mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
⊢ IsIso γ.left.φH.op → IsIso γ.left.φH

pp: case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
⊢ IsIso γ.left.φH → IsIso γ.left.φH.op

Tactic:
· intro h
    apply isIso_of_op

Goal state after:
case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
⊢ IsIso γ.left.φH → IsIso γ.left.φH.op
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: case e_a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.LeftHomologyData
A : C
k : A ⟶ S.X₂
x : A ⟶ S.X₁
hx : k = x ≫ S.f
⊢ h.liftK k ⋯ = x ≫ h.f'

Tactic:
simp only [← cancel_mono h.i, hx, liftK_i, assoc, f'_i]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A K' H' : C
eK : K' ≅ h.K
eH : H' ≅ h.H
⊢ ∀ (j : WalkingParallelPair),
    ((Cocone.precompose (parallelPair.ext (Iso.refl S.X₁) eK.symm ⋯ ⋯).inv).obj (CokernelCofork.ofπ h.π ⋯)).ι.app j ≫
        eH.symm.hom =
      (CokernelCofork.ofπ (eK.hom ≫ h.π ≫ eH.inv) ⋯).ι.app j

Tactic:
by rintro (_ | _) <;> simp [IsKernel.isoKernel]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
inst✝¹ : S₁.HasLeftHomology
inst✝ : S₂.HasLeftHomology
⊢ cyclesMap φ = h₁.cyclesIso.hom ≫ γ.φK ≫ h₂.cyclesIso.inv

Tactic:
by
  dsimp [LeftHomologyData.cyclesIso, cyclesMapIso']
  rw [← γ.cyclesMap'_eq, ← cyclesMap'_comp, ← cyclesMap'_comp, id_comp, comp_id]
  rfl

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ✝ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ IsIso (homologyMap' φ h₁ h₂)

Tactic:
dsimp only [homologyMap']

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ✝ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ IsIso (leftHomologyMap' φ h₁.left h₂.left)
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
inst✝¹ : S₁.HasRightHomology
inst✝ : S₂.HasRightHomology
⊢ rightHomologyMap φ = h₁.rightHomologyIso.hom ≫ γ.φH ≫ h₂.rightHomologyIso.inv

Tactic:
dsimp [RightHomologyData.rightHomologyIso, rightHomologyMapIso']
  rw [← γ.rightHomologyMap'_eq, ← rightHomologyMap'_comp,
    ← rightHomologyMap'_comp, id_comp, comp_id]
  rfl

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: case refine_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
h₁ : IsIso (cyclesMap φ)
h₂ : Epi φ.τ₁
h : S₂.toCycles ≫ inv (cyclesMap φ) ≫ S₁.homologyπ = 0
z : (CokernelCofork.ofπ S₂.homologyπ ⋯).pt ⟶ S₁.homology
hz : S₂.homologyπ ≫ z = inv (cyclesMap φ) ≫ S₁.homologyπ
⊢ homologyMap φ ≫ z = 𝟙 S₁.homology

Tactic:
rw [← cancel_epi S₁.homologyπ, homologyπ_naturality_assoc, hz,
      IsIso.hom_inv_id_assoc, comp_id]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₂ ⟶ S₃
⊢ S₁.g ≫ h.h₂ ≫ ψ.τ₂ + h.h₁ ≫ S₂.f ≫ ψ.τ₂ + φ₂.τ₂ ≫ ψ.τ₂ = S₁.g ≫ h.h₂ ≫ ψ.τ₂ + h.h₁ ≫ ψ.τ₁ ≫ S₃.f + φ₂.τ₂ ≫ ψ.τ₂

Tactic:
ψ.comm₁₂

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
ψ : S₂ ⟶ S₃
⊢ S₁.g ≫ h.h₂ ≫ ψ.τ₂ + h.h₁ ≫ S₂.f ≫ ψ.τ₂ + φ₂.τ₂ ≫ ψ.τ₂ = S₁.g ≫ h.h₂ ≫ ψ.τ₂ + h.h₁ ≫ S₂.f ≫ ψ.τ₂ + φ₂.τ₂ ≫ ψ.τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ leftHomologyMap' φ₁ h₁ h₂ = leftHomologyMap' φ₂ h₁ h₂

Tactic:
by
  rw [h.eq_add_nullHomotopic, leftHomologyMap'_add, leftHomologyMap'_nullHomotopic, add_zero]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Ab

Goal state before:
pp: S : ShortComplex Ab
⊢ S.Exact ↔ (AddCommGrpCat.Hom.hom S.f).range = (AddCommGrpCat.Hom.hom S.g).ker

Tactic:
by
  rw [ab_exact_iff_ker_le_range]
  constructor
  · intro h
    refine le_antisymm ?_ h
    rintro _ ⟨x₁, rfl⟩
    rw [AddMonoidHom.mem_ker, ← ConcreteCategory.comp_apply, S.zero]
    rfl
  · intro h
    rw [h]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

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 : Preadditive C
inst✝ : Preadditive D
S S₁ S₂ : ShortComplex C
s : S₁.Splitting
e : S₁ ≅ S₂
⊢ (e.inv.τ₂ ≫ s.r ≫ e.hom.τ₁) ≫ S₂.f + S₂.g ≫ e.inv.τ₃ ≫ s.s ≫ e.hom.τ₂ = 𝟙 S₂.X₂

Tactic:
have eq := e.inv.τ₂ ≫= s.id =≫ e.hom.τ₂

Goal state after:
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 : Preadditive C
inst✝ : Preadditive D
S S₁ S₂ : ShortComplex C
s : S₁.Splitting
e : S₁ ≅ S₂
eq : e.inv.τ₂ ≫ (s.r ≫ S₁.f + S₁.g ≫ s.s) ≫ e.hom.τ₂ = e.inv.τ₂ ≫ 𝟙 S₁.X₂ ≫ e.hom.τ₂
⊢ (e.inv.τ₂ ≫ s.r ≫ e.hom.τ₁) ≫ S₂.f + S₂.g ≫ e.inv.τ₃ ≫ s.s ≫ e.hom.τ₂ = 𝟙 S₂.X₂
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
B : C
X Y : Subobject B
h : X ≤ Y
⊢ ∀ {Z : C} (g h_1 : Z ⟶ underlying.obj X), g ≫ X.ofLE Y h = h_1 ≫ X.ofLE Y h → g = h_1

Tactic:
intro Z

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ Z✝ : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
B : C
X Y : Subobject B
h : X ≤ Y
Z : C
⊢ ∀ (g h_1 : Z ⟶ underlying.obj X), g ≫ X.ofLE Y h = h_1 ≫ X.ofLE Y h → g = h_1
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Preadditive C
inst✝² : Preadditive D
S : ShortComplex C
inst✝¹ : Balanced C
hS : S.Exact
inst✝ : Epi S.g
⊢ IsColimit (CokernelCofork.ofπ S.g ⋯)

Tactic:
have := hS.hasHomology
  have := hS.isIso_fromOpcycles
  exact IsColimit.ofIsoColimit S.opcyclesIsCokernel
    (Cofork.ext (asIso S.fromOpcycles) (by simp))

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.MonoOver

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasImages C
f : X ⟶ Y
inst✝ : Mono f
⊢ (X_1 : MonoOver X) → («exists» f).obj X_1 ≅ (map f).obj X_1

Tactic:
intro Z

Goal state after:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X Y Z✝ : C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasImages C
f : X ⟶ Y
inst✝ : Mono f
Z : MonoOver X
⊢ («exists» f).obj Z ≅ (map f).obj Z
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: case mpr.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
w : φ.τ₂ ≫ S₂.g = 0
h₁ : { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.Exact
h₂ : Mono φ.τ₂
⊢ h₁.lift S₂.iCycles ⋯ ≫ S₂.liftCycles φ.τ₂ ⋯ = 𝟙 S₂.cycles

Tactic:
← cancel_mono S₂.iCycles,

Goal state after:
case mpr.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
w : φ.τ₂ ≫ S₂.g = 0
h₁ : { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.Exact
h₂ : Mono φ.τ₂
⊢ (h₁.lift S₂.iCycles ⋯ ≫ S₂.liftCycles φ.τ₂ ⋯) ≫ S₂.iCycles = 𝟙 S₂.cycles ≫ S₂.iCycles
Module: Mathlib.CategoryTheory.Abelian.Exact

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
h : S.Exact
⊢ IsLimit (KernelFork.ofι (Abelian.image.ι S.f) ⋯)

Tactic:
rw [exact_iff_kernel_ι_comp_cokernel_π_zero] at h

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
h : kernel.ι S.g ≫ cokernel.π S.f = 0
⊢ IsLimit (KernelFork.ofι (Abelian.image.ι S.f) ⋯)
Module: Mathlib.CategoryTheory.Abelian.Exact

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
h : S.Exact
⊢ IsColimit (CokernelCofork.ofπ (Abelian.coimage.π S.g) ⋯)

Tactic:
rw [exact_iff_kernel_ι_comp_cokernel_π_zero] at h

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
h : kernel.ι S.g ≫ cokernel.π S.f = 0
⊢ IsColimit (CokernelCofork.ofπ (Abelian.coimage.π S.g) ⋯)
Module: Mathlib.CategoryTheory.Abelian.Exact

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
h : kernel.ι S.g ≫ cokernel.π S.f = 0
Z'✝ : C
u : S.X₂ ⟶ Z'✝
hu : S.f ≫ u = 0
⊢ kernel.ι S.g ≫ u = 0

Tactic:
rw [← cokernel.π_desc S.f u hu, ← Category.assoc, h, zero_comp]

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Lattice

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : LocallySmall.{w, v₁, u₁} C
inst✝¹ : WellPowered.{w, v₁, u₁} C
inst✝ : HasWidePullbacks C
A : C
s : Set (Subobject A)
f : Subobject A
hf : f ∈ s
⊢ sInf s ≤ f

Tactic:
by
  fapply le_of_comm
  · exact (underlyingIso _).hom ≫
      Limits.limit.π (wideCospan s)
        (some ⟨equivShrink (Subobject A) f,
          Set.mem_image_of_mem (equivShrink (Subobject A)) hf⟩) ≫
      eqToHom (congr_arg (fun X : Subobject A => (X : C)) (Equiv.symm_apply_apply _ _))
  · dsimp [sInf]
    simp only [Category.assoc, ← underlyingIso_hom_comp_eq_mk,
      Iso.cancel_iso_hom_left]
    convert! limit.w (wideCospan s) (WidePullbackShape.Hom.term _)
    simp

Goal state after:
Module: Mathlib.CategoryTheory.Functor.KanExtension.Pointwise

Goal state before:
pp: C : Type u_1
D : Type u_2
D' : Type u_3
H : Type u_4
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Category.{v_3, u_3} D'
inst✝ : Category.{v_4, u_4} H
L : C ⥤ D
L' : C ⥤ D'
F : C ⥤ H
E E' : L.RightExtension F
Y : D
g₁ g₂ : StructuredArrow Y L
φ : g₁ ⟶ g₂
⊢ (CostructuredArrow.left E).map (g₁.hom ≫ L.map (StructuredArrow.Hom.right φ)) ≫
      (CostructuredArrow.hom E).app g₂.right =
    (CostructuredArrow.left E).map g₁.hom ≫ (CostructuredArrow.hom E).app g₁.right ≫ F.map φ.right

Tactic:
Functor.map_comp,

Goal state after:
C : Type u_1
D : Type u_2
D' : Type u_3
H : Type u_4
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Category.{v_3, u_3} D'
inst✝ : Category.{v_4, u_4} H
L : C ⥤ D
L' : C ⥤ D'
F : C ⥤ H
E E' : L.RightExtension F
Y : D
g₁ g₂ : StructuredArrow Y L
φ : g₁ ⟶ g₂
⊢ ((CostructuredArrow.left E).map g₁.hom ≫ (CostructuredArrow.left E).map (L.map (StructuredArrow.Hom.right φ))) ≫
      (CostructuredArrow.hom E).app g₂.right =
    (CostructuredArrow.left E).map g₁.hom ≫ (CostructuredArrow.hom E).app g₁.right ≫ F.map φ.right
Module: Mathlib.CategoryTheory.Functor.KanExtension.Pointwise

Goal state before:
pp: C : Type u_1
D : Type u_2
D' : Type u_3
H : Type u_4
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Category.{v_3, u_3} D'
inst✝ : Category.{v_4, u_4} H
L : C ⥤ D
L' : C ⥤ D'
F : C ⥤ H
E✝ E'✝ : L.RightExtension F
Y : D
E E' : L.RightExtension F
φ : E ⟶ E'
G : StructuredArrow Y L
⊢ φ.left.app Y ≫ (E'.coneAt Y).π.app G = (E.coneAt Y).π.app G

Tactic:
by
    dsimp
    rw [← CostructuredArrow.w φ]
    simp

Goal state after:
Module: Mathlib.CategoryTheory.Functor.KanExtension.Adjunction

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
L : C ⥤ D
H : Type u_3
inst✝¹ : Category.{v_3, u_3} H
inst✝ : ∀ (F : C ⥤ H), L.HasRightKanExtension F
F : D ⥤ H
G₁ G₂ : C ⥤ H
β : ((whiskeringLeft C D H).obj L).obj F ⟶ G₁
f : G₁ ⟶ G₂
X : C
⊢ ((L.ran.obj G₂).liftOfIsRightKanExtension (L.ranCounit.app G₂) F (β ≫ f)).app (L.obj X) ≫ (L.ranCounit.app G₂).app X =
    (((L.ran.obj G₁).liftOfIsRightKanExtension (L.ranCounit.app G₁) F β).app (L.obj X) ≫ (L.ran.map f).app (L.obj X)) ≫
      (L.ranCounit.app G₂).app X

Tactic:
rw [liftOfIsRightKanExtension_fac_app, NatTrans.comp_app, assoc]

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
H : Type u_3
inst✝¹ : Category.{v_3, u_3} H
inst✝ : ∀ (F : C ⥤ H), L.HasRightKanExtension F
F : D ⥤ H
G₁ G₂ : C ⥤ H
β : ((whiskeringLeft C D H).obj L).obj F ⟶ G₁
f : G₁ ⟶ G₂
X : C
⊢ β.app X ≫ f.app X =
    ((L.ran.obj G₁).liftOfIsRightKanExtension (L.ranCounit.app G₁) F β).app (L.obj X) ≫
      (L.ran.map f).app (L.obj X) ≫ (L.ranCounit.app G₂).app X
Module: Mathlib.CategoryTheory.Functor.KanExtension.Pointwise

Goal state before:
pp: C : Type u_1
D : Type u_2
D' : Type u_3
H : Type u_4
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Category.{v_3, u_3} D'
inst✝ : Category.{v_4, u_4} H
L : C ⥤ D
L' : C ⥤ D'
F : C ⥤ H
E E' : L.RightExtension F
h : E.IsPointwiseRightKanExtension
G : L.RightExtension F
Y₁ Y₂ : D
φ : Y₁ ⟶ Y₂
X : StructuredArrow Y₂ L
⊢ (CostructuredArrow.left G).map φ ≫ (h Y₂).lift (G.coneAt Y₂) ≫ (E.coneAt Y₂).π.app X =
    ((h Y₁).lift (G.coneAt Y₁) ≫ (CostructuredArrow.left E).map φ) ≫ (E.coneAt Y₂).π.app X

Tactic:
(h Y₂).fac (coneAt G Y₂) X

Goal state after:
C : Type u_1
D : Type u_2
D' : Type u_3
H : Type u_4
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Category.{v_3, u_3} D'
inst✝ : Category.{v_4, u_4} H
L : C ⥤ D
L' : C ⥤ D'
F : C ⥤ H
E E' : L.RightExtension F
h : E.IsPointwiseRightKanExtension
G : L.RightExtension F
Y₁ Y₂ : D
φ : Y₁ ⟶ Y₂
X : StructuredArrow Y₂ L
⊢ (CostructuredArrow.left G).map φ ≫ (G.coneAt Y₂).π.app X =
    ((h Y₁).lift (G.coneAt Y₁) ≫ (CostructuredArrow.left E).map φ) ≫ (E.coneAt Y₂).π.app X
Module: Mathlib.CategoryTheory.Limits.Shapes.Countable

Goal state before:
pp: J : Type u_2
inst✝² : Countable J
inst✝¹ : Preorder J
inst✝ : IsFiltered J
j : J
⊢ ∃ n, j ≤ sequentialFunctor_obj J n

Tactic:
obtain ⟨m, h⟩ := (exists_surjective_nat _).choose_spec j
  refine ⟨m + 1, ?_⟩
  simpa only [h] using! leOfHom (IsFilteredOrEmpty.cocone_objs ((exists_surjective_nat _).choose m)
    (sequentialFunctor_obj J m)).choose_spec.choose

Goal state after:
Module: Mathlib.CategoryTheory.Functor.KanExtension.Basic

Goal state before:
pp: C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} C'
inst✝² : Category.{v_3, u_3} H
inst✝¹ : Category.{v_4, u_4} D
inst✝ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : D ⥤ D'
F₀ : C ⥤ H
F₁ : D ⥤ H
F₂ : D' ⥤ H
α : F₀ ⟶ L ⋙ F₁
hα : StructuredArrow.IsUniversal (mk F₁ α)
b : L'.LeftExtension F₁
hb : StructuredArrow.IsUniversal b
y : (L ⋙ L').LeftExtension F₀
u : L'.LeftExtension F₁ := ⋯
a : (precomp₂ L' α).obj b ⟶ y
⊢ StructuredArrow.hom (mk F₁ α) ≫
      ((whiskeringLeft C D H).obj L).map
        (StructuredArrow.hom b ≫ ((whiskeringLeft D D' H).obj L').map (StructuredArrow.Hom.right a)) =
    StructuredArrow.hom (mk F₁ α) ≫
      ((whiskeringLeft C D H).obj L).map
        (StructuredArrow.hom b ≫
          ((whiskeringLeft D D' H).obj L').map (StructuredArrow.Hom.right (StructuredArrow.homMk (hb.desc u) ⋯)))

Tactic:
ext t

Goal state after:
C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} C'
inst✝² : Category.{v_3, u_3} H
inst✝¹ : Category.{v_4, u_4} D
inst✝ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : D ⥤ D'
F₀ : C ⥤ H
F₁ : D ⥤ H
F₂ : D' ⥤ H
α : F₀ ⟶ L ⋙ F₁
hα : StructuredArrow.IsUniversal (mk F₁ α)
b : L'.LeftExtension F₁
hb : StructuredArrow.IsUniversal b
y : (L ⋙ L').LeftExtension F₀
u : L'.LeftExtension F₁ :=
  mk (StructuredArrow.right y)
    (hα.desc
      (mk (L' ⋙ StructuredArrow.right y) (StructuredArrow.hom y ≫ (L.associator L' (StructuredArrow.right y)).hom)))
a : (precomp₂ L' α).obj b ⟶ y
t : C
⊢ (StructuredArrow.hom (mk F₁ α) ≫
          ((whiskeringLeft C D H).obj L).map
            (StructuredArrow.hom b ≫ ((whiskeringLeft D D' H).obj L').map (StructuredArrow.Hom.right a))).app
      t =
    (StructuredArrow.hom (mk F₁ α) ≫
          ((whiskeringLeft C D H).obj L).map
            (StructuredArrow.hom b ≫
              ((whiskeringLeft D D' H).obj L').map
                (StructuredArrow.Hom.right (StructuredArrow.homMk (hb.desc u) ⋯)))).app
      t
Module: Mathlib.CategoryTheory.Functor.KanExtension.Basic

Goal state before:
pp: C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} C'
inst✝² : Category.{v_3, u_3} H
inst✝¹ : Category.{v_4, u_4} D
inst✝ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : D ⥤ D'
F₀ : C ⥤ H
F₁ : D ⥤ H
F₂ : D' ⥤ H
α : F₀ ⟶ L ⋙ F₁
hα : StructuredArrow.IsUniversal (mk F₁ α)
b : L'.LeftExtension F₁
hb : StructuredArrow.IsUniversal ((precomp₂ L' α).obj b)
y : L'.LeftExtension F₁
u : (precomp₂ L' α).obj b ⟶ (precomp₂ L' α).obj y := ⋯
⊢ StructuredArrow.hom (mk F₁ α) ≫
      ((whiskeringLeft C D H).obj L).map
        (StructuredArrow.hom b ≫ ((whiskeringLeft D D' H).obj L').map (StructuredArrow.Hom.right u)) =
    StructuredArrow.hom (mk F₁ α) ≫ ((whiskeringLeft C D H).obj L).map (StructuredArrow.hom y)

Tactic:
ext t

Goal state after:
C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} C'
inst✝² : Category.{v_3, u_3} H
inst✝¹ : Category.{v_4, u_4} D
inst✝ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : D ⥤ D'
F₀ : C ⥤ H
F₁ : D ⥤ H
F₂ : D' ⥤ H
α : F₀ ⟶ L ⋙ F₁
hα : StructuredArrow.IsUniversal (mk F₁ α)
b : L'.LeftExtension F₁
hb : StructuredArrow.IsUniversal ((precomp₂ L' α).obj b)
y : L'.LeftExtension F₁
u : (precomp₂ L' α).obj b ⟶ (precomp₂ L' α).obj y := IsInitial.to hb ((precomp₂ L' α).obj y)
t : C
⊢ (StructuredArrow.hom (mk F₁ α) ≫
          ((whiskeringLeft C D H).obj L).map
            (StructuredArrow.hom b ≫ ((whiskeringLeft D D' H).obj L').map (StructuredArrow.Hom.right u))).app
      t =
    (StructuredArrow.hom (mk F₁ α) ≫ ((whiskeringLeft C D H).obj L).map (StructuredArrow.hom y)).app t
Module: Mathlib.CategoryTheory.Functor.KanExtension.Basic

Goal state before:
pp: C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} C'
inst✝² : Category.{v_3, u_3} H
inst✝¹ : Category.{v_4, u_4} D
inst✝ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : D ⥤ D'
F₀ : C ⥤ H
F₁ : D ⥤ H
F₂ : D' ⥤ H
α : F₀ ⟶ L ⋙ F₁
hα : StructuredArrow.IsUniversal (mk F₁ α)
b : L'.LeftExtension F₁
hb : StructuredArrow.IsUniversal ((precomp₂ L' α).obj b)
y : L'.LeftExtension F₁
u : (precomp₂ L' α).obj b ⟶ (precomp₂ L' α).obj y := ⋯
a : b ⟶ y
⊢ StructuredArrow.hom ((precomp₂ L' α).obj b) ≫
      ((whiskeringLeft C D' H).obj (L ⋙ L')).map (StructuredArrow.Hom.right a) =
    StructuredArrow.hom ((precomp₂ L' α).obj b) ≫
      ((whiskeringLeft C D' H).obj (L ⋙ L')).map (StructuredArrow.Hom.right default)

Tactic:
ext t

Goal state after:
C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} C'
inst✝² : Category.{v_3, u_3} H
inst✝¹ : Category.{v_4, u_4} D
inst✝ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : D ⥤ D'
F₀ : C ⥤ H
F₁ : D ⥤ H
F₂ : D' ⥤ H
α : F₀ ⟶ L ⋙ F₁
hα : StructuredArrow.IsUniversal (mk F₁ α)
b : L'.LeftExtension F₁
hb : StructuredArrow.IsUniversal ((precomp₂ L' α).obj b)
y : L'.LeftExtension F₁
u : (precomp₂ L' α).obj b ⟶ (precomp₂ L' α).obj y := IsInitial.to hb ((precomp₂ L' α).obj y)
a : b ⟶ y
t : C
⊢ (StructuredArrow.hom ((precomp₂ L' α).obj b) ≫
          ((whiskeringLeft C D' H).obj (L ⋙ L')).map (StructuredArrow.Hom.right a)).app
      t =
    (StructuredArrow.hom ((precomp₂ L' α).obj b) ≫
          ((whiskeringLeft C D' H).obj (L ⋙ L')).map (StructuredArrow.Hom.right default)).app
      t
Module: Mathlib.CategoryTheory.Comma.StructuredArrow.Small

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : D
T : C ⥤ D
P : ObjectProperty C
inst✝¹ : ObjectProperty.Small.{v₁, v₁, u₁} P
inst✝ : LocallySmall.{v₁, v₂, u₂} D
this : P.inverseImage (proj S T) = ObjectProperty.ofObj fun f ↦ mk f.snd
⊢ ObjectProperty.Small.{v₁, v₁, max u₁ v₂} (P.inverseImage (proj S T))

Tactic:
by
    rw [this]
    infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Comma.StructuredArrow.Small

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : C ⥤ D
T : D
P : ObjectProperty C
inst✝¹ : ObjectProperty.Small.{v₁, v₁, u₁} P
inst✝ : LocallySmall.{v₁, v₂, u₂} D
this : P.inverseImage (proj S T) = ObjectProperty.ofObj fun f ↦ mk f.snd
⊢ ObjectProperty.Small.{v₁, v₁, max u₁ v₂} (P.inverseImage (proj S T))

Tactic:
by
    rw [this]
    infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Generator.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
P : ObjectProperty C
hP : P.IsSeparating
X : C
inst✝ : HasCoproduct (P.coproductFromFamily X)
Z✝ : C
u v : X ⟶ Z✝
huv : P.coproductFrom X ≫ u = P.coproductFrom X ≫ v
G : C
hG : P G
h : G ⟶ X
⊢ h ≫ u = h ≫ v

Tactic:
by simpa using P.ιCoproductFrom h hG ≫= huv

Goal state after:
Module: Mathlib.CategoryTheory.Adjunction.AdjointFunctorTheorems

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
G : D ⥤ C
inst✝ : G.IsRightAdjoint
A : C
B : D
h : A ⟶ G.obj B
⊢ (fun x ↦ (Adjunction.ofIsRightAdjoint G).unit.app A) PUnit.unit ≫
      G.map (((Adjunction.ofIsRightAdjoint G).homEquiv A B).symm h) =
    h

Tactic:
rw [← Adjunction.homEquiv_unit, Equiv.apply_symm_apply]

Goal state after:
Module: Mathlib.CategoryTheory.Adjunction.AdjointFunctorTheorems

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
G : D ⥤ C
inst✝ : G.IsRightAdjoint
⊢ SolutionSetCondition G

Tactic:
by
  intro A
  refine
    ⟨PUnit, fun _ => G.leftAdjoint.obj A, fun _ => (Adjunction.ofIsRightAdjoint G).unit.app A, ?_⟩
  intro B h
  refine ⟨PUnit.unit, ((Adjunction.ofIsRightAdjoint G).homEquiv _ _).symm h, ?_⟩
  rw [← Adjunction.homEquiv_unit, Equiv.apply_symm_apply]

Goal state after:
Module: Mathlib.CategoryTheory.Generator.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
G : Cᵒᵖ
⊢ IsCodetector (unop G) ↔ IsDetector G

Tactic:
by
  rw [IsDetector, IsCodetector, ← isCodetecting_unop_iff, unop_singleton]

Goal state after:
Module: Mathlib.CategoryTheory.Generator.Basic

Goal state before:
pp: case refine_2
C : Type u₁
inst✝ : Category.{v₁, u₁} C
G : C
h : (yoneda.obj G).ReflectsIsomorphisms
X Y : C
f : X ⟶ Y
hf : ∀ (h : X ⟶ G), ∃! h', f ≫ h' = h
⊢ IsIso f

Tactic:
← isIso_op_iff

Goal state after:
case refine_2
C : Type u₁
inst✝ : Category.{v₁, u₁} C
G : C
h : (yoneda.obj G).ReflectsIsomorphisms
X Y : C
f : X ⟶ Y
hf : ∀ (h : X ⟶ G), ∃! h', f ≫ h' = h
⊢ IsIso f.op
Module: Mathlib.Data.ZMod.IntUnitsPower

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R (Additive ℤˣ)
⊢ Int.instUnitsPow = NPow.toPow

Tactic:
by with_implicit rfl

Goal state after: