ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.isCodetector_op_iff
theorem isCodetector_op_iff (G : C) : IsCodetector (op G) ↔ IsDetector G 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C)
Goal: CategoryTheory.IsCodetector (Opposite.op G) ↔ CategoryTheory.IsDetector G
-- CategoryTheory.isCodetector_unop_iff
theorem isCodetector_unop_iff (G : Cᵒᵖ) : IsCodetector (unop G) ↔ IsDetector G 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : Cᵒᵖ)
Goal: CategoryTheory.IsCodetector (Opposite.unop G) ↔ CategoryTheory.IsDetector G
-- CategoryTheory.isDetector_unop_iff
theorem isDetector_unop_iff (G : Cᵒᵖ) : IsDetector (unop G) ↔ IsCodetector G 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : Cᵒᵖ)
Goal: CategoryTheory.IsDetector (Opposite.unop G) ↔ CategoryTheory.IsCodetector G
-- CategoryTheory.IsDetector.isSeparator
theorem IsDetector.isSeparator [HasEqualizers C] {G : C} : IsDetector G → IsSeparator G 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasEqualizers C] {G : C}
Goal: CategoryTheory.IsDetector G → CategoryTheory.IsSeparator G
-- CategoryTheory.IsCodetector.isCoseparator
theorem IsCodetector.isCoseparator [HasCoequalizers C] {G : C} : IsCodetector G → IsCoseparator G 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasCoequalizers C] {G : C}
Goal: CategoryTheory.IsCodetector G → CategoryTheory.IsCoseparator G
-- CategoryTheory.IsSeparator.isDetector
theorem IsSeparator.isDetector [Balanced C] {G : C} : IsSeparator G → IsDetector G 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Balanced C] {G : C}
Goal: CategoryTheory.IsSeparator G → CategoryTheory.IsDetector G
-- CategoryTheory.IsCospearator.isCodetector
theorem IsCospearator.isCodetector [Balanced C] {G : C} : IsCoseparator G → IsCodetector G 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Balanced C] {G : C}
Goal: CategoryTheory.IsCoseparator G → CategoryTheory.IsCodetector G
-- CategoryTheory.isSeparator_def
theorem isSeparator_def (G : C) :
    IsSeparator G ↔ ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ h : G ⟶ X, h ≫ f = h ≫ g) → f = g 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C)
Goal: CategoryTheory.IsSeparator G ↔
  ∀ ⦃X Y : C⦄ (f g : X ⟶ Y),
    (∀ (h : G ⟶ X), CategoryTheory.CategoryStruct.comp h f = CategoryTheory.CategoryStruct.comp h g) → f = g
-- CategoryTheory.IsSeparator.def
theorem IsSeparator.def {G : C} :
    IsSeparator G → ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ h : G ⟶ X, h ≫ f = h ≫ g) → f = g 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {G : C}
Goal: CategoryTheory.IsSeparator G →
  ∀ ⦃X Y : C⦄ (f g : X ⟶ Y),
    (∀ (h : G ⟶ X), CategoryTheory.CategoryStruct.comp h f = CategoryTheory.CategoryStruct.comp h g) → f = g
-- CategoryTheory.isCoseparator_def
theorem isCoseparator_def (G : C) :
    IsCoseparator G ↔ ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ h : Y ⟶ G, f ≫ h = g ≫ h) → f = g 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C)
Goal: CategoryTheory.IsCoseparator G ↔
  ∀ ⦃X Y : C⦄ (f g : X ⟶ Y),
    (∀ (h : Y ⟶ G), CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g h) → f = g
-- CategoryTheory.IsCoseparator.def
theorem IsCoseparator.def {G : C} :
    IsCoseparator G → ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ h : Y ⟶ G, f ≫ h = g ≫ h) → f = g 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {G : C}
