mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Galois.Topology

Goal state before:
pp: case refine_1
C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCategory C
inst✝ : FiberFunctor F
S U : Set (Aut F)
hU : U ⊆ S
hUopen : IsOpen U
hUone : 1 ∈ U
I : Set C
w✝ : Fintype ↑I
hc : ∀ X ∈ I, IsConnected X
hmem : ∀ (σ : Aut F), (∀ (X : ↑I), σ.hom.app ↑X = 𝟙 (F.obj ↑X)) → σ ∈ U
P : C := ∏ᶜ fun X ↦ ↑X
⊢ ∃ i, True ∧ ↑(MulAction.stabilizer (Aut F) i.pt) ⊆ S

Tactic:
obtain ⟨A, a, hgal, hbij⟩ := exists_galois_representative F P

Goal state after:
case refine_1
C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCategory C
inst✝ : FiberFunctor F
S U : Set (Aut F)
hU : U ⊆ S
hUopen : IsOpen U
hUone : 1 ∈ U
I : Set C
w✝ : Fintype ↑I
hc : ∀ X ∈ I, IsConnected X
hmem : ∀ (σ : Aut F), (∀ (X : ↑I), σ.hom.app ↑X = 𝟙 (F.obj ↑X)) → σ ∈ U
P : C := ∏ᶜ fun X ↦ ↑X
A : C
a : (F.obj A).obj
hgal : IsGalois A
hbij : Function.Bijective fun f ↦ (ConcreteCategory.hom (F.map f)) a
⊢ ∃ i, True ∧ ↑(MulAction.stabilizer (Aut F) i.pt) ⊆ S
Module: Mathlib.CategoryTheory.Galois.Topology

Goal state before:
pp: case refine_1
C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCategory C
inst✝ : FiberFunctor F
S U : Set (Aut F)
hU : U ⊆ S
hUopen : IsOpen U
hUone : 1 ∈ U
I : Set C
w✝ : Fintype ↑I
hc : ∀ X ∈ I, IsConnected X
hmem : ∀ (σ : Aut F), (∀ (X : ↑I), σ.hom.app ↑X = 𝟙 (F.obj ↑X)) → σ ∈ U
P : C := ∏ᶜ fun X ↦ ↑X
A : C
a : (F.obj A).obj
hgal : IsGalois A
hbij : Function.Bijective fun f ↦ (ConcreteCategory.hom (F.map f)) a
t : Aut F
ht : (ConcreteCategory.hom (t.hom.app A)) a = a
this : ∀ (X : ↑I), IsConnected ↑X
⊢ ∀ (X : ↑I), t.hom.app ↑X = 𝟙 (F.obj ↑X)

Tactic:
have (X : I) : Nonempty (F.obj X.val) := nonempty_fiber_of_isConnected F X

Goal state after:
case refine_1
C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCategory C
inst✝ : FiberFunctor F
S U : Set (Aut F)
hU : U ⊆ S
hUopen : IsOpen U
hUone : 1 ∈ U
I : Set C
w✝ : Fintype ↑I
hc : ∀ X ∈ I, IsConnected X
hmem : ∀ (σ : Aut F), (∀ (X : ↑I), σ.hom.app ↑X = 𝟙 (F.obj ↑X)) → σ ∈ U
P : C := ∏ᶜ fun X ↦ ↑X
A : C
a : (F.obj A).obj
hgal : IsGalois A
hbij : Function.Bijective fun f ↦ (ConcreteCategory.hom (F.map f)) a
t : Aut F
ht : (ConcreteCategory.hom (t.hom.app A)) a = a
this✝ : ∀ (X : ↑I), IsConnected ↑X
this : ∀ (X : ↑I), Nonempty (F.obj ↑X).obj
⊢ ∀ (X : ↑I), t.hom.app ↑X = 𝟙 (F.obj ↑X)
Module: Mathlib.CategoryTheory.GradedObject.Braiding

