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 592 | 597 | theorem oangle_left_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arccos (dist p₁ p₂ / dist p₁ p₃) := by |
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm,
angle_eq_arccos_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h),
dist_comm p₁ p₃]
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 601 | 606 | theorem oangle_right_eq_arcsin_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₂ p₃ p₁ = Real.arcsin (dist p₁ p₂ / dist p₁ p₃) := by |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs,
angle_eq_arcsin_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inl (left_ne_of_oangle_eq_pi_div_two h))]
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 610 | 616 | theorem oangle_left_eq_arcsin_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arcsin (dist p₃ p₂ / dist p₁ p₃) := by |
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm,
angle_eq_arcsin_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inr (left_ne_of_oangle_eq_pi_div_two h)),
dist_comm p₁ p... | 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 620 | 625 | theorem oangle_right_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₂ p₃ p₁ = Real.arctan (dist p₁ p₂ / dist p₃ p₂) := by |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs,
angle_eq_arctan_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(right_ne_of_oangle_eq_pi_div_two h)]
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 629 | 634 | theorem oangle_left_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arctan (dist p₃ p₂ / dist p₁ p₂) := by |
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm,
angle_eq_arctan_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(left_ne_of_oangle_eq_pi_div_two h)]
| 2,365 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 638 | 642 | theorem cos_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
Real.Angle.cos (∡ p₂ p₃ p₁) = dist p₃ p₂ / dist p₁ p₃ := by |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
cos_angle_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)]
| 2,365 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 109 | 112 | theorem HasDerivWithinAt.inner {f g : ℝ → E} {f' g' : E} {s : Set ℝ} {x : ℝ}
(hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x) :
HasDerivWithinAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) s x := by |
simpa using (hf.hasFDerivWithinAt.inner 𝕜 hg.hasFDerivWithinAt).hasDerivWithinAt
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 115 | 118 | theorem HasDerivAt.inner {f g : ℝ → E} {f' g' : E} {x : ℝ} :
HasDerivAt f f' x → HasDerivAt g g' x →
HasDerivAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) x := by |
simpa only [← hasDerivWithinAt_univ] using HasDerivWithinAt.inner 𝕜
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 310 | 313 | theorem differentiableWithinAt_euclidean :
DifferentiableWithinAt 𝕜 f t y ↔ ∀ i, DifferentiableWithinAt 𝕜 (fun x => f x i) t y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableWithinAt_iff, differentiableWithinAt_pi]
rfl
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 316 | 319 | theorem differentiableAt_euclidean :
DifferentiableAt 𝕜 f y ↔ ∀ i, DifferentiableAt 𝕜 (fun x => f x i) y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableAt_iff, differentiableAt_pi]
rfl
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 322 | 325 | theorem differentiableOn_euclidean :
DifferentiableOn 𝕜 f t ↔ ∀ i, DifferentiableOn 𝕜 (fun x => f x i) t := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableOn_iff, differentiableOn_pi]
rfl
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 328 | 330 | theorem differentiable_euclidean : Differentiable 𝕜 f ↔ ∀ i, Differentiable 𝕜 fun x => f x i := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiable_iff, differentiable_pi]
rfl
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 333 | 337 | theorem hasStrictFDerivAt_euclidean :
HasStrictFDerivAt f f' y ↔
∀ i, HasStrictFDerivAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasStrictFDerivAt_iff, hasStrictFDerivAt_pi']
rfl
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 340 | 344 | theorem hasFDerivWithinAt_euclidean :
HasFDerivWithinAt f f' t y ↔
∀ i, HasFDerivWithinAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') t y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_pi']
rfl
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 347 | 350 | theorem contDiffWithinAt_euclidean {n : ℕ∞} :
ContDiffWithinAt 𝕜 n f t y ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => f x i) t y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffWithinAt_iff, contDiffWithinAt_pi]
rfl
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 353 | 356 | theorem contDiffAt_euclidean {n : ℕ∞} :
ContDiffAt 𝕜 n f y ↔ ∀ i, ContDiffAt 𝕜 n (fun x => f x i) y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffAt_iff, contDiffAt_pi]
rfl
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 359 | 362 | theorem contDiffOn_euclidean {n : ℕ∞} :
ContDiffOn 𝕜 n f t ↔ ∀ i, ContDiffOn 𝕜 n (fun x => f x i) t := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffOn_iff, contDiffOn_pi]
rfl
| 2,366 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 365 | 367 | theorem contDiff_euclidean {n : ℕ∞} : ContDiff 𝕜 n f ↔ ∀ i, ContDiff 𝕜 n fun x => f x i := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiff_iff, contDiff_pi]
rfl
| 2,366 |
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.AdaptationNote
open Metric Function AffineMap Set AffineSubspace
open scoped Topology RealInnerProductSpace
variable {E F : Type*} [NormedAddCommGrou... | Mathlib/Geometry/Euclidean/Inversion/Calculus.lean | 87 | 108 | theorem hasFDerivAt_inversion (hx : x ≠ c) :
HasFDerivAt (inversion c R)
((R / dist x c) ^ 2 • (reflection (ℝ ∙ (x - c))ᗮ : F →L[ℝ] F)) x := by |
rcases add_left_surjective c x with ⟨x, rfl⟩
have : HasFDerivAt (inversion c R) (?_ : F →L[ℝ] F) (c + x) := by
#adaptation_note /-- nightly-2024-03-16: simp was
simp (config := { unfoldPartialApp := true }) only [inversion] -/
simp only [inversion_def]
simp_rw [dist_eq_norm, div_pow, div_eq_mul_inv... | 2,367 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Calculus.LagrangeMultipliers
import Mathlib.LinearAlgebra.Eigenspace.Basic
#align_import analysis.inner_product_space.rayleigh from "leanprover-co... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 57 | 64 | theorem rayleigh_smul (x : E) {c : 𝕜} (hc : c ≠ 0) :
rayleighQuotient T (c • x) = rayleighQuotient T x := by |
by_cases hx : x = 0
· simp [hx]
have : ‖c‖ ≠ 0 := by simp [hc]
have : ‖x‖ ≠ 0 := by simp [hx]
field_simp [norm_smul, T.reApplyInnerSelf_smul]
ring
| 2,368 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Calculus.LagrangeMultipliers
import Mathlib.LinearAlgebra.Eigenspace.Basic
#align_import analysis.inner_product_space.rayleigh from "leanprover-co... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 67 | 80 | theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
rayleighQuotient T '' {0}ᶜ = rayleighQuotient T '' sphere 0 r := by |
ext a
constructor
· rintro ⟨x, hx : x ≠ 0, hxT⟩
have : ‖x‖ ≠ 0 := by simp [hx]
let c : 𝕜 := ↑‖x‖⁻¹ * r
have : c ≠ 0 := by simp [c, hx, hr.ne']
refine ⟨c • x, ?_, ?_⟩
· field_simp [c, norm_smul, abs_of_pos hr]
· rw [T.rayleigh_smul x this]
exact hxT
· rintro ⟨x, hx, hxT⟩
exact... | 2,368 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Calculus.LagrangeMultipliers
import Mathlib.LinearAlgebra.Eigenspace.Basic
#align_import analysis.inner_product_space.rayleigh from "leanprover-co... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 107 | 114 | theorem _root_.LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ] F}
(hT : (T : F →ₗ[ℝ] F).IsSymmetric) (x₀ : F) :
HasStrictFDerivAt T.reApplyInnerSelf (2 • (innerSL ℝ (T x₀))) x₀ := by |
convert T.hasStrictFDerivAt.inner ℝ (hasStrictFDerivAt_id x₀) using 1
ext y
rw [ContinuousLinearMap.smul_apply, ContinuousLinearMap.comp_apply, fderivInnerCLM_apply,
ContinuousLinearMap.prod_apply, innerSL_apply, id, ContinuousLinearMap.id_apply,
hT.apply_clm x₀ y, real_inner_comm _ x₀, two_smul]
| 2,368 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Calculus.LagrangeMultipliers
import Mathlib.LinearAlgebra.Eigenspace.Basic
#align_import analysis.inner_product_space.rayleigh from "leanprover-co... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 119 | 138 | theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
(hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
∃ a b : ℝ, (a, b) ≠ 0 ∧ a • x₀ + b • T x₀ = 0 := by |
have H : IsLocalExtrOn T.reApplyInnerSelf {x : F | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀ := by
convert hextr
ext x
simp [dist_eq_norm]
-- find Lagrange multipliers for the function `T.re_apply_inner_self` and the
-- hypersurface-defining function `fun x ↦ ‖x‖ ^ 2`
obtain ⟨a, b, h₁, h₂⟩ :=
IsLocalExtrOn.exists... | 2,368 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 68 | 72 | theorem invariant_orthogonalComplement_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
T v ∈ (eigenspace T μ)ᗮ := by |
intro w hw
have : T w = (μ : 𝕜) • w := by rwa [mem_eigenspace_iff] at hw
simp [← hT w, this, inner_smul_left, hv w hw]
| 2,369 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 76 | 79 | theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ = μ := by |
obtain ⟨v, hv₁, hv₂⟩ := hμ.exists_hasEigenvector
rw [mem_eigenspace_iff] at hv₁
simpa [hv₂, inner_smul_left, inner_smul_right, hv₁] using hT v v
| 2,369 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 83 | 91 | theorem orthogonalFamily_eigenspaces :
OrthogonalFamily 𝕜 (fun μ => eigenspace T μ) fun μ => (eigenspace T μ).subtypeₗᵢ := by |
rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
by_cases hv' : v = 0
· simp [hv']
have H := hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector ⟨hv, hv'⟩)
rw [mem_eigenspace_iff] at hv hw
refine Or.resolve_left ?_ hμν.symm
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
| 2,369 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 102 | 105 | theorem orthogonalComplement_iSup_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
T v ∈ (⨆ μ, eigenspace T μ)ᗮ := by |
rw [← Submodule.iInf_orthogonal] at hv ⊢
exact T.iInf_invariant hT.invariant_orthogonalComplement_eigenspace v hv
| 2,369 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 110 | 115 | theorem orthogonalComplement_iSup_eigenspaces (μ : 𝕜) :
eigenspace (T.restrict hT.orthogonalComplement_iSup_eigenspaces_invariant) μ = ⊥ := by |
set p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
refine eigenspace_restrict_eq_bot hT.orthogonalComplement_iSup_eigenspaces_invariant ?_
have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrtho_orthogonal_right _).mono_left (le_iSup _ _)
exact H₂.disjoint
| 2,369 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 125 | 131 | theorem orthogonalComplement_iSup_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)ᗮ = ⊥ := by |
have hT' : IsSymmetric _ :=
hT.restrict_invariant hT.orthogonalComplement_iSup_eigenspaces_invariant
-- a self-adjoint operator on a nontrivial inner product space has an eigenvalue
haveI :=
hT'.subsingleton_of_no_eigenvalue_finiteDimensional hT.orthogonalComplement_iSup_eigenspaces
exact Submodule.eq_... | 2,369 |
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Hermitian
#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 78 | 80 | theorem eigenvectorUnitary_mulVec (j : n) :
eigenvectorUnitary hA *ᵥ Pi.single j 1 = ⇑(hA.eigenvectorBasis j) := by |
simp only [mulVec_single, eigenvectorUnitary_apply, mul_one]
| 2,370 |
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Hermitian
#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 82 | 84 | theorem star_eigenvectorUnitary_mulVec (j : n) :
(star (eigenvectorUnitary hA : Matrix n n 𝕜)) *ᵥ ⇑(hA.eigenvectorBasis j) = Pi.single j 1 := by |
rw [← eigenvectorUnitary_mulVec, mulVec_mulVec, unitary.coe_star_mul_self, one_mulVec]
| 2,370 |
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Hermitian
#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 87 | 100 | theorem star_mul_self_mul_eq_diagonal :
(star (eigenvectorUnitary hA : Matrix n n 𝕜)) * A * (eigenvectorUnitary hA : Matrix n n 𝕜)
= diagonal (RCLike.ofReal ∘ hA.eigenvalues) := by |
apply Matrix.toEuclideanLin.injective
apply Basis.ext (EuclideanSpace.basisFun n 𝕜).toBasis
intro i
simp only [toEuclideanLin_apply, OrthonormalBasis.coe_toBasis, EuclideanSpace.basisFun_apply,
WithLp.equiv_single, ← mulVec_mulVec, eigenvectorUnitary_mulVec, ← mulVec_mulVec,
mulVec_eigenvectorBasis, M... | 2,370 |
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Hermitian
#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 106 | 111 | theorem spectral_theorem :
A = (eigenvectorUnitary hA : Matrix n n 𝕜) * diagonal (RCLike.ofReal ∘ hA.eigenvalues)
* (star (eigenvectorUnitary hA : Matrix n n 𝕜)) := by |
rw [← star_mul_self_mul_eq_diagonal, mul_assoc, mul_assoc,
(Matrix.mem_unitaryGroup_iff).mp (eigenvectorUnitary hA).2, mul_one,
← mul_assoc, (Matrix.mem_unitaryGroup_iff).mp (eigenvectorUnitary hA).2, one_mul]
| 2,370 |
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Hermitian
#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 114 | 119 | theorem eigenvalues_eq (i : n) :
(hA.eigenvalues i) = RCLike.re (Matrix.dotProduct (star ⇑(hA.eigenvectorBasis i))
(A *ᵥ ⇑(hA.eigenvectorBasis i))):= by |
simp only [mulVec_eigenvectorBasis, dotProduct_smul,← EuclideanSpace.inner_eq_star_dotProduct,
inner_self_eq_norm_sq_to_K, RCLike.smul_re, hA.eigenvectorBasis.orthonormal.1 i,
mul_one, algebraMap.coe_one, one_pow, RCLike.one_re]
| 2,370 |
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Hermitian
#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 123 | 126 | theorem det_eq_prod_eigenvalues : det A = ∏ i, (hA.eigenvalues i : 𝕜) := by |
convert congr_arg det hA.spectral_theorem
rw [det_mul_right_comm]
simp
| 2,370 |
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
open scoped ComplexOrder
namespace Matrix
variable {m n R 𝕜 : Type*}
variable [Fintype m] [Fint... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 81 | 87 | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef := by |
classical
rw [(by simp : M = 1 * M * 1), submatrix_mul (he₂ := Function.bijective_id),
submatrix_mul (he₂ := Function.bijective_id), submatrix_id_id]
simpa only [conjTranspose_submatrix, conjTranspose_one] using
conjTranspose_mul_mul_same hM (Matrix.submatrix 1 id e)
| 2,371 |
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
open scoped ComplexOrder
namespace Matrix
variable {m n R 𝕜 : Type*}
variable [Fintype m] [Fint... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 90 | 93 | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef := by |
refine ⟨IsHermitian.transpose hM.1, fun x => ?_⟩
convert hM.2 (star x) using 1
rw [mulVec_transpose, Matrix.dotProduct_mulVec, star_star, dotProduct_comm]
| 2,371 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 52 | 59 | theorem fromBlocks_eq_of_invertible₁₁ (A : Matrix m m α) (B : Matrix m n α) (C : Matrix l m α)
(D : Matrix l n α) [Invertible A] :
fromBlocks A B C D =
fromBlocks 1 0 (C * ⅟ A) 1 * fromBlocks A 0 0 (D - C * ⅟ A * B) *
fromBlocks 1 (⅟ A * B) 0 1 := by |
simp only [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, add_zero, zero_add,
Matrix.one_mul, Matrix.mul_one, invOf_mul_self, Matrix.mul_invOf_self_assoc,
Matrix.mul_invOf_mul_self_cancel, Matrix.mul_assoc, add_sub_cancel]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 100 | 104 | theorem invOf_fromBlocks_zero₂₁_eq (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α)
[Invertible A] [Invertible D] [Invertible (fromBlocks A B 0 D)] :
⅟ (fromBlocks A B 0 D) = fromBlocks (⅟ A) (-(⅟ A * B * ⅟ D)) 0 (⅟ D) := by |
letI := fromBlocksZero₂₁Invertible A B D
convert (rfl : ⅟ (fromBlocks A B 0 D) = _)
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 107 | 111 | theorem invOf_fromBlocks_zero₁₂_eq (A : Matrix m m α) (C : Matrix n m α) (D : Matrix n n α)
[Invertible A] [Invertible D] [Invertible (fromBlocks A 0 C D)] :
⅟ (fromBlocks A 0 C D) = fromBlocks (⅟ A) 0 (-(⅟ D * C * ⅟ A)) (⅟ D) := by |
letI := fromBlocksZero₁₂Invertible A C D
convert (rfl : ⅟ (fromBlocks A 0 C D) = _)
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 390 | 394 | theorem det_fromBlocks₁₁ (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α)
(D : Matrix n n α) [Invertible A] :
(Matrix.fromBlocks A B C D).det = det A * det (D - C * ⅟ A * B) := by |
rw [fromBlocks_eq_of_invertible₁₁ (A := A), det_mul, det_mul, det_fromBlocks_zero₂₁,
det_fromBlocks_zero₂₁, det_fromBlocks_zero₁₂, det_one, det_one, one_mul, one_mul, mul_one]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 398 | 401 | theorem det_fromBlocks_one₁₁ (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) :
(Matrix.fromBlocks 1 B C D).det = det (D - C * B) := by |
haveI : Invertible (1 : Matrix m m α) := invertibleOne
rw [det_fromBlocks₁₁, invOf_one, Matrix.mul_one, det_one, one_mul]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 406 | 413 | theorem det_fromBlocks₂₂ (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α)
(D : Matrix n n α) [Invertible D] :
(Matrix.fromBlocks A B C D).det = det D * det (A - B * ⅟ D * C) := by |
have : fromBlocks A B C D =
(fromBlocks D C B A).submatrix (Equiv.sumComm _ _) (Equiv.sumComm _ _) := by
ext (i j)
cases i <;> cases j <;> rfl
rw [this, det_submatrix_equiv_self, det_fromBlocks₁₁]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 417 | 420 | theorem det_fromBlocks_one₂₂ (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) :
(Matrix.fromBlocks A B C 1).det = det (A - B * C) := by |
haveI : Invertible (1 : Matrix n n α) := invertibleOne
rw [det_fromBlocks₂₂, invOf_one, Matrix.mul_one, det_one, one_mul]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 425 | 430 | theorem det_one_add_mul_comm (A : Matrix m n α) (B : Matrix n m α) :
det (1 + A * B) = det (1 + B * A) :=
calc
det (1 + A * B) = det (fromBlocks 1 (-A) B 1) := by |
rw [det_fromBlocks_one₂₂, Matrix.neg_mul, sub_neg_eq_add]
_ = det (1 + B * A) := by rw [det_fromBlocks_one₁₁, Matrix.mul_neg, sub_neg_eq_add]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 434 | 435 | theorem det_mul_add_one_comm (A : Matrix m n α) (B : Matrix n m α) :
det (A * B + 1) = det (B * A + 1) := by | rw [add_comm, det_one_add_mul_comm, add_comm]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 438 | 440 | theorem det_one_sub_mul_comm (A : Matrix m n α) (B : Matrix n m α) :
det (1 - A * B) = det (1 - B * A) := by |
rw [sub_eq_add_neg, ← Matrix.neg_mul, det_one_add_mul_comm, Matrix.mul_neg, ← sub_eq_add_neg]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 444 | 446 | theorem det_one_add_col_mul_row (u v : m → α) : det (1 + col u * row v) = 1 + v ⬝ᵥ u := by |
rw [det_one_add_mul_comm, det_unique, Pi.add_apply, Pi.add_apply, Matrix.one_apply_eq,
Matrix.row_mul_col_apply]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 454 | 459 | theorem det_add_col_mul_row {A : Matrix m m α} (hA : IsUnit A.det) (u v : m → α) :
(A + col u * row v).det = A.det * (1 + row v * A⁻¹ * col u).det := by |
nth_rewrite 1 [← Matrix.mul_one A]
rwa [← Matrix.mul_nonsing_inv_cancel_left A (col u * row v),
← Matrix.mul_add, det_mul, ← Matrix.mul_assoc, det_one_add_mul_comm,
← Matrix.mul_assoc]
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 482 | 491 | theorem schur_complement_eq₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}
(B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (x : m → 𝕜) (y : n → 𝕜) [Invertible A]
(hA : A.IsHermitian) :
(star (x ⊕ᵥ y)) ᵥ* (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =
(star (x + (A⁻¹ * B) *ᵥ y)) ᵥ* A ⬝ᵥ (x + (A⁻¹ * B) *ᵥ... |
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hA.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 494 | 503 | theorem schur_complement_eq₂₂ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜)
(B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (x : m → 𝕜) (y : n → 𝕜) [Invertible D]
(hD : D.IsHermitian) :
(star (x ⊕ᵥ y)) ᵥ* (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =
(star ((D⁻¹ * Bᴴ) *ᵥ x + y)) ᵥ* D ⬝ᵥ ((D⁻¹ * Bᴴ) *ᵥ x... |
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hD.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
| 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 506 | 519 | theorem IsHermitian.fromBlocks₁₁ [Fintype m] [DecidableEq m] {A : Matrix m m 𝕜} (B : Matrix m n 𝕜)
(D : Matrix n n 𝕜) (hA : A.IsHermitian) :
(Matrix.fromBlocks A B Bᴴ D).IsHermitian ↔ (D - Bᴴ * A⁻¹ * B).IsHermitian := by |
have hBAB : (Bᴴ * A⁻¹ * B).IsHermitian := by
apply isHermitian_conjTranspose_mul_mul
apply hA.inv
rw [isHermitian_fromBlocks_iff]
constructor
· intro h
apply IsHermitian.sub h.2.2.2 hBAB
· intro h
refine ⟨hA, rfl, conjTranspose_conjTranspose B, ?_⟩
rw [← sub_add_cancel D]
apply IsHerm... | 2,372 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 522 | 527 | theorem IsHermitian.fromBlocks₂₂ [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜) (B : Matrix m n 𝕜)
{D : Matrix n n 𝕜} (hD : D.IsHermitian) :
(Matrix.fromBlocks A B Bᴴ D).IsHermitian ↔ (A - B * D⁻¹ * Bᴴ).IsHermitian := by |
rw [← isHermitian_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap]
convert IsHermitian.fromBlocks₁₁ _ _ hD <;> simp
| 2,372 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {𝕜 : Type*} [RCLike 𝕜]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (· < ·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 57 | 66 | theorem LDL.lowerInv_eq_gramSchmidtBasis :
LDL.lowerInv hS =
((Pi.basisFun 𝕜 n).toMatrix
(@gramSchmidtBasis 𝕜 (n → 𝕜) _ (_ : _) (InnerProductSpace.ofMatrix hS.transpose) n _ _ _
(Pi.basisFun 𝕜 n)))ᵀ := by |
letI := NormedAddCommGroup.ofMatrix hS.transpose
letI := InnerProductSpace.ofMatrix hS.transpose
ext i j
rw [LDL.lowerInv, Basis.coePiBasisFun.toMatrix_eq_transpose, coe_gramSchmidtBasis]
rfl
| 2,373 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {𝕜 : Type*} [RCLike 𝕜]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (· < ·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 93 | 97 | theorem LDL.lowerInv_triangular {i j : n} (hij : i < j) : LDL.lowerInv hS i j = 0 := by |
rw [←
@gramSchmidt_triangular 𝕜 (n → 𝕜) _ (_ : _) (InnerProductSpace.ofMatrix hS.transpose) n _ _ _
i j hij (Pi.basisFun 𝕜 n),
Pi.basisFun_repr, LDL.lowerInv]
| 2,373 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {𝕜 : Type*} [RCLike 𝕜]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (· < ·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 102 | 113 | theorem LDL.diag_eq_lowerInv_conj : LDL.diag hS = LDL.lowerInv hS * S * (LDL.lowerInv hS)ᴴ := by |
ext i j
by_cases hij : i = j
· simp only [diag, diagEntries, EuclideanSpace.inner_piLp_equiv_symm, star_star, hij,
diagonal_apply_eq, Matrix.mul_assoc]
rfl
· simp only [LDL.diag, hij, diagonal_apply_ne, Ne, not_false_iff, mul_mul_apply]
rw [conjTranspose, transpose_map, transpose_transpose, dotProd... | 2,373 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {𝕜 : Type*} [RCLike 𝕜]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (· < ·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 123 | 127 | theorem LDL.lower_conj_diag : LDL.lower hS * LDL.diag hS * (LDL.lower hS)ᴴ = S := by |
rw [LDL.lower, conjTranspose_nonsing_inv, Matrix.mul_assoc,
Matrix.inv_mul_eq_iff_eq_mul_of_invertible (LDL.lowerInv hS),
Matrix.mul_inv_eq_iff_eq_mul_of_invertible]
exact LDL.diag_eq_lowerInv_conj hS
| 2,373 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 82 | 84 | theorem closedBall_eq_image (x : E) (r : ℝ) :
closedBall x r = toEuclidean.symm '' Metric.closedBall (toEuclidean x) r := by |
rw [toEuclidean.image_symm_eq_preimage, closedBall_eq_preimage]
| 2,374 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 108 | 110 | theorem nhds_basis_closedBall {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (closedBall x) := by |
rw [toEuclidean.toHomeomorph.nhds_eq_comap x]
exact Metric.nhds_basis_closedBall.comap _
| 2,374 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 117 | 119 | theorem nhds_basis_ball {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (ball x) := by |
rw [toEuclidean.toHomeomorph.nhds_eq_comap x]
exact Metric.nhds_basis_ball.comap _
| 2,374 |
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.M... | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | 68 | 92 | theorem fourierIntegral_half_period_translate {w : V} (hw : w ≠ 0) :
(∫ v : V, 𝐞 (-⟪v, w⟫) • f (v + i w)) = -∫ v : V, 𝐞 (-⟪v, w⟫) • f v := by |
have hiw : ⟪i w, w⟫ = 1 / 2 := by
rw [inner_smul_left, inner_self_eq_norm_sq_to_K, RCLike.ofReal_real_eq_id, id,
RCLike.conj_to_real, ← div_div, div_mul_cancel₀]
rwa [Ne, sq_eq_zero_iff, norm_eq_zero]
have :
(fun v : V => 𝐞 (-⟪v, w⟫) • f (v + i w)) =
fun v : V => (fun x : V => -(𝐞 (-⟪x, w... | 2,375 |
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.M... | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | 96 | 104 | theorem fourierIntegral_eq_half_sub_half_period_translate {w : V} (hw : w ≠ 0)
(hf : Integrable f) :
∫ v : V, 𝐞 (-⟪v, w⟫) • f v = (1 / (2 : ℂ)) • ∫ v : V, 𝐞 (-⟪v, w⟫) • (f v - f (v + i w)) := by |
simp_rw [smul_sub]
rw [integral_sub, fourierIntegral_half_period_translate hw, sub_eq_add_neg, neg_neg, ←
two_smul ℂ _, ← @smul_assoc _ _ _ _ _ _ (IsScalarTower.left ℂ), smul_eq_mul]
· norm_num
exacts [(Real.fourierIntegral_convergent_iff w).2 hf,
(Real.fourierIntegral_convergent_iff w).2 (hf.comp_add_... | 2,375 |
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.M... | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | 111 | 194 | theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
(hf2 : HasCompactSupport f) :
Tendsto (fun w : V => ∫ v : V, 𝐞 (-⟪v, w⟫) • f v) (cocompact V) (𝓝 0) := by |
refine NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => ?_
suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, 𝐞 (-⟪v, w⟫) • f v‖ < ε by
simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
dist_eq_norm', sub_zero]
exact
let ⟨T, hT⟩ := this
⟨T, fun b ... | 2,375 |
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Set.Pointwise.Support
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheo... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 43 | 73 | theorem exists_smooth_tsupport_subset {s : Set E} {x : E} (hs : s ∈ 𝓝 x) :
∃ f : E → ℝ,
tsupport f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ⊤ f ∧ range f ⊆ Icc 0 1 ∧ f x = 1 := by |
obtain ⟨d : ℝ, d_pos : 0 < d, hd : Euclidean.closedBall x d ⊆ s⟩ :=
Euclidean.nhds_basis_closedBall.mem_iff.1 hs
let c : ContDiffBump (toEuclidean x) :=
{ rIn := d / 2
rOut := d
rIn_pos := half_pos d_pos
rIn_lt_rOut := half_lt_self d_pos }
let f : E → ℝ := c ∘ toEuclidean
have f_supp ... | 2,376 |
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Set.Pointwise.Support
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheo... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 78 | 192 | theorem IsOpen.exists_smooth_support_eq {s : Set E} (hs : IsOpen s) :
∃ f : E → ℝ, f.support = s ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1 := by |
/- For any given point `x` in `s`, one can construct a smooth function with support in `s` and
nonzero at `x`. By second-countability, it follows that we may cover `s` with the supports of
countably many such functions, say `g i`.
Then `∑ i, r i • g i` will be the desired function if `r i` is a sequence ... | 2,376 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 97 | 97 | theorem mem_box : x ∈ box n d ↔ ∀ i, x i < d := by | simp only [box, Fintype.mem_piFinset, mem_range]
| 2,377 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 101 | 101 | theorem card_box : (box n d).card = d ^ n := by | simp [box]
| 2,377 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 105 | 105 | theorem box_zero : box (n + 1) 0 = ∅ := by | simp [box]
| 2,377 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 118 | 118 | theorem sphere_zero_right (n k : ℕ) : sphere (n + 1) 0 k = ∅ := by | simp [sphere]
| 2,377 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 125 | 129 | theorem norm_of_mem_sphere {x : Fin n → ℕ} (hx : x ∈ sphere n d k) :
‖(WithLp.equiv 2 _).symm ((↑) ∘ x : Fin n → ℝ)‖ = √↑k := by |
rw [EuclideanSpace.norm_eq]
dsimp
simp_rw [abs_cast, ← cast_pow, ← cast_sum, (mem_filter.1 hx).2]
| 2,377 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 147 | 147 | theorem map_zero (d : ℕ) (a : Fin 0 → ℕ) : map d a = 0 := by | simp [map]
| 2,377 |
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