ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AffineSubspace.wSameSide_smul_vsub_vadd_left
theorem wSameSide_smul_vsub_vadd_left {s : AffineSubspace R P} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) {t : R} (ht : 0 ≤ t) : s.WSameSide (t • (x -ᵥ p₁) +ᵥ p₂) 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} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 ≤ t)
Goal: s.WSameSide (t • (x -ᵥ p₁) +ᵥ p₂) x
-- AffineSubspace.wSameSide_smul_vsub_vadd_right
theorem wSameSide_smul_vsub_vadd_right {s : AffineSubspace R P} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) {t : R} (ht : 0 ≤ t) : s.WSameSide x (t • (x -ᵥ p₁) +ᵥ p₂) 
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} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 ≤ t)
Goal: s.WSameSide x (t • (x -ᵥ p₁) +ᵥ p₂)
-- AffineSubspace.wSameSide_lineMap_left
theorem wSameSide_lineMap_left {s : AffineSubspace R P} {x : P} (y : P) (h : x ∈ s) {t : R}
    (ht : 0 ≤ t) : s.WSameSide (lineMap x y t) 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) (h : x ∈ s) {t : R} (ht : 0 ≤ t)
Goal: s.WSameSide ((AffineMap.lineMap x y) t) y
-- AffineSubspace.wSameSide_lineMap_right
theorem wSameSide_lineMap_right {s : AffineSubspace R P} {x : P} (y : P) (h : x ∈ s) {t : R}
    (ht : 0 ≤ t) : s.WSameSide y (lineMap x y t) 
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) (h : x ∈ s) {t : R} (ht : 0 ≤ t)
Goal: s.WSameSide y ((AffineMap.lineMap x y) t)
-- AffineSubspace.wOppSide_smul_vsub_vadd_left
theorem wOppSide_smul_vsub_vadd_left {s : AffineSubspace R P} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : s.WOppSide (t • (x -ᵥ p₁) +ᵥ p₂) 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} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0)
Goal: s.WOppSide (t • (x -ᵥ p₁) +ᵥ p₂) x
-- AffineSubspace.wOppSide_smul_vsub_vadd_right
theorem wOppSide_smul_vsub_vadd_right {s : AffineSubspace R P} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : s.WOppSide x (t • (x -ᵥ p₁) +ᵥ p₂) 
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} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0)
Goal: s.WOppSide x (t • (x -ᵥ p₁) +ᵥ p₂)
-- AffineSubspace.wOppSide_lineMap_left
theorem wOppSide_lineMap_left {s : AffineSubspace R P} {x : P} (y : P) (h : x ∈ s) {t : R}
    (ht : t ≤ 0) : s.WOppSide (lineMap x y t) 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) (h : x ∈ s) {t : R} (ht : t ≤ 0)
Goal: s.WOppSide ((AffineMap.lineMap x y) t) y
-- AffineSubspace.wOppSide_lineMap_right
theorem wOppSide_lineMap_right {s : AffineSubspace R P} {x : P} (y : P) (h : x ∈ s) {t : R}
    (ht : t ≤ 0) : s.WOppSide y (lineMap x y t) 
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) (h : x ∈ s) {t : R} (ht : t ≤ 0)
Goal: s.WOppSide y ((AffineMap.lineMap x y) t)
-- Wbtw.wSameSide₂₃
theorem _root_.Wbtw.wSameSide₂₃ {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)
    (hx : x ∈ s) : s.WSameSide y z 
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 z : P} (h : Wbtw R x y z) (hx : x ∈ s)
Goal: s.WSameSide y z
-- Wbtw.wSameSide₃₂
theorem _root_.Wbtw.wSameSide₃₂ {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)
    (hx : x ∈ s) : s.WSameSide z 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 z : P} (h : Wbtw R x y z) (hx : x ∈ s)
Goal: s.WSameSide z y
-- Wbtw.wSameSide₁₂
theorem _root_.Wbtw.wSameSide₁₂ {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)
    (hz : z ∈ 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 y z : P} (h : Wbtw R x y z) (hz : z ∈ s)
Goal: s.WSameSide x y
-- Wbtw.wSameSide₂₁
theorem _root_.Wbtw.wSameSide₂₁ {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)
    (hz : z ∈ s) : 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 z : P} (h : Wbtw R x y z) (hz : z ∈ s)
Goal: s.WSameSide y x
-- Wbtw.wOppSide₁₃
theorem _root_.Wbtw.wOppSide₁₃ {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)
    (hy : y ∈ s) : s.WOppSide x z 
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 z : P} (h : Wbtw R x y z) (hy : y ∈ s)
Goal: s.WOppSide x z
-- Wbtw.wOppSide₃₁
theorem _root_.Wbtw.wOppSide₃₁ {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)
    (hy : y ∈ s) : s.WOppSide z 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 z : P} (h : Wbtw R x y z) (hy : y ∈ s)
Goal: s.WOppSide z x
-- AffineSubspace.wOppSide_self_iff
@[simp]
theorem wOppSide_self_iff {s : AffineSubspace R P} {x : P} : s.WOppSide x x ↔ x ∈ s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x : P}
Goal: s.WOppSide x x ↔ x ∈ s
-- AffineSubspace.not_sOppSide_self
theorem not_sOppSide_self (s : AffineSubspace R P) (x : P) : ¬s.SOppSide x x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] (s : AffineSubspace R P) (x : P)
Goal: ¬s.SOppSide x x
-- AffineSubspace.wSameSide_iff_exists_left
theorem wSameSide_iff_exists_left {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
    s.WSameSide x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s)
Goal: s.WSameSide x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
-- AffineSubspace.wSameSide_iff_exists_right
theorem wSameSide_iff_exists_right {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
    s.WSameSide x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s)
Goal: s.WSameSide x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
-- AffineSubspace.sSameSide_iff_exists_left
theorem sSameSide_iff_exists_left {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
    s.SSameSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s)
Goal: s.SSameSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
-- AffineSubspace.sSameSide_iff_exists_right
theorem sSameSide_iff_exists_right {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
    s.SSameSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s)
Goal: s.SSameSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
-- AffineSubspace.wOppSide_iff_exists_left
theorem wOppSide_iff_exists_left {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
    s.WOppSide x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -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] {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s)
Goal: s.WOppSide x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
-- AffineSubspace.wOppSide_iff_exists_right
theorem wOppSide_iff_exists_right {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
    s.WOppSide x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -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] {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s)
Goal: s.WOppSide x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
-- AffineSubspace.sOppSide_iff_exists_left
theorem sOppSide_iff_exists_left {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
    s.SOppSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -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] {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s)
Goal: s.SOppSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
-- AffineSubspace.sOppSide_iff_exists_right
theorem sOppSide_iff_exists_right {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
    s.SOppSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -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] {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s)
Goal: s.SOppSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
-- AffineSubspace.WSameSide.trans
theorem WSameSide.trans {s : AffineSubspace R P} {x y z : P} (hxy : s.WSameSide x y)
    (hyz : s.WSameSide y z) (hy : y ∉ s) : s.WSameSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.WSameSide x y) (hyz : s.WSameSide y z) (hy : y ∉ s)
Goal: s.WSameSide x z
-- AffineSubspace.WSameSide.trans_sSameSide
theorem WSameSide.trans_sSameSide {s : AffineSubspace R P} {x y z : P} (hxy : s.WSameSide x y)
    (hyz : s.SSameSide y z) : s.WSameSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.WSameSide x y) (hyz : s.SSameSide y z)
Goal: s.WSameSide x z
-- AffineSubspace.WSameSide.trans_wOppSide
theorem WSameSide.trans_wOppSide {s : AffineSubspace R P} {x y z : P} (hxy : s.WSameSide x y)
    (hyz : s.WOppSide y z) (hy : y ∉ s) : s.WOppSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.WSameSide x y) (hyz : s.WOppSide y z) (hy : y ∉ s)
Goal: s.WOppSide x z
-- AffineSubspace.WSameSide.trans_sOppSide
theorem WSameSide.trans_sOppSide {s : AffineSubspace R P} {x y z : P} (hxy : s.WSameSide x y)
    (hyz : s.SOppSide y z) : s.WOppSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.WSameSide x y) (hyz : s.SOppSide y z)
Goal: s.WOppSide x z
-- AffineSubspace.SSameSide.trans_wSameSide
theorem SSameSide.trans_wSameSide {s : AffineSubspace R P} {x y z : P} (hxy : s.SSameSide x y)
    (hyz : s.WSameSide y z) : s.WSameSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.SSameSide x y) (hyz : s.WSameSide y z)
Goal: s.WSameSide x z
-- AffineSubspace.SSameSide.trans
theorem SSameSide.trans {s : AffineSubspace R P} {x y z : P} (hxy : s.SSameSide x y)
    (hyz : s.SSameSide y z) : s.SSameSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.SSameSide x y) (hyz : s.SSameSide y z)
Goal: s.SSameSide x z
-- AffineSubspace.SSameSide.trans_wOppSide
theorem SSameSide.trans_wOppSide {s : AffineSubspace R P} {x y z : P} (hxy : s.SSameSide x y)
    (hyz : s.WOppSide y z) : s.WOppSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.SSameSide x y) (hyz : s.WOppSide y z)
Goal: s.WOppSide x z
-- AffineSubspace.SSameSide.trans_sOppSide
theorem SSameSide.trans_sOppSide {s : AffineSubspace R P} {x y z : P} (hxy : s.SSameSide x y)
    (hyz : s.SOppSide y z) : s.SOppSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.SSameSide x y) (hyz : s.SOppSide y z)
Goal: s.SOppSide x z
-- AffineSubspace.WOppSide.trans_wSameSide
theorem WOppSide.trans_wSameSide {s : AffineSubspace R P} {x y z : P} (hxy : s.WOppSide x y)
    (hyz : s.WSameSide y z) (hy : y ∉ s) : s.WOppSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.WOppSide x y) (hyz : s.WSameSide y z) (hy : y ∉ s)
Goal: s.WOppSide x z
-- AffineSubspace.WOppSide.trans_sSameSide
theorem WOppSide.trans_sSameSide {s : AffineSubspace R P} {x y z : P} (hxy : s.WOppSide x y)
    (hyz : s.SSameSide y z) : s.WOppSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.WOppSide x y) (hyz : s.SSameSide y z)
Goal: s.WOppSide x z
-- AffineSubspace.WOppSide.trans
theorem WOppSide.trans {s : AffineSubspace R P} {x y z : P} (hxy : s.WOppSide x y)
    (hyz : s.WOppSide y z) (hy : y ∉ s) : s.WSameSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.WOppSide x y) (hyz : s.WOppSide y z) (hy : y ∉ s)
Goal: s.WSameSide x z
-- AffineSubspace.WOppSide.trans_sOppSide
theorem WOppSide.trans_sOppSide {s : AffineSubspace R P} {x y z : P} (hxy : s.WOppSide x y)
    (hyz : s.SOppSide y z) : s.WSameSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.WOppSide x y) (hyz : s.SOppSide y z)
Goal: s.WSameSide x z
-- AffineSubspace.SOppSide.trans_wSameSide
theorem SOppSide.trans_wSameSide {s : AffineSubspace R P} {x y z : P} (hxy : s.SOppSide x y)
    (hyz : s.WSameSide y z) : s.WOppSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.SOppSide x y) (hyz : s.WSameSide y z)
Goal: s.WOppSide x z
-- AffineSubspace.SOppSide.trans_sSameSide
theorem SOppSide.trans_sSameSide {s : AffineSubspace R P} {x y z : P} (hxy : s.SOppSide x y)
    (hyz : s.SSameSide y z) : s.SOppSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.SOppSide x y) (hyz : s.SSameSide y z)
Goal: s.SOppSide x z
-- AffineSubspace.SOppSide.trans_wOppSide
theorem SOppSide.trans_wOppSide {s : AffineSubspace R P} {x y z : P} (hxy : s.SOppSide x y)
    (hyz : s.WOppSide y z) : s.WSameSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.SOppSide x y) (hyz : s.WOppSide y z)
Goal: s.WSameSide x z
-- AffineSubspace.SOppSide.trans
theorem SOppSide.trans {s : AffineSubspace R P} {x y z : P} (hxy : s.SOppSide x y)
    (hyz : s.SOppSide y z) : s.SSameSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (hxy : s.SOppSide x y) (hyz : s.SOppSide y z)
Goal: s.SSameSide x z
-- AffineSubspace.wSameSide_and_wOppSide_iff
theorem wSameSide_and_wOppSide_iff {s : AffineSubspace R P} {x y : P} :
    s.WSameSide x y ∧ s.WOppSide x y ↔ x ∈ s ∨ y ∈ s 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P}
Goal: s.WSameSide x y ∧ s.WOppSide x y ↔ x ∈ s ∨ y ∈ s
-- AffineSubspace.WSameSide.not_sOppSide
theorem WSameSide.not_sOppSide {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) :
    ¬s.SOppSide 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] {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y)
Goal: ¬s.SOppSide x y
-- AffineSubspace.SSameSide.not_wOppSide
theorem SSameSide.not_wOppSide {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) :
    ¬s.WOppSide 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] {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y)
Goal: ¬s.WOppSide x y
-- AffineSubspace.SSameSide.not_sOppSide
theorem SSameSide.not_sOppSide {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) :
    ¬s.SOppSide 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] {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y)
Goal: ¬s.SOppSide x y
-- AffineSubspace.WOppSide.not_sSameSide
theorem WOppSide.not_sSameSide {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) :
    ¬s.SSameSide 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] {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y)
Goal: ¬s.SSameSide x y
-- AffineSubspace.SOppSide.not_wSameSide
theorem SOppSide.not_wSameSide {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :
    ¬s.WSameSide 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] {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y)
Goal: ¬s.WSameSide x y
-- AffineSubspace.SOppSide.not_sSameSide
theorem SOppSide.not_sSameSide {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :
    ¬s.SSameSide 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] {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y)
Goal: ¬s.SSameSide x y
-- AffineSubspace.wOppSide_iff_exists_wbtw
theorem wOppSide_iff_exists_wbtw {s : AffineSubspace R P} {x y : P} :
    s.WOppSide x y ↔ ∃ p ∈ s, Wbtw R x p y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P}
Goal: s.WOppSide x y ↔ ∃ p ∈ s, Wbtw R x p y
-- AffineSubspace.SOppSide.exists_sbtw
theorem SOppSide.exists_sbtw {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :
    ∃ p ∈ s, Sbtw R x p y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y)
Goal: ∃ p ∈ s, Sbtw R x p y
-- Sbtw.sOppSide_of_not_mem_of_mem
theorem _root_.Sbtw.sOppSide_of_not_mem_of_mem {s : AffineSubspace R P} {x y z : P}
    (h : Sbtw R x y z) (hx : x ∉ s) (hy : y ∈ s) : s.SOppSide x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y z : P} (h : Sbtw R x y z) (hx : x ∉ s) (hy : y ∈ s)
Goal: s.SOppSide x z
-- AffineSubspace.sSameSide_smul_vsub_vadd_left
theorem sSameSide_smul_vsub_vadd_left {s : AffineSubspace R P} {x p₁ p₂ : P} (hx : x ∉ s)
    (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 < t) : s.SSameSide (t • (x -ᵥ p₁) +ᵥ p₂) x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x p₁ p₂ : P} (hx : x ∉ s) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 < t)
Goal: s.SSameSide (t • (x -ᵥ p₁) +ᵥ p₂) x
-- AffineSubspace.sSameSide_smul_vsub_vadd_right
theorem sSameSide_smul_vsub_vadd_right {s : AffineSubspace R P} {x p₁ p₂ : P} (hx : x ∉ s)
    (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 < t) : s.SSameSide x (t • (x -ᵥ p₁) +ᵥ p₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x p₁ p₂ : P} (hx : x ∉ s) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 < t)
Goal: s.SSameSide x (t • (x -ᵥ p₁) +ᵥ p₂)
-- AffineSubspace.sSameSide_lineMap_left
theorem sSameSide_lineMap_left {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R}
    (ht : 0 < t) : s.SSameSide (lineMap x y t) y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R} (ht : 0 < t)
Goal: s.SSameSide ((AffineMap.lineMap x y) t) y
-- AffineSubspace.sSameSide_lineMap_right
theorem sSameSide_lineMap_right {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R}
    (ht : 0 < t) : s.SSameSide y (lineMap x y t) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R} (ht : 0 < t)
Goal: s.SSameSide y ((AffineMap.lineMap x y) t)
-- AffineSubspace.sOppSide_smul_vsub_vadd_left
theorem sOppSide_smul_vsub_vadd_left {s : AffineSubspace R P} {x p₁ p₂ : P} (hx : x ∉ s)
    (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t < 0) : s.SOppSide (t • (x -ᵥ p₁) +ᵥ p₂) x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x p₁ p₂ : P} (hx : x ∉ s) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t < 0)
Goal: s.SOppSide (t • (x -ᵥ p₁) +ᵥ p₂) x
-- AffineSubspace.sOppSide_smul_vsub_vadd_right
theorem sOppSide_smul_vsub_vadd_right {s : AffineSubspace R P} {x p₁ p₂ : P} (hx : x ∉ s)
    (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t < 0) : s.SOppSide x (t • (x -ᵥ p₁) +ᵥ p₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x p₁ p₂ : P} (hx : x ∉ s) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t < 0)
Goal: s.SOppSide x (t • (x -ᵥ p₁) +ᵥ p₂)
-- AffineSubspace.sOppSide_lineMap_left
theorem sOppSide_lineMap_left {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R}
    (ht : t < 0) : s.SOppSide (lineMap x y t) y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R} (ht : t < 0)
Goal: s.SOppSide ((AffineMap.lineMap x y) t) y
-- AffineSubspace.sOppSide_lineMap_right
theorem sOppSide_lineMap_right {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R}
    (ht : t < 0) : s.SOppSide y (lineMap x y t) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedField R] [AddCommGroup V] [Module R V] [AddTorsor V P] {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R} (ht : t < 0)
Goal: s.SOppSide y ((AffineMap.lineMap x y) t)