ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Prod.mk_vadd_mk
@[simp]
theorem mk_vadd_mk (v : G) (v' : G') (p : P) (p' : P') : (v, v') +ᵥ (p, p') = (v +ᵥ p, v' +ᵥ p') 
Arguments: {G : Type u_1} {G' : Type u_2} {P : Type u_3} {P' : Type u_4} [AddGroup G] [AddGroup G'] [AddTorsor G P] [AddTorsor G' P'] (v : G) (v' : G') (p : P) (p' : P')
Goal: (v, v') +ᵥ (p, p') = (v +ᵥ p, v' +ᵥ p')
-- Prod.fst_vsub
@[simp]
theorem fst_vsub (p₁ p₂ : P × P') : (p₁ -ᵥ p₂ : G × G').1 = p₁.1 -ᵥ p₂.1 
Arguments: {G : Type u_1} {G' : Type u_2} {P : Type u_3} {P' : Type u_4} [AddGroup G] [AddGroup G'] [AddTorsor G P] [AddTorsor G' P'] (p₁ p₂ : P × P')
Goal: (p₁ -ᵥ p₂).1 = p₁.1 -ᵥ p₂.1
-- Prod.snd_vsub
@[simp]
theorem snd_vsub (p₁ p₂ : P × P') : (p₁ -ᵥ p₂ : G × G').2 = p₁.2 -ᵥ p₂.2 
Arguments: {G : Type u_1} {G' : Type u_2} {P : Type u_3} {P' : Type u_4} [AddGroup G] [AddGroup G'] [AddTorsor G P] [AddTorsor G' P'] (p₁ p₂ : P × P')
Goal: (p₁ -ᵥ p₂).2 = p₁.2 -ᵥ p₂.2
-- Prod.mk_vsub_mk
@[simp]
theorem mk_vsub_mk (p₁ p₂ : P) (p₁' p₂' : P') :
    ((p₁, p₁') -ᵥ (p₂, p₂') : G × G') = (p₁ -ᵥ p₂, p₁' -ᵥ p₂') 
Arguments: {G : Type u_1} {G' : Type u_2} {P : Type u_3} {P' : Type u_4} [AddGroup G] [AddGroup G'] [AddTorsor G P] [AddTorsor G' P'] (p₁ p₂ : P) (p₁' p₂' : P')
Goal: (p₁, p₁') -ᵥ (p₂, p₂') = (p₁ -ᵥ p₂, p₁' -ᵥ p₂')
-- Equiv.coe_vaddConst
@[simp]
theorem coe_vaddConst (p : P) : ⇑(vaddConst p) = fun v => v +ᵥ p 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (p : P)
Goal: ⇑(Equiv.vaddConst p) = fun v => v +ᵥ p
-- Equiv.coe_vaddConst_symm
@[simp]
theorem coe_vaddConst_symm (p : P) : ⇑(vaddConst p).symm = fun p' => p' -ᵥ p 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (p : P)
Goal: ⇑(Equiv.vaddConst p).symm = fun p' => p' -ᵥ p
-- Equiv.coe_constVSub
@[simp] lemma coe_constVSub (p : P) : ⇑(constVSub p) = (p -ᵥ ·) 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (p : P)
Goal: ⇑(Equiv.constVSub p) = fun x => p -ᵥ x
-- Equiv.coe_constVSub_symm
@[simp]
theorem coe_constVSub_symm (p : P) : ⇑(constVSub p).symm = fun (v : G) => -v +ᵥ p 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (p : P)
Goal: ⇑(Equiv.constVSub p).symm = fun v => -v +ᵥ p
-- Equiv.coe_constVAdd
@[simp] lemma coe_constVAdd (v : G) : ⇑(constVAdd P v) = (v +ᵥ ·) 
Arguments: {G : Type u_1} (P : Type u_2) [AddGroup G] [AddTorsor G P] (v : G)
Goal: ⇑(Equiv.constVAdd P v) = fun x => v +ᵥ x
-- Equiv.constVAdd_zero
@[simp]
theorem constVAdd_zero : constVAdd P (0 : G) = 1 
Arguments: (G : Type u_1) (P : Type u_2) [AddGroup G] [AddTorsor G P]
Goal: Equiv.constVAdd P 0 = 1
-- Equiv.constVAdd_add
@[simp]
theorem constVAdd_add (v₁ v₂ : G) : constVAdd P (v₁ + v₂) = constVAdd P v₁ * constVAdd P v₂ 
Arguments: {G : Type u_1} (P : Type u_2) [AddGroup G] [AddTorsor G P] (v₁ v₂ : G)
Goal: Equiv.constVAdd P (v₁ + v₂) = Equiv.constVAdd P v₁ * Equiv.constVAdd P v₂
-- Equiv.pointReflection_apply
theorem pointReflection_apply (x y : P) : pointReflection x y = x -ᵥ y +ᵥ x 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (x y : P)
Goal: (Equiv.pointReflection x) y = x -ᵥ y +ᵥ x
-- Equiv.pointReflection_vsub_left
@[simp]
theorem pointReflection_vsub_left (x y : P) : pointReflection x y -ᵥ x = x -ᵥ y 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (x y : P)
Goal: (Equiv.pointReflection x) y -ᵥ x = x -ᵥ y
-- Equiv.left_vsub_pointReflection
@[simp]
theorem left_vsub_pointReflection (x y : P) : x -ᵥ pointReflection x y = y -ᵥ x 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (x y : P)
Goal: x -ᵥ (Equiv.pointReflection x) y = y -ᵥ x
-- Equiv.pointReflection_vsub_right
@[simp]
theorem pointReflection_vsub_right (x y : P) : pointReflection x y -ᵥ y = 2 • (x -ᵥ y) 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (x y : P)
Goal: (Equiv.pointReflection x) y -ᵥ y = 2 • (x -ᵥ y)
-- Equiv.right_vsub_pointReflection
@[simp]
theorem right_vsub_pointReflection (x y : P) : y -ᵥ pointReflection x y = 2 • (y -ᵥ x) 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (x y : P)
Goal: y -ᵥ (Equiv.pointReflection x) y = 2 • (y -ᵥ x)
-- Equiv.pointReflection_symm
@[simp]
theorem pointReflection_symm (x : P) : (pointReflection x).symm = pointReflection x 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (x : P)
Goal: Equiv.symm (Equiv.pointReflection x) = Equiv.pointReflection x
-- Equiv.pointReflection_self
@[simp]
theorem pointReflection_self (x : P) : pointReflection x x = x 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (x : P)
Goal: (Equiv.pointReflection x) x = x
-- Equiv.pointReflection_involutive
theorem pointReflection_involutive (x : P) : Involutive (pointReflection x : P → P) 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] (x : P)
Goal: Function.Involutive ⇑(Equiv.pointReflection x)
-- Equiv.pointReflection_fixed_iff_of_injective_bit0
theorem pointReflection_fixed_iff_of_injective_bit0 {x y : P} (h : Injective (2 • · : G → G)) :
    pointReflection x y = y ↔ y = x 
Arguments: {G : Type u_1} {P : Type u_2} [AddGroup G] [AddTorsor G P] {x y : P} (h : Function.Injective fun x => 2 • x)
Goal: (Equiv.pointReflection x) y = y ↔ y = x
-- Equiv.injective_pointReflection_left_of_injective_bit0
theorem injective_pointReflection_left_of_injective_bit0 {G P : Type*} [AddCommGroup G]
    [AddTorsor G P] (h : Injective (2 • · : G → G)) (y : P) :
    Injective fun x : P => pointReflection x y 
Arguments: {G : Type u_3} {P : Type u_4} [AddCommGroup G] [AddTorsor G P] (h : Function.Injective fun x => 2 • x) (y : P)
Goal: Function.Injective fun x => (Equiv.pointReflection x) y
-- AddTorsor.subsingleton_iff
theorem AddTorsor.subsingleton_iff (G P : Type*) [AddGroup G] [AddTorsor G P] :
    Subsingleton G ↔ Subsingleton P 
Arguments: (G : Type u_1) (P : Type u_2) [AddGroup G] [AddTorsor G P]
Goal: Subsingleton G ↔ Subsingleton P
-- HomologicalComplex₂.hasTotal_of_iso
lemma hasTotal_of_iso [K.HasTotal c₁₂] : L.HasTotal c₁₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} {K L : HomologicalComplex₂ C c₁ c₂} (e : K ≅ L) (c₁₂ : ComplexShape I₁₂) [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂]
Goal: L.HasTotal c₁₂
-- HomologicalComplex₂.d₁_eq_zero
lemma d₁_eq_zero (h : ¬ c₁.Rel i₁ (c₁.next i₁)) :
    K.d₁ c₁₂ i₁ i₂ i₁₂ = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂) (h : ¬c₁.Rel i₁ (c₁.next i₁))
Goal: K.d₁ c₁₂ i₁ i₂ i₁₂ = 0
-- HomologicalComplex₂.d₂_eq_zero
lemma d₂_eq_zero (h : ¬ c₂.Rel i₂ (c₂.next i₂)) :
    K.d₂ c₁₂ i₁ i₂ i₁₂ = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂) (h : ¬c₂.Rel i₂ (c₂.next i₂))
Goal: K.d₂ c₁₂ i₁ i₂ i₁₂ = 0
-- HomologicalComplex₂.totalAux.d₁_eq'
lemma d₁_eq' {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂) :
    K.d₁ c₁₂ i₁ i₂ i₁₂ = ComplexShape.ε₁ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ • ((K.d i₁ i₁').f i₂ ≫
      K.toGradedObject.ιMapObjOrZero (ComplexShape.π c₁ c₂ c₁₂) ⟨i₁', i₂⟩ i₁₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂)
Goal: K.d₁ c₁₂ i₁ i₂ i₁₂ =
  c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ i₁').f i₂ ≫ K.toGradedObject.ιMapObjOrZero (c₁.π c₂ c₁₂) (i₁', i₂) i₁₂
-- HomologicalComplex₂.totalAux.d₁_eq
lemma d₁_eq {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂)
    (h' : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁', i₂⟩ = i₁₂) :
    K.d₁ c₁₂ i₁ i₂ i₁₂ = ComplexShape.ε₁ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ • ((K.d i₁ i₁').f i₂ ≫
      K.toGradedObject.ιMapObj (ComplexShape.π c₁ c₂ c₁₂) ⟨i₁', i₂⟩ i₁₂ h') 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂) (h' : c₁.π c₂ c₁₂ (i₁', i₂) = i₁₂)
Goal: K.d₁ c₁₂ i₁ i₂ i₁₂ = c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ i₁').f i₂ ≫ K.toGradedObject.ιMapObj (c₁.π c₂ c₁₂) (i₁', i₂) i₁₂ h'
-- HomologicalComplex₂.totalAux.d₂_eq'
lemma d₂_eq' (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂) :
    K.d₂ c₁₂ i₁ i₂ i₁₂ = ComplexShape.ε₂ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ • ((K.X i₁).d i₂ i₂' ≫
    K.toGradedObject.ιMapObjOrZero (ComplexShape.π c₁ c₂ c₁₂) ⟨i₁, i₂'⟩ i₁₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂)
Goal: K.d₂ c₁₂ i₁ i₂ i₁₂ =
  c₁.ε₂ c₂ c₁₂ (i₁, i₂) • (K.X i₁).d i₂ i₂' ≫ K.toGradedObject.ιMapObjOrZero (c₁.π c₂ c₁₂) (i₁, i₂') i₁₂
-- HomologicalComplex₂.totalAux.d₂_eq
lemma d₂_eq (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂)
    (h' : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂'⟩ = i₁₂) :
    K.d₂ c₁₂ i₁ i₂ i₁₂ = ComplexShape.ε₂ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ • ((K.X i₁).d i₂ i₂' ≫
    K.toGradedObject.ιMapObj (ComplexShape.π c₁ c₂ c₁₂) ⟨i₁, i₂'⟩ i₁₂ h') 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂) (h' : c₁.π c₂ c₁₂ (i₁, i₂') = i₁₂)
Goal: K.d₂ c₁₂ i₁ i₂ i₁₂ = c₁.ε₂ c₂ c₁₂ (i₁, i₂) • (K.X i₁).d i₂ i₂' ≫ K.toGradedObject.ιMapObj (c₁.π c₂ c₁₂) (i₁, i₂') i₁₂ h'
-- HomologicalComplex₂.d₁_eq_zero'
lemma d₁_eq_zero' {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂)
    (h' : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁', i₂⟩ ≠ i₁₂) :
    K.d₁ c₁₂ i₁ i₂ i₁₂ = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂) (h' : c₁.π c₂ c₁₂ (i₁', i₂) ≠ i₁₂)
Goal: K.d₁ c₁₂ i₁ i₂ i₁₂ = 0
-- HomologicalComplex₂.d₂_eq_zero'
lemma d₂_eq_zero' (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂)
    (h' : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂'⟩ ≠ i₁₂) :
    K.d₂ c₁₂ i₁ i₂ i₁₂ = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂) (h' : c₁.π c₂ c₁₂ (i₁, i₂') ≠ i₁₂)
Goal: K.d₂ c₁₂ i₁ i₂ i₁₂ = 0
-- HomologicalComplex₂.totalAux.ιMapObj_D₁
@[reassoc (attr := simp)]
lemma ιMapObj_D₁ (i₁₂ i₁₂' : I₁₂) (i : I₁ × I₂) (h : ComplexShape.π c₁ c₂ c₁₂ i = i₁₂) :
    K.toGradedObject.ιMapObj (ComplexShape.π c₁ c₂ c₁₂) i i₁₂ h ≫ K.D₁ c₁₂ i₁₂ i₁₂' =
      K.d₁ c₁₂ i.1 i.2 i₁₂' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' : I₁₂) (i : I₁ × I₂) (h : c₁.π c₂ c₁₂ i = i₁₂)
Goal: K.toGradedObject.ιMapObj (c₁.π c₂ c₁₂) i i₁₂ h ≫ K.D₁ c₁₂ i₁₂ i₁₂' = K.d₁ c₁₂ i.1 i.2 i₁₂'
-- HomologicalComplex₂.totalAux.ιMapObj_D₂
@[reassoc (attr := simp)]
lemma ιMapObj_D₂ (i₁₂ i₁₂' : I₁₂) (i : I₁ × I₂) (h : ComplexShape.π c₁ c₂ c₁₂ i = i₁₂) :
    K.toGradedObject.ιMapObj (ComplexShape.π c₁ c₂ c₁₂) i i₁₂ h ≫ K.D₂ c₁₂ i₁₂ i₁₂' =
      K.d₂ c₁₂ i.1 i.2 i₁₂' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' : I₁₂) (i : I₁ × I₂) (h : c₁.π c₂ c₁₂ i = i₁₂)
Goal: K.toGradedObject.ιMapObj (c₁.π c₂ c₁₂) i i₁₂ h ≫ K.D₂ c₁₂ i₁₂ i₁₂' = K.d₂ c₁₂ i.1 i.2 i₁₂'
-- HomologicalComplex₂.D₁_shape
lemma D₁_shape (i₁₂ i₁₂' : I₁₂) (h₁₂ : ¬ c₁₂.Rel i₁₂ i₁₂') : K.D₁ c₁₂ i₁₂ i₁₂' = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' : I₁₂) (h₁₂ : ¬c₁₂.Rel i₁₂ i₁₂')
Goal: K.D₁ c₁₂ i₁₂ i₁₂' = 0
-- HomologicalComplex₂.D₂_shape
lemma D₂_shape (i₁₂ i₁₂' : I₁₂) (h₁₂ : ¬ c₁₂.Rel i₁₂ i₁₂') : K.D₂ c₁₂ i₁₂ i₁₂' = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' : I₁₂) (h₁₂ : ¬c₁₂.Rel i₁₂ i₁₂')
Goal: K.D₂ c₁₂ i₁₂ i₁₂' = 0
-- HomologicalComplex₂.D₁_D₁
@[reassoc (attr := simp)]
lemma D₁_D₁ (i₁₂ i₁₂' i₁₂'' : I₁₂) : K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' i₁₂'' : I₁₂)
Goal: K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0
-- HomologicalComplex₂.D₂_D₂
@[reassoc (attr := simp)]
lemma D₂_D₂ (i₁₂ i₁₂' i₁₂'' : I₁₂) : K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' i₁₂'' : I₁₂)
Goal: K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' = 0
-- HomologicalComplex₂.D₂_D₁
@[reassoc (attr := simp)]
lemma D₂_D₁ (i₁₂ i₁₂' i₁₂'' : I₁₂) :
    K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = - K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' i₁₂'' : I₁₂)
Goal: K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = -K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂''
-- HomologicalComplex₂.D₁_D₂
@[reassoc]
lemma D₁_D₂ (i₁₂ i₁₂' i₁₂'' : I₁₂) :
    K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' = - K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' i₁₂'' : I₁₂)
Goal: K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' = -K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂''
-- HomologicalComplex₂.total_d
@[simps d]
noncomputable def total : HomologicalComplex C c₁₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' : I₁₂)
Goal: (K.total c₁₂).d i₁₂ i₁₂' = K.D₁ c₁₂ i₁₂ i₁₂' + K.D₂ c₁₂ i₁₂ i₁₂'
-- HomologicalComplex₂.XXIsoOfEq_hom_ιTotal
@[reassoc (attr := simp)]
lemma XXIsoOfEq_hom_ιTotal {x₁ y₁ : I₁} (h₁ : x₁ = y₁) {x₂ y₂ : I₂} (h₂ : x₂ = y₂)
    (i₁₂ : I₁₂) (h : ComplexShape.π c₁ c₂ c₁₂ (y₁, y₂) = i₁₂) :
    (K.XXIsoOfEq h₁ h₂).hom ≫ K.ιTotal c₁₂ y₁ y₂ i₁₂ h =
      K.ιTotal c₁₂ x₁ x₂ i₁₂ (by rw [h₁, h₂, h]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] {x₁ y₁ : I₁} (h₁ : x₁ = y₁) {x₂ y₂ : I₂} (h₂ : x₂ = y₂) (i₁₂ : I₁₂) (h : c₁.π c₂ c₁₂ (y₁, y₂) = i₁₂)
Goal: (HomologicalComplex₂.XXIsoOfEq C c₁ c₂ K h₁ h₂).hom ≫ K.ιTotal c₁₂ y₁ y₂ i₁₂ h = K.ιTotal c₁₂ x₁ x₂ i₁₂ ⋯
-- HomologicalComplex₂.XXIsoOfEq_inv_ιTotal
@[reassoc (attr := simp)]
lemma XXIsoOfEq_inv_ιTotal {x₁ y₁ : I₁} (h₁ : x₁ = y₁) {x₂ y₂ : I₂} (h₂ : x₂ = y₂)
    (i₁₂ : I₁₂) (h : ComplexShape.π c₁ c₂ c₁₂ (x₁, x₂) = i₁₂) :
    (K.XXIsoOfEq h₁ h₂).inv ≫ K.ιTotal c₁₂ x₁ x₂ i₁₂ h =
      K.ιTotal c₁₂ y₁ y₂ i₁₂ (by rw [← h, h₁, h₂]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] {x₁ y₁ : I₁} (h₁ : x₁ = y₁) {x₂ y₂ : I₂} (h₂ : x₂ = y₂) (i₁₂ : I₁₂) (h : c₁.π c₂ c₁₂ (x₁, x₂) = i₁₂)
Goal: (HomologicalComplex₂.XXIsoOfEq C c₁ c₂ K h₁ h₂).inv ≫ K.ιTotal c₁₂ x₁ x₂ i₁₂ h = K.ιTotal c₁₂ y₁ y₂ i₁₂ ⋯
-- HomologicalComplex₂.ιTotalOrZero_eq
lemma ιTotalOrZero_eq (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂)
    (h : ComplexShape.π c₁ c₂ c₁₂ (i₁, i₂) = i₁₂) :
    K.ιTotalOrZero c₁₂ i₁ i₂ i₁₂ = K.ιTotal c₁₂ i₁ i₂ i₁₂ h 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂) (h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂)
Goal: K.ιTotalOrZero c₁₂ i₁ i₂ i₁₂ = K.ιTotal c₁₂ i₁ i₂ i₁₂ h
-- HomologicalComplex₂.ιTotalOrZero_eq_zero
lemma ιTotalOrZero_eq_zero (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂)
    (h : ComplexShape.π c₁ c₂ c₁₂ (i₁, i₂) ≠ i₁₂) :
    K.ιTotalOrZero c₁₂ i₁ i₂ i₁₂ = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂) (h : c₁.π c₂ c₁₂ (i₁, i₂) ≠ i₁₂)
Goal: K.ιTotalOrZero c₁₂ i₁ i₂ i₁₂ = 0
-- HomologicalComplex₂.ι_D₁
@[reassoc (attr := simp)]
lemma ι_D₁ (i₁₂ i₁₂' : I₁₂) (i₁ : I₁) (i₂ : I₂) (h : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = i₁₂) :
    K.ιTotal c₁₂ i₁ i₂ i₁₂ h ≫ K.D₁ c₁₂ i₁₂ i₁₂' =
      K.d₁ c₁₂ i₁ i₂ i₁₂' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' : I₁₂) (i₁ : I₁) (i₂ : I₂) (h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂)
Goal: K.ιTotal c₁₂ i₁ i₂ i₁₂ h ≫ K.D₁ c₁₂ i₁₂ i₁₂' = K.d₁ c₁₂ i₁ i₂ i₁₂'
-- HomologicalComplex₂.ι_D₂
@[reassoc (attr := simp)]
lemma ι_D₂ (i₁₂ i₁₂' : I₁₂) (i₁ : I₁) (i₂ : I₂)
    (h : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = i₁₂) :
    K.ιTotal c₁₂ i₁ i₂ i₁₂ h ≫ K.D₂ c₁₂ i₁₂ i₁₂' =
      K.d₂ c₁₂ i₁ i₂ i₁₂' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁₂ i₁₂' : I₁₂) (i₁ : I₁) (i₂ : I₂) (h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂)
Goal: K.ιTotal c₁₂ i₁ i₂ i₁₂ h ≫ K.D₂ c₁₂ i₁₂ i₁₂' = K.d₂ c₁₂ i₁ i₂ i₁₂'
-- HomologicalComplex₂.d₁_eq'
lemma d₁_eq' {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂) :
    K.d₁ c₁₂ i₁ i₂ i₁₂ = ComplexShape.ε₁ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ • ((K.d i₁ i₁').f i₂ ≫
      K.ιTotalOrZero c₁₂ i₁' i₂ i₁₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂)
Goal: K.d₁ c₁₂ i₁ i₂ i₁₂ = c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ i₁').f i₂ ≫ K.ιTotalOrZero c₁₂ i₁' i₂ i₁₂
-- HomologicalComplex₂.d₁_eq
lemma d₁_eq {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂)
    (h' : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁', i₂⟩ = i₁₂) :
    K.d₁ c₁₂ i₁ i₂ i₁₂ = ComplexShape.ε₁ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ • ((K.d i₁ i₁').f i₂ ≫
      K.ιTotal c₁₂ i₁' i₂ i₁₂ h') 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂) (h' : c₁.π c₂ c₁₂ (i₁', i₂) = i₁₂)
Goal: K.d₁ c₁₂ i₁ i₂ i₁₂ = c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ i₁').f i₂ ≫ K.ιTotal c₁₂ i₁' i₂ i₁₂ h'
-- HomologicalComplex₂.d₂_eq'
lemma d₂_eq' (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂) :
    K.d₂ c₁₂ i₁ i₂ i₁₂ = ComplexShape.ε₂ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ • ((K.X i₁).d i₂ i₂' ≫
    K.ιTotalOrZero c₁₂ i₁ i₂' i₁₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂)
Goal: K.d₂ c₁₂ i₁ i₂ i₁₂ = c₁.ε₂ c₂ c₁₂ (i₁, i₂) • (K.X i₁).d i₂ i₂' ≫ K.ιTotalOrZero c₁₂ i₁ i₂' i₁₂
-- HomologicalComplex₂.d₂_eq
lemma d₂_eq (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂)
    (h' : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂'⟩ = i₁₂) :
    K.d₂ c₁₂ i₁ i₂ i₁₂ = ComplexShape.ε₂ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ • ((K.X i₁).d i₂ i₂' ≫
    K.ιTotal c₁₂ i₁ i₂' i₁₂ h') 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {I₁ : Type u_2} {I₂ : Type u_3} {I₁₂ : Type u_4} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c₁₂ : ComplexShape I₁₂) [DecidableEq I₁₂] [TotalComplexShape c₁ c₂ c₁₂] [K.HasTotal c₁₂] (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂) (h' : c₁.π c₂ c₁₂ (i₁, i₂') = i₁₂)
Goal: K.d₂ c₁₂ i₁ i₂ i₁₂ = c₁.ε₂ c₂ c₁₂ (i₁, i₂) • (K.X i₁).d i₂ i₂' ≫ K.ιTotal c₁₂ i₁ i₂' i₁₂ h'