mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Sites.CoverPreserving

Goal state before:
pp: case hunique
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
hF₁ : CompatiblePreserving K F
hF₂ : CoverPreserving J K F
G : Sheaf K (Type (max u₁ v₁ u₂ v₂))
X : C
S : Sieve X
hS : S ∈ J X
x : FamilyOfElements (F.op ⋙ G.obj) S.arrows
hx : x.Compatible
y₁ y₂ : (F.op ⋙ G.obj).obj (op X)
hy₁ : x.IsAmalgamation y₁
hy₂ : x.IsAmalgamation y₂
⊢ y₁ = y₂

Tactic:
apply (((isSheaf_iff_isSheaf_of_type _ _).1 G.2).isSeparated _ (hF₂.cover_preserve hS)).ext

Goal state after:
case hunique
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
hF₁ : CompatiblePreserving K F
hF₂ : CoverPreserving J K F
G : Sheaf K (Type (max u₁ v₁ u₂ v₂))
X : C
S : Sieve X
hS : S ∈ J X
x : FamilyOfElements (F.op ⋙ G.obj) S.arrows
hx : x.Compatible
y₁ y₂ : (F.op ⋙ G.obj).obj (op X)
hy₁ : x.IsAmalgamation y₁
hy₂ : x.IsAmalgamation y₂
⊢ ∀ ⦃Y : D⦄ ⦃f : Y ⟶ F.obj X⦄,
    (Sieve.functorPushforward F S).arrows f →
      (ConcreteCategory.hom (G.obj.map f.op)) y₁ = (ConcreteCategory.hom (G.obj.map f.op)) y₂
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
ℱ : Dᵒᵖ ⥤ Type v
ℱ' : Sheaf K (Type v)
α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
X : D
x : ℱ.obj (op X)
⊢ ∀ {Z : D} {W₁ W₂ : C} (iWX₁ : G.obj W₁ ⟶ X) (iWX₂ : G.obj W₂ ⟶ X) (iZW₁ : Z ⟶ G.obj W₁) (iZW₂ : Z ⟶ G.obj W₂),
    iZW₁ ≫ iWX₁ = iZW₂ ≫ iWX₂ →
      (ConcreteCategory.hom (ℱ'.obj.map iZW₁.op))
          ((ConcreteCategory.hom (α.app (op W₁))) ((ConcreteCategory.hom (ℱ.map iWX₁.op)) x)) =
        (ConcreteCategory.hom (ℱ'.obj.map iZW₂.op))
          ((ConcreteCategory.hom (α.app (op W₂))) ((ConcreteCategory.hom (ℱ.map iWX₂.op)) x))

Tactic:
introv e

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
ℱ : Dᵒᵖ ⥤ Type v
ℱ' : Sheaf K (Type v)
α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
X : D
x : ℱ.obj (op X)
Z : D
W₁ W₂ : C
iWX₁ : G.obj W₁ ⟶ X
iWX₂ : G.obj W₂ ⟶ X
iZW₁ : Z ⟶ G.obj W₁
iZW₂ : Z ⟶ G.obj W₂
e : iZW₁ ≫ iWX₁ = iZW₂ ≫ iWX₂
⊢ (ConcreteCategory.hom (ℱ'.obj.map iZW₁.op))
      ((ConcreteCategory.hom (α.app (op W₁))) ((ConcreteCategory.hom (ℱ.map iWX₁.op)) x)) =
    (ConcreteCategory.hom (ℱ'.obj.map iZW₂.op))
      ((ConcreteCategory.hom (α.app (op W₂))) ((ConcreteCategory.hom (ℱ.map iWX₂.op)) x))
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁴ : Category.{v_2, u_2} D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{v_4, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ✝ : Dᵒᵖ ⥤ A
ℱ'✝ ℱ ℱ' : Sheaf K A
i : G.op ⋙ ℱ.obj ≅ G.op ⋙ ℱ'.obj
⊢ ℱ.obj ≅ ℱ'.obj

