mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: case refine_2.refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 isComplex₂_iff._proof_2 ⋯)
      ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact

pp: case refine_2.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 0 1 isComplex₂_iff._proof_2 ⋯)
      ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact

Tactic:
·

Goal state after:
case refine_2.refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 isComplex₂_iff._proof_2 ⋯)
      ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact
Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X Y Z : C
n a a' : ℤ
h : n + a = a'
⊢ a + n = a'

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X Y Z : C
n a : ℤ
f : Opposite.op ((shiftFunctor C a).obj Y) ⟶ (shiftFunctor Cᵒᵖ n).obj (Opposite.op X)
a' : ℤ
h : n + a = a'
⊢ a + n = a'

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X Y Z : C
a' : ℤ
f : ShiftedHom X Y a'
n a : ℤ
h : n + a = a'
⊢ a + n = a'

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X Y Z : C
n m : ℤ
f : ShiftedHom X Y n
g : ShiftedHom Y Z m
q : ℤ
hq : n + m = q
⊢ m + n = q

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: case refine_2.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 0 1 isComplex₂_iff._proof_2 ⋯)
      ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact

Tactic:
·

Goal state after:
case refine_2.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 0 1 isComplex₂_iff._proof_2 ⋯)
      ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact
Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X Y Z : C
n m a a' a'' : ℤ
ha' : n + a = a'
ha'' : m + a' = a''
x : Opposite.op ((shiftFunctor C a).obj Y) ⟶ (shiftFunctor Cᵒᵖ (m + n)).obj (Opposite.op X)
⊢ m + n + a = a''

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X Y : C
n m a a' a'' : ℤ
ha' : n + a = a'
ha'' : m + a' = a''
x : Opposite.op ((shiftFunctor C a).obj Y) ⟶ (shiftFunctor Cᵒᵖ (m + n)).obj (Opposite.op X)
⊢ a + n = a'

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X Y : C
n m a a' a'' : ℤ
ha' : n + a = a'
ha'' : m + a' = a''
x : Opposite.op ((shiftFunctor C a).obj Y) ⟶ (shiftFunctor Cᵒᵖ (m + n)).obj (Opposite.op X)
⊢ n + m = m + n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X Y : C
n m a a' a'' : ℤ
ha' : n + a = a'
ha'' : m + a' = a''
x : Opposite.op ((shiftFunctor C a).obj Y) ⟶ (shiftFunctor Cᵒᵖ (m + n)).obj (Opposite.op X)
⊢ a + n + m = a''

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated

Goal state before:
pp: case mp
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle Cᵒᵖ
⊢ Opposite.unop ((triangleOpEquivalence C).inverse.obj T) ∈ Pretriangulated.distinguishedTriangles →
    ∃ T',
      ∃ (_ : T' ∈ Pretriangulated.distinguishedTriangles),
        Nonempty (T ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T'))

pp: case mpr
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle Cᵒᵖ
⊢ (∃ T',
      ∃ (_ : T' ∈ Pretriangulated.distinguishedTriangles),
        Nonempty (T ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T'))) →
    Opposite.unop ((triangleOpEquivalence C).inverse.obj T) ∈ Pretriangulated.distinguishedTriangles

Tactic:
·

Goal state after:
case mp
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle Cᵒᵖ
⊢ Opposite.unop ((triangleOpEquivalence C).inverse.obj T) ∈ Pretriangulated.distinguishedTriangles →
    ∃ T',
      ∃ (_ : T' ∈ Pretriangulated.distinguishedTriangles),
        Nonempty (T ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T'))
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated

Goal state before:
pp: case mpr
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle Cᵒᵖ
⊢ (∃ T',
      ∃ (_ : T' ∈ Pretriangulated.distinguishedTriangles),
        Nonempty (T ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T'))) →
    Opposite.unop ((triangleOpEquivalence C).inverse.obj T) ∈ Pretriangulated.distinguishedTriangles

Tactic:
·

Goal state after:
case mpr
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle Cᵒᵖ
⊢ (∃ T',
      ∃ (_ : T' ∈ Pretriangulated.distinguishedTriangles),
        Nonempty (T ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T'))) →
    Opposite.unop ((triangleOpEquivalence C).inverse.obj T) ∈ Pretriangulated.distinguishedTriangles
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X Y : Cᵒᵖ
f : X ⟶ Y
⊢ ∃ Z g h, Triangle.mk f g h ∈ distinguishedTriangles C

Tactic:
obtain ⟨Z, g, h, H⟩ := Pretriangulated.distinguished_cocone_triangle₁ f.unop

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X Y : Cᵒᵖ
f : X ⟶ Y
Z : C
g : Z ⟶ Opposite.unop Y
h : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z
H : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles
⊢ ∃ Z g h, Triangle.mk f g h ∈ distinguishedTriangles C
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangle

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ ((inverse C ⋙ functor C).obj T).mor₁ ≫ (Iso.refl ((inverse C ⋙ functor C).obj T).obj₂).hom =
    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₁

pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ ((inverse C ⋙ functor C).obj T).mor₂ ≫ (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃).hom =
    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₂).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₂

pp: case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ ((inverse C ⋙ functor C).obj T).mor₃ ≫ (shiftFunctor Cᵒᵖ 1).map (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁).hom =
    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₃

Tactic:
·

Goal state after:
case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ ((inverse C ⋙ functor C).obj T).mor₁ ≫ (Iso.refl ((inverse C ⋙ functor C).obj T).obj₂).hom =
    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₁
Module: Mathlib.CategoryTheory.Triangulated.Yoneda

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
T : Triangle C
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : Cᵒᵖ
x : unop A ⟶ (shiftFunctor C n₀).obj T.obj₃
⊢ 1 + n₀ = n₁

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangle

Goal state before:
pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ ((inverse C ⋙ functor C).obj T).mor₂ ≫ (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃).hom =
    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₂).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₂

pp: case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ ((inverse C ⋙ functor C).obj T).mor₃ ≫ (shiftFunctor Cᵒᵖ 1).map (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁).hom =
    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₃

Tactic:
·

Goal state after:
case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ ((inverse C ⋙ functor C).obj T).mor₂ ≫ (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃).hom =
    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₂).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₂
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangle

Goal state before:
pp: case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ ((inverse C ⋙ functor C).obj T).mor₃ ≫ (shiftFunctor Cᵒᵖ 1).map (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁).hom =
    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₃

Tactic:
·

Goal state after:
case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ ((inverse C ⋙ functor C).obj T).mor₃ ≫ (shiftFunctor Cᵒᵖ 1).map (Iso.refl ((inverse C ⋙ functor C).obj T).obj₁).hom =
    (Iso.refl ((inverse C ⋙ functor C).obj T).obj₃).hom ≫ ((𝟭 (Triangle Cᵒᵖ)).obj T).mor₃
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X✝ : C
S : ShortComplex C
hS : S.ShortExact
inst✝ : HasDerivedCategory C
X : C
n₀ : ℕ
x : Ext X S.X₃ n₀
n₁ : ℕ
h : n₀ + 1 = n₁
⊢ ↑n₀ + 1 = ↑n₁

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Linear

Goal state before:
pp: case mp
C : Type u₁
inst✝¹² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹¹ : Category.{v₂, u₂} D
inst✝¹⁰ : Preadditive C
inst✝⁹ : Preadditive D
E : Type u_1
inst✝⁸ : Category.{v_1, u_1} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
inst✝⁶ : Preadditive E
R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Linear R C
inst✝³ : Linear R D
inst✝² : Linear R E
inst✝¹ : Functor.Linear R L
F : C ⥤ E
G : D ⥤ E
inst✝ : Lifting L W F G
⊢ Functor.Linear R F → Functor.Linear R G

pp: case mpr
C : Type u₁
inst✝¹² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹¹ : Category.{v₂, u₂} D
inst✝¹⁰ : Preadditive C
inst✝⁹ : Preadditive D
E : Type u_1
inst✝⁸ : Category.{v_1, u_1} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
inst✝⁶ : Preadditive E
R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Linear R C
inst✝³ : Linear R D
inst✝² : Linear R E
inst✝¹ : Functor.Linear R L
F : C ⥤ E
G : D ⥤ E
inst✝ : Lifting L W F G
⊢ Functor.Linear R G → Functor.Linear R F

Tactic:
·

Goal state after:
case mp
C : Type u₁
inst✝¹² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹¹ : Category.{v₂, u₂} D
inst✝¹⁰ : Preadditive C
inst✝⁹ : Preadditive D
E : Type u_1
inst✝⁸ : Category.{v_1, u_1} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
inst✝⁶ : Preadditive E
R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Linear R C
inst✝³ : Linear R D
inst✝² : Linear R E
inst✝¹ : Functor.Linear R L
F : C ⥤ E
G : D ⥤ E
inst✝ : Lifting L W F G
⊢ Functor.Linear R F → Functor.Linear R G
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
P Y : C
n : ℕ
inst✝ : Projective P
e : Ext P Y (n + 1)
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ ↑n + 1 + -(↑n + 1) = 0

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Linear

Goal state before:
pp: case mpr
C : Type u₁
inst✝¹² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹¹ : Category.{v₂, u₂} D
inst✝¹⁰ : Preadditive C
inst✝⁹ : Preadditive D
E : Type u_1
inst✝⁸ : Category.{v_1, u_1} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
inst✝⁶ : Preadditive E
R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Linear R C
inst✝³ : Linear R D
inst✝² : Linear R E
inst✝¹ : Functor.Linear R L
F : C ⥤ E
G : D ⥤ E
inst✝ : Lifting L W F G
⊢ Functor.Linear R G → Functor.Linear R F

Tactic:
·

Goal state after:
case mpr
C : Type u₁
inst✝¹² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹¹ : Category.{v₂, u₂} D
inst✝¹⁰ : Preadditive C
inst✝⁹ : Preadditive D
E : Type u_1
inst✝⁸ : Category.{v_1, u_1} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
inst✝⁶ : Preadditive E
R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Linear R C
inst✝³ : Linear R D
inst✝² : Linear R E
inst✝¹ : Functor.Linear R L
F : C ⥤ E
G : D ⥤ E
inst✝ : Lifting L W F G
⊢ Functor.Linear R G → Functor.Linear R F
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
P Y : C
n : ℕ
inst✝ : Projective P
e : Ext P Y (n + 1)
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ -(↑n + 1) < 0

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : EnoughProjectives C
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this : HasExt C
n : ℕ
hn : ∀ (X Y : C), Small.{w, max u v} (Ext X Y n)
X Y : C
S : ShortComplex C :=
  { X₁ := kernel (Projective.π X), X₂ := Projective.over X, X₃ := X, f := kernel.ι (Projective.π X),
    g := Projective.π X, zero := ⋯ }
hS : S.ShortExact
x₃ : Ext S.X₃ Y (n + 1)
⊢ 1 + n = n + 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ ↑n₀ + 1 = ↑n₁

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y✝ : C
inst✝ : HasDerivedCategory C
Y : C
n₀ : ℕ
x : Ext S.X₁ Y n₀
n₁ : ℕ
h : 1 + n₀ = n₁
⊢ ↑n₀ + 1 = ↑n₁

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X I : C
n : ℕ
inst✝ : Injective I
e : Ext X I (n + 1)
K : CochainComplex C ℤ := (CochainComplex.singleFunctor C 0).obj X
⊢ -↑n ≤ 0

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X I : C
n : ℕ
inst✝ : Injective I
e : Ext X I (n + 1)
K : CochainComplex C ℤ := (CochainComplex.singleFunctor C 0).obj X
this✝ : K.IsStrictlyGE (-↑n)
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ ↑n + 1 + -(↑n + 1) = 0

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X I : C
n : ℕ
inst✝ : Injective I
e : Ext X I (n + 1)
K : CochainComplex C ℤ := (CochainComplex.singleFunctor C 0).obj X
this✝ : K.IsStrictlyGE (-↑n)
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ -(↑n + 1) < -↑n

Tactic:
by lia

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Orthogonal

Goal state before:
pp: case mp
K : Type u_7
V : Type u_9
V₂ : Type u_11
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J : K →+* K
B : V →ₗ[K] V →ₛₗ[J] V₂
x y : V
⊢ (∀ (n : V), (∃ a, a • x = n) → (B n) y = 0) → (B x) y = 0

pp: case mpr
K : Type u_7
V : Type u_9
V₂ : Type u_11
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J : K →+* K
B : V →ₗ[K] V →ₛₗ[J] V₂
x y : V
⊢ (B x) y = 0 → ∀ (n : V), (∃ a, a • x = n) → (B n) y = 0

Tactic:
·

Goal state after:
case mp
K : Type u_7
V : Type u_9
V₂ : Type u_11
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J : K →+* K
B : V →ₗ[K] V →ₛₗ[J] V₂
x y : V
⊢ (∀ (n : V), (∃ a, a • x = n) → (B n) y = 0) → (B x) y = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Orthogonal

Goal state before:
pp: case mpr
K : Type u_7
V : Type u_9
V₂ : Type u_11
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J : K →+* K
B : V →ₗ[K] V →ₛₗ[J] V₂
x y : V
⊢ (B x) y = 0 → ∀ (n : V), (∃ a, a • x = n) → (B n) y = 0

Tactic:
·

Goal state after:
case mpr
K : Type u_7
V : Type u_9
V₂ : Type u_11
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J : K →+* K
B : V →ₗ[K] V →ₛₗ[J] V₂
x y : V
⊢ (B x) y = 0 → ∀ (n : V), (∃ a, a • x = n) → (B n) y = 0
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y : C
n₀ n₁ : ℕ
h : 1 + n₀ = n₁
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ ↑n₀ + 1 = ↑n₁

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: case mp
A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
⊢ Function.Injective ⇑solution.fromQuotient → solution.π.ker = Submodule.span A (Set.range relations.relation)

pp: case mpr
A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
⊢ solution.π.ker = Submodule.span A (Set.range relations.relation) → Function.Injective ⇑solution.fromQuotient

Tactic:
·

Goal state after:
case mp
A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
⊢ Function.Injective ⇑solution.fromQuotient → solution.π.ker = Submodule.span A (Set.range relations.relation)
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: case mpr
A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
⊢ solution.π.ker = Submodule.span A (Set.range relations.relation) → Function.Injective ⇑solution.fromQuotient

Tactic:
·

Goal state after:
case mpr
A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
⊢ solution.π.ker = Submodule.span A (Set.range relations.relation) → Function.Injective ⇑solution.fromQuotient
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: case refine_1
m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Fintype m
inst✝ : Finite n
this : Fintype n
h : ∀ (v : m → R), (∀ (w : n → R), v ⬝ᵥ M *ᵥ w = 0) → v = 0
v : m → R
hv : v ᵥ* M = 0
w : n → R
⊢ v ⬝ᵥ M *ᵥ w = 0

pp: case refine_2
m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Fintype m
inst✝ : Finite n
this : Fintype n
h : ∀ (v : m → R), v ᵥ* M = 0 → v = 0
w : m → R
hw : ∀ (w_1 : n → R), w ⬝ᵥ M *ᵥ w_1 = 0
i : n
⊢ (w ᵥ* M) i = 0 i

Tactic:
·

Goal state after:
case refine_1
m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Fintype m
inst✝ : Finite n
this : Fintype n
h : ∀ (v : m → R), (∀ (w : n → R), v ⬝ᵥ M *ᵥ w = 0) → v = 0
v : m → R
hv : v ᵥ* M = 0
w : n → R
⊢ v ⬝ᵥ M *ᵥ w = 0