Goal: CategoryTheory.IsCoseparator G →
  ∀ ⦃X Y : C⦄ (f g : X ⟶ Y),
    (∀ (h : Y ⟶ G), CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g h) → f = g
-- CategoryTheory.isDetector_def
theorem isDetector_def (G : C) :
    IsDetector G ↔ ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ h : G ⟶ Y, ∃! h', h' ≫ f = h) → IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C)
Goal: CategoryTheory.IsDetector G ↔
  ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : G ⟶ Y), ∃! h', CategoryTheory.CategoryStruct.comp h' f = h) → CategoryTheory.IsIso f
-- CategoryTheory.IsDetector.def
theorem IsDetector.def {G : C} :
    IsDetector G → ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ h : G ⟶ Y, ∃! h', h' ≫ f = h) → IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {G : C}
Goal: CategoryTheory.IsDetector G →
  ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : G ⟶ Y), ∃! h', CategoryTheory.CategoryStruct.comp h' f = h) → CategoryTheory.IsIso f
-- CategoryTheory.isCodetector_def
theorem isCodetector_def (G : C) :
    IsCodetector G ↔ ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ h : X ⟶ G, ∃! h', f ≫ h' = h) → IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C)
Goal: CategoryTheory.IsCodetector G ↔
  ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : X ⟶ G), ∃! h', CategoryTheory.CategoryStruct.comp f h' = h) → CategoryTheory.IsIso f
-- CategoryTheory.IsCodetector.def
theorem IsCodetector.def {G : C} :
    IsCodetector G → ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ h : X ⟶ G, ∃! h', f ≫ h' = h) → IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {G : C}
Goal: CategoryTheory.IsCodetector G →
  ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : X ⟶ G), ∃! h', CategoryTheory.CategoryStruct.comp f h' = h) → CategoryTheory.IsIso f
-- CategoryTheory.isSeparator_iff_faithful_coyoneda_obj
theorem isSeparator_iff_faithful_coyoneda_obj (G : C) :
    IsSeparator G ↔ (coyoneda.obj (op G)).Faithful 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C)
Goal: CategoryTheory.IsSeparator G ↔ (CategoryTheory.coyoneda.obj (Opposite.op G)).Faithful
-- CategoryTheory.isCoseparator_iff_faithful_yoneda_obj
theorem isCoseparator_iff_faithful_yoneda_obj (G : C) : IsCoseparator G ↔ (yoneda.obj G).Faithful 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C)
Goal: CategoryTheory.IsCoseparator G ↔ (CategoryTheory.yoneda.obj G).Faithful
-- CategoryTheory.isSeparator_iff_epi
theorem isSeparator_iff_epi (G : C) [∀ A : C, HasCoproduct fun _ : G ⟶ A => G] :
    IsSeparator G ↔ ∀ A : C, Epi (Sigma.desc fun f : G ⟶ A => f) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C) [∀ (A : C), CategoryTheory.Limits.HasCoproduct fun x => G]
Goal: CategoryTheory.IsSeparator G ↔ ∀ (A : C), CategoryTheory.Epi (CategoryTheory.Limits.Sigma.desc fun f => f)
-- CategoryTheory.isCoseparator_iff_mono
theorem isCoseparator_iff_mono (G : C) [∀ A : C, HasProduct fun _ : A ⟶ G => G] :
    IsCoseparator G ↔ ∀ A : C, Mono (Pi.lift fun f : A ⟶ G => f) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C) [∀ (A : C), CategoryTheory.Limits.HasProduct fun x => G]
Goal: CategoryTheory.IsCoseparator G ↔ ∀ (A : C), CategoryTheory.Mono (CategoryTheory.Limits.Pi.lift fun f => f)
-- CategoryTheory.isSeparator_coprod
theorem isSeparator_coprod (G H : C) [HasBinaryCoproduct G H] :
    IsSeparator (G ⨿ H) ↔ IsSeparating ({G, H} : Set C) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] (G H : C) [CategoryTheory.Limits.HasBinaryCoproduct G H]
Goal: CategoryTheory.IsSeparator (G ⨿ H) ↔ CategoryTheory.IsSeparating {G, H}
-- CategoryTheory.isSeparator_coprod_of_isSeparator_left
theorem isSeparator_coprod_of_isSeparator_left (G H : C) [HasBinaryCoproduct G H]
    (hG : IsSeparator G) : IsSeparator (G ⨿ H) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] (G H : C) [CategoryTheory.Limits.HasBinaryCoproduct G H] (hG : CategoryTheory.IsSeparator G)
Goal: CategoryTheory.IsSeparator (G ⨿ H)
-- CategoryTheory.isSeparator_coprod_of_isSeparator_right
theorem isSeparator_coprod_of_isSeparator_right (G H : C) [HasBinaryCoproduct G H]
    (hH : IsSeparator H) : IsSeparator (G ⨿ H) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] (G H : C) [CategoryTheory.Limits.HasBinaryCoproduct G H] (hH : CategoryTheory.IsSeparator H)
Goal: CategoryTheory.IsSeparator (G ⨿ H)
-- CategoryTheory.isSeparator_sigma
theorem isSeparator_sigma {β : Type w} (f : β → C) [HasCoproduct f] :
    IsSeparator (∐ f) ↔ IsSeparating (Set.range f) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] {β : Type w} (f : β → C) [CategoryTheory.Limits.HasCoproduct f]
Goal: CategoryTheory.IsSeparator (∐ f) ↔ CategoryTheory.IsSeparating (Set.range f)
-- CategoryTheory.isSeparator_sigma_of_isSeparator
theorem isSeparator_sigma_of_isSeparator {β : Type w} (f : β → C) [HasCoproduct f] (b : β)
    (hb : IsSeparator (f b)) : IsSeparator (∐ f) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] {β : Type w} (f : β → C) [CategoryTheory.Limits.HasCoproduct f] (b : β) (hb : CategoryTheory.IsSeparator (f b))
Goal: CategoryTheory.IsSeparator (∐ f)
-- CategoryTheory.isCoseparator_prod
theorem isCoseparator_prod (G H : C) [HasBinaryProduct G H] :
    IsCoseparator (G ⨯ H) ↔ IsCoseparating ({G, H} : Set C) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] (G H : C) [CategoryTheory.Limits.HasBinaryProduct G H]
Goal: CategoryTheory.IsCoseparator (G ⨯ H) ↔ CategoryTheory.IsCoseparating {G, H}
-- CategoryTheory.isCoseparator_prod_of_isCoseparator_left
theorem isCoseparator_prod_of_isCoseparator_left (G H : C) [HasBinaryProduct G H]
    (hG : IsCoseparator G) : IsCoseparator (G ⨯ H) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] (G H : C) [CategoryTheory.Limits.HasBinaryProduct G H] (hG : CategoryTheory.IsCoseparator G)
Goal: CategoryTheory.IsCoseparator (G ⨯ H)
-- CategoryTheory.isCoseparator_prod_of_isCoseparator_right
theorem isCoseparator_prod_of_isCoseparator_right (G H : C) [HasBinaryProduct G H]
    (hH : IsCoseparator H) : IsCoseparator (G ⨯ H) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] (G H : C) [CategoryTheory.Limits.HasBinaryProduct G H] (hH : CategoryTheory.IsCoseparator H)
Goal: CategoryTheory.IsCoseparator (G ⨯ H)
-- CategoryTheory.isCoseparator_pi
theorem isCoseparator_pi {β : Type w} (f : β → C) [HasProduct f] :
    IsCoseparator (∏ᶜ f) ↔ IsCoseparating (Set.range f) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] {β : Type w} (f : β → C) [CategoryTheory.Limits.HasProduct f]
Goal: CategoryTheory.IsCoseparator (∏ᶜ f) ↔ CategoryTheory.IsCoseparating (Set.range f)
-- CategoryTheory.isCoseparator_pi_of_isCoseparator
theorem isCoseparator_pi_of_isCoseparator {β : Type w} (f : β → C) [HasProduct f] (b : β)
    (hb : IsCoseparator (f b)) : IsCoseparator (∏ᶜ f) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] {β : Type w} (f : β → C) [CategoryTheory.Limits.HasProduct f] (b : β) (hb : CategoryTheory.IsCoseparator (f b))
Goal: CategoryTheory.IsCoseparator (∏ᶜ f)
-- CategoryTheory.isDetector_iff_reflectsIsomorphisms_coyoneda_obj
theorem isDetector_iff_reflectsIsomorphisms_coyoneda_obj (G : C) :
    IsDetector G ↔ (coyoneda.obj (op G)).ReflectsIsomorphisms 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C)
Goal: CategoryTheory.IsDetector G ↔ (CategoryTheory.coyoneda.obj (Opposite.op G)).ReflectsIsomorphisms
-- CategoryTheory.isCodetector_iff_reflectsIsomorphisms_yoneda_obj
theorem isCodetector_iff_reflectsIsomorphisms_yoneda_obj (G : C) :
    IsCodetector G ↔ (yoneda.obj G).ReflectsIsomorphisms 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (G : C)
Goal: CategoryTheory.IsCodetector G ↔ (CategoryTheory.yoneda.obj G).ReflectsIsomorphisms
-- CategoryTheory.wellPowered_of_isDetector
theorem wellPowered_of_isDetector [HasPullbacks C] (G : C) (hG : IsDetector G) : WellPowered C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasPullbacks C] (G : C) (hG : CategoryTheory.IsDetector G)
Goal: CategoryTheory.WellPowered C
-- CategoryTheory.Triangulated.TStructure.GE_one_le
structure TStructure where
    LE (n : ℤ) : C → Prop
    GE (n : ℤ) : C → Prop
  LE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (LE n) := by infer_instance
  GE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (GE n) := by infer_instance
  LE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : LE n X) : LE n' (X⟦a⟧)
  GE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : GE n X) : GE n' (X⟦a⟧)
  zero' ⦃X Y : C⦄ (f : X ⟶ Y) (hX : LE 0 X) (hY : GE 1 Y) : f = 0
  LE_zero_le : LE 0 ≤ LE 1
  GE_one_le : GE 1 ≤ GE 0
  exists_triangle_zero_one (A : C) : ∃ (X Y : C) (_ : LE 0 X) (_ : GE 1 Y)
    (f : X ⟶ A) (g : A ⟶ Y) (h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (self : CategoryTheory.Triangulated.TStructure C)
Goal: self.GE 1 ≤ self.GE 0
-- CategoryTheory.Triangulated.TStructure.LE_shift
structure TStructure where
    LE (n : ℤ) : C → Prop
    GE (n : ℤ) : C → Prop
  LE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (LE n) := by infer_instance
  GE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (GE n) := by infer_instance
  LE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : LE n X) : LE n' (X⟦a⟧)
  GE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : GE n X) : GE n' (X⟦a⟧)
  zero' ⦃X Y : C⦄ (f : X ⟶ Y) (hX : LE 0 X) (hY : GE 1 Y) : f = 0
  LE_zero_le : LE 0 ≤ LE 1
  GE_one_le : GE 1 ≤ GE 0
  exists_triangle_zero_one (A : C) : ∃ (X Y : C) (_ : LE 0 X) (_ : GE 1 Y)
    (f : X ⟶ A) (g : A ⟶ Y) (h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (self : CategoryTheory.Triangulated.TStructure C) (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : self.LE n X)
Goal: self.LE n' ((CategoryTheory.shiftFunctor C a).obj X)
-- CategoryTheory.Triangulated.TStructure.GE_closedUnderIsomorphisms
structure TStructure where
    LE (n : ℤ) : C → Prop
    GE (n : ℤ) : C → Prop
  LE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (LE n) := by infer_instance
  GE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (GE n) := by infer_instance
  LE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : LE n X) : LE n' (X⟦a⟧)
  GE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : GE n X) : GE n' (X⟦a⟧)
  zero' ⦃X Y : C⦄ (f : X ⟶ Y) (hX : LE 0 X) (hY : GE 1 Y) : f = 0
  LE_zero_le : LE 0 ≤ LE 1
  GE_one_le : GE 1 ≤ GE 0
  exists_triangle_zero_one (A : C) : ∃ (X Y : C) (_ : LE 0 X) (_ : GE 1 Y)
    (f : X ⟶ A) (g : A ⟶ Y) (h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (self : CategoryTheory.Triangulated.TStructure C) (n : ℤ)
Goal: CategoryTheory.ClosedUnderIsomorphisms (self.GE n)
-- CategoryTheory.Triangulated.TStructure.GE_shift
structure TStructure where
    LE (n : ℤ) : C → Prop
    GE (n : ℤ) : C → Prop
  LE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (LE n) := by infer_instance
  GE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (GE n) := by infer_instance
  LE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : LE n X) : LE n' (X⟦a⟧)
  GE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : GE n X) : GE n' (X⟦a⟧)
  zero' ⦃X Y : C⦄ (f : X ⟶ Y) (hX : LE 0 X) (hY : GE 1 Y) : f = 0
  LE_zero_le : LE 0 ≤ LE 1
  GE_one_le : GE 1 ≤ GE 0
  exists_triangle_zero_one (A : C) : ∃ (X Y : C) (_ : LE 0 X) (_ : GE 1 Y)
    (f : X ⟶ A) (g : A ⟶ Y) (h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (self : CategoryTheory.Triangulated.TStructure C) (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : self.GE n X)
Goal: self.GE n' ((CategoryTheory.shiftFunctor C a).obj X)
-- CategoryTheory.Triangulated.TStructure.LE_closedUnderIsomorphisms
structure TStructure where
    LE (n : ℤ) : C → Prop
    GE (n : ℤ) : C → Prop
  LE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (LE n) := by infer_instance
  GE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (GE n) := by infer_instance
  LE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : LE n X) : LE n' (X⟦a⟧)
  GE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : GE n X) : GE n' (X⟦a⟧)
  zero' ⦃X Y : C⦄ (f : X ⟶ Y) (hX : LE 0 X) (hY : GE 1 Y) : f = 0
  LE_zero_le : LE 0 ≤ LE 1
  GE_one_le : GE 1 ≤ GE 0
  exists_triangle_zero_one (A : C) : ∃ (X Y : C) (_ : LE 0 X) (_ : GE 1 Y)
    (f : X ⟶ A) (g : A ⟶ Y) (h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (self : CategoryTheory.Triangulated.TStructure C) (n : ℤ)
Goal: CategoryTheory.ClosedUnderIsomorphisms (self.LE n)
-- CategoryTheory.Triangulated.TStructure.exists_triangle_zero_one
structure TStructure where
    LE (n : ℤ) : C → Prop
    GE (n : ℤ) : C → Prop
  LE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (LE n) := by infer_instance
  GE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (GE n) := by infer_instance
  LE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : LE n X) : LE n' (X⟦a⟧)
  GE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : GE n X) : GE n' (X⟦a⟧)
  zero' ⦃X Y : C⦄ (f : X ⟶ Y) (hX : LE 0 X) (hY : GE 1 Y) : f = 0
  LE_zero_le : LE 0 ≤ LE 1
  GE_one_le : GE 1 ≤ GE 0
  exists_triangle_zero_one (A : C) : ∃ (X Y : C) (_ : LE 0 X) (_ : GE 1 Y)
    (f : X ⟶ A) (g : A ⟶ Y) (h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (self : CategoryTheory.Triangulated.TStructure C) (A : C)
Goal: ∃ X Y,
  ∃ (_ : self.LE 0 X) (_ : self.GE 1 Y),
    ∃ f g h, CategoryTheory.Pretriangulated.Triangle.mk f g h ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Triangulated.TStructure.LE_zero_le
structure TStructure where
    LE (n : ℤ) : C → Prop
    GE (n : ℤ) : C → Prop
  LE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (LE n) := by infer_instance
  GE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (GE n) := by infer_instance
  LE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : LE n X) : LE n' (X⟦a⟧)
  GE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : GE n X) : GE n' (X⟦a⟧)
  zero' ⦃X Y : C⦄ (f : X ⟶ Y) (hX : LE 0 X) (hY : GE 1 Y) : f = 0
  LE_zero_le : LE 0 ≤ LE 1
  GE_one_le : GE 1 ≤ GE 0
  exists_triangle_zero_one (A : C) : ∃ (X Y : C) (_ : LE 0 X) (_ : GE 1 Y)
    (f : X ⟶ A) (g : A ⟶ Y) (h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (self : CategoryTheory.Triangulated.TStructure C)
Goal: self.LE 0 ≤ self.LE 1
-- CategoryTheory.Triangulated.TStructure.zero'
structure TStructure where
    LE (n : ℤ) : C → Prop
    GE (n : ℤ) : C → Prop
  LE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (LE n) := by infer_instance
  GE_closedUnderIsomorphisms (n : ℤ) : ClosedUnderIsomorphisms (GE n) := by infer_instance
  LE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : LE n X) : LE n' (X⟦a⟧)
  GE_shift (n a n' : ℤ) (h : a + n' = n) (X : C) (hX : GE n X) : GE n' (X⟦a⟧)
  zero' ⦃X Y : C⦄ (f : X ⟶ Y) (hX : LE 0 X) (hY : GE 1 Y) : f = 0
  LE_zero_le : LE 0 ≤ LE 1
  GE_one_le : GE 1 ≤ GE 0
  exists_triangle_zero_one (A : C) : ∃ (X Y : C) (_ : LE 0 X) (_ : GE 1 Y)
    (f : X ⟶ A) (g : A ⟶ Y) (h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (self : CategoryTheory.Triangulated.TStructure C) ⦃X Y : C⦄ (f : X ⟶ Y) (hX : self.LE 0 X) (hY : self.GE 1 Y)
Goal: f = 0
-- CategoryTheory.Triangulated.TStructure.exists_triangle
lemma exists_triangle (A : C) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁) :
    ∃ (X Y : C) (_ : t.LE n₀ X) (_ : t.GE n₁ Y) (f : X ⟶ A) (g : A ⟶ Y)
      (h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (t : CategoryTheory.Triangulated.TStructure C) (A : C) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁)
Goal: ∃ X Y,
  ∃ (_ : t.LE n₀ X) (_ : t.GE n₁ Y),
    ∃ f g h, CategoryTheory.Pretriangulated.Triangle.mk f g h ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Triangulated.TStructure.predicateShift_LE
lemma predicateShift_LE (a n n' : ℤ) (hn' : a + n = n') :
    (PredicateShift (t.LE n) a) = t.LE n' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (t : CategoryTheory.Triangulated.TStructure C) (a n n' : ℤ) (hn' : a + n = n')
Goal: CategoryTheory.PredicateShift (t.LE n) a = t.LE n'
-- CategoryTheory.Triangulated.TStructure.predicateShift_GE
lemma predicateShift_GE (a n n' : ℤ) (hn' : a + n = n') :
    (PredicateShift (t.GE n) a) = t.GE n' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (t : CategoryTheory.Triangulated.TStructure C) (a n n' : ℤ) (hn' : a + n = n')
Goal: CategoryTheory.PredicateShift (t.GE n) a = t.GE n'
-- CategoryTheory.Triangulated.TStructure.LE_monotone
lemma LE_monotone : Monotone t.LE 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (t : CategoryTheory.Triangulated.TStructure C)
Goal: Monotone t.LE
-- CategoryTheory.Triangulated.TStructure.GE_antitone
lemma GE_antitone : Antitone t.GE 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (t : CategoryTheory.Triangulated.TStructure C)
Goal: Antitone t.GE
-- CategoryTheory.Triangulated.TStructure.IsLE.le
class IsLE (X : C) (n : ℤ) : Prop where
  le : t.LE n X
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] {t : CategoryTheory.Triangulated.TStructure C} {X : C} {n : ℤ} [self : t.IsLE X n]
Goal: t.LE n X