ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- map_le_lineMap_iff_slope_le_slope_left
theorem map_le_lineMap_iff_slope_le_slope_left (h : 0 < r * (b - a)) :
    f c ≤ lineMap (f a) (f b) r ↔ slope f a c ≤ slope f a b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (h : 0 < r * (b - a))
Goal: f ((AffineMap.lineMap a b) r) ≤ (AffineMap.lineMap (f a) (f b)) r ↔ slope f a ((AffineMap.lineMap a b) r) ≤ slope f a b
-- lineMap_le_map_iff_slope_le_slope_left
theorem lineMap_le_map_iff_slope_le_slope_left (h : 0 < r * (b - a)) :
    lineMap (f a) (f b) r ≤ f c ↔ slope f a b ≤ slope f a c 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (h : 0 < r * (b - a))
Goal: (AffineMap.lineMap (f a) (f b)) r ≤ f ((AffineMap.lineMap a b) r) ↔ slope f a b ≤ slope f a ((AffineMap.lineMap a b) r)
-- map_lt_lineMap_iff_slope_lt_slope_left
theorem map_lt_lineMap_iff_slope_lt_slope_left (h : 0 < r * (b - a)) :
    f c < lineMap (f a) (f b) r ↔ slope f a c < slope f a b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (h : 0 < r * (b - a))
Goal: f ((AffineMap.lineMap a b) r) < (AffineMap.lineMap (f a) (f b)) r ↔ slope f a ((AffineMap.lineMap a b) r) < slope f a b
-- lineMap_lt_map_iff_slope_lt_slope_left
theorem lineMap_lt_map_iff_slope_lt_slope_left (h : 0 < r * (b - a)) :
    lineMap (f a) (f b) r < f c ↔ slope f a b < slope f a c 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (h : 0 < r * (b - a))
Goal: (AffineMap.lineMap (f a) (f b)) r < f ((AffineMap.lineMap a b) r) ↔ slope f a b < slope f a ((AffineMap.lineMap a b) r)
-- map_le_lineMap_iff_slope_le_slope_right
theorem map_le_lineMap_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :
    f c ≤ lineMap (f a) (f b) r ↔ slope f a b ≤ slope f c b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (h : 0 < (1 - r) * (b - a))
Goal: f ((AffineMap.lineMap a b) r) ≤ (AffineMap.lineMap (f a) (f b)) r ↔ slope f a b ≤ slope f ((AffineMap.lineMap a b) r) b
-- lineMap_le_map_iff_slope_le_slope_right
theorem lineMap_le_map_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :
    lineMap (f a) (f b) r ≤ f c ↔ slope f c b ≤ slope f a b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (h : 0 < (1 - r) * (b - a))
Goal: (AffineMap.lineMap (f a) (f b)) r ≤ f ((AffineMap.lineMap a b) r) ↔ slope f ((AffineMap.lineMap a b) r) b ≤ slope f a b
-- map_lt_lineMap_iff_slope_lt_slope_right
theorem map_lt_lineMap_iff_slope_lt_slope_right (h : 0 < (1 - r) * (b - a)) :
    f c < lineMap (f a) (f b) r ↔ slope f a b < slope f c b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (h : 0 < (1 - r) * (b - a))
Goal: f ((AffineMap.lineMap a b) r) < (AffineMap.lineMap (f a) (f b)) r ↔ slope f a b < slope f ((AffineMap.lineMap a b) r) b
-- lineMap_lt_map_iff_slope_lt_slope_right
theorem lineMap_lt_map_iff_slope_lt_slope_right (h : 0 < (1 - r) * (b - a)) :
    lineMap (f a) (f b) r < f c ↔ slope f c b < slope f a b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (h : 0 < (1 - r) * (b - a))
Goal: (AffineMap.lineMap (f a) (f b)) r < f ((AffineMap.lineMap a b) r) ↔ slope f ((AffineMap.lineMap a b) r) b < slope f a b
-- map_le_lineMap_iff_slope_le_slope
theorem map_le_lineMap_iff_slope_le_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
    f c ≤ lineMap (f a) (f b) r ↔ slope f a c ≤ slope f c b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1)
Goal: f ((AffineMap.lineMap a b) r) ≤ (AffineMap.lineMap (f a) (f b)) r ↔
  slope f a ((AffineMap.lineMap a b) r) ≤ slope f ((AffineMap.lineMap a b) r) b
-- lineMap_le_map_iff_slope_le_slope
theorem lineMap_le_map_iff_slope_le_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
    lineMap (f a) (f b) r ≤ f c ↔ slope f c b ≤ slope f a c 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1)
Goal: (AffineMap.lineMap (f a) (f b)) r ≤ f ((AffineMap.lineMap a b) r) ↔
  slope f ((AffineMap.lineMap a b) r) b ≤ slope f a ((AffineMap.lineMap a b) r)
-- map_lt_lineMap_iff_slope_lt_slope
theorem map_lt_lineMap_iff_slope_lt_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
    f c < lineMap (f a) (f b) r ↔ slope f a c < slope f c b 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1)
Goal: f ((AffineMap.lineMap a b) r) < (AffineMap.lineMap (f a) (f b)) r ↔
  slope f a ((AffineMap.lineMap a b) r) < slope f ((AffineMap.lineMap a b) r) b
-- lineMap_lt_map_iff_slope_lt_slope
theorem lineMap_lt_map_iff_slope_lt_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
    lineMap (f a) (f b) r < f c ↔ slope f c b < slope f a c 
Arguments: {k : Type u_1} {E : Type u_2} [LinearOrderedField k] [OrderedAddCommGroup E] [Module k E] [OrderedSMul k E] {f : k → E} {a b r : k} (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1)
Goal: (AffineMap.lineMap (f a) (f b)) r < f ((AffineMap.lineMap a b) r) ↔
  slope f ((AffineMap.lineMap a b) r) b < slope f a ((AffineMap.lineMap a b) r)
-- AffineMap.map_midpoint
@[simp]
theorem AffineMap.map_midpoint (f : P →ᵃ[R] P') (a b : P) :
    f (midpoint R a b) = midpoint R (f a) (f b) 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] (f : P →ᵃ[R] P') (a b : P)
Goal: f (midpoint R a b) = midpoint R (f a) (f b)
-- AffineEquiv.map_midpoint
@[simp]
theorem AffineEquiv.map_midpoint (f : P ≃ᵃ[R] P') (a b : P) :
    f (midpoint R a b) = midpoint R (f a) (f b) 
Arguments: {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] (f : P ≃ᵃ[R] P') (a b : P)
Goal: f (midpoint R a b) = midpoint R (f a) (f b)
-- AffineEquiv.pointReflection_midpoint_left
theorem AffineEquiv.pointReflection_midpoint_left (x y : P) :
    pointReflection R (midpoint R x y) x = y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: (AffineEquiv.pointReflection R (midpoint R x y)) x = y
-- Equiv.pointReflection_midpoint_left
@[simp] -- Porting note: added variant with `Equiv.pointReflection` for `simp`
theorem Equiv.pointReflection_midpoint_left (x y : P) :
    (Equiv.pointReflection (midpoint R x y)) x = y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: (Equiv.pointReflection (midpoint R x y)) x = y
-- midpoint_comm
theorem midpoint_comm (x y : P) : midpoint R x y = midpoint R y x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: midpoint R x y = midpoint R y x
-- AffineEquiv.pointReflection_midpoint_right
theorem AffineEquiv.pointReflection_midpoint_right (x y : P) :
    pointReflection R (midpoint R x y) y = x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: (AffineEquiv.pointReflection R (midpoint R x y)) y = x
-- Equiv.pointReflection_midpoint_right
@[simp] -- Porting note: added variant with `Equiv.pointReflection` for `simp`
theorem Equiv.pointReflection_midpoint_right (x y : P) :
    (Equiv.pointReflection (midpoint R x y)) y = x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: (Equiv.pointReflection (midpoint R x y)) y = x
-- midpoint_vsub_midpoint
theorem midpoint_vsub_midpoint (p₁ p₂ p₃ p₄ : P) :
    midpoint R p₁ p₂ -ᵥ midpoint R p₃ p₄ = midpoint R (p₁ -ᵥ p₃) (p₂ -ᵥ p₄) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ p₃ p₄ : P)
Goal: midpoint R p₁ p₂ -ᵥ midpoint R p₃ p₄ = midpoint R (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)
-- midpoint_vadd_midpoint
theorem midpoint_vadd_midpoint (v v' : V) (p p' : P) :
    midpoint R v v' +ᵥ midpoint R p p' = midpoint R (v +ᵥ p) (v' +ᵥ p') 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (v v' : V) (p p' : P)
Goal: midpoint R v v' +ᵥ midpoint R p p' = midpoint R (v +ᵥ p) (v' +ᵥ p')
-- midpoint_eq_iff
theorem midpoint_eq_iff {x y z : P} : midpoint R x y = z ↔ pointReflection R z x = y 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P}
Goal: midpoint R x y = z ↔ (AffineEquiv.pointReflection R z) x = y
-- midpoint_pointReflection_left
@[simp]
theorem midpoint_pointReflection_left (x y : P) :
    midpoint R (Equiv.pointReflection x y) y = x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: midpoint R ((Equiv.pointReflection x) y) y = x
-- midpoint_pointReflection_right
@[simp]
theorem midpoint_pointReflection_right (x y : P) :
    midpoint R y (Equiv.pointReflection x y) = x 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P)
Goal: midpoint R y ((Equiv.pointReflection x) y) = x
-- midpoint_vsub_left
@[simp]
theorem midpoint_vsub_left (p₁ p₂ : P) : midpoint R p₁ p₂ -ᵥ p₁ = (⅟ 2 : R) • (p₂ -ᵥ p₁) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ : P)
Goal: midpoint R p₁ p₂ -ᵥ p₁ = ⅟2 • (p₂ -ᵥ p₁)
-- midpoint_vsub_right
@[simp]
theorem midpoint_vsub_right (p₁ p₂ : P) : midpoint R p₁ p₂ -ᵥ p₂ = (⅟ 2 : R) • (p₁ -ᵥ p₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ : P)
Goal: midpoint R p₁ p₂ -ᵥ p₂ = ⅟2 • (p₁ -ᵥ p₂)
-- left_vsub_midpoint
@[simp]
theorem left_vsub_midpoint (p₁ p₂ : P) : p₁ -ᵥ midpoint R p₁ p₂ = (⅟ 2 : R) • (p₁ -ᵥ p₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ : P)
Goal: p₁ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₁ -ᵥ p₂)
-- right_vsub_midpoint
@[simp]
theorem right_vsub_midpoint (p₁ p₂ : P) : p₂ -ᵥ midpoint R p₁ p₂ = (⅟ 2 : R) • (p₂ -ᵥ p₁) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ : P)
Goal: p₂ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₂ -ᵥ p₁)
-- midpoint_vsub
theorem midpoint_vsub (p₁ p₂ p : P) :
    midpoint R p₁ p₂ -ᵥ p = (⅟ 2 : R) • (p₁ -ᵥ p) + (⅟ 2 : R) • (p₂ -ᵥ p) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ p : P)
Goal: midpoint R p₁ p₂ -ᵥ p = ⅟2 • (p₁ -ᵥ p) + ⅟2 • (p₂ -ᵥ p)
-- vsub_midpoint
theorem vsub_midpoint (p₁ p₂ p : P) :
    p -ᵥ midpoint R p₁ p₂ = (⅟ 2 : R) • (p -ᵥ p₁) + (⅟ 2 : R) • (p -ᵥ p₂) 
Arguments: {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ p : P)
Goal: p -ᵥ midpoint R p₁ p₂ = ⅟2 • (p -ᵥ p₁) + ⅟2 • (p -ᵥ p₂)
-- midpoint_sub_left
@[simp]
theorem midpoint_sub_left (v₁ v₂ : V) : midpoint R v₁ v₂ - v₁ = (⅟ 2 : R) • (v₂ - v₁) 
Arguments: {R : Type u_1} {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (v₁ v₂ : V)
Goal: midpoint R v₁ v₂ - v₁ = ⅟2 • (v₂ - v₁)
-- midpoint_sub_right
@[simp]
theorem midpoint_sub_right (v₁ v₂ : V) : midpoint R v₁ v₂ - v₂ = (⅟ 2 : R) • (v₁ - v₂) 
Arguments: {R : Type u_1} {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (v₁ v₂ : V)
Goal: midpoint R v₁ v₂ - v₂ = ⅟2 • (v₁ - v₂)
-- left_sub_midpoint
@[simp]
theorem left_sub_midpoint (v₁ v₂ : V) : v₁ - midpoint R v₁ v₂ = (⅟ 2 : R) • (v₁ - v₂) 
Arguments: {R : Type u_1} {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (v₁ v₂ : V)
Goal: v₁ - midpoint R v₁ v₂ = ⅟2 • (v₁ - v₂)
-- right_sub_midpoint
@[simp]
theorem right_sub_midpoint (v₁ v₂ : V) : v₂ - midpoint R v₁ v₂ = (⅟ 2 : R) • (v₂ - v₁) 
Arguments: {R : Type u_1} {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (v₁ v₂ : V)
Goal: v₂ - midpoint R v₁ v₂ = ⅟2 • (v₂ - v₁)
-- midpoint_eq_left_iff
@[simp]
theorem midpoint_eq_left_iff {x y : P} : midpoint R x y = x ↔ x = y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P}
Goal: midpoint R x y = x ↔ x = y
-- left_eq_midpoint_iff
@[simp]
theorem left_eq_midpoint_iff {x y : P} : x = midpoint R x y ↔ x = y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P}
Goal: x = midpoint R x y ↔ x = y
-- midpoint_eq_right_iff
@[simp]
theorem midpoint_eq_right_iff {x y : P} : midpoint R x y = y ↔ x = y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P}
Goal: midpoint R x y = y ↔ x = y
-- right_eq_midpoint_iff
@[simp]
theorem right_eq_midpoint_iff {x y : P} : y = midpoint R x y ↔ x = y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P}
Goal: y = midpoint R x y ↔ x = y
-- midpoint_eq_midpoint_iff_vsub_eq_vsub
theorem midpoint_eq_midpoint_iff_vsub_eq_vsub {x x' y y' : P} :
    midpoint R x y = midpoint R x' y' ↔ x -ᵥ x' = y' -ᵥ y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x x' y y' : P}
Goal: midpoint R x y = midpoint R x' y' ↔ x -ᵥ x' = y' -ᵥ y
-- midpoint_eq_iff'
theorem midpoint_eq_iff' {x y z : P} : midpoint R x y = z ↔ Equiv.pointReflection z x = y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P}
Goal: midpoint R x y = z ↔ (Equiv.pointReflection z) x = y
-- midpoint_unique
theorem midpoint_unique (R' : Type*) [Ring R'] [Invertible (2 : R')] [Module R' V] (x y : P) :
    midpoint R x y = midpoint R' x y 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (R' : Type u_6) [Ring R'] [Invertible 2] [Module R' V] (x y : P)
Goal: midpoint R x y = midpoint R' x y
-- midpoint_self
@[simp]
theorem midpoint_self (x : P) : midpoint R x x = x 
Arguments: (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P)
Goal: midpoint R x x = x
-- midpoint_add_self
@[simp]
theorem midpoint_add_self (x y : V) : midpoint R x y + midpoint R x y = x + y 
Arguments: (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V)
Goal: midpoint R x y + midpoint R x y = x + y
-- midpoint_zero_add
theorem midpoint_zero_add (x y : V) : midpoint R 0 (x + y) = midpoint R x y 
Arguments: (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V)
Goal: midpoint R 0 (x + y) = midpoint R x y
-- midpoint_eq_smul_add
theorem midpoint_eq_smul_add (x y : V) : midpoint R x y = (⅟ 2 : R) • (x + y) 
Arguments: (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V)
Goal: midpoint R x y = ⅟2 • (x + y)
-- midpoint_self_neg
@[simp]
theorem midpoint_self_neg (x : V) : midpoint R x (-x) = 0 
Arguments: (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x : V)
Goal: midpoint R x (-x) = 0
-- midpoint_neg_self
@[simp]
theorem midpoint_neg_self (x : V) : midpoint R (-x) x = 0 
Arguments: (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x : V)
Goal: midpoint R (-x) x = 0
-- midpoint_sub_add
@[simp]
theorem midpoint_sub_add (x y : V) : midpoint R (x - y) (x + y) = x 
Arguments: (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V)
Goal: midpoint R (x - y) (x + y) = x
-- midpoint_add_sub
@[simp]
theorem midpoint_add_sub (x y : V) : midpoint R (x + y) (x - y) = x 
Arguments: (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V)
Goal: midpoint R (x + y) (x - y) = x
-- AddMonoidHom.coe_ofMapMidpoint
@[simp]
theorem coe_ofMapMidpoint (f : E → F) (h0 : f 0 = 0)
    (hm : ∀ x y, f (midpoint R x y) = midpoint R' (f x) (f y)) :
    ⇑(ofMapMidpoint R R' f h0 hm) = f 
Arguments: (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [Ring R] [Invertible 2] [AddCommGroup E] [Module R E] [Ring R'] [Invertible 2] [AddCommGroup F] [Module R' F] (f : E → F) (h0 : f 0 = 0) (hm : ∀ (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y))
Goal: ⇑(AddMonoidHom.ofMapMidpoint R R' f h0 hm) = f
-- AffineBasis.ind'
structure AffineBasis (ι : Type u₁) (k : Type u₂) {V : Type u₃} (P : Type u₄) [AddCommGroup V]
  [AffineSpace V P] [Ring k] [Module k V] where
  protected toFun : ι → P
  protected ind' : AffineIndependent k toFun
  protected tot' : affineSpan k (range toFun) = ⊤
Arguments: {ι : Type u₁} {k : Type u₂} {V : Type u₃} {P : Type u₄} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (self : AffineBasis ι k P)
Goal: AffineIndependent k self.toFun
-- AffineBasis.tot'
structure AffineBasis (ι : Type u₁) (k : Type u₂) {V : Type u₃} (P : Type u₄) [AddCommGroup V]
  [AffineSpace V P] [Ring k] [Module k V] where
  protected toFun : ι → P
  protected ind' : AffineIndependent k toFun
  protected tot' : affineSpan k (range toFun) = ⊤
Arguments: {ι : Type u₁} {k : Type u₂} {V : Type u₃} {P : Type u₄} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (self : AffineBasis ι k P)
Goal: affineSpan k (Set.range self.toFun) = ⊤
-- AffineBasis.ext
@[ext]
theorem ext {b₁ b₂ : AffineBasis ι k P} (h : (b₁ : ι → P) = b₂) : b₁ = b₂ 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] {b₁ b₂ : AffineBasis ι k P} (h : ⇑b₁ = ⇑b₂)
Goal: b₁ = b₂
-- AffineBasis.ind
theorem ind : AffineIndependent k b 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P)
Goal: AffineIndependent k ⇑b
-- AffineBasis.tot
theorem tot : affineSpan k (range b) = ⊤ 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P)
Goal: affineSpan k (Set.range ⇑b) = ⊤
-- AffineBasis.nonempty
protected theorem nonempty : Nonempty ι 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P)
Goal: Nonempty ι
-- AffineBasis.coe_reindex
@[simp, norm_cast]
theorem coe_reindex : ⇑(b.reindex e) = b ∘ e.symm 
Arguments: {ι : Type u_1} {ι' : Type u_2} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (e : ι ≃ ι')
Goal: ⇑(b.reindex e) = ⇑b ∘ ⇑e.symm
-- AffineBasis.reindex_apply
@[simp]
theorem reindex_apply (i' : ι') : b.reindex e i' = b (e.symm i') 
Arguments: {ι : Type u_1} {ι' : Type u_2} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (e : ι ≃ ι') (i' : ι')
Goal: (b.reindex e) i' = b (e.symm i')
-- AffineBasis.reindex_refl
@[simp]
theorem reindex_refl : b.reindex (Equiv.refl _) = b 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P)
Goal: b.reindex (Equiv.refl ι) = b
-- AffineBasis.basisOf_apply
@[simp]
theorem basisOf_apply (i : ι) (j : { j : ι // j ≠ i }) : b.basisOf i j = b ↑j -ᵥ b i 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (i : ι) (j : { j // j ≠ i })
Goal: (b.basisOf i) j = b ↑j -ᵥ b i
-- AffineBasis.basisOf_reindex
@[simp]
theorem basisOf_reindex (i : ι') :
    (b.reindex e).basisOf i =
      (b.basisOf <| e.symm i).reindex (e.subtypeEquiv fun _ => e.eq_symm_apply.not) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (e : ι ≃ ι') (i : ι')
Goal: (b.reindex e).basisOf i = (b.basisOf (e.symm i)).reindex (e.subtypeEquiv ⋯)
-- AffineBasis.linear_eq_sumCoords
@[simp]
theorem linear_eq_sumCoords (i : ι) : (b.coord i).linear = -(b.basisOf i).sumCoords 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (i : ι)
Goal: (b.coord i).linear = -(b.basisOf i).sumCoords
-- AffineBasis.coord_reindex
@[simp]
theorem coord_reindex (i : ι') : (b.reindex e).coord i = b.coord (e.symm i) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (e : ι ≃ ι') (i : ι')
Goal: (b.reindex e).coord i = b.coord (e.symm i)
-- AffineBasis.coord_apply_eq
@[simp]
theorem coord_apply_eq (i : ι) : b.coord i (b i) = 1 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (i : ι)
Goal: (b.coord i) (b i) = 1
-- AffineBasis.coord_apply_ne
@[simp]
theorem coord_apply_ne (h : i ≠ j) : b.coord i (b j) = 0 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) {i j : ι} (h : i ≠ j)
Goal: (b.coord i) (b j) = 0
-- AffineBasis.coord_apply
theorem coord_apply [DecidableEq ι] (i j : ι) : b.coord i (b j) = if i = j then 1 else 0 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [DecidableEq ι] (i j : ι)
Goal: (b.coord i) (b j) = if i = j then 1 else 0
-- AffineBasis.coord_apply_combination_of_mem
@[simp]
theorem coord_apply_combination_of_mem (hi : i ∈ s) {w : ι → k} (hw : s.sum w = 1) :
    b.coord i (s.affineCombination k b w) = w i 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) {s : Finset ι} {i : ι} (hi : i ∈ s) {w : ι → k} (hw : s.sum w = 1)
Goal: (b.coord i) ((Finset.affineCombination k s ⇑b) w) = w i
-- AffineBasis.coord_apply_combination_of_not_mem
@[simp]
theorem coord_apply_combination_of_not_mem (hi : i ∉ s) {w : ι → k} (hw : s.sum w = 1) :
    b.coord i (s.affineCombination k b w) = 0 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) {s : Finset ι} {i : ι} (hi : i ∉ s) {w : ι → k} (hw : s.sum w = 1)
Goal: (b.coord i) ((Finset.affineCombination k s ⇑b) w) = 0
-- AffineBasis.sum_coord_apply_eq_one
@[simp]
theorem sum_coord_apply_eq_one [Fintype ι] (q : P) : ∑ i, b.coord i q = 1 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Fintype ι] (q : P)
Goal: ∑ i : ι, (b.coord i) q = 1