mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝ : Group G
t₁ t₂ : TopologicalSpace G
h₁ : IsTopologicalGroup G
h₂ : IsTopologicalGroup G
⊢ IsTopologicalGroup G

Tactic:
rw [inf_eq_iInf]

Goal state after:
G : Type w
inst✝ : Group G
t₁ t₂ : TopologicalSpace G
h₁ : IsTopologicalGroup G
h₂ : IsTopologicalGroup G
⊢ IsTopologicalGroup G
Module: Mathlib.Topology.Algebra.ContinuousMonoidHom

Goal state before:
pp: A : Type u_2
E : Type u_6
inst✝⁴ : Monoid A
inst✝³ : TopologicalSpace A
inst✝² : CommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousMul E
f : A →ₜ* E
a : A
n : ℕ
ih : (f ^ n) a = f a ^ n
⊢ (f ^ n) a * f a = f a ^ n * f a

Tactic:
ih

Goal state after:
A : Type u_2
E : Type u_6
inst✝⁴ : Monoid A
inst✝³ : TopologicalSpace A
inst✝² : CommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousMul E
f : A →ₜ* E
a : A
n : ℕ
ih : (f ^ n) a = f a ^ n
⊢ f a ^ n * f a = f a ^ n * f a
Module: Mathlib.Topology.Algebra.Field

Goal state before:
pp: α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : T1Space 𝕜
inst✝² : Field 𝕜
inst✝¹ : ContinuousInv₀ 𝕜
inst✝ : ContinuousMul 𝕜
hS : IsPreconnected S
hf : ContinuousOn f S
hg : ContinuousOn g S
hsq : EqOn (f ^ 2) (g ^ 2) S
hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0
y : α
hy : y ∈ S
hy' : f y = g y
x : α
hx : x ∈ S
⊢ f x = g x

Tactic:
by
  rcases hS.eq_or_eq_neg_of_sq_eq hf hg @hsq @hg_ne with (h | h)
  · exact h hx
  · rw [h _, Pi.neg_apply, neg_eq_iff_add_eq_zero, ← two_mul, mul_eq_zero,
      (iff_of_eq (iff_false _)).2 (hg_ne _)] at hy' ⊢ <;> assumption

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: R : Type u_1
𝕜 : Type u_2
inst✝⁵ : NonUnitalNonAssocRing R
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalAddGroup R
norm : R → 𝕜
norm_nonneg : ∀ (x : R), 0 ≤ norm x
norm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y
nhds_basis : (𝓝 0).HasBasis (fun x ↦ 0 < x) fun ε ↦ {x | norm x < ε}
⊢ IsTopologicalRing R

Tactic:
by
  have h0 : ∀ f : R → R, ∀ c ≥ (0 : 𝕜), (∀ x, norm (f x) ≤ c * norm x) →
      Tendsto f (𝓝 0) (𝓝 0) := by
    refine fun f c c0 hf ↦ (nhds_basis.tendsto_iff nhds_basis).2 fun ε ε0 ↦ ?_
    rcases exists_pos_mul_lt ε0 c with ⟨δ, δ0, hδ⟩
    refine ⟨δ, δ0, fun x hx ↦ (hf _).trans_lt ?_⟩
    exact (mul_le_mul_of_nonneg_left (le_of_lt hx) c0).trans_lt hδ
  apply IsTopologicalRing.of_addGroup_of_nhds_zero
  case hmul =>
    refine ((nhds_basis.prod nhds_basis).tendsto_iff nhds_basis).2 fun ε ε0 ↦ ?_
    refine ⟨(1, ε), ⟨one_pos, ε0⟩, fun (x, y) ⟨hx, hy⟩ => ?_⟩
    simp only at *
    calc norm (x * y) ≤ norm x * norm y := norm_mul_le _ _
    _ < ε := (mul_le_of_le_one_left (norm_nonneg _) hx.le).trans_lt hy
  case hmul_left => exact fun x => h0 _ (norm x) (norm_nonneg _) (norm_mul_le x)
  case hmul_right =>
    exact fun y => h0 (· * y) (norm y) (norm_nonneg y) fun x =>
      (norm_mul_le x y).trans_eq (mul_comm _ _)

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n = 0 ∧ c = d ∨ n < 0 ∧ d = 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)

Tactic:
rcases h with h | h
    · simp only [h.left, h.right, zpow_zero, mul_one]
      exact tendsto_const_nhds
    · exact h.2.symm ▸ tendsto_const_mul_zpow_atTop_zero h.1

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.EpiMono

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderQuotients
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hS : S.ShortExact
h₂ : P S.X₂
⊢ P S.X₃

Tactic:
by
  have := hS.epi_g
  exact P.prop_of_epi S.g h₂

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
hx : ¬x ≡ a [PMOD p]
⊢ IsOpen[inst✝¹] {x | (fun x ↦ toIcoMod hp a x = toIocMod hp a x) x}

Tactic:
Ico_eq_locus_Ioc_eq_iUnion_Ioo

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
hx : ¬x ≡ a [PMOD p]
⊢ IsOpen[inst✝¹] (⋃ z, Ioo (a + z • p) (a + p + z • p))
Module: Mathlib.Algebra.Module.CharacterModule

Goal state before:
pp: R : Type uR
inst✝³ : CommRing R
A : Type uA
inst✝² : AddCommGroup A
A' : Type u_1
inst✝¹ : AddCommGroup A'
B : Type uB
inst✝ : AddCommGroup B
a : A
x : ℤ
⊢ x ∈ (LinearMap.toSpanSingleton ℤ A a).ker ↔ x • a = 0

Tactic:
LinearMap.mem_ker,

Goal state after:
R : Type uR
inst✝³ : CommRing R
A : Type uA
inst✝² : AddCommGroup A
A' : Type u_1
inst✝¹ : AddCommGroup A'
B : Type uB
inst✝ : AddCommGroup B
a : A
x : ℤ
⊢ (LinearMap.toSpanSingleton ℤ A a) x = 0 ↔ x • a = 0
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : AddCommGroup 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsOrderedAddMonoid 𝕜
hp : 0 < p
x : 𝕜
hx : 0 < x
⊢ ↑x = 0 ↔ ∃ n, n • p = x

Tactic:
rw [coe_eq_zero_iff]

Goal state after:
𝕜 : Type u_1
inst✝² : AddCommGroup 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsOrderedAddMonoid 𝕜
hp : 0 < p
x : 𝕜
hx : 0 < x
⊢ (∃ n, n • p = x) ↔ ∃ n, n • p = x
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
⊢ QuotientAddGroup.mk '' Ioc a (a + p) = univ

Tactic:
rw [image_eq_range]
  exact (equivIoc p a).symm.range_eq_univ

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
inst✝³ : Field 𝕜
p q : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
inst✝ : FloorRing 𝕜
n : ℤ
x : AddCircle p
hn : ↑n ≠ 0
⊢ (QuotientAddGroup.mk' (zmultiples p)) (n • ((↑n)⁻¹ * ↑((equivIco p 0) x))) = x

Tactic:
← smul_mul_assoc,

Goal state after:
𝕜 : Type u_1
B : Type u_2
inst✝³ : Field 𝕜
p q : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
inst✝ : FloorRing 𝕜
n : ℤ
x : AddCircle p
hn : ↑n ≠ 0
⊢ (QuotientAddGroup.mk' (zmultiples p)) (n • (↑n)⁻¹ * ↑((equivIco p 0) x)) = x
Module: Mathlib.CategoryTheory.Preadditive.CommGrp_

Goal state before:
pp: case refine_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : CartesianMonoidalCategory C
inst✝ : BraidedCategory C
x✝ : CommGrp C
⊢ fst ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X +
      snd ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X =
    MonObj.mul

Tactic:
apply monoidal_hom_ext

Goal state after:
case refine_2.h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : CartesianMonoidalCategory C
inst✝ : BraidedCategory C
x✝ : CommGrp C
⊢ lift (𝟙 ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X) 0 ≫
      (fst ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X +
        snd ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X) =
    lift (𝟙 ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X) 0 ≫ MonObj.mul

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : CartesianMonoidalCategory C
inst✝ : BraidedCategory C
x✝ : CommGrp C
⊢ lift 0 (𝟙 ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X) ≫
      (fst ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X +
        snd ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X) =
    lift 0 (𝟙 ((CommGrp.forget C ⋙ toCommGrp C).obj x✝).X) ≫ MonObj.mul
Module: Mathlib.Algebra.Category.ModuleCat.Subobject

Goal state before:
pp: R : Type u
inst✝² : Ring R
L M N : ModuleCat R
f : L ⟶ M
inst✝¹ : HasImage f
g : underlying.obj (imageSubobject f) ⟶ N
inst✝ : HasCokernel g
x y : ↑N
l : ↑L
w : x = y + (ConcreteCategory.hom g) ((ConcreteCategory.hom (factorThruImageSubobject f)) l)
⊢ (ConcreteCategory.hom (cokernel.π g)) x = (ConcreteCategory.hom (cokernel.π g)) y

Tactic:
subst w
  simp

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
X Y Z : Type u
z : Z
x : X
y : Y
⊢ (Hom.hom ((free R).map (α_ X Y Z).hom))
      ((Hom.hom (μIso R (X ⊗ Y) Z).hom)
        ((Hom.hom ((μIso R X Y).hom ▷ (free R).obj Z))
          ((Hom.hom ((μIso R X Y).inv ▷ (free R).obj Z)) ((Hom.hom (μIso R (X ⊗ Y) Z).inv) (freeMk ((x, y), z)))))) =
    (Hom.hom (μIso R X (Y ⊗ Z)).hom)
      ((Hom.hom ((free R).obj X ◁ (μIso R Y Z).hom))
        ((Hom.hom (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom)
          ((Hom.hom ((μIso R X Y).inv ▷ (free R).obj Z)) ((Hom.hom (μIso R (X ⊗ Y) Z).inv) (freeMk ((x, y), z))))))

Tactic:
rw [μIso_inv_freeMk, MonoidalCategory.whiskerRight_apply, μIso_inv_freeMk,
          MonoidalCategory.whiskerRight_apply, μIso_hom_freeMk_tmul_freeMk,
          μIso_hom_freeMk_tmul_freeMk, free_map_apply,
          CategoryTheory.associator_hom_apply, MonoidalCategory.associator_hom_apply,
          MonoidalCategory.whiskerLeft_apply, μIso_hom_freeMk_tmul_freeMk,
          μIso_hom_freeMk_tmul_freeMk]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
A : C
x₂ : A ⟶ S.P
hx₂ : x₂ ≫ pullback.snd S.L₁.g S.v₀₁.τ₃ = 0
⊢ x₂ ≫ pullback.fst S.L₁.g S.v₀₁.τ₃ ≫ S.L₁.g = 0

Tactic:
pullback.condition,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
A : C
x₂ : A ⟶ S.P
hx₂ : x₂ ≫ pullback.snd S.L₁.g S.v₀₁.τ₃ = 0
⊢ x₂ ≫ pullback.snd S.L₁.g S.v₀₁.τ₃ ≫ S.v₀₁.τ₃ = 0
Module: Mathlib.Algebra.Homology.ShortComplex.SnakeLemma

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : SnakeInput C
A : C
x₂ : A ⟶ S.P
hx₂ : x₂ ≫ pullback.snd S.L₁.g S.v₀₁.τ₃ = 0
A' : C
π : A' ⟶ A
hπ : Epi π
x₁ : A' ⟶ S.L₁.X₁
fac : π ≫ x₂ ≫ pullback.fst S.L₁.g S.v₀₁.τ₃ = x₁ ≫ S.L₁.f
⊢ (π ≫ x₂) ≫ pullback.snd S.L₁.g S.v₀₁.τ₃ = (x₁ ≫ S.L₀'.f) ≫ pullback.snd S.L₁.g S.v₀₁.τ₃

Tactic:
by simp [L₀', hx₂]

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

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u
inst✝⁶ : Category.{v, u} D
inst✝⁵ : Preadditive D
inst✝⁴ : Linear R D
F G : Free R C ⥤ D
inst✝³ : F.Additive
inst✝² : Functor.Linear R F
inst✝¹ : G.Additive
inst✝ : Functor.Linear R G
α : embedding R C ⋙ F ≅ embedding R C ⋙ G
X Y : Free R C
⊢ ∀ (f : X ⟶ Y), F.map f ≫ (α.app Y).hom = (α.app X).hom ≫ G.map f

Tactic:
f

Goal state after:
R : Type u_1
inst✝⁸ : CommRing R
C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u
inst✝⁶ : Category.{v, u} D
inst✝⁵ : Preadditive D
inst✝⁴ : Linear R D
F G : Free R C ⥤ D
inst✝³ : F.Additive
inst✝² : Functor.Linear R F
inst✝¹ : G.Additive
inst✝ : Functor.Linear R G
α : embedding R C ⋙ F ≅ embedding R C ⋙ G
X Y : Free R C
f : X ⟶ Y
⊢ F.map f ≫ (α.app Y).hom = (α.app X).hom ≫ G.map f
Module: Mathlib.Algebra.Category.ModuleCat.Biproducts

Goal state before:
pp: R : Type u
A : Type uA
M : Type uM
B : Type uB
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup A
inst✝⁴ : AddCommGroup B
inst✝³ : AddCommGroup M
inst✝² : Module R A
inst✝¹ : Module R B
inst✝ : Module R M
j : A →ₗ[R] M
g : M →ₗ[R] B
f : M →ₗ[R] A
hg : Function.Surjective ⇑g
exac : j.range = g.ker
h : f ∘ₗ j = LinearMap.id
⊢ Function.Surjective ⇑(↑ULift.moduleEquiv.symm ∘ₗ g ∘ₗ ↑ULift.moduleEquiv)

Tactic:
simpa using hg

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

Goal state before:
pp: X✝ Y✝ : WalkingReflexivePair
⊢ ∀ (f : X✝.Hom Y✝), (Hom.id X✝).comp f = f

Tactic:
f

Goal state after:
X✝ Y✝ : WalkingReflexivePair
f : X✝.Hom Y✝
⊢ (Hom.id X✝).comp f = f
Module: Mathlib.CategoryTheory.Limits.Shapes.Reflexive

Goal state before:
pp: X✝ Y✝ : WalkingReflexivePair
⊢ ∀ (f : X✝.Hom Y✝), f.comp (Hom.id Y✝) = f

Tactic:
f

Goal state after:
X✝ Y✝ : WalkingReflexivePair
f : X✝.Hom Y✝
⊢ f.comp (Hom.id Y✝) = f
Module: Mathlib.CategoryTheory.Limits.Shapes.Reflexive

Goal state before:
pp: W✝ X✝ Y✝ Z✝ : WalkingReflexivePair
⊢ ∀ (f : W✝.Hom X✝) (g : X✝.Hom Y✝) (h : Y✝.Hom Z✝), (f.comp g).comp h = f.comp (g.comp h)

Tactic:
f

Goal state after:
W✝ X✝ Y✝ Z✝ : WalkingReflexivePair
f : W✝.Hom X✝
⊢ ∀ (g : X✝.Hom Y✝) (h : Y✝.Hom Z✝), (f.comp g).comp h = f.comp (g.comp h)
Module: Mathlib.CategoryTheory.Monad.Adjunction

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
inst✝ : Coreflective R
X : (coreflectorAdjunction R).toComonad.Coalgebra
⊢ (coreflectorAdjunction R).counit.app X.A ≫ X.a = 𝟙 ((coreflector R ⋙ R).obj X.A)

Tactic:
dsimp only [Functor.id_obj]
        rw [← (coreflectorAdjunction R).counit_naturality]
        dsimp only [Functor.comp_obj, Adjunction.toMonad_coe]
        rw [counit_obj_eq_map_counit, ← Functor.map_comp, ← Functor.map_comp]
        simpa using congrArg (fun t ↦ R.map ((coreflector R).map t)) X.counit

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

Goal state before:
pp: case succ
C : Type u_1
inst✝ : Category.{v_1, u_1} C
n m : ℕ
F✝ G✝ : ComposableArrows C n
F G : ComposableArrows C (n + 1)
α : F.obj' 0 ⋯ ⟶ G.obj' 0 ⋯
β : F.δ₀ ⟶ G.δ₀
w : F.map' 0 1 homMk₁._proof_4 ⋯ ≫ app' β 0 ⋯ = α ≫ G.map' 0 1 homMk₁._proof_4 ⋯
i : ℕ
hi : i + 1 < n + 1
⊢ (F.map' (i + 1) (i + 1 + 1) ⋯ hi ≫
      match ⟨i + 1 + 1, ⋯⟩ with
      | ⟨0, isLt⟩ => α
      | ⟨i.succ, hi⟩ => app' β i ⋯) =
    (match ⟨i + 1, ⋯⟩ with
      | ⟨0, isLt⟩ => α
      | ⟨i.succ, hi⟩ => app' β i ⋯) ≫
      G.map' (i + 1) (i + 1 + 1) ⋯ hi

Tactic:
exact naturality' β i (i + 1)

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

Goal state before:
pp: X✝ Y✝ Z✝ : WalkingParallelPair
⊢ ∀ (f : X✝ ⟶ Y✝) (g : Y✝ ⟶ Z✝),
    (match Z✝, X✝, f ≫ g with
      | .(one), .(zero), WalkingParallelPairHom.left => Hom.left
      | .(one), .(zero), WalkingParallelPairHom.right => Hom.right
      | x, .(x), WalkingParallelPairHom.id .(x) =>
        Hom.id
          (match x with
          | one => WalkingReflexivePair.zero
          | zero => WalkingReflexivePair.one)) =
      (match Y✝, X✝, f with
        | .(one), .(zero), WalkingParallelPairHom.left => Hom.left
        | .(one), .(zero), WalkingParallelPairHom.right => Hom.right
        | x, .(x), WalkingParallelPairHom.id .(x) =>
          Hom.id
            (match x with
            | one => WalkingReflexivePair.zero
            | zero => WalkingReflexivePair.one)) ≫
        match Z✝, Y✝, g with
        | .(one), .(zero), WalkingParallelPairHom.left => Hom.left
        | .(one), .(zero), WalkingParallelPairHom.right => Hom.right
        | x, .(x), WalkingParallelPairHom.id .(x) =>
          Hom.id
            (match x with
            | one => WalkingReflexivePair.zero
            | zero => WalkingReflexivePair.one)

Tactic:
f

Goal state after:
X✝ Y✝ Z✝ : WalkingParallelPair
f : X✝ ⟶ Y✝
⊢ ∀ (g : Y✝ ⟶ Z✝),
    (match Z✝, X✝, f ≫ g with
      | .(one), .(zero), WalkingParallelPairHom.left => Hom.left
      | .(one), .(zero), WalkingParallelPairHom.right => Hom.right
      | x, .(x), WalkingParallelPairHom.id .(x) =>
        Hom.id
          (match x with
          | one => WalkingReflexivePair.zero
          | zero => WalkingReflexivePair.one)) =
      (match Y✝, X✝, f with
        | .(one), .(zero), WalkingParallelPairHom.left => Hom.left
        | .(one), .(zero), WalkingParallelPairHom.right => Hom.right
        | x, .(x), WalkingParallelPairHom.id .(x) =>
          Hom.id
            (match x with
            | one => WalkingReflexivePair.zero
            | zero => WalkingReflexivePair.one)) ≫
        match Z✝, Y✝, g with
        | .(one), .(zero), WalkingParallelPairHom.left => Hom.left
        | .(one), .(zero), WalkingParallelPairHom.right => Hom.right
        | x, .(x), WalkingParallelPairHom.id .(x) =>
          Hom.id
            (match x with
            | one => WalkingReflexivePair.zero
            | zero => WalkingReflexivePair.one)
Module: Mathlib.RingTheory.Ideal.IsPrimary

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
hi : I.radical.IsMaximal
x y : R
hxy : x * y ∈ I
h : ¬I + span {y} = ⊤
⊢ x ∈ I ∨ y ∈ I.radical

Tactic:
obtain ⟨m, hm, hy⟩ := exists_le_maximal (I + span {y}) h
      rw [add_le_iff, span_singleton_le_iff_mem, ← hm.isPrime.radical_le_iff] at hy
      exact Or.inr (hi.eq_of_le hm.ne_top hy.1 ▸ hy.2)

Goal state after:
Module: Mathlib.RingTheory.Ideal.MinimalPrime.Basic

Goal state before:
pp: case ih.refine_1
R : Type u_1
inst✝¹ : CommSemiring R
I J : Ideal R
inst✝ : J.IsPrime
e : I ≤ J
S : Set (Ideal R)ᵒᵈ := {p | IsPrime p ∧ I ≤ OrderDual.ofDual p}
c : Set (Ideal R)
hc : c ⊆ S
hc' : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
J' : (Ideal R)ᵒᵈ
hJ' : J' ∈ c
⊢ I ≤ OrderDual.ofDual (OrderDual.toDual (sInf c))

Tactic:
OrderDual.ofDual_toDual,

Goal state after:
case ih.refine_1
R : Type u_1
inst✝¹ : CommSemiring R
I J : Ideal R
inst✝ : J.IsPrime
e : I ≤ J
S : Set (Ideal R)ᵒᵈ := {p | IsPrime p ∧ I ≤ OrderDual.ofDual p}
c : Set (Ideal R)
hc : c ⊆ S
hc' : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
J' : (Ideal R)ᵒᵈ
hJ' : J' ∈ c
⊢ I ≤ sInf c
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R : Type u_1
M : Type u_3
N : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : Module R M
inst✝¹ : Module R N
p : Submonoid R
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
x : M
s : ↥p
a✝ : Function.uncurry (IsLocalizedModule.mk' f) (x, s) ∈ ⊤
⊢ Function.uncurry (IsLocalizedModule.mk' f) (x, s) ∈ localized₀ p f ⊤

Tactic:
exact ⟨x, trivial, s, rfl⟩

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι₁ : Sort u_3
f : ι₁ → Submodule R M
ι₂ : Sort u_4
g : ι₂ → Set M
⊢ (⨅ i, f i).colon (⋃ j, g j) = ⨅ i, ⨅ j, (f i).colon (g j)

Tactic:
by
  aesop (add simp mem_colon)

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N₁ N₂ : Submodule R M
S : Set M
h : S ⊆ ↑N₂
⊢ (N₁ ⊓ N₂).colon S = N₁.colon S

Tactic:
by
  aesop (add simp mem_colon)

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
S : Set M
⊢ N.colon S = ⊤ ↔ S ⊆ ↑N

Tactic:
by
  aesop (add simp [mem_colon, Ideal.eq_top_iff_one])

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N₁ N₂ : Submodule R M
S : Set M
⊢ (N₁ ⊓ N₂).colon S = N₁.colon S ⊓ N₂.colon S

Tactic:
by
  aesop (add simp mem_colon)

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set M
ι : Sort u_3
f : ι → Submodule R M
⊢ (⨅ i, f i).colon S = ⨅ i, (f i).colon S

Tactic:
by
  aesop (add simp mem_colon)

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set M
ι : Type u_3
s : Finset ι
f : ι → Submodule R M
⊢ (s.inf f).colon S = s.inf fun i ↦ (f i).colon S

Tactic:
by
  aesop (add simp mem_colon)

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set M
⊢ ⊤.colon S = ⊤

Tactic:
by
  aesop (add simp mem_colon)

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
S₁ S₂ : Set M
⊢ N.colon (S₁ ∪ S₂) = N.colon S₁ ⊓ N.colon S₂

Tactic:
by
  aesop (add simp mem_colon)

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
ι : Sort u_3
f : ι → Set M
⊢ N.colon (⋃ i, f i) = ⨅ i, N.colon (f i)

Tactic:
by
  aesop (add simp mem_colon)

Goal state after:
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
⊢ N.colon ∅ = ⊤

Tactic:
by
  aesop (add simp mem_colon)

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R✝ : Type u_1
S✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
inst✝²¹ : CommSemiring R✝
inst✝²⁰ : CommSemiring S✝
inst✝¹⁹ : AddCommMonoid M✝
inst✝¹⁸ : AddCommMonoid N✝
inst✝¹⁷ : Module R✝ M✝
inst✝¹⁶ : Module R✝ N✝
inst✝¹⁵ : Algebra R✝ S✝
inst✝¹⁴ : Module S✝ N✝
inst✝¹³ : IsScalarTower R✝ S✝ N✝
p✝ : Submonoid R✝
inst✝¹² : IsLocalization p✝ S✝
f✝ : M✝ →ₗ[R✝] N✝
inst✝¹¹ : IsLocalizedModule p✝ f✝
M'✝¹ M'' : Submodule R✝ M✝
R : Type u_5
S : Type u_6
M : Type u_7
N : Type u_8
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M'✝ M' : Submodule R M
x : ↥p
m : N ⧸ localized' S p f M'
⊢ ↑x • IsLocalization.mk' S 1 x • m = m

Tactic:
← smul_assoc,

Goal state after:
R✝ : Type u_1
S✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
inst✝²¹ : CommSemiring R✝
inst✝²⁰ : CommSemiring S✝
inst✝¹⁹ : AddCommMonoid M✝
inst✝¹⁸ : AddCommMonoid N✝
inst✝¹⁷ : Module R✝ M✝
inst✝¹⁶ : Module R✝ N✝
inst✝¹⁵ : Algebra R✝ S✝
inst✝¹⁴ : Module S✝ N✝
inst✝¹³ : IsScalarTower R✝ S✝ N✝
p✝ : Submonoid R✝
inst✝¹² : IsLocalization p✝ S✝
f✝ : M✝ →ₗ[R✝] N✝
inst✝¹¹ : IsLocalizedModule p✝ f✝
M'✝¹ M'' : Submodule R✝ M✝
R : Type u_5
S : Type u_6
M : Type u_7
N : Type u_8
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M'✝ M' : Submodule R M
x : ↥p
m : N ⧸ localized' S p f M'
⊢ (↑x • IsLocalization.mk' S 1 x) • m = m
Module: Mathlib.RingTheory.Ideal.MinimalPrime.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
I p : Ideal R
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : P.LiesOver p
hI : p ∈ I.minimalPrimes
J : Ideal S
hJP : J ≤ P
hJ : map (Quotient.mk (map (algebraMap R S) p)) P ∈ (map (Quotient.mk (map (algebraMap R S) p)) J).minimalPrimes
q : Ideal S
x✝ : (fun q ↦ q.IsPrime ∧ map (algebraMap R S) I ⊔ J ≤ q) q
hqle : q ≤ P
left✝ : q.IsPrime
hleq : map (algebraMap R S) I ≤ q ∧ J ≤ q
⊢ map (algebraMap R S) p ≤ q

Tactic:
Ideal.map_le_iff_le_comap

Goal state after:
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
I p : Ideal R
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : P.LiesOver p
hI : p ∈ I.minimalPrimes
J : Ideal S
hJP : J ≤ P
hJ : map (Quotient.mk (map (algebraMap R S) p)) P ∈ (map (Quotient.mk (map (algebraMap R S) p)) J).minimalPrimes
q : Ideal S
x✝ : (fun q ↦ q.IsPrime ∧ map (algebraMap R S) I ⊔ J ≤ q) q
hqle : q ≤ P
left✝ : q.IsPrime
hleq : map (algebraMap R S) I ≤ q ∧ J ≤ q
⊢ p ≤ comap (algebraMap R S) q