ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearMap.BilinForm.orthogonal_orthogonal
lemma orthogonal_orthogonal (hB : B.Nondegenerate) (hB₀ : B.IsRefl) (W : Submodule K V) :
    B.orthogonal (B.orthogonal W) = W 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} (hB : B.Nondegenerate) (hB₀ : B.IsRefl) (W : Submodule K V)
Goal: B.orthogonal (B.orthogonal W) = W
-- LinearMap.BilinForm.isCompl_orthogonal_iff_disjoint
lemma isCompl_orthogonal_iff_disjoint (hB₀ : B.IsRefl) :
    IsCompl W (B.orthogonal W) ↔ Disjoint W (B.orthogonal W) 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} {W : Submodule K V} (hB₀ : B.IsRefl)
Goal: IsCompl W (B.orthogonal W) ↔ Disjoint W (B.orthogonal W)
-- LinearMap.BilinForm.isCompl_orthogonal_of_restrict_nondegenerate
theorem isCompl_orthogonal_of_restrict_nondegenerate
    (b₁ : B.IsRefl) (b₂ : (B.restrict W).Nondegenerate) : IsCompl W (B.orthogonal W) 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} {W : Submodule K V} (b₁ : B.IsRefl) (b₂ : (B.restrict W).Nondegenerate)
Goal: IsCompl W (B.orthogonal W)
-- LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal
theorem restrict_nondegenerate_iff_isCompl_orthogonal
    (b₁ : B.IsRefl) : (B.restrict W).Nondegenerate ↔ IsCompl W (B.orthogonal W) 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} {W : Submodule K V} (b₁ : B.IsRefl)
Goal: (B.restrict W).Nondegenerate ↔ IsCompl W (B.orthogonal W)
-- LinearMap.BilinForm.orthogonal_eq_top_iff
lemma orthogonal_eq_top_iff (b₁ : B.IsRefl) (b₂ : (B.restrict W).Nondegenerate) :
    B.orthogonal W = ⊤ ↔ W = ⊥ 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} {W : Submodule K V} (b₁ : B.IsRefl) (b₂ : (B.restrict W).Nondegenerate)
Goal: B.orthogonal W = ⊤ ↔ W = ⊥
-- LinearMap.BilinForm.eq_top_of_restrict_nondegenerate_of_orthogonal_eq_bot
lemma eq_top_of_restrict_nondegenerate_of_orthogonal_eq_bot
    (b₁ : B.IsRefl) (b₂ : (B.restrict W).Nondegenerate) (b₃ : B.orthogonal W = ⊥) :
    W = ⊤ 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} {W : Submodule K V} (b₁ : B.IsRefl) (b₂ : (B.restrict W).Nondegenerate) (b₃ : B.orthogonal W = ⊥)
Goal: W = ⊤
-- LinearMap.BilinForm.orthogonal_eq_bot_iff
lemma orthogonal_eq_bot_iff
    (b₁ : B.IsRefl) (b₂ : (B.restrict W).Nondegenerate) (b₃ : B.Nondegenerate) :
    B.orthogonal W = ⊥ ↔ W = ⊤ 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} {W : Submodule K V} (b₁ : B.IsRefl) (b₂ : (B.restrict W).Nondegenerate) (b₃ : B.Nondegenerate)
Goal: B.orthogonal W = ⊥ ↔ W = ⊤
-- LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict
lemma inf_orthogonal_self_le_ker_restrict (b₁ : B.IsRefl) :
    W ⊓ B.orthogonal W ≤ (LinearMap.ker <| B.restrict W).map W.subtype 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] {B : LinearMap.BilinForm K V} {W : Submodule K V} (b₁ : B.IsRefl)
Goal: W ⊓ B.orthogonal W ≤ Submodule.map W.subtype (LinearMap.ker (B.restrict W))
-- LinearMap.BilinForm.restrict_nondegenerate_orthogonal_spanSingleton
theorem restrict_nondegenerate_orthogonal_spanSingleton (B : BilinForm K V) (b₁ : B.Nondegenerate)
    (b₂ : B.IsRefl) {x : V} (hx : ¬B.IsOrtho x x) :
    Nondegenerate <| B.restrict <| B.orthogonal (K ∙ x) 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] (B : LinearMap.BilinForm K V) (b₁ : B.Nondegenerate) (b₂ : B.IsRefl) {x : V} (hx : ¬B.IsOrtho x x)
Goal: (B.restrict (B.orthogonal (Submodule.span K {x}))).Nondegenerate
-- SameRay.zero_left
@[simp]
theorem zero_left (y : M) : SameRay R 0 y 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (y : M)
Goal: SameRay R 0 y
-- SameRay.zero_right
@[simp]
theorem zero_right (x : M) : SameRay R x 0 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (x : M)
Goal: SameRay R x 0
-- SameRay.of_subsingleton
@[nontriviality]
theorem of_subsingleton [Subsingleton M] (x y : M) : SameRay R x y 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] [Subsingleton M] (x y : M)
Goal: SameRay R x y
-- SameRay.of_subsingleton'
@[nontriviality]
theorem of_subsingleton' [Subsingleton R] (x y : M) : SameRay R x y 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] [Subsingleton R] (x y : M)
Goal: SameRay R x y
-- SameRay.refl
@[refl]
theorem refl (x : M) : SameRay R x x 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (x : M)
Goal: SameRay R x x
-- SameRay.rfl
protected theorem rfl : SameRay R x x 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x : M}
Goal: SameRay R x x
-- SameRay.symm
@[symm]
theorem symm (h : SameRay R x y) : SameRay R y x 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y : M} (h : SameRay R x y)
Goal: SameRay R y x
-- SameRay.exists_pos
theorem exists_pos (h : SameRay R x y) (hx : x ≠ 0) (hy : y ≠ 0) :
    ∃ r₁ r₂ : R, 0 < r₁ ∧ 0 < r₂ ∧ r₁ • x = r₂ • y 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y : M} (h : SameRay R x y) (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ∃ r₁ r₂, 0 < r₁ ∧ 0 < r₂ ∧ r₁ • x = r₂ • y
-- SameRay.sameRay_comm
theorem sameRay_comm : SameRay R x y ↔ SameRay R y x 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y : M}
Goal: SameRay R x y ↔ SameRay R y x
-- SameRay.trans
theorem trans (hxy : SameRay R x y) (hyz : SameRay R y z) (hy : y = 0 → x = 0 ∨ z = 0) :
    SameRay R x z 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y z : M} (hxy : SameRay R x y) (hyz : SameRay R y z) (hy : y = 0 → x = 0 ∨ z = 0)
Goal: SameRay R x z
-- SameRay.sameRay_nonneg_smul_right
lemma sameRay_nonneg_smul_right (v : M) (h : 0 ≤ a) : SameRay R v (a • v) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {S : Type u_5} [OrderedCommSemiring S] [Algebra S R] [Module S M] [SMulPosMono S R] [IsScalarTower S R M] {a : S} (v : M) (h : 0 ≤ a)
Goal: SameRay R v (a • v)
-- SameRay.sameRay_nonneg_smul_left
lemma sameRay_nonneg_smul_left (v : M) (ha : 0 ≤ a) : SameRay R (a • v) v 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {S : Type u_5} [OrderedCommSemiring S] [Algebra S R] [Module S M] [SMulPosMono S R] [IsScalarTower S R M] {a : S} (v : M) (ha : 0 ≤ a)
Goal: SameRay R (a • v) v
-- SameRay.sameRay_pos_smul_right
lemma sameRay_pos_smul_right (v : M) (ha : 0 < a) : SameRay R v (a • v) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {S : Type u_5} [OrderedCommSemiring S] [Algebra S R] [Module S M] [SMulPosMono S R] [IsScalarTower S R M] {a : S} (v : M) (ha : 0 < a)
Goal: SameRay R v (a • v)
-- SameRay.sameRay_pos_smul_left
lemma sameRay_pos_smul_left (v : M) (ha : 0 < a) : SameRay R (a • v) v 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {S : Type u_5} [OrderedCommSemiring S] [Algebra S R] [Module S M] [SMulPosMono S R] [IsScalarTower S R M] {a : S} (v : M) (ha : 0 < a)
Goal: SameRay R (a • v) v
-- SameRay.nonneg_smul_right
lemma nonneg_smul_right (h : SameRay R x y) (ha : 0 ≤ a) : SameRay R x (a • y) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y : M} {S : Type u_5} [OrderedCommSemiring S] [Algebra S R] [Module S M] [SMulPosMono S R] [IsScalarTower S R M] {a : S} (h : SameRay R x y) (ha : 0 ≤ a)
Goal: SameRay R x (a • y)
-- SameRay.nonneg_smul_left
lemma nonneg_smul_left (h : SameRay R x y) (ha : 0 ≤ a) : SameRay R (a • x) y 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y : M} {S : Type u_5} [OrderedCommSemiring S] [Algebra S R] [Module S M] [SMulPosMono S R] [IsScalarTower S R M] {a : S} (h : SameRay R x y) (ha : 0 ≤ a)
Goal: SameRay R (a • x) y
-- SameRay.pos_smul_right
theorem pos_smul_right (h : SameRay R x y) (ha : 0 < a) : SameRay R x (a • y) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y : M} {S : Type u_5} [OrderedCommSemiring S] [Algebra S R] [Module S M] [SMulPosMono S R] [IsScalarTower S R M] {a : S} (h : SameRay R x y) (ha : 0 < a)
Goal: SameRay R x (a • y)
-- SameRay.pos_smul_left
theorem pos_smul_left (h : SameRay R x y) (hr : 0 < a) : SameRay R (a • x) y 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y : M} {S : Type u_5} [OrderedCommSemiring S] [Algebra S R] [Module S M] [SMulPosMono S R] [IsScalarTower S R M] {a : S} (h : SameRay R x y) (hr : 0 < a)
Goal: SameRay R (a • x) y
-- SameRay.map
theorem map (f : M →ₗ[R] N) (h : SameRay R x y) : SameRay R (f x) (f y) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {x y : M} (f : M →ₗ[R] N) (h : SameRay R x y)
Goal: SameRay R (f x) (f y)
-- Function.Injective.sameRay_map_iff
theorem _root_.Function.Injective.sameRay_map_iff
    {F : Type*} [FunLike F M N] [LinearMapClass F R M N]
    {f : F} (hf : Function.Injective f) :
    SameRay R (f x) (f y) ↔ SameRay R x y 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {x y : M} {F : Type u_6} [FunLike F M N] [LinearMapClass F R M N] {f : F} (hf : Function.Injective ⇑f)
Goal: SameRay R (f x) (f y) ↔ SameRay R x y
-- SameRay.sameRay_map_iff
@[simp]
theorem sameRay_map_iff (e : M ≃ₗ[R] N) : SameRay R (e x) (e y) ↔ SameRay R x y 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {x y : M} (e : M ≃ₗ[R] N)
Goal: SameRay R (e x) (e y) ↔ SameRay R x y
-- SameRay.smul
theorem smul {S : Type*} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M]
    (h : SameRay R x y) (s : S) : SameRay R (s • x) (s • y) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y : M} {S : Type u_6} [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] (h : SameRay R x y) (s : S)
Goal: SameRay R (s • x) (s • y)
-- SameRay.add_left
theorem add_left (hx : SameRay R x z) (hy : SameRay R y z) : SameRay R (x + y) z 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y z : M} (hx : SameRay R x z) (hy : SameRay R y z)
Goal: SameRay R (x + y) z
-- SameRay.add_right
theorem add_right (hy : SameRay R x y) (hz : SameRay R x z) : SameRay R x (y + z) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {x y z : M} (hy : SameRay R x y) (hz : SameRay R x z)
Goal: SameRay R x (y + z)
-- equiv_iff_sameRay
theorem equiv_iff_sameRay {v₁ v₂ : RayVector R M} : v₁ ≈ v₂ ↔ SameRay R (v₁ : M) v₂ 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {v₁ v₂ : RayVector R M}
Goal: v₁ ≈ v₂ ↔ SameRay R ↑v₁ ↑v₂
-- Module.Ray.ind
theorem Module.Ray.ind {C : Module.Ray R M → Prop} (h : ∀ (v) (hv : v ≠ 0), C (rayOfNeZero R v hv))
    (x : Module.Ray R M) : C x 
Arguments: (R : Type u_1) [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {C : Module.Ray R M → Prop} (h : ∀ (v : M) (hv : v ≠ 0), C (rayOfNeZero R v hv)) (x : Module.Ray R M)
Goal: C x
-- ray_eq_iff
theorem ray_eq_iff {v₁ v₂ : M} (hv₁ : v₁ ≠ 0) (hv₂ : v₂ ≠ 0) :
    rayOfNeZero R _ hv₁ = rayOfNeZero R _ hv₂ ↔ SameRay R v₁ v₂ 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {v₁ v₂ : M} (hv₁ : v₁ ≠ 0) (hv₂ : v₂ ≠ 0)
Goal: rayOfNeZero R v₁ hv₁ = rayOfNeZero R v₂ hv₂ ↔ SameRay R v₁ v₂
-- ray_pos_smul
@[simp]
theorem ray_pos_smul {v : M} (h : v ≠ 0) {r : R} (hr : 0 < r) (hrv : r • v ≠ 0) :
    rayOfNeZero R (r • v) hrv = rayOfNeZero R v h 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {v : M} (h : v ≠ 0) {r : R} (hr : 0 < r) (hrv : r • v ≠ 0)
Goal: rayOfNeZero R (r • v) hrv = rayOfNeZero R v h
-- Module.Ray.map_apply
@[simp]
theorem Module.Ray.map_apply (e : M ≃ₗ[R] N) (v : M) (hv : v ≠ 0) :
    Module.Ray.map e (rayOfNeZero _ v hv) = rayOfNeZero _ (e v) (e.map_ne_zero_iff.2 hv) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (e : M ≃ₗ[R] N) (v : M) (hv : v ≠ 0)
Goal: (Module.Ray.map e) (rayOfNeZero R v hv) = rayOfNeZero R (e v) ⋯
-- Module.Ray.map_refl
@[simp]
theorem Module.Ray.map_refl : (Module.Ray.map <| LinearEquiv.refl R M) = Equiv.refl _ 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M]
Goal: Module.Ray.map (LinearEquiv.refl R M) = Equiv.refl (Module.Ray R M)
-- Module.Ray.map_symm
@[simp]
theorem Module.Ray.map_symm (e : M ≃ₗ[R] N) : (Module.Ray.map e).symm = Module.Ray.map e.symm 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (e : M ≃ₗ[R] N)
Goal: (Module.Ray.map e).symm = Module.Ray.map e.symm
-- Module.Ray.linearEquiv_smul_eq_map
@[simp]
theorem Module.Ray.linearEquiv_smul_eq_map (e : M ≃ₗ[R] M) (v : Module.Ray R M) :
    e • v = Module.Ray.map e v 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (e : M ≃ₗ[R] M) (v : Module.Ray R M)
Goal: e • v = (Module.Ray.map e) v
-- smul_rayOfNeZero
@[simp]
theorem smul_rayOfNeZero (g : G) (v : M) (hv) :
    g • rayOfNeZero R v hv = rayOfNeZero R (g • v) ((smul_ne_zero_iff_ne _).2 hv) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {G : Type u_5} [Group G] [DistribMulAction G M] [SMulCommClass R G M] (g : G) (v : M) (hv : v ≠ 0)
Goal: g • rayOfNeZero R v hv = rayOfNeZero R (g • v) ⋯
-- Module.Ray.units_smul_of_pos
theorem units_smul_of_pos (u : Rˣ) (hu : 0 < (u.1 : R)) (v : Module.Ray R M) : u • v = v 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (u : Rˣ) (hu : 0 < ↑u) (v : Module.Ray R M)
Goal: u • v = v
-- Module.Ray.someRayVector_ray
@[simp]
theorem someRayVector_ray (x : Module.Ray R M) : (⟦x.someRayVector⟧ : Module.Ray R M) = x 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (x : Module.Ray R M)
Goal: ⟦x.someRayVector⟧ = x
-- Module.Ray.someVector_ne_zero
@[simp]
theorem someVector_ne_zero (x : Module.Ray R M) : x.someVector ≠ 0 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (x : Module.Ray R M)
Goal: x.someVector ≠ 0
-- Module.Ray.someVector_ray
@[simp]
theorem someVector_ray (x : Module.Ray R M) : rayOfNeZero R _ x.someVector_ne_zero = x 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (x : Module.Ray R M)
Goal: rayOfNeZero R x.someVector ⋯ = x
-- sameRay_neg_iff
@[simp]
theorem sameRay_neg_iff : SameRay R (-x) (-y) ↔ SameRay R x y 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M}
Goal: SameRay R (-x) (-y) ↔ SameRay R x y
-- sameRay_neg_swap
theorem sameRay_neg_swap : SameRay R (-x) y ↔ SameRay R x (-y) 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M}
Goal: SameRay R (-x) y ↔ SameRay R x (-y)
-- eq_zero_of_sameRay_neg_smul_right
theorem eq_zero_of_sameRay_neg_smul_right [NoZeroSMulDivisors R M] {r : R} (hr : r < 0)
    (h : SameRay R x (r • x)) : x = 0 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {x : M} [NoZeroSMulDivisors R M] {r : R} (hr : r < 0) (h : SameRay R x (r • x))
Goal: x = 0
-- eq_zero_of_sameRay_self_neg
theorem eq_zero_of_sameRay_self_neg [NoZeroSMulDivisors R M] (h : SameRay R x (-x)) : x = 0 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {x : M} [NoZeroSMulDivisors R M] (h : SameRay R x (-x))
Goal: x = 0
-- RayVector.coe_neg
@[simp, norm_cast]
theorem coe_neg {R : Type*} (v : RayVector R M) : ↑(-v) = -(v : M) 
Arguments: {M : Type u_2} [AddCommGroup M] {R : Type u_4} (v : RayVector R M)
Goal: ↑(-v) = -↑v
-- RayVector.equiv_neg_iff
@[simp]
theorem equiv_neg_iff {v₁ v₂ : RayVector R M} : -v₁ ≈ -v₂ ↔ v₁ ≈ v₂ 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {v₁ v₂ : RayVector R M}
Goal: -v₁ ≈ -v₂ ↔ v₁ ≈ v₂
-- neg_rayOfNeZero
@[simp]
theorem neg_rayOfNeZero (v : M) (h : v ≠ 0) :
    -rayOfNeZero R _ h = rayOfNeZero R (-v) (neg_ne_zero.2 h) 
Arguments: (R : Type u_1) [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (v : M) (h : v ≠ 0)
Goal: -rayOfNeZero R v h = rayOfNeZero R (-v) ⋯
-- Module.Ray.ne_neg_self
theorem ne_neg_self [NoZeroSMulDivisors R M] (x : Module.Ray R M) : x ≠ -x 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] (x : Module.Ray R M)
Goal: x ≠ -x
-- Module.Ray.neg_units_smul
theorem neg_units_smul (u : Rˣ) (v : Module.Ray R M) : -u • v = -(u • v) 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (u : Rˣ) (v : Module.Ray R M)
Goal: -u • v = -(u • v)
-- Module.Ray.units_smul_of_neg
theorem units_smul_of_neg (u : Rˣ) (hu : u.1 < 0) (v : Module.Ray R M) : u • v = -v 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (u : Rˣ) (hu : ↑u < 0) (v : Module.Ray R M)
Goal: u • v = -v
-- Module.Ray.map_neg
@[simp]
protected theorem map_neg (f : M ≃ₗ[R] N) (v : Module.Ray R M) : map f (-v) = -map f v 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} {N : Type u_3} [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (f : M ≃ₗ[R] N) (v : Module.Ray R M)
Goal: (Module.Ray.map f) (-v) = -(Module.Ray.map f) v
-- sameRay_of_mem_orbit
theorem sameRay_of_mem_orbit {v₁ v₂ : M} (h : v₁ ∈ MulAction.orbit ↥(Units.posSubgroup R) v₂) :
    SameRay R v₁ v₂ 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {v₁ v₂ : M} (h : v₁ ∈ MulAction.orbit (↥(Units.posSubgroup R)) v₂)
Goal: SameRay R v₁ v₂
-- units_inv_smul
@[simp]
theorem units_inv_smul (u : Rˣ) (v : Module.Ray R M) : u⁻¹ • v = u • v 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (u : Rˣ) (v : Module.Ray R M)
Goal: u⁻¹ • v = u • v
-- sameRay_smul_right_iff
@[simp]
theorem sameRay_smul_right_iff {v : M} {r : R} : SameRay R v (r • v) ↔ 0 ≤ r ∨ v = 0 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {v : M} {r : R}
Goal: SameRay R v (r • v) ↔ 0 ≤ r ∨ v = 0
-- sameRay_smul_right_iff_of_ne
theorem sameRay_smul_right_iff_of_ne {v : M} (hv : v ≠ 0) {r : R} (hr : r ≠ 0) :
    SameRay R v (r • v) ↔ 0 < r 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {v : M} (hv : v ≠ 0) {r : R} (hr : r ≠ 0)
Goal: SameRay R v (r • v) ↔ 0 < r
-- sameRay_smul_left_iff
@[simp]
theorem sameRay_smul_left_iff {v : M} {r : R} : SameRay R (r • v) v ↔ 0 ≤ r ∨ v = 0 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {v : M} {r : R}
Goal: SameRay R (r • v) v ↔ 0 ≤ r ∨ v = 0
-- sameRay_smul_left_iff_of_ne
theorem sameRay_smul_left_iff_of_ne {v : M} (hv : v ≠ 0) {r : R} (hr : r ≠ 0) :
    SameRay R (r • v) v ↔ 0 < r 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {v : M} (hv : v ≠ 0) {r : R} (hr : r ≠ 0)
Goal: SameRay R (r • v) v ↔ 0 < r
-- sameRay_neg_smul_right_iff
@[simp]
theorem sameRay_neg_smul_right_iff {v : M} {r : R} : SameRay R (-v) (r • v) ↔ r ≤ 0 ∨ v = 0 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {v : M} {r : R}
Goal: SameRay R (-v) (r • v) ↔ r ≤ 0 ∨ v = 0
-- sameRay_neg_smul_right_iff_of_ne
theorem sameRay_neg_smul_right_iff_of_ne {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0) :
    SameRay R (-v) (r • v) ↔ r < 0 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0)
Goal: SameRay R (-v) (r • v) ↔ r < 0
-- sameRay_neg_smul_left_iff
@[simp]
theorem sameRay_neg_smul_left_iff {v : M} {r : R} : SameRay R (r • v) (-v) ↔ r ≤ 0 ∨ v = 0 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {v : M} {r : R}
Goal: SameRay R (r • v) (-v) ↔ r ≤ 0 ∨ v = 0
-- sameRay_neg_smul_left_iff_of_ne
theorem sameRay_neg_smul_left_iff_of_ne {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0) :
    SameRay R (r • v) (-v) ↔ r < 0 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0)
Goal: SameRay R (r • v) (-v) ↔ r < 0
-- units_smul_eq_self_iff
@[simp]
theorem units_smul_eq_self_iff {u : Rˣ} {v : Module.Ray R M} : u • v = v ↔ 0 < u.1 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {u : Rˣ} {v : Module.Ray R M}
Goal: u • v = v ↔ 0 < ↑u
-- units_smul_eq_neg_iff
@[simp]
theorem units_smul_eq_neg_iff {u : Rˣ} {v : Module.Ray R M} : u • v = -v ↔ u.1 < 0 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {u : Rˣ} {v : Module.Ray R M}
Goal: u • v = -v ↔ ↑u < 0
-- sameRay_or_sameRay_neg_iff_not_linearIndependent
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {x y : M}
Goal: SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y]
-- sameRay_or_ne_zero_and_sameRay_neg_iff_not_linearIndependent
theorem sameRay_or_ne_zero_and_sameRay_neg_iff_not_linearIndependent {x y : M} :
    SameRay R x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {x y : M}
Goal: SameRay R x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y]
-- SameRay.exists_pos_left
theorem exists_pos_left (h : SameRay R x y) (hx : x ≠ 0) (hy : y ≠ 0) :
    ∃ r : R, 0 < r ∧ r • x = y 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (h : SameRay R x y) (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ∃ r, 0 < r ∧ r • x = y
-- SameRay.exists_pos_right
theorem exists_pos_right (h : SameRay R x y) (hx : x ≠ 0) (hy : y ≠ 0) :
    ∃ r : R, 0 < r ∧ x = r • y 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (h : SameRay R x y) (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ∃ r, 0 < r ∧ x = r • y
-- SameRay.exists_nonneg_left
theorem exists_nonneg_left (h : SameRay R x y) (hx : x ≠ 0) : ∃ r : R, 0 ≤ r ∧ r • x = y 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {x y : M} (h : SameRay R x y) (hx : x ≠ 0)
Goal: ∃ r, 0 ≤ r ∧ r • x = y