ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- sbtw_iff_right_ne_and_left_mem_image_Ioi
theorem sbtw_iff_right_ne_and_left_mem_image_Ioi {x y z : P} :
    Sbtw R x y z ↔ z ≠ y ∧ x ∈ lineMap z y '' Set.Ioi (1 : R) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P}
Goal: Sbtw R x y z ↔ z ≠ y ∧ x ∈ ⇑(AffineMap.lineMap z y) '' Set.Ioi 1
-- Sbtw.left_mem_image_Ioi
theorem Sbtw.left_mem_image_Ioi {x y z : P} (h : Sbtw R x y z) :
    x ∈ lineMap z y '' Set.Ioi (1 : R) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Sbtw R x y z)
Goal: x ∈ ⇑(AffineMap.lineMap z y) '' Set.Ioi 1
-- Sbtw.left_mem_affineSpan
theorem Sbtw.left_mem_affineSpan {x y z : P} (h : Sbtw R x y z) : x ∈ line[R, z, y] 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Sbtw R x y z)
Goal: x ∈ affineSpan R {z, y}
-- wbtw_smul_vadd_smul_vadd_of_nonneg_of_le
theorem wbtw_smul_vadd_smul_vadd_of_nonneg_of_le (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁)
    (hr₂ : r₁ ≤ r₂) : Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁) (hr₂ : r₁ ≤ r₂)
Goal: Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x)
-- wbtw_or_wbtw_smul_vadd_of_nonneg
theorem wbtw_or_wbtw_smul_vadd_of_nonneg (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁) (hr₂ : 0 ≤ r₂) :
    Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ Wbtw R x (r₂ • v +ᵥ x) (r₁ • v +ᵥ x) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁) (hr₂ : 0 ≤ r₂)
Goal: Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ Wbtw R x (r₂ • v +ᵥ x) (r₁ • v +ᵥ x)
-- wbtw_smul_vadd_smul_vadd_of_nonpos_of_le
theorem wbtw_smul_vadd_smul_vadd_of_nonpos_of_le (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0)
    (hr₂ : r₂ ≤ r₁) : Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0) (hr₂ : r₂ ≤ r₁)
Goal: Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x)
-- wbtw_or_wbtw_smul_vadd_of_nonpos
theorem wbtw_or_wbtw_smul_vadd_of_nonpos (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0) (hr₂ : r₂ ≤ 0) :
    Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ Wbtw R x (r₂ • v +ᵥ x) (r₁ • v +ᵥ x) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0) (hr₂ : r₂ ≤ 0)
Goal: Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ Wbtw R x (r₂ • v +ᵥ x) (r₁ • v +ᵥ x)
-- wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg
theorem wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0)
    (hr₂ : 0 ≤ r₂) : Wbtw R (r₁ • v +ᵥ x) x (r₂ • v +ᵥ x) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0) (hr₂ : 0 ≤ r₂)
Goal: Wbtw R (r₁ • v +ᵥ x) x (r₂ • v +ᵥ x)
-- wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos
theorem wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁)
    (hr₂ : r₂ ≤ 0) : Wbtw R (r₁ • v +ᵥ x) x (r₂ • v +ᵥ x) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁) (hr₂ : r₂ ≤ 0)
Goal: Wbtw R (r₁ • v +ᵥ x) x (r₂ • v +ᵥ x)
-- Wbtw.trans_left_right
theorem Wbtw.trans_left_right {w x y z : P} (h₁ : Wbtw R w y z) (h₂ : Wbtw R w x y) :
    Wbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {w x y z : P} (h₁ : Wbtw R w y z) (h₂ : Wbtw R w x y)
Goal: Wbtw R x y z
-- Wbtw.trans_right_left
theorem Wbtw.trans_right_left {w x y z : P} (h₁ : Wbtw R w x z) (h₂ : Wbtw R x y z) :
    Wbtw R w x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {w x y z : P} (h₁ : Wbtw R w x z) (h₂ : Wbtw R x y z)
Goal: Wbtw R w x y
-- Sbtw.trans_left_right
theorem Sbtw.trans_left_right {w x y z : P} (h₁ : Sbtw R w y z) (h₂ : Sbtw R w x y) :
    Sbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {w x y z : P} (h₁ : Sbtw R w y z) (h₂ : Sbtw R w x y)
Goal: Sbtw R x y z
-- Sbtw.trans_right_left
theorem Sbtw.trans_right_left {w x y z : P} (h₁ : Sbtw R w x z) (h₂ : Sbtw R x y z) :
    Sbtw R w x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {w x y z : P} (h₁ : Sbtw R w x z) (h₂ : Sbtw R x y z)
Goal: Sbtw R w x y
-- Wbtw.collinear
theorem Wbtw.collinear {x y z : P} (h : Wbtw R x y z) : Collinear R ({x, y, z} : Set P) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Wbtw R x y z)
Goal: Collinear R {x, y, z}
-- Collinear.wbtw_or_wbtw_or_wbtw
theorem Collinear.wbtw_or_wbtw_or_wbtw {x y z : P} (h : Collinear R ({x, y, z} : Set P)) :
    Wbtw R x y z ∨ Wbtw R y z x ∨ Wbtw R z x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Collinear R {x, y, z})
Goal: Wbtw R x y z ∨ Wbtw R y z x ∨ Wbtw R z x y
-- wbtw_iff_sameRay_vsub
theorem wbtw_iff_sameRay_vsub {x y z : P} : Wbtw R x y z ↔ SameRay R (y -ᵥ x) (z -ᵥ y) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P}
Goal: Wbtw R x y z ↔ SameRay R (y -ᵥ x) (z -ᵥ y)
-- wbtw_pointReflection
theorem wbtw_pointReflection (x y : P) : Wbtw R y x (pointReflection R x y) 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: Wbtw R y x ((AffineEquiv.pointReflection R x) y)
-- sbtw_pointReflection_of_ne
theorem sbtw_pointReflection_of_ne {x y : P} (h : x ≠ y) : Sbtw R y x (pointReflection R x y) 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P} (h : x ≠ y)
Goal: Sbtw R y x ((AffineEquiv.pointReflection R x) y)
-- wbtw_midpoint
theorem wbtw_midpoint (x y : P) : Wbtw R x (midpoint R x y) y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: Wbtw R x (midpoint R x y) y
-- sbtw_midpoint_of_ne
theorem sbtw_midpoint_of_ne {x y : P} (h : x ≠ y) : Sbtw R x (midpoint R x y) y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P} (h : x ≠ y)
Goal: Sbtw R x (midpoint R x y) y
-- AffineSubspace.WSameSide.map
theorem WSameSide.map {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) (f : P →ᵃ[R] P') :
    (s.map f).WSameSide (f x) (f y) 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) (f : P →ᵃ[R] P')
Goal: (AffineSubspace.map f s).WSameSide (f x) (f y)
-- Function.Injective.wSameSide_map_iff
theorem _root_.Function.Injective.wSameSide_map_iff {s : AffineSubspace R P} {x y : P}
    {f : P →ᵃ[R] P'} (hf : Function.Injective f) :
    (s.map f).WSameSide (f x) (f y) ↔ s.WSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f)
Goal: (AffineSubspace.map f s).WSameSide (f x) (f y) ↔ s.WSameSide x y
-- Function.Injective.sSameSide_map_iff
theorem _root_.Function.Injective.sSameSide_map_iff {s : AffineSubspace R P} {x y : P}
    {f : P →ᵃ[R] P'} (hf : Function.Injective f) :
    (s.map f).SSameSide (f x) (f y) ↔ s.SSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f)
Goal: (AffineSubspace.map f s).SSameSide (f x) (f y) ↔ s.SSameSide x y
-- AffineEquiv.wSameSide_map_iff
@[simp]
theorem _root_.AffineEquiv.wSameSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
    (s.map ↑f).WSameSide (f x) (f y) ↔ s.WSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P')
Goal: (AffineSubspace.map (↑f) s).WSameSide (f x) (f y) ↔ s.WSameSide x y
-- AffineEquiv.sSameSide_map_iff
@[simp]
theorem _root_.AffineEquiv.sSameSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
    (s.map ↑f).SSameSide (f x) (f y) ↔ s.SSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P')
Goal: (AffineSubspace.map (↑f) s).SSameSide (f x) (f y) ↔ s.SSameSide x y
-- AffineSubspace.WOppSide.map
theorem WOppSide.map {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) (f : P →ᵃ[R] P') :
    (s.map f).WOppSide (f x) (f y) 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) (f : P →ᵃ[R] P')
Goal: (AffineSubspace.map f s).WOppSide (f x) (f y)
-- Function.Injective.wOppSide_map_iff
theorem _root_.Function.Injective.wOppSide_map_iff {s : AffineSubspace R P} {x y : P}
    {f : P →ᵃ[R] P'} (hf : Function.Injective f) :
    (s.map f).WOppSide (f x) (f y) ↔ s.WOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f)
Goal: (AffineSubspace.map f s).WOppSide (f x) (f y) ↔ s.WOppSide x y
-- Function.Injective.sOppSide_map_iff
theorem _root_.Function.Injective.sOppSide_map_iff {s : AffineSubspace R P} {x y : P}
    {f : P →ᵃ[R] P'} (hf : Function.Injective f) :
    (s.map f).SOppSide (f x) (f y) ↔ s.SOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f)
Goal: (AffineSubspace.map f s).SOppSide (f x) (f y) ↔ s.SOppSide x y
-- AffineEquiv.wOppSide_map_iff
@[simp]
theorem _root_.AffineEquiv.wOppSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
    (s.map ↑f).WOppSide (f x) (f y) ↔ s.WOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P')
Goal: (AffineSubspace.map (↑f) s).WOppSide (f x) (f y) ↔ s.WOppSide x y
-- AffineEquiv.sOppSide_map_iff
@[simp]
theorem _root_.AffineEquiv.sOppSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
    (s.map ↑f).SOppSide (f x) (f y) ↔ s.SOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P')
Goal: (AffineSubspace.map (↑f) s).SOppSide (f x) (f y) ↔ s.SOppSide x y
-- AffineSubspace.WSameSide.nonempty
theorem WSameSide.nonempty {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) :
    (s : Set P).Nonempty 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y)
Goal: (↑s).Nonempty
-- AffineSubspace.SSameSide.nonempty
theorem SSameSide.nonempty {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) :
    (s : Set P).Nonempty 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y)
Goal: (↑s).Nonempty
-- AffineSubspace.WOppSide.nonempty
theorem WOppSide.nonempty {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) :
    (s : Set P).Nonempty 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y)
Goal: (↑s).Nonempty
-- AffineSubspace.SOppSide.nonempty
theorem SOppSide.nonempty {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :
    (s : Set P).Nonempty 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y)
Goal: (↑s).Nonempty
-- AffineSubspace.SSameSide.wSameSide
theorem SSameSide.wSameSide {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) :
    s.WSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y)
Goal: s.WSameSide x y
-- AffineSubspace.SSameSide.left_not_mem
theorem SSameSide.left_not_mem {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) : x ∉ s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y)
Goal: x ∉ s
-- AffineSubspace.SSameSide.right_not_mem
theorem SSameSide.right_not_mem {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) : y ∉ s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y)
Goal: y ∉ s
-- AffineSubspace.SOppSide.wOppSide
theorem SOppSide.wOppSide {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :
    s.WOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y)
Goal: s.WOppSide x y
-- AffineSubspace.SOppSide.left_not_mem
theorem SOppSide.left_not_mem {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) : x ∉ s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y)
Goal: x ∉ s
-- AffineSubspace.SOppSide.right_not_mem
theorem SOppSide.right_not_mem {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) : y ∉ s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y)
Goal: y ∉ s
-- AffineSubspace.wSameSide_comm
theorem wSameSide_comm {s : AffineSubspace R P} {x y : P} : s.WSameSide x y ↔ s.WSameSide y x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P}
Goal: s.WSameSide x y ↔ s.WSameSide y x
-- AffineSubspace.sSameSide_comm
theorem sSameSide_comm {s : AffineSubspace R P} {x y : P} : s.SSameSide x y ↔ s.SSameSide y x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P}
Goal: s.SSameSide x y ↔ s.SSameSide y x
-- AffineSubspace.wOppSide_comm
theorem wOppSide_comm {s : AffineSubspace R P} {x y : P} : s.WOppSide x y ↔ s.WOppSide y x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P}
Goal: s.WOppSide x y ↔ s.WOppSide y x
-- AffineSubspace.sOppSide_comm
theorem sOppSide_comm {s : AffineSubspace R P} {x y : P} : s.SOppSide x y ↔ s.SOppSide y x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P}
Goal: s.SOppSide x y ↔ s.SOppSide y x
-- AffineSubspace.not_wSameSide_bot
theorem not_wSameSide_bot (x y : P) : ¬(⊥ : AffineSubspace R P).WSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: ¬⊥.WSameSide x y
-- AffineSubspace.not_sSameSide_bot
theorem not_sSameSide_bot (x y : P) : ¬(⊥ : AffineSubspace R P).SSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: ¬⊥.SSameSide x y
-- AffineSubspace.not_wOppSide_bot
theorem not_wOppSide_bot (x y : P) : ¬(⊥ : AffineSubspace R P).WOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: ¬⊥.WOppSide x y
-- AffineSubspace.not_sOppSide_bot
theorem not_sOppSide_bot (x y : P) : ¬(⊥ : AffineSubspace R P).SOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: ¬⊥.SOppSide x y
-- AffineSubspace.wSameSide_self_iff
@[simp]
theorem wSameSide_self_iff {s : AffineSubspace R P} {x : P} :
    s.WSameSide x x ↔ (s : Set P).Nonempty 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P}
Goal: s.WSameSide x x ↔ (↑s).Nonempty
-- AffineSubspace.sSameSide_self_iff
theorem sSameSide_self_iff {s : AffineSubspace R P} {x : P} :
    s.SSameSide x x ↔ (s : Set P).Nonempty ∧ x ∉ s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P}
Goal: s.SSameSide x x ↔ (↑s).Nonempty ∧ x ∉ s
-- AffineSubspace.wSameSide_of_left_mem
theorem wSameSide_of_left_mem {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s) :
    s.WSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s)
Goal: s.WSameSide x y
-- AffineSubspace.wSameSide_of_right_mem
theorem wSameSide_of_right_mem {s : AffineSubspace R P} (x : P) {y : P} (hy : y ∈ s) :
    s.WSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} (x : P) {y : P} (hy : y ∈ s)
Goal: s.WSameSide x y
-- AffineSubspace.wOppSide_of_left_mem
theorem wOppSide_of_left_mem {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s) :
    s.WOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s)
Goal: s.WOppSide x y
-- AffineSubspace.wOppSide_of_right_mem
theorem wOppSide_of_right_mem {s : AffineSubspace R P} (x : P) {y : P} (hy : y ∈ s) :
    s.WOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} (x : P) {y : P} (hy : y ∈ s)
Goal: s.WOppSide x y
-- AffineSubspace.wSameSide_vadd_left_iff
theorem wSameSide_vadd_left_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
    s.WSameSide (v +ᵥ x) y ↔ s.WSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction)
Goal: s.WSameSide (v +ᵥ x) y ↔ s.WSameSide x y
-- AffineSubspace.wSameSide_vadd_right_iff
theorem wSameSide_vadd_right_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
    s.WSameSide x (v +ᵥ y) ↔ s.WSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction)
Goal: s.WSameSide x (v +ᵥ y) ↔ s.WSameSide x y
-- AffineSubspace.sSameSide_vadd_left_iff
theorem sSameSide_vadd_left_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
    s.SSameSide (v +ᵥ x) y ↔ s.SSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction)
Goal: s.SSameSide (v +ᵥ x) y ↔ s.SSameSide x y
-- AffineSubspace.sSameSide_vadd_right_iff
theorem sSameSide_vadd_right_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
    s.SSameSide x (v +ᵥ y) ↔ s.SSameSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction)
Goal: s.SSameSide x (v +ᵥ y) ↔ s.SSameSide x y
-- AffineSubspace.wOppSide_vadd_left_iff
theorem wOppSide_vadd_left_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
    s.WOppSide (v +ᵥ x) y ↔ s.WOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction)
Goal: s.WOppSide (v +ᵥ x) y ↔ s.WOppSide x y
-- AffineSubspace.wOppSide_vadd_right_iff
theorem wOppSide_vadd_right_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
    s.WOppSide x (v +ᵥ y) ↔ s.WOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction)
Goal: s.WOppSide x (v +ᵥ y) ↔ s.WOppSide x y
-- AffineSubspace.sOppSide_vadd_left_iff
theorem sOppSide_vadd_left_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
    s.SOppSide (v +ᵥ x) y ↔ s.SOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction)
Goal: s.SOppSide (v +ᵥ x) y ↔ s.SOppSide x y
-- AffineSubspace.sOppSide_vadd_right_iff
theorem sOppSide_vadd_right_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
    s.SOppSide x (v +ᵥ y) ↔ s.SOppSide x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction)
Goal: s.SOppSide x (v +ᵥ y) ↔ s.SOppSide x y