Tactic:
have : ∀ X : Dᵒᵖ, IsIso ((sheafHom i.hom).app X) := by
    intro X
    rw [← isIso_iff_of_reflects_iso _ yoneda]
    use (sheafYonedaHom i.inv).app X
    constructor <;> ext x : 2 <;>
      simp only [sheafHom, NatTrans.comp_app, NatTrans.id_app, Functor.map_preimage]
    · exact ((Types.presheafIso (isoOver i (unop x))).app X).hom_inv_id
    · exact ((Types.presheafIso (isoOver i (unop x))).app X).inv_hom_id
  haveI : IsIso (sheafHom i.hom) := by apply NatIso.isIso_of_isIso_app
  apply asIso (sheafHom i.hom)

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X i' = E.X i

Tactic:
hii'

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X i' = E.X i'
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X i = E.X i'

Tactic:
hii'

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X i' = E.X i'
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X j = E.X j'

Tactic:
by rw [hjj']

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Canonical

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P✝ : Cᵒᵖ ⥤ Type w
X✝ : C
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type w
X : C
S : Sieve X
hS : S ∈ {S | ∀ (Y : C) (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
R : Sieve X
hR :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f → Sieve.pullback f R ∈ {S | ∀ (Y_1 : C) (f : Y_1 ⟶ Y), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
Z : C
g : Z ⟶ X
⊢ Presieve.IsSheafFor P (Sieve.pullback g R).arrows

Tactic:
refine Presieve.isSheafFor_trans P (pullback g S) _ (hS Z g) ?_ ?_

Goal state after:
case refine_1
C : Type u
inst✝ : Category.{v, u} C
P✝ : Cᵒᵖ ⥤ Type w
X✝ : C
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type w
X : C
S : Sieve X
hS : S ∈ {S | ∀ (Y : C) (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
R : Sieve X
hR :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f → Sieve.pullback f R ∈ {S | ∀ (Y_1 : C) (f : Y_1 ⟶ Y), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
Z : C
g : Z ⟶ X
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ Z⦄,
    (Sieve.pullback g R).arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.pullback g S)).arrows

case refine_2
C : Type u
inst✝ : Category.{v, u} C
P✝ : Cᵒᵖ ⥤ Type w
X✝ : C
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type w
X : C
S : Sieve X
hS : S ∈ {S | ∀ (Y : C) (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
R : Sieve X
hR :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f → Sieve.pullback f R ∈ {S | ∀ (Y_1 : C) (f : Y_1 ⟶ Y), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
Z : C
g : Z ⟶ X
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ Z⦄,
    (Sieve.pullback g S).arrows f → Presieve.IsSheafFor P (Sieve.pullback f (Sieve.pullback g R)).arrows
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst✝ : IsDenseSubsite J K G
Y : Sheaf J A
⊢ IsIso ((G.sheafAdjunctionCocontinuous A J K).counit.app Y)

Tactic:
apply +allowSynthFailures ReflectsIsomorphisms.reflects (sheafToPresheaf J A)

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst✝ : IsDenseSubsite J K G
Y : Sheaf J A
⊢ IsIso ((sheafToPresheaf J A).map ((G.sheafAdjunctionCocontinuous A J K).counit.app Y))
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology

Goal state before:
pp: case mpr
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁴ : Category.{v_2, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type v
inst✝³ : Category.{u, v} A
inst✝² : G.LocallyCoverDense K
inst✝¹ : G.IsLocallyFull K
inst✝ : G.IsLocallyFaithful K
X : C
Y : Over X
T : ↑(J ((Over.forget X).obj Y))
Z : C
f : Z ⟶ (Over.forget X).obj Y
hf : (↑T).arrows f
⊢ (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f

Tactic:
exact ⟨Over.mk (f ≫ Y.hom), Over.homMk f, 𝟙 _, hf, (Category.id_comp _).symm⟩

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Equivalence

Goal state before:
pp: case mp
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
K : GrothendieckTopology D
e : C ≌ D
inst✝¹ : e.functor.IsCocontinuous J K
inst✝ : e.inverse.IsCocontinuous K J
X : C
S : Sieve X
H : Sieve.functorPushforward e.functor S ∈ K (e.functor.obj X)
⊢ Sieve.functorPullback e.functor (Sieve.functorPushforward e.functor S) ≤ S

Tactic:
(Sieve.fullyFaithfulFunctorGaloisCoinsertion e.functor X).u_l_eq S

Goal state after:
case mp
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
K : GrothendieckTopology D
e : C ≌ D
inst✝¹ : e.functor.IsCocontinuous J K
inst✝ : e.inverse.IsCocontinuous K J
X : C
S : Sieve X
H : Sieve.functorPushforward e.functor S ∈ K (e.functor.obj X)
⊢ S ≤ S
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
A : Type u_1
inst✝² : Category.{v_1, u_1} A
S : C
E : PreOneHypercover S
F : PreOneHypercover S
G✝ : PreOneHypercover S
inst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
G : PreOneHypercover S
f : G.Hom E
g : G.Hom F
i✝ j✝ : G.I₀
k : G.I₁ i✝ j✝
⊢ f.h₁ k ≫ E.p₁ (f.s₁ k, g.s₁ k).1 ≫ E.f (__spread✝⁻⁰.s₀ i✝).1 =
    g.h₁ k ≫ F.p₁ (f.s₁ k, g.s₁ k).2 ≫ F.f (__spread✝⁻⁰.s₀ i✝).2

Tactic:
by
    rw [f.w₁₁_assoc k, g.w₁₁_assoc k]
    simp

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Opens

Goal state before:
pp: X : TopCat
U : (Opens ↑X)ᵒᵖ
⊢ (map (𝟙 X)).obj (unop U) = unop U

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Topology.Category.TopCat.OpenNhds

Goal state before:
pp: X : TopCat
x : ↑X
U : (OpenNhds x)ᵒᵖ
⊢ (map (𝟙 X) x).obj (unop U) = unop U

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.Sites

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : TopCat
f : X ⟶ Y
F : Presheaf C Y
h : IsOpenEmbedding ⇑(ConcreteCategory.hom f)
hF : F.IsSheaf
⊢ IsSheaf (h.functor.op ⋙ F)

Tactic:
have := h.functor_isContinuous
  exact Functor.op_comp_isSheaf _ _ _ ⟨_, hF⟩

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.Sites

Goal state before:
pp: case refine_1.cone_objs
C : Type u
inst✝ : Category.{v, u} C
X Y : TopCat
f : X ⟶ Y
F : TopCat.Presheaf C Y
U : Opens ↑X
V W : StructuredArrow U (Opens.map f)
⊢ ∃ W_1 x x, True

Tactic:
exact ⟨⟨⟨PUnit.unit⟩, V.right ⊓ W.right, homOfLE <| le_inf V.hom.le W.hom.le⟩,
        StructuredArrow.homMk (homOfLE inf_le_left),
        StructuredArrow.homMk (homOfLE inf_le_right), trivial⟩

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
ι : Type u_2
U : ι → Opens ↑X
V : OpensLeCover U
A B : StructuredArrow V (pairwiseToOpensLeCover U)
⊢ ∃ l, List.IsChain Zag (A :: l) ∧ (A :: l).getLast ⋯ = B

Tactic:
rcases A with ⟨⟨⟨⟩⟩, ⟨i⟩ | ⟨i, j⟩, a⟩

Goal state after:
case single
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
ι : Type u_2
U : ι → Opens ↑X
V : OpensLeCover U
B : StructuredArrow V (pairwiseToOpensLeCover U)
i : ι
a : (Functor.fromPUnit V).obj { as := PUnit.unit } ⟶ (pairwiseToOpensLeCover U).obj (single i)
⊢ ∃ l,
    List.IsChain Zag ({ left := { as := PUnit.unit }, right := single i, hom := a } :: l) ∧
      ({ left := { as := PUnit.unit }, right := single i, hom := a } :: l).getLast ⋯ = B

case pair
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
ι : Type u_2
U : ι → Opens ↑X
V : OpensLeCover U
B : StructuredArrow V (pairwiseToOpensLeCover U)
i j : ι
a : (Functor.fromPUnit V).obj { as := PUnit.unit } ⟶ (pairwiseToOpensLeCover U).obj (Pairwise.pair i j)
⊢ ∃ l,
    List.IsChain Zag ({ left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } :: l) ∧
      ({ left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } :: l).getLast ⋯ = B
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : TopCat
ℱ : Presheaf C X
x : ↑X
⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom

Tactic:
ext
  simp only [stalkPushforward, germ, colim_map, ι_colimMap_assoc, whiskerRight_app]
  erw [CategoryTheory.Functor.map_id]
  simp [stalkFunctor]

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Stalk

Goal state before:
pp: C : Type u_1
inst✝² : SmallCategory C
inst✝¹ : IsFiltered C
R : C ⥤ RingCat
M : C ⥤ Ab
inst✝ : (i : C) → Module ↑(R.obj i) ↑(M.obj i)
H :
  ∀ {i j : C} (f : i ⟶ j) (r : ↑(R.obj i)) (m : ↑(M.obj i)),
    (ConcreteCategory.hom (M.map f)) (r • m) = (ConcreteCategory.hom (R.map f)) r • (ConcreteCategory.hom (M.map f)) m
m : ↑(AddCommGrpCat.FilteredColimits.colimit M)
⊢ ∀ (a : (j : C) × (((M ⋙ forget₂ AddCommGrpCat AddGrpCat) ⋙ forget₂ AddGrpCat AddMonCat) ⋙ forget AddMonCat).obj j),
    1 •
        Quot.mk
          (((M ⋙ forget₂ AddCommGrpCat AddGrpCat) ⋙ forget₂ AddGrpCat AddMonCat) ⋙ forget AddMonCat).ColimitTypeRel a =
      Quot.mk (((M ⋙ forget₂ AddCommGrpCat AddGrpCat) ⋙ forget₂ AddGrpCat AddMonCat) ⋙ forget AddMonCat).ColimitTypeRel
        a

Tactic:
by
    rintro ⟨V, b⟩
    refine Functor.ιColimitType_eq_of_map_eq_map _ _ _ (𝟙 _) (rightToMax _ _) ?_
    dsimp
    simp only [map_one, ← ConcreteCategory.comp_apply, ← Functor.map_comp, one_smul,
      Category.comp_id]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.PiProd

Goal state before:
pp: R : Type u_1
inst✝⁹ : Semiring R
M₁ : Type u_2
inst✝⁸ : TopologicalSpace M₁
inst✝⁷ : AddCommMonoid M₁
inst✝⁶ : Module R M₁
M₂ : Type u_3
inst✝⁵ : TopologicalSpace M₂
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₂
M₃ : Type u_4
inst✝² : TopologicalSpace M₃
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
f : M₁ →L[R] M₂
g : M₁ →L[R] M₃
⊢ (↑(f.prod g)).ker = (↑f).ker ⊓ (↑g).ker

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ModuleTopology

Goal state before:
pp: case mp
R : Type u_1
S : Type u_2
τR : TopologicalSpace R
τS : TopologicalSpace S
inst✝⁸ : Semiring R
inst✝⁷ : Semiring S
σ : R →+* S
σ' : S →+* R
inst✝⁶ : RingHomInvPair σ σ'
inst✝⁵ : RingHomInvPair σ' σ
A : Type u_3
inst✝⁴ : AddCommMonoid A
inst✝³ : Module R A
τA : TopologicalSpace A
inst✝² : IsModuleTopology R A
B' : Type u_5
inst✝¹ : AddCommMonoid B'
inst✝ : Module S B'
τB' : TopologicalSpace B'
hσ : Continuous ⇑σ
hσ' : Continuous ⇑σ'
e : A ≃SL[σ] B'
g : A →ₛₗ[σ] B' := ⋯
g' : B' →ₛₗ[σ'] A := ⋯
h : A →+ B' := ⋯
h' : B' →+ A := ⋯
τ : TopologicalSpace B'
⊢ (∃ x ∈ {t | ContinuousSMul R A ∧ ContinuousAdd A}, TopologicalSpace.induced (⇑e.toHomeomorph.symm) x = τ) →
    τ ∈ {t | ContinuousSMul S B' ∧ ContinuousAdd B'}

pp: case mpr
R : Type u_1
S : Type u_2
τR : TopologicalSpace R
τS : TopologicalSpace S
inst✝⁸ : Semiring R
inst✝⁷ : Semiring S
σ : R →+* S
σ' : S →+* R
inst✝⁶ : RingHomInvPair σ σ'
inst✝⁵ : RingHomInvPair σ' σ
A : Type u_3
inst✝⁴ : AddCommMonoid A
inst✝³ : Module R A
τA : TopologicalSpace A
inst✝² : IsModuleTopology R A
B' : Type u_5
inst✝¹ : AddCommMonoid B'
inst✝ : Module S B'
τB' : TopologicalSpace B'
hσ : Continuous ⇑σ
hσ' : Continuous ⇑σ'
e : A ≃SL[σ] B'
g : A →ₛₗ[σ] B' := ⋯
g' : B' →ₛₗ[σ'] A := ⋯
h : A →+ B' := ⋯
h' : B' →+ A := ⋯
τ : TopologicalSpace B'
⊢ τ ∈ {t | ContinuousSMul S B' ∧ ContinuousAdd B'} →
    ∃ x ∈ {t | ContinuousSMul R A ∧ ContinuousAdd A}, TopologicalSpace.induced (⇑e.toHomeomorph.symm) x = τ

Tactic:
· rintro ⟨σ, ⟨hσ1, hσ2⟩, rfl⟩
      exact ⟨continuousSMul_inducedₛₗ g' hσ', continuousAdd_induced h'⟩

Goal state after:
case mpr
R : Type u_1
S : Type u_2
τR : TopologicalSpace R
τS : TopologicalSpace S
inst✝⁸ : Semiring R
inst✝⁷ : Semiring S
σ : R →+* S
σ' : S →+* R
inst✝⁶ : RingHomInvPair σ σ'
inst✝⁵ : RingHomInvPair σ' σ
A : Type u_3
inst✝⁴ : AddCommMonoid A
inst✝³ : Module R A
τA : TopologicalSpace A
inst✝² : IsModuleTopology R A
B' : Type u_5
inst✝¹ : AddCommMonoid B'
inst✝ : Module S B'
τB' : TopologicalSpace B'
hσ : Continuous ⇑σ
hσ' : Continuous ⇑σ'
e : A ≃SL[σ] B'
g : A →ₛₗ[σ] B' := ⋯
g' : B' →ₛₗ[σ'] A := ⋯
h : A →+ B' := ⋯
h' : B' →+ A := ⋯
τ : TopologicalSpace B'
⊢ τ ∈ {t | ContinuousSMul S B' ∧ ContinuousAdd B'} →
    ∃ x ∈ {t | ContinuousSMul R A ∧ ContinuousAdd A}, TopologicalSpace.induced (⇑e.toHomeomorph.symm) x = τ
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic

Goal state before:
pp: R₁ : Type u_1
inst✝³ : Semiring R₁
M₁ : Type u_4
inst✝² : TopologicalSpace M₁
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R₁ M₁
T : M₁ →L[R₁] M₁
hT : IsUnit T
⊢ IsHomeomorph ⇑T

Tactic:
obtain ⟨T, rfl⟩ := hT

Goal state after:
R₁ : Type u_1
inst✝³ : Semiring R₁
M₁ : Type u_4
inst✝² : TopologicalSpace M₁
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R₁ M₁
T : (M₁ →L[R₁] M₁)ˣ
⊢ IsHomeomorph ⇑↑T
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic

Goal state before:
pp: case succ
R : Type u_1
inst✝² : Ring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
c : R
n : ℕ
ihn : toSpanSingleton R c ^ n = toSpanSingleton R (c ^ n)
⊢ toSpanSingleton R c ^ (n + 1) = toSpanSingleton R (c ^ (n + 1))

Tactic:
| succ n ihn =>

Goal state after:
Module: Mathlib.Algebra.Category.Semigrp.Basic

Goal state before:
pp: M N : MagmaCat
e : M ≅ N
x : ↑M
⊢ (ConcreteCategory.hom e.inv) ((ConcreteCategory.hom e.hom) x) = x

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.Category.Semigrp.Basic

Goal state before:
pp: M N : MagmaCat
e : M ≅ N
s : ↑N
⊢ (ConcreteCategory.hom e.hom) ((ConcreteCategory.hom e.inv) s) = s

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.Category.Semigrp.Basic

Goal state before:
pp: X Y : Semigrp
e : X ≅ Y
x : ↑X
⊢ (ConcreteCategory.hom e.inv) ((ConcreteCategory.hom e.hom) x) = x

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.Category.Semigrp.Basic

Goal state before:
pp: X Y : Semigrp
e : X ≅ Y
s : ↑Y
⊢ (ConcreteCategory.hom e.hom) ((ConcreteCategory.hom e.inv) s) = s

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Equiv

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : Semiring R
M : Type u_2
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
M₂ : Type u_3
inst✝⁶ : TopologicalSpace M₂
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
ι : Type u_4
φ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (φ i)
inst✝² : (i : ι) → AddCommMonoid (φ i)
inst✝¹ : (i : ι) → Module R (φ i)
I J : Set ι
inst✝ : DecidablePred fun i ↦ i ∈ I
hd : Disjoint I J
hu : Set.univ ⊆ I ∪ J
i : ι
⊢ Continuous[Pi.topologicalSpace, inst✝³ i] fun a ↦
    (LinearMap.pi fun i ↦ if h : i ∈ I then LinearMap.proj ⟨i, h⟩ else 0) a i

Tactic:
by
        dsimp
        split_ifs <;> [apply continuous_apply; exact continuous_zero]

Goal state after:
Module: Mathlib.CategoryTheory.Comma.CardinalArrow

Goal state before:
pp: C : Type u
inst✝¹ : SmallCategory C
inst✝ : FinCategory C
⊢ Nonempty (FinCategory C)

Tactic:
exact ⟨inferInstance⟩

Goal state after:
Module: Mathlib.CategoryTheory.Limits.MorphismProperty

Goal state before:
pp: case inst
T : Type u_1
inst✝⁷ : Category.{v_1, u_1} T
P : MorphismProperty T
A : Type u_2
inst✝⁶ : Category.{v_2, u_2} A
L : A ⥤ T
X : T
inst✝⁵ : P.IsStableUnderComposition
inst✝⁴ : P.IsStableUnderBaseChange
inst✝³ : P.HasOfPostcompProperty P
inst✝² : HasPullbacks A
inst✝¹ : HasPullbacks T
inst✝ : PreservesLimitsOfShape WalkingCospan L
⊢ (commaObj L (Functor.fromPUnit X) P).IsClosedUnderLimitsOfShape WalkingCospan

Tactic:
exact CostructuredArrow.closedUnderLimitsOfShape_walkingCospan _ _

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.OverAdjunction

Goal state before:
pp: T : Type u_1
inst✝⁷ : Category.{v_1, u_1} T
P Q : MorphismProperty T
inst✝⁶ : Q.IsMultiplicative
X Y Z : T
f : X ⟶ Y
g : Y ⟶ Z
inst✝⁵ : P.IsStableUnderBaseChangeAlong f
inst✝⁴ : P.IsStableUnderBaseChangeAlong g
inst✝³ : P.HasPullbacksAlong f
inst✝² : P.HasPullbacksAlong g
inst✝¹ : Q.RespectsIso
inst✝ : Q.IsStableUnderBaseChange
A : P.Over Q Z
⊢ ((pullbackComp f g (f ≫ g) ⋯).hom.app A).left ≫ pullback.fst (pullback.snd A.hom g) f ≫ pullback.fst A.hom g =
    pullback.fst A.hom (f ≫ g)

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.CharP.MixedCharZero

Goal state before:
pp: case mpr
R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
⊢ (∃ I, I.IsMaximal ∧ CharP (R ⧸ I) p) → ∃ I, I ≠ ⊤ ∧ CharP (R ⧸ I) p

Tactic:
intro ⟨I, hI_max, h_charP⟩
    use I
    exact ⟨Ideal.IsMaximal.ne_top hI_max, h_charP⟩

Goal state after:
Module: Mathlib.Algebra.CharP.MixedCharZero

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
h : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)
p : ℕ
p_pos : p > 0
⊢ ¬MixedCharZero R p

Tactic:
by_contra hp_mixedChar

Goal state after:
R : Type u_1
inst✝ : CommRing R
h : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)
p : ℕ
p_pos : p > 0
hp_mixedChar : MixedCharZero R p
⊢ False
Module: Mathlib.Data.Seq.Computation

Goal state before:
pp: case some
α : Type u
s : Computation α
a val✝ : α
f0 : s.val 0 = some val✝
h :
  (match some val✝ with
    | none => Sum.inr s.tail
    | some a => Sum.inl a) =
    Sum.inl a
n : ℕ
⊢ s.val n = (pure a).val n

Tactic:
induction n with
    | zero => injection h with h'; rwa [h'] at f0
    | succ n IH => exact s.2 IH

Goal state after:
Module: Mathlib.Data.Seq.Computation

Goal state before:
pp: α : Type u
R : Computation α → Computation α → Prop
bisim : IsBisimulation R
s₁ s₂ : Computation α
r✝¹ : R s₁ s₂
t₁ t₂ : Stream' (Option α)
e : ∃ s s', s.val = t₁ ∧ s'.val = t₂ ∧ R s s'
s s' : Computation α
r✝ : R s s'
this : s.head = s'.head ∧ R s.tail s'.tail
r : R s.tail s'.tail
⊢ s.tail.val = s.val.tail

Tactic:
by cases s; rfl

Goal state after:
Module: Mathlib.Data.Stream.Init

Goal state before:
pp: case right
α : Type u
s₁ s' s : Stream' α
h : s' = s.even ⋈ s.odd
⊢ (fun s' s ↦ s' = s.even ⋈ s.odd) (s.even ⋈ s.odd).tail s.tail

Tactic:
simp [odd_eq, odd_eq, tail_interleave, tail_even]

Goal state after:
Module: Mathlib.Data.Stream.Init

Goal state before:
pp: α : Type u
s : Stream' α
⊢ [s.head] ++ₛ s.tail = s

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Data.Stream.Init

Goal state before:
pp: α : Type u
n : ℕ
x : List α
a : Stream' α
h : n ≤ x.length
i✝ : ℕ
h₁✝ : i✝ < (take n (x ++ₛ a)).length
h₂✝ : i✝ < (List.take n x).length
⊢ (take n (x ++ₛ a))[i✝] = (List.take n x)[i✝]

Tactic:
List.getElem_take,

Goal state after:
α : Type u
n : ℕ
x : List α
a : Stream' α
h : n ≤ x.length
i✝ : ℕ
h₁✝ : i✝ < (take n (x ++ₛ a)).length
h₂✝ : i✝ < (List.take n x).length
⊢ (take n (x ++ₛ a))[i✝] = x[i✝]
Module: Mathlib.Data.Seq.Computation

Goal state before:
pp: α : Type u
s : Computation α
h✝ : s.Terminates
a : α
h : s.get = a
⊢ s.get ∈ s

Tactic:
apply get_mem

Goal state after:
Module: Mathlib.Data.Seq.Basic

Goal state before:
pp: case refine_2
α : Type u
s : Seq α
n : ℕ
hn : s.get? n = none
⊢ (∃ n, s.get? n = none) ∧ ∃ m ≤ n, s.get? m = none

Tactic:
exact ⟨⟨n, hn⟩, ⟨n, le_rfl, hn⟩⟩

Goal state after:
Module: Mathlib.Data.Seq.Computation

Goal state before:
pp: case think
α : Type u
β : Type v
f : α → β
s✝ : Computation α
c₁ c₂ : Computation β
h : c₁ = c₂ ∨ ∃ s, c₁ = s.bind (pure ∘ f) ∧ c₂ = map f s
s : Computation α
⊢ BisimO (fun c₁ c₂ ↦ c₁ = c₂ ∨ ∃ s, c₁ = s.bind (pure ∘ f) ∧ c₂ = map f s) (s.think.bind (pure ∘ f)).destruct
    (map f s.think).destruct

Tactic:
simpa using Or.inr ⟨s, rfl, rfl⟩

Goal state after: