mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : HasFiniteProducts A
inst✝ : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) (ι.map f₂) f₃ ∈ distinguishedTriangles
hT' : Triangle.mk 0 (𝟙 (ι.obj X₃)) 0 ∈ distinguishedTriangles
⊢ ι.map
      (Cofork.π (CokernelCofork.ofπ (πQ (ι.map f₂) (𝟙 (ι.obj X₃))) ⋯) ≫
        (Iso.refl (CokernelCofork.ofπ (πQ (ι.map f₂) (𝟙 (ι.obj X₃))) ⋯).pt).hom) =
    ι.map (Cofork.π (CokernelCofork.ofπ f₂ ⋯))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : HasFiniteProducts A
inst✝ : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) (ι.map f₂) f₃ ∈ distinguishedTriangles
⊢ IsColimit (CokernelCofork.ofπ f₂ ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst✝ : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ (shiftFunctor C 1).obj (ι.obj X₂)
hT : Triangle.mk (ι.map π) f₂ f₃ ∈ distinguishedTriangles
⊢ admissibleMorphism ι π

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst✝ : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ (shiftFunctor C 1).obj (ι.obj X₂)
hT : Triangle.mk (ι.map π) f₂ f₃ ∈ distinguishedTriangles
this : admissibleMorphism ι π
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₁
β : X₁ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
⊢ Cofork.π (CokernelCofork.ofπ (πQ f₂ β) ⋯) ≫ 𝟙 Q = Cofork.π (CokernelCofork.ofπ (πQ f₂ β) ⋯) ≫ 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a b : ℤ
f : WithBotTop.coe a ⟶ WithBotTop.coe b
⊢ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst✝ : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ (shiftFunctor C 1).obj (ι.obj X₂)
hT : Triangle.mk (ι.map π) f₂ f₃ ∈ distinguishedTriangles
this : admissibleMorphism ι π
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₁
β : X₁ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
hQ : 𝟙 Q = 0
⊢ ι.map (𝟙 Q) = 0

Tactic:
hQ,

Goal state after:
C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst✝ : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ (shiftFunctor C 1).obj (ι.obj X₂)
hT : Triangle.mk (ι.map π) f₂ f₃ ∈ distinguishedTriangles
this : admissibleMorphism ι π
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₁
β : X₁ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
hQ : 𝟙 Q = 0
⊢ ι.map 0 = 0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst✝ : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ (shiftFunctor C 1).obj (ι.obj X₂)
hT : Triangle.mk (ι.map π) f₂ f₃ ∈ distinguishedTriangles
this : admissibleMorphism ι π
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₁
β : X₁ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
hQ : 𝟙 Q = 0
⊢ IsZero (Triangle.mk α β γ).obj₃

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
x y : EInt
f : x ⟶ y
⊢ WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst✝ : Epi π
⊢ ∃ X₁ i δ, Triangle.mk (ι.map i) (ι.map π) δ ∈ distinguishedTriangles

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : EInt
⊢ WithBotTop.rec (motive := fun {x} ↦
      (x ⟶ n) → (WithBotTop.rec 0 t.truncLT (𝟭 C) x ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) n))
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊥ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊥)) (fun b f ↦ 0) (fun f ↦ 0) n f)
      (fun a f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (WithBotTop.coe a ⟶ y) →
            (WithBotTop.rec 0 t.truncLT (𝟭 C) (WithBotTop.coe a) ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y))
          (fun f ↦ 0) (fun b f ↦ t.natTransTruncLTOfLE a b ⋯) (fun f ↦ t.truncLTι a) n f)
      (fun f ↦
        WithBotTop.rec (motive := fun {y} ↦
          (⊤ ⟶ y) → (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤ ⟶ WithBotTop.rec 0 t.truncLT (𝟭 C) y)) (fun f ↦ 0) (fun b f ↦ 0)
          (fun f ↦ 𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) ⊤)) n f)
      n (𝟙 n) =
    𝟙 (WithBotTop.rec 0 t.truncLT (𝟭 C) n)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : Preadditive C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁵ : Preadditive A
inst✝⁴ : ι.Full
inst✝³ : ι.Faithful
inst✝² : HasFiniteProducts A
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
inst✝ : IsTriangulated C
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
⊢ admissibleMorphism ι f₁

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CommMonoid β
s A : Set α
t : Set β
f : α → β
hf : IsMulFreimanIso 2 s t f
hAs : A ⊆ s
⊢ ThreeGPFree (f '' A) ↔ ThreeGPFree A

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CommMonoid β
s : Set α
t : Set β
f : α → β
hf : IsMulFreimanIso 2 s t f
⊢ ThreeGPFree s ↔ ThreeGPFree t

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : CommMonoid α
inst✝² : CommMonoid β
s : Set α
inst✝¹ : FunLike F α β
inst✝ : MulHomClass F α β
f : F
hf : InjOn (⇑f) (s * s)
h : ThreeGPFree s
a : α
ha : a ∈ s
b : α
hb : b ∈ s
c : α
hc : c ∈ s
habc : f a * f c = f b * f b
⊢ f (a * c) = f (b * b)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : CommMonoid α
inst✝² : CommMonoid β
s : Set α
inst✝¹ : FunLike F α β
inst✝ : MulHomClass F α β
f : F
hf : InjOn (⇑f) (s * s)
h : ThreeGPFree s
⊢ ThreeGPFree (⇑f '' s)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoid α
inst✝ : IsCancelMul α
s : Set α
hs : ThreeGPFree s
⊢ ∀ ⦃a : α⦄, a ∈ s → ∀ ⦃b : α⦄, b ∈ s → ∀ ⦃c : α⦄, c ∈ s → a * c = b * b → b = c

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : Preadditive C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁵ : Preadditive A
inst✝⁴ : ι.Full
inst✝³ : ι.Faithful
inst✝² : HasFiniteProducts A
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
inst✝ : IsTriangulated C
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
this✝ : admissibleMorphism ι f₁
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
this : Epi (πQ f₂ β)
I : A
i : I ⟶ X₂
δ : ι.obj Q ⟶ (shiftFunctor C 1).obj (ι.obj I)
hI : Triangle.mk (ι.map i) (ι.map (πQ f₂ β)) δ ∈ distinguishedTriangles
H : Octahedron ⋯ ⋯ ⋯ ⋯
m₁ : X₁ ⟶ I
hm₁ : (shiftFunctor C 1).map (ι.map m₁) = H.m₁
m₃ : ι.obj I ⟶ (shiftFunctor C 1).obj (ι.obj K)
hm₃ : (shiftFunctor C 1).map m₃ = H.m₃
⊢ Triangle.mk (ι.map (ιK f₃ α)) (ι.map m₁) (-m₃) ∈ distinguishedTriangles

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : Preadditive C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁵ : Preadditive A
inst✝⁴ : ι.Full
inst✝³ : ι.Faithful
inst✝² : HasFiniteProducts A
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
inst✝ : IsTriangulated C
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
this✝ : admissibleMorphism ι f₁
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
this : Epi (πQ f₂ β)
I : A
i : I ⟶ X₂
δ : ι.obj Q ⟶ (shiftFunctor C 1).obj (ι.obj I)
hI : Triangle.mk (ι.map i) (ι.map (πQ f₂ β)) δ ∈ distinguishedTriangles
H : Octahedron ⋯ ⋯ ⋯ ⋯
m₁ : X₁ ⟶ I
hm₁ : (shiftFunctor C 1).map (ι.map m₁) = H.m₁
m₃ : ι.obj I ⟶ (shiftFunctor C 1).obj (ι.obj K)
hm₃ : (shiftFunctor C 1).map m₃ = H.m₃
Hmem : Triangle.mk (ι.map (ιK f₃ α)) (ι.map m₁) (-m₃) ∈ distinguishedTriangles
⊢ (ι ⋙ shiftFunctor C 1).map (m₁ ≫ i) = (ι ⋙ shiftFunctor C 1).map f₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹² : Category.{v_1, u_1} C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : Preadditive C
inst✝⁹ : HasShift C ℤ
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁵ : Preadditive A
inst✝⁴ : ι.Full
inst✝³ : ι.Faithful
inst✝² : HasFiniteProducts A
inst✝¹ : ι.Additive
hA : admissibleMorphism ι = ⊤
inst✝ : IsTriangulated C
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
⊢ Nonempty (Abelian.AbelianStruct f₁)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CommMonoid β
A : Set α
B : Set β
f₁ f₂ : α → β
n : ℕ
hf₁ : IsMulFreimanHom n A B f₁
h : EqOn f₁ f₂ A
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h' : s.prod = t.prod
⊢ (map f₂ s).prod = (map f₂ t).prod

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: R : Type u_1
α : Type u_2
β : Type u_3
r : α → β → Prop
s : Finset α
t : Finset β
inst✝¹ : CommMonoid R
f : α → β → R
inst✝ : (a : α) → (b : β) → Decidable (r a b)
⊢ ∏ a ∈ s, ∏ b ∈ bipartiteAbove r t a, f a b = ∏ b ∈ t, ∏ a ∈ bipartiteBelow r s b, f a b

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
s : Finset α
t : Finset β
inst✝ : (a : α) → (b : β) → Decidable (r a b)
⊢ ∑ a ∈ s, #(bipartiteAbove r t a) = ∑ b ∈ t, #(bipartiteBelow r s b)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: R : Type u_1
α : Type u_2
β : Type u_3
r : α → β → Prop
s : Finset α
t : Finset β
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : IsOrderedRing R
m n : R
inst✝ : (a : α) → (b : β) → Decidable (r a b)
hm : ∀ a ∈ s, m ≤ ↑(#(bipartiteAbove r t a))
hn : ∀ b ∈ t, ↑(#(bipartiteBelow r s b)) ≤ n
⊢ ∑ a ∈ s, ↑(#(bipartiteAbove r t a)) = ∑ b ∈ t, ↑(#(bipartiteBelow r s b))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: R : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
r : α → β → Prop
s : Finset α
t : Finset β
m n : R
inst✝ : (a : α) → (b : β) → Decidable (r a b)
hs : s.Nonempty
hm : ∀ a ∈ s, m < ↑(#(bipartiteAbove r t a))
hn : ∀ b ∈ t, ↑(#(bipartiteBelow r s b)) ≤ n
⊢ #s • m = ∑ _a ∈ s, m

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CommMonoid β
A : Set α
B : Set β
f₁ f₂ : α → β
n : ℕ
hf₁ : IsMulFreimanIso n A B f₁
h : EqOn f₁ f₂ A
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ (map f₂ s).prod = (map f₂ t).prod ↔ s.prod = t.prod

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: R : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
r : α → β → Prop
s : Finset α
t : Finset β
m n : R
inst✝ : (a : α) → (b : β) → Decidable (r a b)
hs : s.Nonempty
hm : ∀ a ∈ s, m < ↑(#(bipartiteAbove r t a))
hn : ∀ b ∈ t, ↑(#(bipartiteBelow r s b)) ≤ n
⊢ ∑ a ∈ s, ↑(#(bipartiteAbove r t a)) = ∑ b ∈ t, ↑(#(bipartiteBelow r s b))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CommMonoid β
A : Set α
B : Set β
f : α → β
n : ℕ
g : β → α
hg₁ : MapsTo g B A
hg₂ : RightInvOn g f B
hf : IsMulFreimanIso n A B f
s t : Multiset β
hsB : ∀ ⦃x : β⦄, x ∈ s → x ∈ B
htB : ∀ ⦃x : β⦄, x ∈ t → x ∈ B
hs : s.card = n
ht : t.card = n
⊢ (map g s).card = n

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoid α
inst✝ : IsCancelMul α
s : Set α
a : α
⊢ ThreeGPFree (insert a s) ↔
    ThreeGPFree s ∧
      (∀ ⦃b : α⦄, b ∈ s → ∀ ⦃c : α⦄, c ∈ s → a * c = b * b → a = b) ∧
        ∀ ⦃b : α⦄, b ∈ s → ∀ ⦃c : α⦄, c ∈ s → b * c = a * a → b = a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CommMonoid β
A : Set α
B : Set β
f : α → β
n : ℕ
g : β → α
hg₁ : MapsTo g B A
hg₂ : RightInvOn g f B
hf : IsMulFreimanIso n A B f
s t : Multiset β
hsB : ∀ ⦃x : β⦄, x ∈ s → x ∈ B
htB : ∀ ⦃x : β⦄, x ∈ t → x ∈ B
hs : s.card = n
ht : t.card = n
⊢ (map g t).card = n

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoid α
inst✝ : IsCancelMul α
s : Set α
a : α
hs : ThreeGPFree s
b : α
hb : b ∈ s
c : α
hc : c ∈ s
d : α
hd : d ∈ s
h : (fun x ↦ a • x) b * (fun x ↦ a • x) d = (fun x ↦ a • x) c * (fun x ↦ a • x) c
⊢ b * d = c * c

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoid α
inst✝ : IsCancelMul α
s : Set α
a : α
hs : ThreeGPFree s
⊢ ThreeGPFree (a • s)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: R : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
r : α → β → Prop
s : Finset α
t : Finset β
m n : R
inst✝ : (a : α) → (b : β) → Decidable (r a b)
ht : t.Nonempty
hm : ∀ a ∈ s, m ≤ ↑(#(bipartiteAbove r t a))
hn : ∀ b ∈ t, ↑(#(bipartiteBelow r s b)) < n
⊢ ∑ a ∈ s, ↑(#(bipartiteAbove r t a)) = ∑ b ∈ t, ↑(#(bipartiteBelow r s b))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoid α
inst✝ : IsCancelMul α
s : Set α
a : α
hs : ThreeGPFree (a • s)
b : α
hb : b ∈ s
c : α
hc : c ∈ s
d : α
hd : d ∈ s
h : b * d = c * c
⊢ a • b * a • d = a * c * (a * c)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝² : CommMonoid α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedCancelMonoid α
s : Set α
a : α
hs : ∀ i ∈ s, i < a
⊢ ThreeGPFree (insert a s) ↔ ThreeGPFree s ∧ ∀ ⦃b : α⦄, b ∈ s → ∀ ⦃c : α⦄, c ∈ s → a * c = b * b → a = b

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : IsCancelMulZero α
inst✝ : NoZeroDivisors α
s : Set α
a : α
hs : ThreeGPFree s
ha : a ≠ 0
b : α
hb : b ∈ s
c : α
hc : c ∈ s
d : α
hd : d ∈ s
h : (fun x ↦ a • x) b * (fun x ↦ a • x) d = (fun x ↦ a • x) c * (fun x ↦ a • x) c
⊢ b * d = c * c

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
s : Finset α
t : Finset β
hs : ∀ a ∈ s, ∃ b ∈ t, r a b
ht : ∀ b ∈ t, {a | a ∈ s ∧ r a b}.Subsingleton
a : α
h : a ∈ s
⊢ (bipartiteAbove r t a).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : IsCancelMulZero α
inst✝ : NoZeroDivisors α
s : Set α
a : α
hs : ThreeGPFree s
ha : a ≠ 0
⊢ ThreeGPFree (a • s)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
s : Finset α
t : Finset β
hs : ∀ a ∈ s, ∃ b ∈ t, r a b
ht : ∀ b ∈ t, {a | a ∈ s ∧ r a b}.Subsingleton
b : β
h : b ∈ t
⊢ ∀ a ∈ bipartiteBelow r s b, ∀ b_1 ∈ bipartiteBelow r s b, a = b_1

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : IsCancelMulZero α
inst✝ : NoZeroDivisors α
s : Set α
a : α
ha : a ≠ 0
hs : ThreeGPFree (a • s)
b : α
hb : b ∈ s
c : α
hc : c ∈ s
d : α
hd : d ∈ s
h : b * d = c * c
⊢ a • b * a • d = a * c * (a * c)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
s : Finset α
t : Finset β
hs : ∀ a ∈ s, ∃ b ∈ t, r a b
ht : ∀ b ∈ t, {a | a ∈ s ∧ r a b}.Subsingleton
⊢ #s ≤ #t

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: s : Set ℕ
⊢ ThreeAPFree s ↔ ∀ ⦃a : ℕ⦄, a ∈ s → ∀ ⦃b : ℕ⦄, b ∈ s → ∀ ⦃c : ℕ⦄, c ∈ s → a + c = b + b → a = c

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
B : α → Finset β
s : Finset α
⊢ ∑ j ∈ s, #(B j) = ∑ x ∈ s.biUnion B, #{j | j ∈ s ∧ x ∈ B j}

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
r : α → β → Prop
h₁ : LeftTotal r
h₂ : LeftUnique r
⊢ ∀ a ∈ univ, ∃ b ∈ univ, r a b

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
r : α → β → Prop
h₁ : RightTotal r
h₂ : RightUnique r
⊢ ∀ b ∈ univ, ∃ a ∈ univ, r a b

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : DecidableEq α
inst✝² : Monoid α
inst✝¹ : DecidableEq β
inst✝ : Monoid β
s t : Finset α
⊢ Monotone fun s ↦ Nat.findGreatest (fun m ↦ ∃ t ⊆ s, #t = m ∧ ThreeGPFree ↑t) #s

Tactic:
by

Goal state after: