ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Bicategory.isRightAdjoint_TFAE
theorem isRightAdjoint_TFAE (u : b ⟶ a) :
    List.TFAE [
      IsRightAdjoint u,
      HasAbsLeftKanLift u (𝟙 a),
      ∃ _ : HasLeftKanLift u (𝟙 a), LanLift.CommuteWith u (𝟙 a) u] 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} (u : b ⟶ a)
Goal: [CategoryTheory.Bicategory.IsRightAdjoint u,
    CategoryTheory.Bicategory.HasAbsLeftKanLift u (CategoryTheory.CategoryStruct.id a),
    ∃ (x : CategoryTheory.Bicategory.HasLeftKanLift u (CategoryTheory.CategoryStruct.id a)),
      CategoryTheory.Bicategory.LanLift.CommuteWith u (CategoryTheory.CategoryStruct.id a) u].TFAE
-- _private.Mathlib.Algebra.Polynomial.Identities.0.Polynomial.poly_binom_aux2
private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
    f.eval (x + y) =
      f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) 
Arguments: {R : Type u} [CommRing R] (f : Polynomial R) (x y : R)
Goal: Polynomial.eval (x + y) f =
  f.sum fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2)
-- _private.Mathlib.Algebra.Polynomial.Identities.0.Polynomial.poly_binom_aux3
private theorem poly_binom_aux3 (f : R[X]) (x y : R) :
    f.eval (x + y) =
      ((f.sum fun e a => a * x ^ e) + f.sum fun e a => a * e * x ^ (e - 1) * y) +
        f.sum fun e a => a * (polyBinomAux1 x y e a).val * y ^ 2 
Arguments: {R : Type u} [CommRing R] (f : Polynomial R) (x y : R)
Goal: Polynomial.eval (x + y) f =
  ((f.sum fun e a => a * x ^ e) + f.sum fun e a => a * ↑e * x ^ (e - 1) * y) +
    f.sum fun e a => a * ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2
-- AddAction.stabilizer_empty
@[to_additive (attr := simp)]
lemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α]
Goal: AddAction.stabilizer G ∅ = ⊤
-- MulAction.stabilizer_empty
@[to_additive (attr := simp)]
lemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α]
Goal: MulAction.stabilizer G ∅ = ⊤
-- MulAction.stabilizer_univ
@[to_additive (attr := simp)]
lemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α]
Goal: MulAction.stabilizer G Set.univ = ⊤
-- AddAction.stabilizer_univ
@[to_additive (attr := simp)]
lemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α]
Goal: AddAction.stabilizer G Set.univ = ⊤
-- AddAction.stabilizer_singleton
@[to_additive (attr := simp)]
lemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] (b : α)
Goal: AddAction.stabilizer G {b} = AddAction.stabilizer G b
-- MulAction.stabilizer_singleton
@[to_additive (attr := simp)]
lemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] (b : α)
Goal: MulAction.stabilizer G {b} = MulAction.stabilizer G b
-- AddAction.mem_stabilizer_set
@[to_additive]
lemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {a : G} {s : Set α}
Goal: a ∈ AddAction.stabilizer G s ↔ ∀ (b : α), a +ᵥ b ∈ s ↔ b ∈ s
-- MulAction.mem_stabilizer_set
@[to_additive]
lemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {a : G} {s : Set α}
Goal: a ∈ MulAction.stabilizer G s ↔ ∀ (b : α), a • b ∈ s ↔ b ∈ s
-- AddAction.map_stabilizer_le
@[to_additive]
lemma map_stabilizer_le (f : G →* H) (s : Set G) :
    (stabilizer G s).map f ≤ stabilizer H (f '' s) 
Arguments: {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] (f : G →+ H) (s : Set G)
Goal: AddSubgroup.map f (AddAction.stabilizer G s) ≤ AddAction.stabilizer H (⇑f '' s)
-- MulAction.map_stabilizer_le
@[to_additive]
lemma map_stabilizer_le (f : G →* H) (s : Set G) :
    (stabilizer G s).map f ≤ stabilizer H (f '' s) 
Arguments: {G : Type u_1} {H : Type u_2} [Group G] [Group H] (f : G →* H) (s : Set G)
Goal: Subgroup.map f (MulAction.stabilizer G s) ≤ MulAction.stabilizer H (⇑f '' s)
-- AddAction.stabilizer_add_self
@[to_additive (attr := simp)]
lemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s 
Arguments: {G : Type u_1} [AddGroup G] (s : Set G)
Goal: ↑(AddAction.stabilizer G s) + s = s
-- MulAction.stabilizer_mul_self
@[to_additive (attr := simp)]
lemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s 
Arguments: {G : Type u_1} [Group G] (s : Set G)
Goal: ↑(MulAction.stabilizer G s) * s = s
-- MulAction.stabilizer_subgroup
@[to_additive (attr := simp)]
lemma stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s 
Arguments: {G : Type u_1} [Group G] (s : Subgroup G)
Goal: MulAction.stabilizer G ↑s = s
-- AddAction.stabilizer_addSubgroup
@[to_additive (attr := simp)]
lemma stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s 
Arguments: {G : Type u_1} [AddGroup G] (s : AddSubgroup G)
Goal: AddAction.stabilizer G ↑s = s
-- MulAction.stabilizer_op_subgroup
@[to_additive (attr := simp)]
lemma stabilizer_op_subgroup (s : Subgroup G) : stabilizer Gᵐᵒᵖ (s : Set G) = s.op 
Arguments: {G : Type u_1} [Group G] (s : Subgroup G)
Goal: MulAction.stabilizer Gᵐᵒᵖ ↑s = s.op
-- AddAction.stabilizer_op_addSubgroup
@[to_additive (attr := simp)]
lemma stabilizer_op_subgroup (s : Subgroup G) : stabilizer Gᵐᵒᵖ (s : Set G) = s.op 
Arguments: {G : Type u_1} [AddGroup G] (s : AddSubgroup G)
Goal: AddAction.stabilizer Gᵃᵒᵖ ↑s = s.op
-- AddAction.stabilizer_addSubgroup_op
@[to_additive (attr := simp)]
lemma stabilizer_subgroup_op (s : Subgroup Gᵐᵒᵖ) : stabilizer G (s : Set Gᵐᵒᵖ) = s.unop 
Arguments: {G : Type u_1} [AddGroup G] (s : AddSubgroup Gᵃᵒᵖ)
Goal: AddAction.stabilizer G ↑s = s.unop
-- MulAction.stabilizer_subgroup_op
@[to_additive (attr := simp)]
lemma stabilizer_subgroup_op (s : Subgroup Gᵐᵒᵖ) : stabilizer G (s : Set Gᵐᵒᵖ) = s.unop 
Arguments: {G : Type u_1} [Group G] (s : Subgroup Gᵐᵒᵖ)
Goal: MulAction.stabilizer G ↑s = s.unop
-- MulAction.stabilizer_coe_finset
@[to_additive (attr := simp)]
lemma stabilizer_coe_finset (s : Finset α) : stabilizer G (s : Set α) = stabilizer G s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [DecidableEq α] (s : Finset α)
Goal: MulAction.stabilizer G ↑s = MulAction.stabilizer G s
-- AddAction.stabilizer_coe_finset
@[to_additive (attr := simp)]
lemma stabilizer_coe_finset (s : Finset α) : stabilizer G (s : Set α) = stabilizer G s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [DecidableEq α] (s : Finset α)
Goal: AddAction.stabilizer G ↑s = AddAction.stabilizer G s
-- AddAction.stabilizer_finset_empty
@[to_additive (attr := simp)]
lemma stabilizer_finset_empty : stabilizer G (∅ : Finset α) = ⊤ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [DecidableEq α]
Goal: AddAction.stabilizer G ∅ = ⊤
-- MulAction.stabilizer_finset_empty
@[to_additive (attr := simp)]
lemma stabilizer_finset_empty : stabilizer G (∅ : Finset α) = ⊤ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [DecidableEq α]
Goal: MulAction.stabilizer G ∅ = ⊤
-- AddAction.stabilizer_finset_univ
@[to_additive (attr := simp)]
lemma stabilizer_finset_univ [Fintype α] : stabilizer G (Finset.univ : Finset α) = ⊤ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [DecidableEq α] [Fintype α]
Goal: AddAction.stabilizer G Finset.univ = ⊤
-- MulAction.stabilizer_finset_univ
@[to_additive (attr := simp)]
lemma stabilizer_finset_univ [Fintype α] : stabilizer G (Finset.univ : Finset α) = ⊤ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [DecidableEq α] [Fintype α]
Goal: MulAction.stabilizer G Finset.univ = ⊤
-- AddAction.stabilizer_finset_singleton
@[to_additive (attr := simp)]
lemma stabilizer_finset_singleton (b : α) : stabilizer G ({b} : Finset α) = stabilizer G b 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [DecidableEq α] (b : α)
Goal: AddAction.stabilizer G {b} = AddAction.stabilizer G b
-- MulAction.stabilizer_finset_singleton
@[to_additive (attr := simp)]
lemma stabilizer_finset_singleton (b : α) : stabilizer G ({b} : Finset α) = stabilizer G b 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [DecidableEq α] (b : α)
Goal: MulAction.stabilizer G {b} = MulAction.stabilizer G b
-- MulAction.mem_stabilizer_finset
@[to_additive]
lemma mem_stabilizer_finset {s : Finset α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {a : G} [DecidableEq α] {s : Finset α}
Goal: a ∈ MulAction.stabilizer G s ↔ ∀ (b : α), a • b ∈ s ↔ b ∈ s
-- AddAction.mem_stabilizer_finset
@[to_additive]
lemma mem_stabilizer_finset {s : Finset α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {a : G} [DecidableEq α] {s : Finset α}
Goal: a ∈ AddAction.stabilizer G s ↔ ∀ (b : α), a +ᵥ b ∈ s ↔ b ∈ s
-- MulAction.mem_stabilizer_finset_iff_subset_smul_finset
@[to_additive]
lemma mem_stabilizer_finset_iff_subset_smul_finset {s : Finset α} :
    a ∈ stabilizer G s ↔ s ⊆ a • s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {a : G} [DecidableEq α] {s : Finset α}
Goal: a ∈ MulAction.stabilizer G s ↔ s ⊆ a • s
-- AddAction.mem_stabilizer_finset_iff_subset_vadd_finset
@[to_additive]
lemma mem_stabilizer_finset_iff_subset_smul_finset {s : Finset α} :
    a ∈ stabilizer G s ↔ s ⊆ a • s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {a : G} [DecidableEq α] {s : Finset α}
Goal: a ∈ AddAction.stabilizer G s ↔ s ⊆ a +ᵥ s
-- AddAction.mem_stabilizer_finset_iff_vadd_finset_subset
@[to_additive]
lemma mem_stabilizer_finset_iff_smul_finset_subset {s : Finset α} :
    a ∈ stabilizer G s ↔ a • s ⊆ s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {a : G} [DecidableEq α] {s : Finset α}
Goal: a ∈ AddAction.stabilizer G s ↔ a +ᵥ s ⊆ s
-- MulAction.mem_stabilizer_finset_iff_smul_finset_subset
@[to_additive]
lemma mem_stabilizer_finset_iff_smul_finset_subset {s : Finset α} :
    a ∈ stabilizer G s ↔ a • s ⊆ s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {a : G} [DecidableEq α] {s : Finset α}
Goal: a ∈ MulAction.stabilizer G s ↔ a • s ⊆ s
-- MulAction.mem_stabilizer_finset'
@[to_additive]
lemma mem_stabilizer_finset' {s : Finset α} : a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {a : G} [DecidableEq α] {s : Finset α}
Goal: a ∈ MulAction.stabilizer G s ↔ ∀ ⦃b : α⦄, b ∈ s → a • b ∈ s
-- AddAction.mem_stabilizer_finset'
@[to_additive]
lemma mem_stabilizer_finset' {s : Finset α} : a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {a : G} [DecidableEq α] {s : Finset α}
Goal: a ∈ AddAction.stabilizer G s ↔ ∀ ⦃b : α⦄, b ∈ s → a +ᵥ b ∈ s
-- AddAction.mem_stabilizer_set_iff_subset_vadd_set
@[to_additive]
lemma mem_stabilizer_set_iff_subset_smul_set {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ s ⊆ a • s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {a : G} {s : Set α} (hs : s.Finite)
Goal: a ∈ AddAction.stabilizer G s ↔ s ⊆ a +ᵥ s
-- MulAction.mem_stabilizer_set_iff_subset_smul_set
@[to_additive]
lemma mem_stabilizer_set_iff_subset_smul_set {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ s ⊆ a • s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {a : G} {s : Set α} (hs : s.Finite)
Goal: a ∈ MulAction.stabilizer G s ↔ s ⊆ a • s
-- AddAction.mem_stabilizer_set_iff_vadd_set_subset
@[to_additive]
lemma mem_stabilizer_set_iff_smul_set_subset {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ a • s ⊆ s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {a : G} {s : Set α} (hs : s.Finite)
Goal: a ∈ AddAction.stabilizer G s ↔ a +ᵥ s ⊆ s
-- MulAction.mem_stabilizer_set_iff_smul_set_subset
@[to_additive]
lemma mem_stabilizer_set_iff_smul_set_subset {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ a • s ⊆ s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {a : G} {s : Set α} (hs : s.Finite)
Goal: a ∈ MulAction.stabilizer G s ↔ a • s ⊆ s
-- AddAction.mem_stabilizer_set'
@[to_additive]
lemma mem_stabilizer_set' {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] {a : G} {s : Set α} (hs : s.Finite)
Goal: a ∈ AddAction.stabilizer G s ↔ ∀ ⦃b : α⦄, b ∈ s → a +ᵥ b ∈ s
-- MulAction.mem_stabilizer_set'
@[to_additive]
lemma mem_stabilizer_set' {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] {a : G} {s : Set α} (hs : s.Finite)
Goal: a ∈ MulAction.stabilizer G s ↔ ∀ ⦃b : α⦄, b ∈ s → a • b ∈ s
-- MulAction.mul_stabilizer_self
@[to_additive (attr := simp)]
lemma mul_stabilizer_self : s * stabilizer G s = s 
Arguments: {G : Type u_1} [CommGroup G] (s : Set G)
Goal: s * ↑(MulAction.stabilizer G s) = s
-- AddAction.add_stabilizer_self
@[to_additive (attr := simp)]
lemma mul_stabilizer_self : s * stabilizer G s = s 
Arguments: {G : Type u_1} [AddCommGroup G] (s : Set G)
Goal: s + ↑(AddAction.stabilizer G s) = s
-- MulAction.stabilizer_image_coe_quotient
@[to_additive]
lemma stabilizer_image_coe_quotient : stabilizer Q (q '' s) = ⊥ 
Arguments: {G : Type u_1} [CommGroup G] (s : Set G)
Goal: MulAction.stabilizer (G ⧸ MulAction.stabilizer G s) (QuotientGroup.mk '' s) = ⊥
-- AddAction.stabilizer_image_coe_quotient
@[to_additive]
lemma stabilizer_image_coe_quotient : stabilizer Q (q '' s) = ⊥ 
Arguments: {G : Type u_1} [AddCommGroup G] (s : Set G)
Goal: AddAction.stabilizer (G ⧸ AddAction.stabilizer G s) (QuotientAddGroup.mk '' s) = ⊥
-- CategoryTheory.Functor.mapTriangle_map_hom₁
@[simps]
def mapTriangle : Triangle C ⥤ Triangle D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ]
Goal: ∀ {X Y : CategoryTheory.Pretriangulated.Triangle C} (f : X ⟶ Y), (F.mapTriangle.map f).hom₁ = F.map f.hom₁
-- CategoryTheory.Functor.mapTriangle_map_hom₂
@[simps]
def mapTriangle : Triangle C ⥤ Triangle D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ]
Goal: ∀ {X Y : CategoryTheory.Pretriangulated.Triangle C} (f : X ⟶ Y), (F.mapTriangle.map f).hom₂ = F.map f.hom₂
-- CategoryTheory.Functor.mapTriangle_obj
@[simps]
def mapTriangle : Triangle C ⥤ Triangle D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] (T : CategoryTheory.Pretriangulated.Triangle C)
Goal: F.mapTriangle.obj T =
  CategoryTheory.Pretriangulated.Triangle.mk (F.map T.mor₁) (F.map T.mor₂)
    (CategoryTheory.CategoryStruct.comp (F.map T.mor₃) ((F.commShiftIso 1).hom.app T.obj₁))
-- CategoryTheory.Functor.mapTriangle_map_hom₃
@[simps]
def mapTriangle : Triangle C ⥤ Triangle D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ]
Goal: ∀ {X Y : CategoryTheory.Pretriangulated.Triangle C} (f : X ⟶ Y), (F.mapTriangle.map f).hom₃ = F.map f.hom₃
-- CategoryTheory.Functor.mapTriangleCommShiftIso_hom_app_hom₃
@[simps!]
noncomputable def mapTriangleCommShiftIso (n : ℤ) :
    Triangle.shiftFunctor C n ⋙ F.mapTriangle ≅ F.mapTriangle ⋙ Triangle.shiftFunctor D n 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (n : ℤ) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCommShiftIso n).hom.app X).hom₃ = (F.commShiftIso n).hom.app X.obj₃
-- CategoryTheory.Functor.mapTriangleCommShiftIso_inv_app_hom₁
@[simps!]
noncomputable def mapTriangleCommShiftIso (n : ℤ) :
    Triangle.shiftFunctor C n ⋙ F.mapTriangle ≅ F.mapTriangle ⋙ Triangle.shiftFunctor D n 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (n : ℤ) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCommShiftIso n).inv.app X).hom₁ = (F.commShiftIso n).inv.app X.obj₁
-- CategoryTheory.Functor.mapTriangleCommShiftIso_hom_app_hom₁
@[simps!]
noncomputable def mapTriangleCommShiftIso (n : ℤ) :
    Triangle.shiftFunctor C n ⋙ F.mapTriangle ≅ F.mapTriangle ⋙ Triangle.shiftFunctor D n 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (n : ℤ) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCommShiftIso n).hom.app X).hom₁ = (F.commShiftIso n).hom.app X.obj₁
-- CategoryTheory.Functor.mapTriangleCommShiftIso_hom_app_hom₂
@[simps!]
noncomputable def mapTriangleCommShiftIso (n : ℤ) :
    Triangle.shiftFunctor C n ⋙ F.mapTriangle ≅ F.mapTriangle ⋙ Triangle.shiftFunctor D n 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (n : ℤ) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCommShiftIso n).hom.app X).hom₂ = (F.commShiftIso n).hom.app X.obj₂
-- CategoryTheory.Functor.mapTriangleCommShiftIso_inv_app_hom₃
@[simps!]
noncomputable def mapTriangleCommShiftIso (n : ℤ) :
    Triangle.shiftFunctor C n ⋙ F.mapTriangle ≅ F.mapTriangle ⋙ Triangle.shiftFunctor D n 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (n : ℤ) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCommShiftIso n).inv.app X).hom₃ = (F.commShiftIso n).inv.app X.obj₃
-- CategoryTheory.Functor.mapTriangleCommShiftIso_inv_app_hom₂
@[simps!]
noncomputable def mapTriangleCommShiftIso (n : ℤ) :
    Triangle.shiftFunctor C n ⋙ F.mapTriangle ≅ F.mapTriangle ⋙ Triangle.shiftFunctor D n 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (n : ℤ) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCommShiftIso n).inv.app X).hom₂ = (F.commShiftIso n).inv.app X.obj₂
-- CategoryTheory.Functor.mapTriangleRotateIso_hom_app_hom₂
@[simps!]
def mapTriangleRotateIso :
    F.mapTriangle ⋙ Pretriangulated.rotate D ≅
      Pretriangulated.rotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleRotateIso.hom.app X).hom₂ = CategoryTheory.CategoryStruct.id (F.obj X.obj₃)
-- CategoryTheory.Functor.mapTriangleRotateIso_inv_app_hom₃
@[simps!]
def mapTriangleRotateIso :
    F.mapTriangle ⋙ Pretriangulated.rotate D ≅
      Pretriangulated.rotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleRotateIso.inv.app X).hom₃ = (F.commShiftIso 1).hom.app X.obj₁
-- CategoryTheory.Functor.mapTriangleRotateIso_hom_app_hom₃
@[simps!]
def mapTriangleRotateIso :
    F.mapTriangle ⋙ Pretriangulated.rotate D ≅
      Pretriangulated.rotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleRotateIso.hom.app X).hom₃ = (F.commShiftIso 1).inv.app X.obj₁
-- CategoryTheory.Functor.mapTriangleRotateIso_inv_app_hom₁
@[simps!]
def mapTriangleRotateIso :
    F.mapTriangle ⋙ Pretriangulated.rotate D ≅
      Pretriangulated.rotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleRotateIso.inv.app X).hom₁ = CategoryTheory.CategoryStruct.id (F.obj X.obj₂)
-- CategoryTheory.Functor.mapTriangleRotateIso_inv_app_hom₂
@[simps!]
def mapTriangleRotateIso :
    F.mapTriangle ⋙ Pretriangulated.rotate D ≅
      Pretriangulated.rotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleRotateIso.inv.app X).hom₂ = CategoryTheory.CategoryStruct.id (F.obj X.obj₃)
-- CategoryTheory.Functor.mapTriangleRotateIso_hom_app_hom₁
@[simps!]
def mapTriangleRotateIso :
    F.mapTriangle ⋙ Pretriangulated.rotate D ≅
      Pretriangulated.rotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleRotateIso.hom.app X).hom₁ = CategoryTheory.CategoryStruct.id (F.obj X.obj₂)
-- CategoryTheory.Functor.mapTriangleInvRotateIso_hom_app_hom₁
@[simps!]
noncomputable def mapTriangleInvRotateIso [F.Additive] :
    F.mapTriangle ⋙ Pretriangulated.invRotate D ≅
      Pretriangulated.invRotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleInvRotateIso.hom.app X).hom₁ = (F.commShiftIso (-1)).inv.app X.obj₃