Goal state before:
pp: I : Type u_1
inst✝²⁰ : AddCommMonoid I
C : Type u_2
inst✝¹⁹ : Category.{v_1, u_2} C
inst✝¹⁸ : MonoidalCategory C
X Y Z : GradedObject I C
inst✝¹⁷ : BraidedCategory C
inst✝¹⁶ : X.HasTensor Y
inst✝¹⁵ : Y.HasTensor X
inst✝¹⁴ : Y.HasTensor Z
inst✝¹³ : Z.HasTensor X
inst✝¹² : X.HasTensor Z
inst✝¹¹ : (tensorObj X Y).HasTensor Z
inst✝¹⁰ : X.HasTensor (tensorObj Y Z)
inst✝⁹ : (tensorObj Y Z).HasTensor X
inst✝⁸ : Y.HasTensor (tensorObj Z X)
inst✝⁷ : (tensorObj Y X).HasTensor Z
inst✝⁶ : Y.HasTensor (tensorObj X Z)
inst✝⁵ : X.HasGoodTensor₁₂Tensor Y Z
inst✝⁴ : X.HasGoodTensorTensor₂₃ Y Z
inst✝³ : Y.HasGoodTensor₁₂Tensor Z X
inst✝² : Y.HasGoodTensorTensor₂₃ Z X
inst✝¹ : Y.HasGoodTensor₁₂Tensor X Z
inst✝ : Y.HasGoodTensorTensor₂₃ X Z
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| MonoidalCategoryStruct.tensorHom ((β_ (X i₁) (Y i₂)).hom ≫ ιTensorObj Y X i₂ i₁ (i₁ + i₂) ⋯) (𝟙 Z i₃) ≫
    ιTensorObj (tensorObj Y X) Z (i₁ + i₂) i₃ k ⋯ ≫
      (associator Y X Z).hom k ≫
        whiskerLeft Y (fun k ↦ tensorObjDesc fun i j hij ↦ (β_ (X i) (Z j)).hom ≫ ιTensorObj Z X j i k ⋯) k

Tactic:
categoryOfGradedObjects_id,

Goal state after:
I : Type u_1
inst✝²⁰ : AddCommMonoid I
C : Type u_2
inst✝¹⁹ : Category.{v_1, u_2} C
inst✝¹⁸ : MonoidalCategory C
X Y Z : GradedObject I C
inst✝¹⁷ : BraidedCategory C
inst✝¹⁶ : X.HasTensor Y
inst✝¹⁵ : Y.HasTensor X
inst✝¹⁴ : Y.HasTensor Z
inst✝¹³ : Z.HasTensor X
inst✝¹² : X.HasTensor Z
inst✝¹¹ : (tensorObj X Y).HasTensor Z
inst✝¹⁰ : X.HasTensor (tensorObj Y Z)
inst✝⁹ : (tensorObj Y Z).HasTensor X
inst✝⁸ : Y.HasTensor (tensorObj Z X)
inst✝⁷ : (tensorObj Y X).HasTensor Z
inst✝⁶ : Y.HasTensor (tensorObj X Z)
inst✝⁵ : X.HasGoodTensor₁₂Tensor Y Z
inst✝⁴ : X.HasGoodTensorTensor₂₃ Y Z
inst✝³ : Y.HasGoodTensor₁₂Tensor Z X
inst✝² : Y.HasGoodTensorTensor₂₃ Z X
inst✝¹ : Y.HasGoodTensor₁₂Tensor X Z
inst✝ : Y.HasGoodTensorTensor₂₃ X Z
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| MonoidalCategoryStruct.tensorHom ((β_ (X i₁) (Y i₂)).hom ≫ ιTensorObj Y X i₂ i₁ (i₁ + i₂) ⋯) (𝟙 (Z i₃)) ≫
    ιTensorObj (tensorObj Y X) Z (i₁ + i₂) i₃ k ⋯ ≫
      (associator Y X Z).hom k ≫
        whiskerLeft Y (fun k ↦ tensorObjDesc fun i j hij ↦ (β_ (X i) (Z j)).hom ≫ ιTensorObj Z X j i k ⋯) k
Module: Mathlib.CategoryTheory.GradedObject.Braiding

Goal state before:
pp: I : Type u_1
inst✝²⁰ : AddCommMonoid I
C : Type u_2
inst✝¹⁹ : Category.{v_1, u_2} C
inst✝¹⁸ : MonoidalCategory C
X Y Z : GradedObject I C
inst✝¹⁷ : BraidedCategory C
inst✝¹⁶ : X.HasTensor Y
inst✝¹⁵ : Y.HasTensor X
inst✝¹⁴ : Y.HasTensor Z
inst✝¹³ : Z.HasTensor X
inst✝¹² : X.HasTensor Z
inst✝¹¹ : (tensorObj X Y).HasTensor Z
inst✝¹⁰ : X.HasTensor (tensorObj Y Z)
inst✝⁹ : (tensorObj Y Z).HasTensor X
inst✝⁸ : Y.HasTensor (tensorObj Z X)
inst✝⁷ : (tensorObj Y X).HasTensor Z
inst✝⁶ : Y.HasTensor (tensorObj X Z)
inst✝⁵ : X.HasGoodTensor₁₂Tensor Y Z
inst✝⁴ : X.HasGoodTensorTensor₂₃ Y Z
inst✝³ : Y.HasGoodTensor₁₂Tensor Z X
inst✝² : Y.HasGoodTensorTensor₂₃ Z X
inst✝¹ : Y.HasGoodTensor₁₂Tensor X Z
inst✝ : Y.HasGoodTensorTensor₂₃ X Z
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| MonoidalCategoryStruct.whiskerRight (β_ (X i₁) (Y i₂)).hom (Z i₃) ≫
    (MonoidalCategoryStruct.associator (Y i₂) (X i₁) (Z i₃)).hom ≫
      MonoidalCategoryStruct.whiskerLeft (Y i₂) (ιTensorObj X Z i₁ i₃ (i₁ + i₃) ⋯) ≫
        MonoidalCategoryStruct.tensorHom (𝟙 Y i₂)
            (tensorObjDesc fun i j hij ↦ (β_ (X i) (Z j)).hom ≫ ιTensorObj Z X j i (i₁ + i₃) ⋯) ≫
          ιTensorObj Y (tensorObj Z X) i₂ (i₁ + i₃) k ⋯

Tactic:
categoryOfGradedObjects_id,

Goal state after:
I : Type u_1
inst✝²⁰ : AddCommMonoid I
C : Type u_2
inst✝¹⁹ : Category.{v_1, u_2} C
inst✝¹⁸ : MonoidalCategory C
X Y Z : GradedObject I C
inst✝¹⁷ : BraidedCategory C
inst✝¹⁶ : X.HasTensor Y
inst✝¹⁵ : Y.HasTensor X
inst✝¹⁴ : Y.HasTensor Z
inst✝¹³ : Z.HasTensor X
inst✝¹² : X.HasTensor Z
inst✝¹¹ : (tensorObj X Y).HasTensor Z
inst✝¹⁰ : X.HasTensor (tensorObj Y Z)
inst✝⁹ : (tensorObj Y Z).HasTensor X
inst✝⁸ : Y.HasTensor (tensorObj Z X)
inst✝⁷ : (tensorObj Y X).HasTensor Z
inst✝⁶ : Y.HasTensor (tensorObj X Z)
inst✝⁵ : X.HasGoodTensor₁₂Tensor Y Z
inst✝⁴ : X.HasGoodTensorTensor₂₃ Y Z
inst✝³ : Y.HasGoodTensor₁₂Tensor Z X
inst✝² : Y.HasGoodTensorTensor₂₃ Z X
inst✝¹ : Y.HasGoodTensor₁₂Tensor X Z
inst✝ : Y.HasGoodTensorTensor₂₃ X Z
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| MonoidalCategoryStruct.whiskerRight (β_ (X i₁) (Y i₂)).hom (Z i₃) ≫
    (MonoidalCategoryStruct.associator (Y i₂) (X i₁) (Z i₃)).hom ≫
      MonoidalCategoryStruct.whiskerLeft (Y i₂) (ιTensorObj X Z i₁ i₃ (i₁ + i₃) ⋯) ≫
        MonoidalCategoryStruct.tensorHom (𝟙 (Y i₂))
            (tensorObjDesc fun i j hij ↦ (β_ (X i) (Z j)).hom ≫ ιTensorObj Z X j i (i₁ + i₃) ⋯) ≫
          ιTensorObj Y (tensorObj Z X) i₂ (i₁ + i₃) k ⋯
Module: Mathlib.CategoryTheory.Galois.EssSurj

Goal state before:
pp: C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCategory C
inst✝ : FiberFunctor F
V : OpenSubgroup (Aut F)
I : Set C
hf : Fintype ↑I
hc : ∀ X ∈ I, IsConnected X
hi : ∀ (σ : Aut F), (∀ (X : ↑I), σ.hom.app ↑X = 𝟙 (F.obj ↑X)) → σ ∈ ↑V
this : ∀ (X : ↑I), IsConnected ↑X
⊢ ∃ X, Nonempty ((functorToAction F).obj X ≅ Action.FintypeCat.ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ ↑V)))

Tactic:
have (X : I) : Nonempty (F.obj X.val) := nonempty_fiber_of_isConnected F X

Goal state after:
C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCategory C
inst✝ : FiberFunctor F
V : OpenSubgroup (Aut F)
I : Set C
hf : Fintype ↑I
hc : ∀ X ∈ I, IsConnected X
hi : ∀ (σ : Aut F), (∀ (X : ↑I), σ.hom.app ↑X = 𝟙 (F.obj ↑X)) → σ ∈ ↑V
this✝ : ∀ (X : ↑I), IsConnected ↑X
this : ∀ (X : ↑I), Nonempty (F.obj ↑X).obj
⊢ ∃ X, Nonempty ((functorToAction F).obj X ≅ Action.FintypeCat.ofMulAction (Aut F) (FintypeCat.of (Aut F ⧸ ↑V)))
Module: Mathlib.CategoryTheory.Groupoid.Subgroupoid

Goal state before:
pp: C : Type u
inst✝ : Groupoid C
S T : Subgroupoid C
⊢ (∀ (a : C) (b : (d : C) × (a ⟶ d)), ⟨a, b⟩ ∈ S → ⟨a, b⟩ ∈ T) ↔ ∀ {c d : C}, S.arrows c d ⊆ T.arrows c d

Tactic:
exact forall_congr' fun c => Sigma.forall

Goal state after:
Module: Mathlib.CategoryTheory.IsoCat

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
e : IsoCat C D
f : IsoCat D E
⊢ (f.inverse ⋙ e.inverse) ⋙ e.functor ⋙ f.functor = 𝟭 E

Tactic:
rw [Functor.assoc, ← Functor.assoc e.inverse, e.counit_eq, Functor.id_comp]
    exact f.counit_eq

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Braiding

Goal state before:
pp: I : Type u_1
inst✝²⁰ : AddCommMonoid I
C : Type u_2
inst✝¹⁹ : Category.{v_1, u_2} C
inst✝¹⁸ : MonoidalCategory C
X Y Z : GradedObject I C
inst✝¹⁷ : BraidedCategory C
inst✝¹⁶ : X.HasTensor Y
inst✝¹⁵ : Y.HasTensor Z
inst✝¹⁴ : Z.HasTensor X
inst✝¹³ : Z.HasTensor Y
inst✝¹² : X.HasTensor Z
inst✝¹¹ : (tensorObj X Y).HasTensor Z
inst✝¹⁰ : X.HasTensor (tensorObj Y Z)
inst✝⁹ : Z.HasTensor (tensorObj X Y)
inst✝⁸ : (tensorObj Z X).HasTensor Y
inst✝⁷ : X.HasTensor (tensorObj Z Y)
inst✝⁶ : (tensorObj X Z).HasTensor Y
inst✝⁵ : X.HasGoodTensor₁₂Tensor Y Z
inst✝⁴ : X.HasGoodTensorTensor₂₃ Y Z
inst✝³ : Z.HasGoodTensor₁₂Tensor X Y
inst✝² : Z.HasGoodTensorTensor₂₃ X Y
inst✝¹ : X.HasGoodTensor₁₂Tensor Z Y
inst✝ : X.HasGoodTensorTensor₂₃ Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| MonoidalCategoryStruct.tensorHom (𝟙 X i₁) ((β_ (Y i₂) (Z i₃)).hom ≫ ιTensorObj Z Y i₃ i₂ (i₂ + i₃) ⋯) ≫
    ιTensorObj X (tensorObj Z Y) i₁ (i₂ + i₃) k ⋯ ≫
      (associator X Z Y).inv k ≫
        whiskerRight (fun k ↦ tensorObjDesc fun i j hij ↦ (β_ (X i) (Z j)).hom ≫ ιTensorObj Z X j i k ⋯) Y k

Tactic:
categoryOfGradedObjects_id,

Goal state after:
I : Type u_1
inst✝²⁰ : AddCommMonoid I
C : Type u_2
inst✝¹⁹ : Category.{v_1, u_2} C
inst✝¹⁸ : MonoidalCategory C
X Y Z : GradedObject I C
inst✝¹⁷ : BraidedCategory C
inst✝¹⁶ : X.HasTensor Y
inst✝¹⁵ : Y.HasTensor Z
inst✝¹⁴ : Z.HasTensor X
inst✝¹³ : Z.HasTensor Y
inst✝¹² : X.HasTensor Z
inst✝¹¹ : (tensorObj X Y).HasTensor Z
inst✝¹⁰ : X.HasTensor (tensorObj Y Z)
inst✝⁹ : Z.HasTensor (tensorObj X Y)
inst✝⁸ : (tensorObj Z X).HasTensor Y
inst✝⁷ : X.HasTensor (tensorObj Z Y)
inst✝⁶ : (tensorObj X Z).HasTensor Y
inst✝⁵ : X.HasGoodTensor₁₂Tensor Y Z
inst✝⁴ : X.HasGoodTensorTensor₂₃ Y Z
inst✝³ : Z.HasGoodTensor₁₂Tensor X Y
inst✝² : Z.HasGoodTensorTensor₂₃ X Y
inst✝¹ : X.HasGoodTensor₁₂Tensor Z Y
inst✝ : X.HasGoodTensorTensor₂₃ Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| MonoidalCategoryStruct.tensorHom (𝟙 (X i₁)) ((β_ (Y i₂) (Z i₃)).hom ≫ ιTensorObj Z Y i₃ i₂ (i₂ + i₃) ⋯) ≫
    ιTensorObj X (tensorObj Z Y) i₁ (i₂ + i₃) k ⋯ ≫
      (associator X Z Y).inv k ≫
        whiskerRight (fun k ↦ tensorObjDesc fun i j hij ↦ (β_ (X i) (Z j)).hom ≫ ιTensorObj Z X j i k ⋯) Y k
Module: Mathlib.CategoryTheory.GradedObject.Braiding

Goal state before:
pp: I : Type u_1
inst✝²⁰ : AddCommMonoid I
C : Type u_2
inst✝¹⁹ : Category.{v_1, u_2} C
inst✝¹⁸ : MonoidalCategory C
X Y Z : GradedObject I C
inst✝¹⁷ : BraidedCategory C
inst✝¹⁶ : X.HasTensor Y
inst✝¹⁵ : Y.HasTensor Z
inst✝¹⁴ : Z.HasTensor X
inst✝¹³ : Z.HasTensor Y
inst✝¹² : X.HasTensor Z
inst✝¹¹ : (tensorObj X Y).HasTensor Z
inst✝¹⁰ : X.HasTensor (tensorObj Y Z)
inst✝⁹ : Z.HasTensor (tensorObj X Y)
inst✝⁸ : (tensorObj Z X).HasTensor Y
inst✝⁷ : X.HasTensor (tensorObj Z Y)
inst✝⁶ : (tensorObj X Z).HasTensor Y
inst✝⁵ : X.HasGoodTensor₁₂Tensor Y Z
inst✝⁴ : X.HasGoodTensorTensor₂₃ Y Z
inst✝³ : Z.HasGoodTensor₁₂Tensor X Y
inst✝² : Z.HasGoodTensorTensor₂₃ X Y
inst✝¹ : X.HasGoodTensor₁₂Tensor Z Y
inst✝ : X.HasGoodTensorTensor₂₃ Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| MonoidalCategoryStruct.whiskerLeft (X i₁) (β_ (Y i₂) (Z i₃)).hom ≫
    (MonoidalCategoryStruct.associator (X i₁) (Z i₃) (Y i₂)).inv ≫
      MonoidalCategoryStruct.whiskerRight (ιTensorObj X Z i₁ i₃ (i₁ + i₃) ⋯) (Y i₂) ≫
        MonoidalCategoryStruct.tensorHom
            (tensorObjDesc fun i j hij ↦ (β_ (X i) (Z j)).hom ≫ ιTensorObj Z X j i (i₁ + i₃) ⋯) (𝟙 Y i₂) ≫
          ιTensorObj (tensorObj Z X) Y (i₁ + i₃) i₂ k ⋯

Tactic:
categoryOfGradedObjects_id,

Goal state after:
I : Type u_1
inst✝²⁰ : AddCommMonoid I
C : Type u_2
inst✝¹⁹ : Category.{v_1, u_2} C
inst✝¹⁸ : MonoidalCategory C
X Y Z : GradedObject I C
inst✝¹⁷ : BraidedCategory C
inst✝¹⁶ : X.HasTensor Y
inst✝¹⁵ : Y.HasTensor Z
inst✝¹⁴ : Z.HasTensor X
inst✝¹³ : Z.HasTensor Y
inst✝¹² : X.HasTensor Z
inst✝¹¹ : (tensorObj X Y).HasTensor Z
inst✝¹⁰ : X.HasTensor (tensorObj Y Z)
inst✝⁹ : Z.HasTensor (tensorObj X Y)
inst✝⁸ : (tensorObj Z X).HasTensor Y
inst✝⁷ : X.HasTensor (tensorObj Z Y)
inst✝⁶ : (tensorObj X Z).HasTensor Y
inst✝⁵ : X.HasGoodTensor₁₂Tensor Y Z
inst✝⁴ : X.HasGoodTensorTensor₂₃ Y Z
inst✝³ : Z.HasGoodTensor₁₂Tensor X Y
inst✝² : Z.HasGoodTensorTensor₂₃ X Y
inst✝¹ : X.HasGoodTensor₁₂Tensor Z Y
inst✝ : X.HasGoodTensorTensor₂₃ Z Y
k i₁ i₂ i₃ : I
h : i₁ + i₂ + i₃ = k
| MonoidalCategoryStruct.whiskerLeft (X i₁) (β_ (Y i₂) (Z i₃)).hom ≫
    (MonoidalCategoryStruct.associator (X i₁) (Z i₃) (Y i₂)).inv ≫
      MonoidalCategoryStruct.whiskerRight (ιTensorObj X Z i₁ i₃ (i₁ + i₃) ⋯) (Y i₂) ≫
        MonoidalCategoryStruct.tensorHom
            (tensorObjDesc fun i j hij ↦ (β_ (X i) (Z j)).hom ≫ ιTensorObj Z X j i (i₁ + i₃) ⋯) (𝟙 (Y i₂)) ≫
          ιTensorObj (tensorObj Z X) Y (i₁ + i₃) i₂ k ⋯
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
X Y : C
b : BinaryBicone X Y
h : b.IsBilimit
⊢ IsPushout b.fst b.snd 0 0

Tactic:
refine IsPushout.of_left ?_ (by simp) (IsPushout.inl_snd' h)

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
X Y : C
b : BinaryBicone X Y
h : b.IsBilimit
⊢ IsPushout (b.inl ≫ b.fst) 0 0 (0 ≫ 0)
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic

Goal state before:
pp: A : Type u₁
B : Type u₂
C : Type u₃
inst✝² : Category.{v₁, u₁} A
inst✝¹ : Category.{v₂, u₂} B
inst✝ : Category.{v₃, u₃} C
F : A ⥤ B
G : C ⥤ B
x y : F ⊡ G
f g : x ⟶ y
hₗ : f.fst = g.fst
hᵣ : f.snd = g.snd
⊢ f = g

Tactic:
apply Hom.ext

Goal state after:
case fst
A : Type u₁
B : Type u₂
C : Type u₃
inst✝² : Category.{v₁, u₁} A
inst✝¹ : Category.{v₂, u₂} B
inst✝ : Category.{v₃, u₃} C
F : A ⥤ B
G : C ⥤ B
x y : F ⊡ G
f g : x ⟶ y
hₗ : f.fst = g.fst
hᵣ : f.snd = g.snd
⊢ f.fst = g.fst

case snd
A : Type u₁
B : Type u₂
C : Type u₃
inst✝² : Category.{v₁, u₁} A
inst✝¹ : Category.{v₂, u₂} B
inst✝ : Category.{v₃, u₃} C
F : A ⥤ B
G : C ⥤ B
x y : F ⊡ G
f g : x ⟶ y
hₗ : f.fst = g.fst
hᵣ : f.snd = g.snd
⊢ f.snd = g.snd
Module: Mathlib.CategoryTheory.Monoidal.Multifunctor

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : MonoidalCategory C
D : Type u_2
inst✝¹ : Category.{v_2, u_2} D
inst✝ : MonoidalCategory D
F₁ F₂ : C ⥤ D
f : F₁ ⟶ F₂
X₁ : C
⊢ ∀ ⦃X Y : C⦄ (f_1 : X ⟶ Y),
    ((curriedTensorPre F₁).obj X₁).map f_1 ≫ (f.app X₁ ⊗ₘ f.app Y) =
      (f.app X₁ ⊗ₘ f.app X) ≫ ((curriedTensorPre F₂).obj X₁).map f_1

Tactic:
simp [← id_tensorHom]

Goal state after:
Module: Mathlib.CategoryTheory.MarkovCategory.Positive

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : MonoidalCategory C
inst✝¹ : PositiveCategory C
X Y : C
f : X ⟶ Y
inst✝ : IsIso f
⊢ Deterministic (f ≫ inv f)

Tactic:
IsIso.hom_inv_id

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : MonoidalCategory C
inst✝¹ : PositiveCategory C
X Y : C
f : X ⟶ Y
inst✝ : IsIso f
⊢ Deterministic (𝟙 X)
Module: Mathlib.CategoryTheory.LocallyCartesianClosed.Sections

Goal state before:
pp: case h₂
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : CartesianMonoidalCategory C
I : C
inst✝² : Closed I
inst✝¹ : ChosenPullbacksAlong (curryRightUnitorHom I)
inst✝ : BraidedCategory C
A : C
X' X : Over I
u : (toOver I).obj A ⟶ X'
g : X' ⟶ X
⊢ ChosenPullbacksAlong.lift (curry ((β_ I A).hom ≫ Hom.left u ≫ Hom.left g)) (toUnit A) ⋯ ≫
      ChosenPullbacksAlong.snd ((ihom I).map X.hom) (curry (ρ_ I).hom) =
    (ChosenPullbacksAlong.lift (curry ((β_ I A).hom ≫ Hom.left u)) (toUnit A) ⋯ ≫ (sections I).map g) ≫
      ChosenPullbacksAlong.snd ((ihom I).map X.hom) (curry (ρ_ I).hom)

Tactic:
· simp

Goal state after:
Module: Mathlib.CategoryTheory.Monad.Monadicity

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₁, u₂} D
G : D ⥤ C
F : C ⥤ D
adj : F ⊣ G
inst✝ : CreatesColimitOfIsSplitPair G
I : ∀ {A B : D} (f g : A ⟶ B) [G.IsSplitPair f g], HasColimit (parallelPair f g ⋙ G)
this : HasCoequalizerOfIsSplitPair G
⊢ ∀ {A B : D} (f g : A ⟶ B) [G.IsSplitPair f g], PreservesColimit (parallelPair f g) G

Tactic:
by intros; infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Monad.Monadicity

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₁, u₂} D
G : D ⥤ C
F : C ⥤ D
adj : F ⊣ G
inst✝ : CreatesColimitOfIsSplitPair G
I : ∀ {A B : D} (f g : A ⟶ B) [G.IsSplitPair f g], HasColimit (parallelPair f g ⋙ G)
this✝ : HasCoequalizerOfIsSplitPair G
this : PreservesColimitOfIsSplitPair G
⊢ ∀ {A B : D} (f g : A ⟶ B) [G.IsSplitPair f g], ReflectsColimit (parallelPair f g) G

Tactic:
by intros; infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Action.Basic

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : MonoidalCategory C
inst✝ : MonoidalRightAction C D
z z' : D
f : z ⟶ z'
x y : C
⊢ f ⊵ᵣ (x ⊗ y) = (αᵣ z x y).hom ≫ (f ⊵ᵣ x) ⊵ᵣ y ≫ (αᵣ z' x y).inv

Tactic:
by
  simp only [← actionHom_id]
  rw [← Category.assoc, ← actionAssocIso_hom_naturality]
  simp

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Mod

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
M : C
inst✝¹ : MonObj M
X : C
inst✝ : ModObj M X
H : ∀ (Z : C) (x : Z ⟶ X) (m : Z ⟶ M), lift m x ≫ leftSMul M X = lift (m • x) x
h : ∀ (Z : C) (f g : Z ⟶ X) (m : Z ⟶ M) (x : Z ⟶ X), lift m x ≫ leftSMul M X = lift f g ↔ x = g ∧ m • x = f
Z : C
f g : Z ⟶ X
a : { a // a.2 = f ∧ a.1 • a.2 = g }
⊢ (↑a).1 • f = g

Tactic:
by grind

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Mod

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
M : C
inst✝¹ : MonObj M
X : C
inst✝ : ModObj M X
H : ∀ (Z : C) (x : Z ⟶ X) (m : Z ⟶ M), lift m x ≫ leftSMul M X = lift (m • x) x
h : ∀ (Z : C) (f g : Z ⟶ X) (m : Z ⟶ M) (x : Z ⟶ X), lift m x ≫ leftSMul M X = lift f g ↔ x = g ∧ m • x = f
Z : C
f g : Z ⟶ X
a : { b // b • f = g }
⊢ (↑a, f).2 = f ∧ (↑a, f).1 • (↑a, f).2 = g

Tactic:
by grind

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Normal

Goal state before:
pp: case refine_2
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : CartesianMonoidalCategory C
G H : C
inst✝⁴ : GrpObj G
inst✝³ : GrpObj H
φ : H ⟶ G
inst✝² : IsMonHom φ
P : C
p : G ⟶ P
inst✝¹ : GrpObj P
inst✝ : IsMonHom p
h : IsPullback φ (toUnit H) p η
⊢ h.lift (G ◁ φ ≫ conj G) (toUnit (G ⊗ H)) ⋯ ≫ φ = G ◁ φ ≫ conj G

Tactic:
· simp

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Closed.Enrichment

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : MonoidalClosed C
X₁ X₂ : C
f : X₁ ⟶ X₂
Y : C
⊢ eHomWhiskerRight C f Y = (pre f).app Y

Tactic:
change (λ_ _).inv ≫ curry' f ▷ _ ≫ comp X₁ X₂ Y = _
  rw [curry'_whiskerRight_comp, Iso.inv_hom_id_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic

Goal state before:
pp: case e_a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalClosed C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₂' F₃ F₃' : J ⥤ C
g : F₂ ⟶ functorEnrichedHom C F₁ F₃
j j' : J
φ : j ⟶ j'
⊢ enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) ((Under.map φ).obj (Under.mk (𝟙 j'))) ≫
      eHomWhiskerRight C (F₁.map φ) (F₃.obj j') =
    enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) (Under.mk (𝟙 j)) ≫ (ihom (F₁.obj j)).map (F₃.map φ)

Tactic:
← enrichedOrdinaryCategorySelf_eHomWhiskerLeft

Goal state after:
case e_a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalClosed C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₂' F₃ F₃' : J ⥤ C
g : F₂ ⟶ functorEnrichedHom C F₁ F₃
j j' : J
φ : j ⟶ j'
⊢ enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) ((Under.map φ).obj (Under.mk (𝟙 j'))) ≫
      eHomWhiskerRight C (F₁.map φ) (F₃.obj j') =
    enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) (Under.mk (𝟙 j)) ≫
      eHomWhiskerLeft C (F₁.obj j) (F₃.map φ)
Module: Mathlib.CategoryTheory.Subterminal

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : C
inst✝ : HasTerminal C
X✝ Y✝ : Subterminals C
f : X✝ ⟶ Y✝
⊢ f.hom ≫ MonoOver.arrow { obj := Over.mk (terminal.from Y✝.obj), property := ⋯ } =
    MonoOver.arrow { obj := Over.mk (terminal.from X✝.obj), property := ⋯ }

Tactic:
ext1 ⟨⟨⟩⟩

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Closed.InternalCurrying

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : MonoidalCategory C
x y z : C
inst✝² : Closed x
inst✝¹ : Closed y
inst✝ : Closed (x ⊗ y)
⊢ ihomCurry x y z ≫ ihomUncurry x y z = 𝟙 ((ihom (x ⊗ y)).obj z)

Tactic:
apply uncurry_injective
  rw [uncurry_natural_left, uncurry_id_eq_ev, uncurry_ihomUncurry]
  dsimp
  rw [associator_naturality_right_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc]
  simp [← dsimp% uncurry_eq, uncurry_uncurry_ihomCurry]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Closed.Ideal

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁵ : CartesianMonoidalCategory C
inst✝⁴ : Reflective i
inst✝³ : MonoidalClosed C
inst✝² : CartesianMonoidalCategory D
inst✝¹ : ExponentialIdeal i
inst✝ : BraidedCategory C
A B : C
⊢ (bijection i A B ((reflector i).obj (A ⊗ B))) (𝟙 ((reflector i).obj (A ⊗ B))) ≫ prodComparison (reflector i) A B =
    𝟙 ((reflector i).obj A ⊗ (reflector i).obj B)

Tactic:
rw [← bijection_natural, Category.id_comp, ← bijection_symm_apply_id,
        Equiv.apply_symm_apply]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
V : Type u₂
inst✝³ : Category.{v₂, u₂} V
inst✝² : MonoidalCategory C
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalClosed V
F G H : C ⥤ V
ℌ : DayConvolutionInternalHom F G H
G' H' : C ⥤ V
f : G ⟶ G'
ℌ' : DayConvolutionInternalHom F G' H'
c j : C
⊢ Wedge.IsLimit.lift (ℌ'.isLimitWedge c) (fun j ↦ ℌ.π c j ≫ (ihom (F.obj j)).map (f.app (j ⊗ c))) ⋯ ≫ ℌ'.π c j =
    ℌ.π c j ≫ (ihom (F.obj j)).map (f.app (j ⊗ c))

Tactic:
← Wedge.mk_ι
      (F := dayConvolutionInternalHomDiagramFunctor F |>.obj _ |>.obj c)
      (H'.obj c) (ℌ'.π c) (ℌ'.hπ c),

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
V : Type u₂
inst✝³ : Category.{v₂, u₂} V
inst✝² : MonoidalCategory C
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalClosed V
F G H : C ⥤ V
ℌ : DayConvolutionInternalHom F G H
G' H' : C ⥤ V
f : G ⟶ G'
ℌ' : DayConvolutionInternalHom F G' H'
c j : C
⊢ Wedge.IsLimit.lift (ℌ'.isLimitWedge c) (fun j ↦ ℌ.π c j ≫ (ihom (F.obj j)).map (f.app (j ⊗ c))) ⋯ ≫
      Multifork.ι (Wedge.mk (H'.obj c) (ℌ'.π c) ⋯) j =
    ℌ.π c j ≫ (ihom (F.obj j)).map (f.app (j ⊗ c))
Module: Mathlib.CategoryTheory.Monoidal.Free.Coherence

Goal state before:
pp: case mk.id
C : Type u
X Y X✝ : F C
⊢ (fun n ↦ X✝.normalizeObj n) = X✝.normalizeObj

Tactic:
funext; rfl

Goal state after: