mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
P P' Q Q' X Y : C
b : BinaryBicone X Y
x✝ : Discrete WalkingPair
as : WalkingPair
⊢ b.toBicone.toCocone.ι.app { as := as } ≫ (Iso.refl b.toBicone.toCocone.pt).hom = b.toCocone.ι.app { as := as }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type u_1
inst✝¹ : Finite J
f : J → C
inst✝ : HasProduct f
⊢ HasBiproduct f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type u_1
inst✝¹ : Finite J
f : J → C
inst✝ : HasCoproduct f
⊢ HasBiproduct f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
⊢ b.ι WalkingPair.left ≫ b.π WalkingPair.left = 𝟙 X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Fintype J
f : J → C
inst✝ : HasBiproduct f
T : C
g : (j : J) → T ⟶ f j
⊢ lift g = ∑ j, g j ≫ ι f j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
⊢ b.ι WalkingPair.left ≫ b.π WalkingPair.right = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
⊢ b.ι WalkingPair.right ≫ b.π WalkingPair.left = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Fintype J
f : J → C
inst✝ : HasBiproduct f
T : C
g : (j : J) → f j ⟶ T
⊢ desc g = ∑ j, π f j ≫ g j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
⊢ b.ι WalkingPair.right ≫ b.π WalkingPair.right = 𝟙 Y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Fintype J
f : J → C
inst✝ : HasBiproduct f
T U : C
g : (j : J) → T ⟶ f j
h : (j : J) → f j ⟶ U
⊢ lift g ≫ desc h = ∑ j, g j ≫ h j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Fintype J
inst✝ : HasFiniteBiproducts C
f g : J → C
h : (j : J) → f j ⟶ g j
⊢ map h = ∑ j, π f j ≫ h j ≫ ι g j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
j : Discrete WalkingPair
⊢ b.toBinaryBicone.toCone.π.app j = (Iso.refl b.toBinaryBicone.toCone.pt).hom ≫ b.toCone.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
J : Type
inst✝² : Fintype J
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : J → C
g : K → C
P : C
x : (j : J) → P ⟶ f j
m : (j : J) → (k : K) → f j ⟶ g k
⊢ lift x ≫ matrix m = lift fun k ↦ ∑ j, x j ≫ m j k

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
J K : Type
inst✝² : Finite J
inst✝¹ : HasFiniteBiproducts C
inst✝ : Fintype K
f : J → C
g : K → C
m : (j : J) → (k : K) → f j ⟶ g k
P : C
x : (k : K) → g k ⟶ P
⊢ matrix m ≫ desc x = desc fun j ↦ ∑ k, m j k ≫ x k

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
j : Discrete WalkingPair
⊢ b.toBinaryBicone.toCocone.ι.app j ≫ (Iso.refl b.toBinaryBicone.toCocone.pt).hom = b.toCocone.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
J K : Type
inst✝² : Finite J
inst✝¹ : HasFiniteBiproducts C
inst✝ : Finite K
f : J → C
g h : K → C
m : (j : J) → (k : K) → f j ⟶ g k
n : (k : K) → g k ⟶ h k
⊢ matrix m ≫ map n = matrix fun j k ↦ m j k ≫ n k

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
J K : Type
inst✝² : Finite J
inst✝¹ : HasFiniteBiproducts C
inst✝ : Finite K
f g : J → C
h : K → C
m : (k : J) → f k ⟶ g k
n : (j : J) → (k : K) → g j ⟶ h k
⊢ map m ≫ matrix n = matrix fun j k ↦ m j ≫ n j k

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
P Q P' Q' : C
b : BinaryBicone P Q
hb : b.IsBilimit
eP : P ≅ P'
eQ : Q ≅ Q'
⊢ BinaryFan.fst ((Cone.postcompose (mapPairIso eP eQ).hom).obj b.toCone) =
    (Iso.refl ((Cone.postcompose (mapPairIso eP eQ).hom).obj b.toCone).pt).hom ≫ BinaryFan.fst (b.ofIso eP eQ).toCone

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
P Q P' Q' : C
b : BinaryBicone P Q
hb : b.IsBilimit
eP : P ≅ P'
eQ : Q ≅ Q'
⊢ BinaryFan.snd ((Cone.postcompose (mapPairIso eP eQ).hom).obj b.toCone) =
    (Iso.refl ((Cone.postcompose (mapPairIso eP eQ).hom).obj b.toCone).pt).hom ≫ BinaryFan.snd (b.ofIso eP eQ).toCone

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
P Q P' Q' : C
b : BinaryBicone P Q
hb : b.IsBilimit
eP : P ≅ P'
eQ : Q ≅ Q'
⊢ IsLimit (b.ofIso eP eQ).toCone

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
β γ : Type
inst✝² : Finite β
ε : β ≃ γ
f : γ → C
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct (f ∘ ⇑ε)
b b' : β
h : ¬b' = b
⊢ ε b' ≠ ε b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
P Q P' Q' : C
b : BinaryBicone P Q
hb : b.IsBilimit
eP : P ≅ P'
eQ : Q ≅ Q'
⊢ BinaryCofan.inl ((Cocone.precompose (mapPairIso eP eQ).inv).obj b.toCocone) ≫
      (Iso.refl ((Cocone.precompose (mapPairIso eP eQ).inv).obj b.toCocone).pt).hom =
    BinaryCofan.inl (b.ofIso eP eQ).toCocone

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
β γ : Type
inst✝² : Finite β
ε : β ≃ γ
f : γ → C
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct (f ∘ ⇑ε)
⊢ ((desc fun b ↦ ι f (ε b)) ≫ lift fun b ↦ π f (ε b)) = 𝟙 (⨁ f ∘ ⇑ε)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J✝ : Type w
C✝ : Type uC
inst✝⁷ : Category.{uC', uC} C✝
inst✝⁶ : HasZeroMorphisms C✝
D : Type uD
inst✝⁵ : Category.{uD', uD} D
inst✝⁴ : HasZeroMorphisms D
F : J✝ → C✝
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
i : J
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct (Subtype.restrict (fun j ↦ j ≠ i) f)
s : Fork (π f i) 0
⊢ (s.ι ≫ toSubtype f fun j ↦ j ≠ i) ≫ Fork.ι (KernelFork.ofι (fromSubtype f fun j ↦ j ≠ i) ⋯) = s.ι

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
P Q P' Q' : C
b : BinaryBicone P Q
hb : b.IsBilimit
eP : P ≅ P'
eQ : Q ≅ Q'
⊢ BinaryCofan.inr ((Cocone.precompose (mapPairIso eP eQ).inv).obj b.toCocone) ≫
      (Iso.refl ((Cocone.precompose (mapPairIso eP eQ).inv).obj b.toCocone).pt).hom =
    BinaryCofan.inr (b.ofIso eP eQ).toCocone

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
P Q P' Q' : C
b : BinaryBicone P Q
hb : b.IsBilimit
eP : P ≅ P'
eQ : Q ≅ Q'
⊢ IsColimit (b.ofIso eP eQ).toCocone

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
β γ : Type
inst✝² : Finite β
ε : β ≃ γ
f : γ → C
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct (f ∘ ⇑ε)
⊢ ((lift fun b ↦ π f (ε b)) ≫ desc fun b ↦ ι f (ε b)) = 𝟙 (⨁ f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : BinaryBicone X Y
x✝ : b.toBicone.IsBilimit
h : IsLimit b.toBicone.toCone
h' : IsColimit b.toBicone.toCocone
⊢ (fun h ↦ { isLimit := b.toBiconeIsLimit.symm h.isLimit, isColimit := b.toBiconeIsColimit.symm h.isColimit })
      ((fun h ↦ { isLimit := b.toBiconeIsLimit h.isLimit, isColimit := b.toBiconeIsColimit h.isColimit })
        { isLimit := h, isColimit := h' }) =
    { isLimit := h, isColimit := h' }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : BinaryBicone X Y
x✝ : b.IsBilimit
h : IsLimit b.toCone
h' : IsColimit b.toCocone
⊢ (fun h ↦ { isLimit := b.toBiconeIsLimit h.isLimit, isColimit := b.toBiconeIsColimit h.isColimit })
      ((fun h ↦ { isLimit := b.toBiconeIsLimit.symm h.isLimit, isColimit := b.toBiconeIsColimit.symm h.isColimit })
        { isLimit := h, isColimit := h' }) =
    { isLimit := h, isColimit := h' }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
b : BinaryBicone X Y
total : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.pt
s : Cone (pair X Y)
j : Discrete WalkingPair
⊢ (BinaryFan.fst s ≫ b.inl + BinaryFan.snd s ≫ b.inr) ≫ b.toCone.π.app j = s.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
x✝ : b.toBinaryBicone.IsBilimit
h : IsLimit b.toBinaryBicone.toCone
h' : IsColimit b.toBinaryBicone.toCocone
⊢ (fun h ↦
        { isLimit := b.toBinaryBiconeIsLimit.symm h.isLimit, isColimit := b.toBinaryBiconeIsColimit.symm h.isColimit })
      ((fun h ↦ { isLimit := b.toBinaryBiconeIsLimit h.isLimit, isColimit := b.toBinaryBiconeIsColimit h.isColimit })
        { isLimit := h, isColimit := h' }) =
    { isLimit := h, isColimit := h' }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
x✝ : b.IsBilimit
h : IsLimit b.toCone
h' : IsColimit b.toCocone
⊢ (fun h ↦ { isLimit := b.toBinaryBiconeIsLimit h.isLimit, isColimit := b.toBinaryBiconeIsColimit h.isColimit })
      ((fun h ↦
          { isLimit := b.toBinaryBiconeIsLimit.symm h.isLimit,
            isColimit := b.toBinaryBiconeIsColimit.symm h.isColimit })
        { isLimit := h, isColimit := h' }) =
    { isLimit := h, isColimit := h' }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J✝ : Type w
C✝ : Type uC
inst✝⁷ : Category.{uC', uC} C✝
inst✝⁶ : HasZeroMorphisms C✝
D : Type uD
inst✝⁵ : Category.{uD', uD} D
inst✝⁴ : HasZeroMorphisms D
F : J✝ → C✝
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
i : J
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct (Subtype.restrict (fun j ↦ j ≠ i) f)
s : Fork (π f i) 0
⊢ ∀ {m : s.pt ⟶ (KernelFork.ofι (fromSubtype f fun j ↦ j ≠ i) ⋯).pt},
    m ≫ Fork.ι (KernelFork.ofι (fromSubtype f fun j ↦ j ≠ i) ⋯) = s.ι → m = s.ι ≫ toSubtype f fun j ↦ j ≠ i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J✝ : Type w
C✝ : Type uC
inst✝⁷ : Category.{uC', uC} C✝
inst✝⁶ : HasZeroMorphisms C✝
D : Type uD
inst✝⁵ : Category.{uD', uD} D
inst✝⁴ : HasZeroMorphisms D
F : J✝ → C✝
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
i : J
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct (Subtype.restrict (fun j ↦ j ≠ i) f)
⊢ (ι f i ≫ toSubtype f fun j ↦ j ≠ i) = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
b : BinaryBicone X Y
total : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.pt
s : Cone (pair X Y)
m : s.pt ⟶ b.toCone.pt
h : ∀ (j : Discrete WalkingPair), m ≫ b.toCone.π.app j = s.π.app j
⊢ m = BinaryFan.fst s ≫ b.inl + BinaryFan.snd s ≫ b.inr

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryBiproduct X Y
⊢ (isoProd X Y).hom = prod.lift fst snd

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryBiproduct X Y
⊢ (isoProd X Y).inv = lift prod.fst prod.snd

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
b : BinaryBicone X Y
total : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.pt
s : Cocone (pair X Y)
j : Discrete WalkingPair
⊢ b.toCocone.ι.app j ≫ (b.fst ≫ BinaryCofan.inl s + b.snd ≫ BinaryCofan.inr s) = s.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryBiproduct X Y
⊢ (isoCoprod X Y).inv = coprod.desc inl inr

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryBiproduct X Y
⊢ (biprod.isoCoprod X Y).hom = biprod.desc coprod.inl coprod.inr

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
b : BinaryBicone X Y
total : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.pt
s : Cocone (pair X Y)
m : b.toCocone.pt ⟶ s.pt
h : ∀ (j : Discrete WalkingPair), b.toCocone.ι.app j ≫ m = s.ι.app j
⊢ m = b.fst ≫ BinaryCofan.inl s + b.snd ≫ BinaryCofan.inr s

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J✝ : Type w
C✝ : Type uC
inst✝⁷ : Category.{uC', uC} C✝
inst✝⁶ : HasZeroMorphisms C✝
D : Type uD
inst✝⁵ : Category.{uD', uD} D
inst✝⁴ : HasZeroMorphisms D
F : J✝ → C✝
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
i : J
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct (Subtype.restrict (fun j ↦ j ≠ i) f)
s : Cofork (ι f i) 0
⊢ Cofork.π (CokernelCofork.ofπ (toSubtype f fun j ↦ j ≠ i) ⋯) ≫ (fromSubtype f fun j ↦ j ≠ i) ≫ s.π = s.π

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
b : BinaryBicone X Y
i : b.IsBilimit
j : Discrete WalkingPair
⊢ (b.fst ≫ b.inl + b.snd ≫ b.inr) ≫ b.toCone.π.app j = 𝟙 b.pt ≫ b.toCone.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
t : BinaryBicone X Y
ht : IsLimit t.toCone
⊢ t.inl = BinaryFan.IsLimit.lift ht (𝟙 X) 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
W X Y Z : C
inst✝¹ : HasBinaryBiproduct W X
inst✝ : HasBinaryBiproduct Y Z
f : W ⟶ Y
g : X ⟶ Z
⊢ map f g = map' f g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
W X Y Z : C
inst✝¹ : HasBinaryBiproduct W X
inst✝ : HasBinaryBiproduct Y Z
f : W ⟶ Y
g : X ⟶ Z
⊢ inl ≫ map f g = f ≫ inl

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
t : BinaryBicone X Y
ht : IsLimit t.toCone
⊢ t.inr = BinaryFan.IsLimit.lift ht 0 (𝟙 Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
W X Y Z : C
inst✝¹ : HasBinaryBiproduct W X
inst✝ : HasBinaryBiproduct Y Z
f : W ⟶ Y
g : X ⟶ Z
⊢ inr ≫ map f g = g ≫ inr

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
t : BinaryBicone X Y
ht : IsLimit t.toCone
⊢ t.fst ≫ t.inl + t.snd ≫ t.inr = 𝟙 t.pt

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
t : Cone (pair X Y)
ht : IsLimit t
⊢ ((BinaryBicone.ofLimitCone ht).fst ≫ (BinaryBicone.ofLimitCone ht).inl +
        (BinaryBicone.ofLimitCone ht).snd ≫ (BinaryBicone.ofLimitCone ht).inr) ≫
      BinaryFan.fst t =
    𝟙 (BinaryBicone.ofLimitCone ht).pt ≫ BinaryFan.fst t

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
t : Cone (pair X Y)
ht : IsLimit t
⊢ ((BinaryBicone.ofLimitCone ht).fst ≫ (BinaryBicone.ofLimitCone ht).inl +
        (BinaryBicone.ofLimitCone ht).snd ≫ (BinaryBicone.ofLimitCone ht).inr) ≫
      BinaryFan.snd t =
    𝟙 (BinaryBicone.ofLimitCone ht).pt ≫ BinaryFan.snd t

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J✝ : Type w
C✝ : Type uC
inst✝⁷ : Category.{uC', uC} C✝
inst✝⁶ : HasZeroMorphisms C✝
D : Type uD
inst✝⁵ : Category.{uD', uD} D
inst✝⁴ : HasZeroMorphisms D
F : J✝ → C✝
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
i : J
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct (Subtype.restrict (fun j ↦ j ≠ i) f)
s : Cofork (ι f i) 0
⊢ ∀ {m : (CokernelCofork.ofπ (toSubtype f fun j ↦ j ≠ i) ⋯).pt ⟶ s.pt},
    Cofork.π (CokernelCofork.ofπ (toSubtype f fun j ↦ j ≠ i) ⋯) ≫ m = s.π → m = (fromSubtype f fun j ↦ j ≠ i) ≫ s.π

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
t : BinaryBicone X Y
ht : IsColimit t.toCocone
⊢ t.fst = BinaryCofan.IsColimit.desc ht (𝟙 X) 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
t : BinaryBicone X Y
ht : IsColimit t.toCocone
⊢ t.snd = BinaryCofan.IsColimit.desc ht 0 (𝟙 Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryBiproduct X Y
b : BinaryBicone X Y
hb : b.IsBilimit
⊢ (hb.isLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit X Y)).inv = desc b.inl b.inr

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J✝ : Type w
C✝ : Type uC
inst✝⁷ : Category.{uC', uC} C✝
inst✝⁶ : HasZeroMorphisms C✝
D : Type uD
inst✝⁵ : Category.{uD', uD} D
inst✝⁴ : HasZeroMorphisms D
F : J✝ → C✝
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : K → Prop
⊢ biproduct.fromSubtype f pᶜ ≫ biproduct.toSubtype f p = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
t : BinaryBicone X Y
ht : IsColimit t.toCocone
⊢ t.fst ≫ t.inl + t.snd ≫ t.inr = 𝟙 t.pt

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q X Y : C
inst✝ : HasBinaryBiproduct X Y
b : BinaryBicone X Y
hb : b.IsBilimit
⊢ lift b.fst b.snd ≫ desc b.inl b.inr = 𝟙 b.pt

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : C
t : Cocone (pair X Y)
ht : IsColimit t
j : Discrete WalkingPair
⊢ t.ι.app j ≫ (Iso.refl t.pt).hom = (BinaryBicone.ofColimitCocone ht).toCocone.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q X Y : C
inst✝ : HasBinaryBiproduct X Y
b : BinaryBicone X Y
hb : b.IsBilimit
⊢ desc b.inl b.inr ≫ lift b.fst b.snd = 𝟙 (X ⊞ Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
X Y : C
inst✝ : HasBinaryBiproduct X Y
⊢ fst ≫ inl + snd ≫ inr = 𝟙 (X ⊞ Y)

Tactic:
by

Goal state after: