ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Function.Injective.wbtw_map_iff
theorem Function.Injective.wbtw_map_iff {x y z : P} {f : P →ᵃ[R] P'} (hf : Function.Injective f) :
    Wbtw R (f x) (f y) (f z) ↔ Wbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {x y z : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f)
Goal: Wbtw R (f x) (f y) (f z) ↔ Wbtw R x y z
-- Function.Injective.sbtw_map_iff
theorem Function.Injective.sbtw_map_iff {x y z : P} {f : P →ᵃ[R] P'} (hf : Function.Injective f) :
    Sbtw R (f x) (f y) (f z) ↔ Sbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {x y z : P} {f : P →ᵃ[R] P'} (hf : Function.Injective ⇑f)
Goal: Sbtw R (f x) (f y) (f z) ↔ Sbtw R x y z
-- AffineEquiv.wbtw_map_iff
@[simp]
theorem AffineEquiv.wbtw_map_iff {x y z : P} (f : P ≃ᵃ[R] P') :
    Wbtw R (f x) (f y) (f z) ↔ Wbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {x y z : P} (f : P ≃ᵃ[R] P')
Goal: Wbtw R (f x) (f y) (f z) ↔ Wbtw R x y z
-- AffineEquiv.sbtw_map_iff
@[simp]
theorem AffineEquiv.sbtw_map_iff {x y z : P} (f : P ≃ᵃ[R] P') :
    Sbtw R (f x) (f y) (f z) ↔ Sbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] {x y z : P} (f : P ≃ᵃ[R] P')
Goal: Sbtw R (f x) (f y) (f z) ↔ Sbtw R x y z
-- wbtw_const_vadd_iff
@[simp]
theorem wbtw_const_vadd_iff {x y z : P} (v : V) :
    Wbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) ↔ Wbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (v : V)
Goal: Wbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) ↔ Wbtw R x y z
-- wbtw_vadd_const_iff
@[simp]
theorem wbtw_vadd_const_iff {x y z : V} (p : P) :
    Wbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) ↔ Wbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : V} (p : P)
Goal: Wbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) ↔ Wbtw R x y z
-- wbtw_const_vsub_iff
@[simp]
theorem wbtw_const_vsub_iff {x y z : P} (p : P) :
    Wbtw R (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) ↔ Wbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (p : P)
Goal: Wbtw R (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) ↔ Wbtw R x y z
-- wbtw_vsub_const_iff
@[simp]
theorem wbtw_vsub_const_iff {x y z : P} (p : P) :
    Wbtw R (x -ᵥ p) (y -ᵥ p) (z -ᵥ p) ↔ Wbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (p : P)
Goal: Wbtw R (x -ᵥ p) (y -ᵥ p) (z -ᵥ p) ↔ Wbtw R x y z
-- sbtw_const_vadd_iff
@[simp]
theorem sbtw_const_vadd_iff {x y z : P} (v : V) :
    Sbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) ↔ Sbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (v : V)
Goal: Sbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) ↔ Sbtw R x y z
-- sbtw_vadd_const_iff
@[simp]
theorem sbtw_vadd_const_iff {x y z : V} (p : P) :
    Sbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) ↔ Sbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : V} (p : P)
Goal: Sbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) ↔ Sbtw R x y z
-- sbtw_const_vsub_iff
@[simp]
theorem sbtw_const_vsub_iff {x y z : P} (p : P) :
    Sbtw R (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) ↔ Sbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (p : P)
Goal: Sbtw R (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) ↔ Sbtw R x y z
-- sbtw_vsub_const_iff
@[simp]
theorem sbtw_vsub_const_iff {x y z : P} (p : P) :
    Sbtw R (x -ᵥ p) (y -ᵥ p) (z -ᵥ p) ↔ Sbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (p : P)
Goal: Sbtw R (x -ᵥ p) (y -ᵥ p) (z -ᵥ p) ↔ Sbtw R x y z
-- Sbtw.wbtw
theorem Sbtw.wbtw {x y z : P} (h : Sbtw R x y z) : Wbtw R x y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Sbtw R x y z)
Goal: Wbtw R x y z
-- Sbtw.ne_left
theorem Sbtw.ne_left {x y z : P} (h : Sbtw R x y z) : y ≠ x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Sbtw R x y z)
Goal: y ≠ x
-- Sbtw.left_ne
theorem Sbtw.left_ne {x y z : P} (h : Sbtw R x y z) : x ≠ y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Sbtw R x y z)
Goal: x ≠ y
-- Sbtw.ne_right
theorem Sbtw.ne_right {x y z : P} (h : Sbtw R x y z) : y ≠ z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Sbtw R x y z)
Goal: y ≠ z
-- Sbtw.right_ne
theorem Sbtw.right_ne {x y z : P} (h : Sbtw R x y z) : z ≠ y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Sbtw R x y z)
Goal: z ≠ y
-- Sbtw.mem_image_Ioo
theorem Sbtw.mem_image_Ioo {x y z : P} (h : Sbtw R x y z) :
    y ∈ lineMap x z '' Set.Ioo (0 : R) 1 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Sbtw R x y z)
Goal: y ∈ ⇑(AffineMap.lineMap x z) '' Set.Ioo 0 1
-- Wbtw.mem_affineSpan
theorem Wbtw.mem_affineSpan {x y z : P} (h : Wbtw R x y z) : y ∈ line[R, x, z] 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Wbtw R x y z)
Goal: y ∈ affineSpan R {x, z}
-- wbtw_comm
theorem wbtw_comm {x y z : P} : Wbtw R x y z ↔ Wbtw R z y x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P}
Goal: Wbtw R x y z ↔ Wbtw R z y x
-- sbtw_comm
theorem sbtw_comm {x y z : P} : Sbtw R x y z ↔ Sbtw R z y x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P}
Goal: Sbtw R x y z ↔ Sbtw R z y x
-- wbtw_self_left
@[simp]
theorem wbtw_self_left (x y : P) : Wbtw R x x y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: Wbtw R x x y
-- wbtw_self_right
@[simp]
theorem wbtw_self_right (x y : P) : Wbtw R x y y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: Wbtw R x y y
-- wbtw_self_iff
@[simp]
theorem wbtw_self_iff {x y : P} : Wbtw R x y x ↔ y = x 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P}
Goal: Wbtw R x y x ↔ y = x
-- not_sbtw_self_left
@[simp]
theorem not_sbtw_self_left (x y : P) : ¬Sbtw R x x y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: ¬Sbtw R x x y
-- not_sbtw_self_right
@[simp]
theorem not_sbtw_self_right (x y : P) : ¬Sbtw R x y y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: ¬Sbtw R x y y
-- Wbtw.left_ne_right_of_ne_left
theorem Wbtw.left_ne_right_of_ne_left {x y z : P} (h : Wbtw R x y z) (hne : y ≠ x) : x ≠ z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Wbtw R x y z) (hne : y ≠ x)
Goal: x ≠ z
-- Wbtw.left_ne_right_of_ne_right
theorem Wbtw.left_ne_right_of_ne_right {x y z : P} (h : Wbtw R x y z) (hne : y ≠ z) : x ≠ z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Wbtw R x y z) (hne : y ≠ z)
Goal: x ≠ z
-- Sbtw.left_ne_right
theorem Sbtw.left_ne_right {x y z : P} (h : Sbtw R x y z) : x ≠ z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Sbtw R x y z)
Goal: x ≠ z
-- sbtw_iff_mem_image_Ioo_and_ne
theorem sbtw_iff_mem_image_Ioo_and_ne [NoZeroSMulDivisors R V] {x y z : P} :
    Sbtw R x y z ↔ y ∈ lineMap x z '' Set.Ioo (0 : R) 1 ∧ x ≠ z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y z : P}
Goal: Sbtw R x y z ↔ y ∈ ⇑(AffineMap.lineMap x z) '' Set.Ioo 0 1 ∧ x ≠ z
-- not_sbtw_self
@[simp]
theorem not_sbtw_self (x y : P) : ¬Sbtw R x y x 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: ¬Sbtw R x y x
-- wbtw_swap_left_iff
theorem wbtw_swap_left_iff [NoZeroSMulDivisors R V] {x y : P} (z : P) :
    Wbtw R x y z ∧ Wbtw R y x z ↔ x = y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y : P} (z : P)
Goal: Wbtw R x y z ∧ Wbtw R y x z ↔ x = y
-- wbtw_swap_right_iff
theorem wbtw_swap_right_iff [NoZeroSMulDivisors R V] (x : P) {y z : P} :
    Wbtw R x y z ∧ Wbtw R x z y ↔ y = z 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] (x : P) {y z : P}
Goal: Wbtw R x y z ∧ Wbtw R x z y ↔ y = z
-- wbtw_rotate_iff
theorem wbtw_rotate_iff [NoZeroSMulDivisors R V] (x : P) {y z : P} :
    Wbtw R x y z ∧ Wbtw R z x y ↔ x = y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] (x : P) {y z : P}
Goal: Wbtw R x y z ∧ Wbtw R z x y ↔ x = y
-- Wbtw.swap_left_iff
theorem Wbtw.swap_left_iff [NoZeroSMulDivisors R V] {x y z : P} (h : Wbtw R x y z) :
    Wbtw R y x z ↔ x = y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y z : P} (h : Wbtw R x y z)
Goal: Wbtw R y x z ↔ x = y
-- Wbtw.swap_right_iff
theorem Wbtw.swap_right_iff [NoZeroSMulDivisors R V] {x y z : P} (h : Wbtw R x y z) :
    Wbtw R x z y ↔ y = z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y z : P} (h : Wbtw R x y z)
Goal: Wbtw R x z y ↔ y = z
-- Wbtw.rotate_iff
theorem Wbtw.rotate_iff [NoZeroSMulDivisors R V] {x y z : P} (h : Wbtw R x y z) :
    Wbtw R z x y ↔ x = y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y z : P} (h : Wbtw R x y z)
Goal: Wbtw R z x y ↔ x = y
-- Sbtw.not_swap_left
theorem Sbtw.not_swap_left [NoZeroSMulDivisors R V] {x y z : P} (h : Sbtw R x y z) :
    ¬Wbtw R y x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y z : P} (h : Sbtw R x y z)
Goal: ¬Wbtw R y x z
-- Sbtw.not_swap_right
theorem Sbtw.not_swap_right [NoZeroSMulDivisors R V] {x y z : P} (h : Sbtw R x y z) :
    ¬Wbtw R x z y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y z : P} (h : Sbtw R x y z)
Goal: ¬Wbtw R x z y
-- Sbtw.not_rotate
theorem Sbtw.not_rotate [NoZeroSMulDivisors R V] {x y z : P} (h : Sbtw R x y z) : ¬Wbtw R z x y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y z : P} (h : Sbtw R x y z)
Goal: ¬Wbtw R z x y
-- wbtw_lineMap_iff
@[simp]
theorem wbtw_lineMap_iff [NoZeroSMulDivisors R V] {x y : P} {r : R} :
    Wbtw R x (lineMap x y r) y ↔ x = y ∨ r ∈ Set.Icc (0 : R) 1 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y : P} {r : R}
Goal: Wbtw R x ((AffineMap.lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1
-- sbtw_lineMap_iff
@[simp]
theorem sbtw_lineMap_iff [NoZeroSMulDivisors R V] {x y : P} {r : R} :
    Sbtw R x (lineMap x y r) y ↔ x ≠ y ∧ r ∈ Set.Ioo (0 : R) 1 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {x y : P} {r : R}
Goal: Sbtw R x ((AffineMap.lineMap x y) r) y ↔ x ≠ y ∧ r ∈ Set.Ioo 0 1
-- wbtw_mul_sub_add_iff
@[simp]
theorem wbtw_mul_sub_add_iff [NoZeroDivisors R] {x y r : R} :
    Wbtw R x (r * (y - x) + x) y ↔ x = y ∨ r ∈ Set.Icc (0 : R) 1 
Arguments: {R : Type u_1} [OrderedRing R] [NoZeroDivisors R] {x y r : R}
Goal: Wbtw R x (r * (y - x) + x) y ↔ x = y ∨ r ∈ Set.Icc 0 1
-- sbtw_mul_sub_add_iff
@[simp]
theorem sbtw_mul_sub_add_iff [NoZeroDivisors R] {x y r : R} :
    Sbtw R x (r * (y - x) + x) y ↔ x ≠ y ∧ r ∈ Set.Ioo (0 : R) 1 
Arguments: {R : Type u_1} [OrderedRing R] [NoZeroDivisors R] {x y r : R}
Goal: Sbtw R x (r * (y - x) + x) y ↔ x ≠ y ∧ r ∈ Set.Ioo 0 1
-- wbtw_zero_one_iff
@[simp]
theorem wbtw_zero_one_iff {x : R} : Wbtw R 0 x 1 ↔ x ∈ Set.Icc (0 : R) 1 
Arguments: {R : Type u_1} [OrderedRing R] {x : R}
Goal: Wbtw R 0 x 1 ↔ x ∈ Set.Icc 0 1
-- wbtw_one_zero_iff
@[simp]
theorem wbtw_one_zero_iff {x : R} : Wbtw R 1 x 0 ↔ x ∈ Set.Icc (0 : R) 1 
Arguments: {R : Type u_1} [OrderedRing R] {x : R}
Goal: Wbtw R 1 x 0 ↔ x ∈ Set.Icc 0 1
-- sbtw_zero_one_iff
@[simp]
theorem sbtw_zero_one_iff {x : R} : Sbtw R 0 x 1 ↔ x ∈ Set.Ioo (0 : R) 1 
Arguments: {R : Type u_1} [OrderedRing R] {x : R}
Goal: Sbtw R 0 x 1 ↔ x ∈ Set.Ioo 0 1
-- sbtw_one_zero_iff
@[simp]
theorem sbtw_one_zero_iff {x : R} : Sbtw R 1 x 0 ↔ x ∈ Set.Ioo (0 : R) 1 
Arguments: {R : Type u_1} [OrderedRing R] {x : R}
Goal: Sbtw R 1 x 0 ↔ x ∈ Set.Ioo 0 1
-- Wbtw.trans_left
theorem Wbtw.trans_left {w x y z : P} (h₁ : Wbtw R w y z) (h₂ : Wbtw R w x y) : Wbtw R w x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing 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 w x z
-- Wbtw.trans_right
theorem Wbtw.trans_right {w x y z : P} (h₁ : Wbtw R w x z) (h₂ : Wbtw R x y z) : Wbtw R w y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing 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 y z
-- Wbtw.trans_sbtw_left
theorem Wbtw.trans_sbtw_left [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w y z)
    (h₂ : Sbtw R w x y) : Sbtw R w x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w y z) (h₂ : Sbtw R w x y)
Goal: Sbtw R w x z
-- Wbtw.trans_sbtw_right
theorem Wbtw.trans_sbtw_right [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w x z)
    (h₂ : Sbtw R x y z) : Sbtw R w y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w x z) (h₂ : Sbtw R x y z)
Goal: Sbtw R w y z
-- Sbtw.trans_left
theorem Sbtw.trans_left [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w y z)
    (h₂ : Sbtw R w x y) : Sbtw R w x z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w y z) (h₂ : Sbtw R w x y)
Goal: Sbtw R w x z
-- Sbtw.trans_right
theorem Sbtw.trans_right [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w x z)
    (h₂ : Sbtw R x y z) : Sbtw R w y z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w x z) (h₂ : Sbtw R x y z)
Goal: Sbtw R w y z
-- Wbtw.trans_left_ne
theorem Wbtw.trans_left_ne [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w y z)
    (h₂ : Wbtw R w x y) (h : y ≠ z) : x ≠ z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w y z) (h₂ : Wbtw R w x y) (h : y ≠ z)
Goal: x ≠ z
-- Wbtw.trans_right_ne
theorem Wbtw.trans_right_ne [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w x z)
    (h₂ : Wbtw R x y z) (h : w ≠ x) : w ≠ y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w x z) (h₂ : Wbtw R x y z) (h : w ≠ x)
Goal: w ≠ y
-- Sbtw.trans_wbtw_left_ne
theorem Sbtw.trans_wbtw_left_ne [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w y z)
    (h₂ : Wbtw R w x y) : x ≠ z 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w y z) (h₂ : Wbtw R w x y)
Goal: x ≠ z
-- Sbtw.trans_wbtw_right_ne
theorem Sbtw.trans_wbtw_right_ne [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w x z)
    (h₂ : Wbtw R x y z) : w ≠ y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w x z) (h₂ : Wbtw R x y z)
Goal: w ≠ y
-- Sbtw.affineCombination_of_mem_affineSpan_pair
theorem Sbtw.affineCombination_of_mem_affineSpan_pair [NoZeroDivisors R] [NoZeroSMulDivisors R V]
    {ι : Type*} {p : ι → P} (ha : AffineIndependent R p) {w w₁ w₂ : ι → R} {s : Finset ι}
    (hw : ∑ i ∈ s, w i = 1) (hw₁ : ∑ i ∈ s, w₁ i = 1) (hw₂ : ∑ i ∈ s, w₂ i = 1)
    (h : s.affineCombination R p w ∈
      line[R, s.affineCombination R p w₁, s.affineCombination R p w₂])
    {i : ι} (his : i ∈ s) (hs : Sbtw R (w₁ i) (w i) (w₂ i)) :
    Sbtw R (s.affineCombination R p w₁) (s.affineCombination R p w)
      (s.affineCombination R p w₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [OrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroDivisors R] [NoZeroSMulDivisors R V] {ι : Type u_6} {p : ι → P} (ha : AffineIndependent R p) {w w₁ w₂ : ι → R} {s : Finset ι} (hw : ∑ i ∈ s, w i = 1) (hw₁ : ∑ i ∈ s, w₁ i = 1) (hw₂ : ∑ i ∈ s, w₂ i = 1) (h :
  (Finset.affineCombination R s p) w ∈
    affineSpan R {(Finset.affineCombination R s p) w₁, (Finset.affineCombination R s p) w₂}) {i : ι} (his : i ∈ s) (hs : Sbtw R (w₁ i) (w i) (w₂ i))
Goal: Sbtw R ((Finset.affineCombination R s p) w₁) ((Finset.affineCombination R s p) w) ((Finset.affineCombination R s p) w₂)
-- Wbtw.sameRay_vsub
theorem Wbtw.sameRay_vsub {x y z : P} (h : 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} [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} (h : Wbtw R x y z)
Goal: SameRay R (y -ᵥ x) (z -ᵥ y)
-- Wbtw.sameRay_vsub_left
theorem Wbtw.sameRay_vsub_left {x y z : P} (h : Wbtw R x y z) : SameRay R (y -ᵥ x) (z -ᵥ x) 
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 z : P} (h : Wbtw R x y z)
Goal: SameRay R (y -ᵥ x) (z -ᵥ x)
-- Wbtw.sameRay_vsub_right
theorem Wbtw.sameRay_vsub_right {x y z : P} (h : Wbtw R x y z) : SameRay R (z -ᵥ x) (z -ᵥ 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 z : P} (h : Wbtw R x y z)
Goal: SameRay R (z -ᵥ x) (z -ᵥ y)
-- sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [LinearOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [NoZeroSMulDivisors R V] {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P} (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃)) (h₁' : p ∈ affineSpan R {t.points i₁, p₁}) (h₂' : p ∈ affineSpan R {t.points i₂, p₂})
Goal: Sbtw R (t.points i₁) p p₁
-- wbtw_iff_left_eq_or_right_mem_image_Ici
theorem wbtw_iff_left_eq_or_right_mem_image_Ici {x y z : P} :
    Wbtw R x y z ↔ x = y ∨ z ∈ lineMap x y '' Set.Ici (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: Wbtw R x y z ↔ x = y ∨ z ∈ ⇑(AffineMap.lineMap x y) '' Set.Ici 1
-- Wbtw.right_mem_image_Ici_of_left_ne
theorem Wbtw.right_mem_image_Ici_of_left_ne {x y z : P} (h : Wbtw R x y z) (hne : x ≠ y) :
    z ∈ lineMap x y '' Set.Ici (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 : Wbtw R x y z) (hne : x ≠ y)
Goal: z ∈ ⇑(AffineMap.lineMap x y) '' Set.Ici 1
-- Wbtw.right_mem_affineSpan_of_left_ne
theorem Wbtw.right_mem_affineSpan_of_left_ne {x y z : P} (h : Wbtw R x y z) (hne : x ≠ y) :
    z ∈ line[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 z : P} (h : Wbtw R x y z) (hne : x ≠ y)
Goal: z ∈ affineSpan R {x, y}
-- sbtw_iff_left_ne_and_right_mem_image_Ioi
theorem sbtw_iff_left_ne_and_right_mem_image_Ioi {x y z : P} :
    Sbtw R x y z ↔ x ≠ y ∧ z ∈ lineMap x 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 ↔ x ≠ y ∧ z ∈ ⇑(AffineMap.lineMap x y) '' Set.Ioi 1
-- Sbtw.right_mem_image_Ioi
theorem Sbtw.right_mem_image_Ioi {x y z : P} (h : Sbtw R x y z) :
    z ∈ lineMap x 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: z ∈ ⇑(AffineMap.lineMap x y) '' Set.Ioi 1
-- Sbtw.right_mem_affineSpan
theorem Sbtw.right_mem_affineSpan {x y z : P} (h : Sbtw R x y z) : z ∈ line[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 z : P} (h : Sbtw R x y z)
Goal: z ∈ affineSpan R {x, y}
-- wbtw_iff_right_eq_or_left_mem_image_Ici
theorem wbtw_iff_right_eq_or_left_mem_image_Ici {x y z : P} :
    Wbtw R x y z ↔ z = y ∨ x ∈ lineMap z y '' Set.Ici (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: Wbtw R x y z ↔ z = y ∨ x ∈ ⇑(AffineMap.lineMap z y) '' Set.Ici 1
-- Wbtw.left_mem_image_Ici_of_right_ne
theorem Wbtw.left_mem_image_Ici_of_right_ne {x y z : P} (h : Wbtw R x y z) (hne : z ≠ y) :
    x ∈ lineMap z y '' Set.Ici (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 : Wbtw R x y z) (hne : z ≠ y)
Goal: x ∈ ⇑(AffineMap.lineMap z y) '' Set.Ici 1
-- Wbtw.left_mem_affineSpan_of_right_ne
theorem Wbtw.left_mem_affineSpan_of_right_ne {x y z : P} (h : Wbtw R x y z) (hne : z ≠ y) :
    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 : Wbtw R x y z) (hne : z ≠ y)
Goal: x ∈ affineSpan R {z, y}