formal/lean/mathlib/linear_algebra/ray.lean

/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import linear_algebra.basic

/-!
# Rays in modules

This file defines rays in modules.

## Main definitions

* `same_ray`: two vectors belong to the same ray if they are proportional with a nonnegative
  coefficient.

* `module.ray` is a type for the equivalence class of nonzero vectors in a module with some
common positive multiple.
-/

noncomputable theory

open_locale big_operators

section ordered_comm_semiring

variables (R : Type*) [ordered_comm_semiring R]
variables {M : Type*} [add_comm_monoid M] [module R M]
variables {N : Type*} [add_comm_monoid N] [module R N]
variables (ι : Type*) [decidable_eq ι]

/-- Two vectors are in the same ray if either one of them is zero or some positive multiples of them
are equal (in the typical case over a field, this means one of them is a nonnegative multiple of
the other). -/
def same_ray (v₁ v₂ : M) : Prop :=
v₁ = 0 ∨ v₂ = 0 ∨ ∃ (r₁ r₂ : R), 0 < r₁ ∧ 0 < r₂ ∧ r₁ • v₁ = r₂ • v₂

variables {R}

namespace same_ray

variables {x y z : M}

@[simp] lemma zero_left (y : M) : same_ray R 0 y := or.inl rfl

@[simp] lemma zero_right (x : M) : same_ray R x 0 := or.inr $ or.inl rfl

@[nontriviality] lemma of_subsingleton [subsingleton M] (x y : M) : same_ray R x y :=
by { rw [subsingleton.elim x 0], exact zero_left _ }

@[nontriviality] lemma of_subsingleton' [subsingleton R] (x y : M) : same_ray R x y :=
by { haveI := module.subsingleton R M, exact of_subsingleton x y }

/-- `same_ray` is reflexive. -/
@[refl] lemma refl (x : M) : same_ray R x x :=
begin
  nontriviality R,
  exact or.inr (or.inr $ ⟨1, 1, zero_lt_one, zero_lt_one, rfl⟩)
end

protected lemma rfl : same_ray R x x := refl _

/-- `same_ray` is symmetric. -/
@[symm] lemma symm (h : same_ray R x y) : same_ray R y x :=
(or.left_comm.1 h).imp_right $ or.imp_right $ λ ⟨r₁, r₂, h₁, h₂, h⟩, ⟨r₂, r₁, h₂, h₁, h.symm⟩

/-- If `x` and `y` are nonzero vectors on the same ray, then there exist positive numbers `r₁ r₂`
such that `r₁ • x = r₂ • y`. -/
lemma exists_pos (h : same_ray R x y) (hx : x ≠ 0) (hy : y ≠ 0) :
  ∃ r₁ r₂ : R, 0 < r₁ ∧ 0 < r₂ ∧ r₁ • x = r₂ • y :=
(h.resolve_left hx).resolve_left hy

lemma _root_.same_ray_comm : same_ray R x y ↔ same_ray R y x :=
⟨same_ray.symm, same_ray.symm⟩

/-- `same_ray` is transitive unless the vector in the middle is zero and both other vectors are
nonzero. -/
lemma trans (hxy : same_ray R x y) (hyz : same_ray R y z) (hy : y = 0 → x = 0 ∨ z = 0) :
  same_ray R x z :=
begin
  rcases eq_or_ne x 0 with rfl|hx, { exact zero_left z },
  rcases eq_or_ne z 0 with rfl|hz, { exact zero_right x },
  rcases eq_or_ne y 0 with rfl|hy, { exact (hy rfl).elim (λ h, (hx h).elim) (λ h, (hz h).elim) },
  rcases hxy.exists_pos hx hy with ⟨r₁, r₂, hr₁, hr₂, h₁⟩,
  rcases hyz.exists_pos hy hz with ⟨r₃, r₄, hr₃, hr₄, h₂⟩,
  refine or.inr (or.inr $ ⟨r₃ * r₁, r₂ * r₄, mul_pos hr₃ hr₁, mul_pos hr₂ hr₄, _⟩),
  rw [mul_smul, mul_smul, h₁, ← h₂, smul_comm]
end

/-- A vector is in the same ray as a nonnegative multiple of itself. -/
lemma _root_.same_ray_nonneg_smul_right (v : M) {r : R} (h : 0 ≤ r) : same_ray R v (r • v) :=
or.inr $ h.eq_or_lt.imp (λ h, h ▸ zero_smul R v) $
  λ h, ⟨r, 1, h, by { nontriviality R, exact zero_lt_one }, (one_smul _ _).symm⟩

/-- A vector is in the same ray as a positive multiple of itself. -/
lemma _root_.same_ray_pos_smul_right (v : M) {r : R} (h : 0 < r) : same_ray R v (r • v) :=
same_ray_nonneg_smul_right v h.le

/-- A vector is in the same ray as a nonnegative multiple of one it is in the same ray as. -/
lemma nonneg_smul_right {r : R} (h : same_ray R x y) (hr : 0 ≤ r) : same_ray R x (r • y) :=
h.trans (same_ray_nonneg_smul_right y hr) $ λ hy, or.inr $ by rw [hy, smul_zero]

/-- A vector is in the same ray as a positive multiple of one it is in the same ray as. -/
lemma pos_smul_right {r : R} (h : same_ray R x y) (hr : 0 < r) : same_ray R x (r • y) :=
h.nonneg_smul_right hr.le

/-- A nonnegative multiple of a vector is in the same ray as that vector. -/
lemma _root_.same_ray_nonneg_smul_left (v : M) {r : R} (h : 0 ≤ r) : same_ray R (r • v) v :=
(same_ray_nonneg_smul_right v h).symm

/-- A positive multiple of a vector is in the same ray as that vector. -/
lemma _root_.same_ray_pos_smul_left (v : M) {r : R} (h : 0 < r) : same_ray R (r • v) v :=
same_ray_nonneg_smul_left v h.le

/-- A nonnegative multiple of a vector is in the same ray as one it is in the same ray as. -/
lemma nonneg_smul_left {r : R} (h : same_ray R x y) (hr : 0 ≤ r) : same_ray R (r • x) y :=
(h.symm.nonneg_smul_right hr).symm

/-- A positive multiple of a vector is in the same ray as one it is in the same ray as. -/
lemma pos_smul_left {r : R} (h : same_ray R x y) (hr : 0 < r) : same_ray R (r • x) y :=
h.nonneg_smul_left hr.le

/-- If two vectors are on the same ray then they remain so after applying a linear map. -/
lemma map (f : M →ₗ[R] N) (h : same_ray R x y) : same_ray R (f x) (f y) :=
h.imp (λ hx, by rw [hx, map_zero]) $ or.imp (λ hy, by rw [hy, map_zero]) $
  λ ⟨r₁, r₂, hr₁, hr₂, h⟩, ⟨r₁, r₂, hr₁, hr₂, by rw [←f.map_smul, ←f.map_smul, h]⟩

/-- The images of two vectors under a linear equivalence are on the same ray if and only if the
original vectors are on the same ray. -/
@[simp] lemma _root_.same_ray_map_iff (e : M ≃ₗ[R] N) : same_ray R (e x) (e y) ↔ same_ray R x y :=
⟨λ h, by simpa using same_ray.map e.symm.to_linear_map h, same_ray.map e.to_linear_map⟩

/-- If two vectors are on the same ray then both scaled by the same action are also on the same
ray. -/
lemma smul {S : Type*} [monoid S] [distrib_mul_action S M] [smul_comm_class R S M]
  (h : same_ray R x y) (s : S) : same_ray R (s • x) (s • y) :=
h.map (s • (linear_map.id : M →ₗ[R] M))

/-- If `x` and `y` are on the same ray as `z`, then so is `x + y`. -/
lemma add_left (hx : same_ray R x z) (hy : same_ray R y z) : same_ray R (x + y) z :=
begin
  rcases eq_or_ne x 0 with rfl|hx₀, { rwa zero_add },
  rcases eq_or_ne y 0 with rfl|hy₀, { rwa add_zero },
  rcases eq_or_ne z 0 with rfl|hz₀, { apply zero_right },
  rcases hx.exists_pos hx₀ hz₀ with ⟨rx, rz₁, hrx, hrz₁, Hx⟩,
  rcases hy.exists_pos hy₀ hz₀ with ⟨ry, rz₂, hry, hrz₂, Hy⟩,
  refine or.inr (or.inr ⟨rx * ry, ry * rz₁ + rx * rz₂, mul_pos hrx hry, _, _⟩),
  { apply_rules [add_pos, mul_pos] },
  { simp only [mul_smul, smul_add, add_smul, ← Hx, ← Hy],
    rw smul_comm }
end

/-- If `y` and `z` are on the same ray as `x`, then so is `y + z`. -/
lemma add_right (hy : same_ray R x y) (hz : same_ray R x z) : same_ray R x (y + z) :=
(hy.symm.add_left hz.symm).symm

end same_ray

/-- Nonzero vectors, as used to define rays. This type depends on an unused argument `R` so that
`ray_vector.setoid` can be an instance. -/
@[nolint unused_arguments has_inhabited_instance]
def ray_vector (R M : Type*) [has_zero M] := {v : M // v ≠ 0}

instance ray_vector.has_coe {R M : Type*} [has_zero M] :
  has_coe (ray_vector R M) M := coe_subtype

instance {R M : Type*} [has_zero M] [nontrivial M] : nonempty (ray_vector R M) :=
let ⟨x, hx⟩ := exists_ne (0 : M) in ⟨⟨x, hx⟩⟩

variables (R M)

/-- The setoid of the `same_ray` relation for the subtype of nonzero vectors. -/
instance : setoid (ray_vector R M) :=
{ r := λ x y, same_ray R (x : M) y,
  iseqv := ⟨λ x, same_ray.refl _, λ x y h, h.symm,
    λ x y z hxy hyz, hxy.trans hyz $ λ hy, (y.2 hy).elim⟩ }

/-- A ray (equivalence class of nonzero vectors with common positive multiples) in a module. -/
@[nolint has_inhabited_instance]
def module.ray := quotient (ray_vector.setoid R M)

variables {R M}

/-- Equivalence of nonzero vectors, in terms of same_ray. -/
lemma equiv_iff_same_ray {v₁ v₂ : ray_vector R M} :
  v₁ ≈ v₂ ↔ same_ray R (v₁ : M) v₂ :=
iff.rfl

variables (R)

/-- The ray given by a nonzero vector. -/
protected def ray_of_ne_zero (v : M) (h : v ≠ 0) : module.ray R M := ⟦⟨v, h⟩⟧

/-- An induction principle for `module.ray`, used as `induction x using module.ray.ind`. -/
lemma module.ray.ind {C : module.ray R M → Prop}
  (h : ∀ v (hv : v ≠ 0), C (ray_of_ne_zero R v hv)) (x : module.ray R M) : C x :=
quotient.ind (subtype.rec $ by exact h) x

variable {R}

instance [nontrivial M] : nonempty (module.ray R M) :=
nonempty.map quotient.mk infer_instance

/-- The rays given by two nonzero vectors are equal if and only if those vectors
satisfy `same_ray`. -/
lemma ray_eq_iff {v₁ v₂ : M} (hv₁ : v₁ ≠ 0) (hv₂ : v₂ ≠ 0) :
  ray_of_ne_zero R _ hv₁ = ray_of_ne_zero R _ hv₂ ↔ same_ray R v₁ v₂ :=
quotient.eq

/-- The ray given by a positive multiple of a nonzero vector. -/
@[simp] lemma ray_pos_smul {v : M} (h : v ≠ 0) {r : R} (hr : 0 < r)
  (hrv : r • v ≠ 0) : ray_of_ne_zero R (r • v) hrv = ray_of_ne_zero R v h :=
(ray_eq_iff _ _).2 $ same_ray_pos_smul_left v hr

/-- An equivalence between modules implies an equivalence between ray vectors. -/
def ray_vector.map_linear_equiv (e : M ≃ₗ[R] N) : ray_vector R M ≃ ray_vector R N :=
equiv.subtype_equiv e.to_equiv $ λ _, e.map_ne_zero_iff.symm

/-- An equivalence between modules implies an equivalence between rays. -/
def module.ray.map (e : M ≃ₗ[R] N) : module.ray R M ≃ module.ray R N :=
quotient.congr (ray_vector.map_linear_equiv e) $ λ ⟨a, ha⟩ ⟨b, hb⟩, (same_ray_map_iff _).symm

@[simp] lemma module.ray.map_apply (e : M ≃ₗ[R] N) (v : M) (hv : v ≠ 0) :
  module.ray.map e (ray_of_ne_zero _ v hv) = ray_of_ne_zero _ (e v) (e.map_ne_zero_iff.2 hv) := rfl

@[simp] lemma module.ray.map_refl : (module.ray.map $ linear_equiv.refl R M) = equiv.refl _ :=
equiv.ext $ module.ray.ind R $ λ _ _, rfl

@[simp] lemma module.ray.map_symm (e : M ≃ₗ[R] N) :
  (module.ray.map e).symm = module.ray.map e.symm := rfl

section action
variables {G : Type*} [group G] [distrib_mul_action G M]

/-- Any invertible action preserves the non-zeroness of ray vectors. This is primarily of interest
when `G = Rˣ` -/
instance {R : Type*} : mul_action G (ray_vector R M) :=
{ smul := λ r, (subtype.map ((•) r) $ λ a, (smul_ne_zero_iff_ne _).2),
  mul_smul := λ a b m, subtype.ext $ mul_smul a b _,
  one_smul := λ m, subtype.ext $ one_smul _ _ }

variables [smul_comm_class R G M]

/-- Any invertible action preserves the non-zeroness of rays. This is primarily of interest when
`G = Rˣ` -/
instance : mul_action G (module.ray R M) :=
{ smul := λ r, quotient.map ((•) r) (λ a b h, h.smul _),
  mul_smul := λ a b, quotient.ind $ by exact(λ m, congr_arg quotient.mk $ mul_smul a b _),
  one_smul := quotient.ind $ by exact (λ m, congr_arg quotient.mk $ one_smul _ _), }

/-- The action via `linear_equiv.apply_distrib_mul_action` corresponds to `module.ray.map`. -/
@[simp] lemma module.ray.linear_equiv_smul_eq_map (e : M ≃ₗ[R] M) (v : module.ray R M) :
  e • v = module.ray.map e v := rfl

@[simp] lemma smul_ray_of_ne_zero (g : G) (v : M) (hv) :
  g • ray_of_ne_zero R v hv = ray_of_ne_zero R (g • v) ((smul_ne_zero_iff_ne _).2 hv) := rfl

end action

namespace module.ray

/-- Scaling by a positive unit is a no-op. -/
lemma units_smul_of_pos (u : Rˣ) (hu : 0 < (u : R)) (v : module.ray R M) :
  u • v = v :=
begin
  induction v using module.ray.ind,
  rw [smul_ray_of_ne_zero, ray_eq_iff],
  exact same_ray_pos_smul_left _ hu
end

/-- An arbitrary `ray_vector` giving a ray. -/
def some_ray_vector (x : module.ray R M) : ray_vector R M := quotient.out x

/-- The ray of `some_ray_vector`. -/
@[simp] lemma some_ray_vector_ray (x : module.ray R M) :
  (⟦x.some_ray_vector⟧ : module.ray R M) = x :=
quotient.out_eq _

/-- An arbitrary nonzero vector giving a ray. -/
def some_vector (x : module.ray R M) : M := x.some_ray_vector

/-- `some_vector` is nonzero. -/
@[simp] lemma some_vector_ne_zero (x : module.ray R M) : x.some_vector ≠ 0 :=
x.some_ray_vector.property

/-- The ray of `some_vector`. -/
@[simp] lemma some_vector_ray (x : module.ray R M) :
  ray_of_ne_zero R _ x.some_vector_ne_zero = x :=
(congr_arg _ (subtype.coe_eta _ _) : _).trans x.out_eq

end module.ray

end ordered_comm_semiring

section ordered_comm_ring

variables {R : Type*} [ordered_comm_ring R]
variables {M N : Type*} [add_comm_group M] [add_comm_group N] [module R M] [module R N] {x y : M}

/-- `same_ray.neg` as an `iff`. -/
@[simp] lemma same_ray_neg_iff : same_ray R (-x) (-y) ↔ same_ray R x y :=
by simp only [same_ray, neg_eq_zero, smul_neg, neg_inj]

alias same_ray_neg_iff ↔ same_ray.of_neg same_ray.neg

lemma same_ray_neg_swap : same_ray R (-x) y ↔ same_ray R x (-y) :=
by rw [← same_ray_neg_iff, neg_neg]

lemma eq_zero_of_same_ray_neg_smul_right [no_zero_smul_divisors R M] {r : R} (hr : r < 0)
  (h : same_ray R x (r • x)) :
  x = 0 :=
begin
  rcases h with rfl|h₀|⟨r₁, r₂, hr₁, hr₂, h⟩,
  { refl },
  { simpa [hr.ne] using h₀ },
  { rw [← sub_eq_zero, smul_smul, ← sub_smul, smul_eq_zero] at h,
    refine h.resolve_left (ne_of_gt $ sub_pos.2 _),
    exact (mul_neg_of_pos_of_neg hr₂ hr).trans hr₁ }
end

/-- If a vector is in the same ray as its negation, that vector is zero. -/
lemma eq_zero_of_same_ray_self_neg [no_zero_smul_divisors R M] (h : same_ray R x (-x)) :
  x = 0 :=
begin
  nontriviality M, haveI : nontrivial R := module.nontrivial R M,
  refine eq_zero_of_same_ray_neg_smul_right (neg_lt_zero.2 (@one_pos R _ _)) _,
  rwa [neg_one_smul]
end

namespace ray_vector

/-- Negating a nonzero vector. -/
instance {R : Type*} : has_neg (ray_vector R M) := ⟨λ v, ⟨-v, neg_ne_zero.2 v.prop⟩⟩

/-- Negating a nonzero vector commutes with coercion to the underlying module. -/
@[simp, norm_cast] lemma coe_neg {R : Type*} (v : ray_vector R M) : ↑(-v) = -(v : M) := rfl

/-- Negating a nonzero vector twice produces the original vector. -/
instance {R : Type*} : has_involutive_neg (ray_vector R M) :=
{ neg := has_neg.neg,
  neg_neg := λ v, by rw [subtype.ext_iff, coe_neg, coe_neg, neg_neg] }

/-- If two nonzero vectors are equivalent, so are their negations. -/
@[simp] lemma equiv_neg_iff {v₁ v₂ : ray_vector R M} : -v₁ ≈ -v₂ ↔ v₁ ≈ v₂ :=
same_ray_neg_iff

end ray_vector

variables (R)

/-- Negating a ray. -/
instance : has_neg (module.ray R M) :=
⟨quotient.map (λ v, -v) (λ v₁ v₂, ray_vector.equiv_neg_iff.2)⟩

/-- The ray given by the negation of a nonzero vector. -/
@[simp] lemma neg_ray_of_ne_zero (v : M) (h : v ≠ 0) :
  -(ray_of_ne_zero R _ h) = ray_of_ne_zero R (-v) (neg_ne_zero.2 h) :=
rfl

namespace module.ray

variables {R}

/-- Negating a ray twice produces the original ray. -/
instance : has_involutive_neg (module.ray R M) :=
{ neg := has_neg.neg,
  neg_neg := λ x, quotient.ind (λ a, congr_arg quotient.mk $ neg_neg _) x }

variables {R M}

/-- A ray does not equal its own negation. -/
lemma ne_neg_self [no_zero_smul_divisors R M] (x : module.ray R M) : x ≠ -x :=
begin
  induction x using module.ray.ind with x hx,
  rw [neg_ray_of_ne_zero, ne.def, ray_eq_iff],
  exact mt eq_zero_of_same_ray_self_neg hx
end

lemma neg_units_smul (u : Rˣ) (v : module.ray R M) : (-u) • v = - (u • v) :=
begin
  induction v using module.ray.ind,
  simp only [smul_ray_of_ne_zero, units.smul_def, units.coe_neg, neg_smul, neg_ray_of_ne_zero]
end

/-- Scaling by a negative unit is negation. -/
lemma units_smul_of_neg (u : Rˣ) (hu : (u : R) < 0) (v : module.ray R M) :
  u • v = -v :=
begin
  rw [← neg_inj, neg_neg, ← neg_units_smul, units_smul_of_pos],
  rwa [units.coe_neg, right.neg_pos_iff]
end

end module.ray

end ordered_comm_ring

section linear_ordered_comm_ring

variables {R : Type*} [linear_ordered_comm_ring R]
variables {M : Type*} [add_comm_group M] [module R M]

/-- `same_ray` follows from membership of `mul_action.orbit` for the `units.pos_subgroup`. -/
lemma same_ray_of_mem_orbit {v₁ v₂ : M} (h : v₁ ∈ mul_action.orbit (units.pos_subgroup R) v₂) :
  same_ray R v₁ v₂ :=
begin
  rcases h with ⟨⟨r, hr : 0 < (r : R)⟩, (rfl : r • v₂ = v₁)⟩,
  exact same_ray_pos_smul_left _ hr
end

/-- Scaling by an inverse unit is the same as scaling by itself. -/
@[simp] lemma units_inv_smul (u : Rˣ) (v : module.ray R M) :
  u⁻¹ • v = u • v :=
calc u⁻¹ • v = (u * u) • u⁻¹ • v :
  eq.symm $ (u⁻¹ • v).units_smul_of_pos _ $ mul_self_pos.2 u.ne_zero
... = u • v : by rw [mul_smul, smul_inv_smul]

section
variables [no_zero_smul_divisors R M]

@[simp] lemma same_ray_smul_right_iff {v : M} {r : R} :
  same_ray R v (r • v) ↔ 0 ≤ r ∨ v = 0 :=
⟨λ hrv, or_iff_not_imp_left.2 $ λ hr, eq_zero_of_same_ray_neg_smul_right (not_le.1 hr) hrv,
  or_imp_distrib.2 ⟨same_ray_nonneg_smul_right v, λ h, h.symm ▸ same_ray.zero_left _⟩⟩

/-- A nonzero vector is in the same ray as a multiple of itself if and only if that multiple
is positive. -/
lemma same_ray_smul_right_iff_of_ne {v : M} (hv : v ≠ 0) {r : R} (hr : r ≠ 0) :
  same_ray R v (r • v) ↔ 0 < r :=
by simp only [same_ray_smul_right_iff, hv, or_false, hr.symm.le_iff_lt]

@[simp] lemma same_ray_smul_left_iff {v : M} {r : R} : same_ray R (r • v) v ↔ 0 ≤ r ∨ v = 0 :=
same_ray_comm.trans same_ray_smul_right_iff

/-- A multiple of a nonzero vector is in the same ray as that vector if and only if that multiple
is positive. -/
lemma same_ray_smul_left_iff_of_ne {v : M} (hv : v ≠ 0) {r : R} (hr : r ≠ 0) :
  same_ray R (r • v) v ↔ 0 < r :=
same_ray_comm.trans (same_ray_smul_right_iff_of_ne hv hr)

@[simp] lemma same_ray_neg_smul_right_iff {v : M} {r : R} :
  same_ray R (-v) (r • v) ↔ r ≤ 0 ∨ v = 0 :=
by rw [← same_ray_neg_iff, neg_neg, ← neg_smul, same_ray_smul_right_iff, neg_nonneg]

lemma same_ray_neg_smul_right_iff_of_ne {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0) :
  same_ray R (-v) (r • v) ↔ r < 0 :=
by simp only [same_ray_neg_smul_right_iff, hv, or_false, hr.le_iff_lt]

@[simp] lemma same_ray_neg_smul_left_iff {v : M} {r : R} :
  same_ray R (r • v) (-v) ↔ r ≤ 0 ∨ v = 0 :=
same_ray_comm.trans same_ray_neg_smul_right_iff

lemma same_ray_neg_smul_left_iff_of_ne {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0) :
  same_ray R (r • v) (-v) ↔ r < 0 :=
same_ray_comm.trans $ same_ray_neg_smul_right_iff_of_ne hv hr

@[simp] lemma units_smul_eq_self_iff {u : Rˣ} {v : module.ray R M} :
  u • v = v ↔ (0 : R) < u :=
begin
  induction v using module.ray.ind with v hv,
  simp only [smul_ray_of_ne_zero, ray_eq_iff, units.smul_def,
    same_ray_smul_left_iff_of_ne hv u.ne_zero]
end

@[simp] lemma units_smul_eq_neg_iff {u : Rˣ} {v : module.ray R M} :
  u • v = -v ↔ ↑u < (0 : R) :=
by rw [← neg_inj, neg_neg, ← module.ray.neg_units_smul, units_smul_eq_self_iff, units.coe_neg,
  neg_pos]

end

end linear_ordered_comm_ring

namespace same_ray

variables {R : Type*} [linear_ordered_field R]
variables {M : Type*} [add_comm_group M] [module R M] {x y v₁ v₂ : M}

lemma exists_pos_left (h : same_ray R x y) (hx : x ≠ 0) (hy : y ≠ 0) :
  ∃ r : R, 0 < r ∧ r • x = y :=
let ⟨r₁, r₂, hr₁, hr₂, h⟩ := h.exists_pos hx hy in
  ⟨r₂⁻¹ * r₁, mul_pos (inv_pos.2 hr₂) hr₁, by rw [mul_smul, h, inv_smul_smul₀ hr₂.ne']⟩

lemma exists_pos_right (h : same_ray R x y) (hx : x ≠ 0) (hy : y ≠ 0) :
  ∃ r : R, 0 < r ∧ x = r • y :=
(h.symm.exists_pos_left hy hx).imp $ λ _, and.imp_right eq.symm

/-- If a vector `v₂` is on the same ray as a nonzero vector `v₁`, then it is equal to `c • v₁` for
some nonnegative `c`. -/
lemma exists_nonneg_left (h : same_ray R x y) (hx : x ≠ 0) : ∃ r : R, 0 ≤ r ∧ r • x = y :=
begin
  obtain rfl | hy := eq_or_ne y 0,
  { exact ⟨0, le_rfl, zero_smul _ _⟩ },
  { exact (h.exists_pos_left hx hy).imp (λ _, and.imp_left le_of_lt) }
end

/-- If a vector `v₁` is on the same ray as a nonzero vector `v₂`, then it is equal to `c • v₂` for
some nonnegative `c`. -/
lemma exists_nonneg_right (h : same_ray R x y) (hy : y ≠ 0) : ∃ r : R, 0 ≤ r ∧ x = r • y :=
(h.symm.exists_nonneg_left hy).imp $ λ _, and.imp_right eq.symm

/-- If vectors `v₁` and `v₂` are on the same ray, then for some nonnegative `a b`, `a + b = 1`, we
have `v₁ = a • (v₁ + v₂)` and `v₂ = b • (v₁ + v₂)`. -/
lemma exists_eq_smul_add (h : same_ray R v₁ v₂) :
  ∃ a b : R, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ v₁ = a • (v₁ + v₂) ∧ v₂ = b • (v₁ + v₂) :=
begin
  rcases h with rfl|rfl|⟨r₁, r₂, h₁, h₂, H⟩,
  { use [0, 1], simp },
  { use [1, 0], simp },
  { have h₁₂ : 0 < r₁ + r₂, from add_pos h₁ h₂,
    refine ⟨r₂ / (r₁ + r₂), r₁ / (r₁ + r₂), div_nonneg h₂.le h₁₂.le, div_nonneg h₁.le h₁₂.le,
      _, _, _⟩,
    { rw [← add_div, add_comm, div_self h₁₂.ne'] },
    { rw [div_eq_inv_mul, mul_smul, smul_add, ← H, ← add_smul, add_comm r₂,
        inv_smul_smul₀ h₁₂.ne'] },
    { rw [div_eq_inv_mul, mul_smul, smul_add, H, ← add_smul, add_comm r₂,
        inv_smul_smul₀ h₁₂.ne'] } }
end

/-- If vectors `v₁` and `v₂` are on the same ray, then they are nonnegative multiples of the same
vector. Actually, this vector can be assumed to be `v₁ + v₂`, see `same_ray.exists_eq_smul_add`. -/
lemma exists_eq_smul (h : same_ray R v₁ v₂) :
  ∃ (u : M) (a b : R), 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ v₁ = a • u ∧ v₂ = b • u :=
⟨v₁ + v₂, h.exists_eq_smul_add⟩

end same_ray