mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 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]
⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝[<] x ⊔ 𝓝[≥] x, toIcoDiv hp a x_1 = toIcoDiv hp a x

Tactic:
Filter.eventually_sup

Goal state after:
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]
⊢ (∀ᶠ (x_1 : 𝕜) in 𝓝[<] x, toIcoDiv hp a x_1 = toIcoDiv hp a x) ∧
    ∀ᶠ (x_1 : 𝕜) in 𝓝[≥] x, toIcoDiv hp a x_1 = toIcoDiv hp a x
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 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]
⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝[≤] x ⊔ 𝓝[>] x, toIocDiv hp a x_1 = toIocDiv hp a x

Tactic:
Filter.eventually_sup

Goal state after:
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]
⊢ (∀ᶠ (x_1 : 𝕜) in 𝓝[≤] x, toIocDiv hp a x_1 = toIocDiv hp a x) ∧
    ∀ᶠ (x_1 : 𝕜) in 𝓝[>] x, toIocDiv hp a x_1 = toIocDiv hp a x
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
this : ∀ x ∈ s, ∃ y, x ⋖ y
⊢ {x | ∃ y, x ⋖ y}.Countable

Tactic:
choose! y hy using this

Goal state after:
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
⊢ {x | ∃ y, x ⋖ y}.Countable
Module: Mathlib.Algebra.Module.CharacterModule

Goal state before:
pp: case neg
A : Type uA
inst✝ : AddCommGroup A
a : A
n : ℤ
h✝ : ¬addOrderOf a = 0
hn : 1 = addOrderOf a
⊢ a = 0

Tactic:
rwa [eq_comm, AddMonoid.addOrderOf_eq_one_iff] at hn

Goal state after:
Module: Mathlib.Algebra.Category.Grp.LeftExactFunctor

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasFiniteBiproducts C
F : C ⥤ AddCommGrpCat
inst✝ : PreservesFiniteLimits F
X : C
this✝¹ : (F ⋙ forget AddCommGrpCat).Braided := Functor.Braided.ofChosenFiniteProducts (F ⋙ forget AddCommGrpCat)
this✝ : F.Monoidal := Functor.Monoidal.ofChosenFiniteProducts F
this : F.Additive
x✝ :
  (commGrpTypeEquivalenceCommGrp.inverse.obj (toCommGrp.obj (F.obj X))).X ⊗
    (commGrpTypeEquivalenceCommGrp.inverse.obj (toCommGrp.obj (F.obj X))).X
⊢ (ConcreteCategory.hom (μ ≫ Multiplicative.toAdd.toIso.hom)).toFun x✝ =
    (ConcreteCategory.hom
          ((Multiplicative.toAdd.toIso.hom ⊗ₘ Multiplicative.toAdd.toIso.hom) ≫
            Functor.LaxMonoidal.μ (forget AddCommGrpCat) (F.obj X) (F.obj X) ≫
              (forget AddCommGrpCat).map
                (SemiCartesianMonoidalCategory.fst (F.obj X) (F.obj X) +
                  SemiCartesianMonoidalCategory.snd (F.obj X) (F.obj X)))).toFun
      x✝

Tactic:
simp only [types_tensorObj_def, TypeCat.Fun.toFun_apply, CategoryTheory.comp_apply,
    Equiv.toIso_hom_hom_apply, Functor.comp_obj, hom_add, tensor_apply, TypeCat.hom_ofHom,
    TypeCat.Fun.coe_mk, AddMonoidHom.add_apply]

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasFiniteBiproducts C
F : C ⥤ AddCommGrpCat
inst✝ : PreservesFiniteLimits F
X : C
this✝¹ : (F ⋙ forget AddCommGrpCat).Braided := Functor.Braided.ofChosenFiniteProducts (F ⋙ forget AddCommGrpCat)
this✝ : F.Monoidal := Functor.Monoidal.ofChosenFiniteProducts F
this : F.Additive
x✝ :
  (commGrpTypeEquivalenceCommGrp.inverse.obj (toCommGrp.obj (F.obj X))).X ⊗
    (commGrpTypeEquivalenceCommGrp.inverse.obj (toCommGrp.obj (F.obj X))).X
⊢ Multiplicative.toAdd ((ConcreteCategory.hom μ) x✝) =
    (Hom.hom (SemiCartesianMonoidalCategory.fst (F.obj X) (F.obj X)))
        ((ConcreteCategory.hom (Functor.LaxMonoidal.μ (forget AddCommGrpCat) (F.obj X) (F.obj X)))
          (Multiplicative.toAdd x✝.1, Multiplicative.toAdd x✝.2)) +
      (Hom.hom (SemiCartesianMonoidalCategory.snd (F.obj X) (F.obj X)))
        ((ConcreteCategory.hom (Functor.LaxMonoidal.μ (forget AddCommGrpCat) (F.obj X) (F.obj X)))
          (Multiplicative.toAdd x✝.1, Multiplicative.toAdd x✝.2))
Module: Mathlib.CategoryTheory.Abelian.Refinements

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
A : C
γ : A ⟶ S.homology
A' : C
π : A' ⟶ A
hπ : Epi π
z : A' ⟶ S.cycles
hz : π ≫ γ = z ≫ S.homologyπ
⊢ π ≫ γ = S.liftCycles (z ≫ S.iCycles) ⋯ ≫ S.homologyπ

Tactic:
hz

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex C
A : C
γ : A ⟶ S.homology
A' : C
π : A' ⟶ A
hπ : Epi π
z : A' ⟶ S.cycles
hz : π ≫ γ = z ≫ S.homologyπ
⊢ z ≫ S.homologyπ = S.liftCycles (z ≫ S.iCycles) ⋯ ≫ S.homologyπ
Module: Mathlib.Algebra.Category.ModuleCat.Adjunctions

Goal state before:
pp: R : Type u
inst✝ : CommRing R
⊢ ↟(Finsupp.lapply PUnit.unit) ≫ ↟(Finsupp.lsingle PUnit.unit) = 𝟙 ((free R).obj (𝟙_ (Type u)))

Tactic:
ext ⟨⟩

Goal state after:
R : Type u
inst✝ : CommRing R
⊢ (ConcreteCategory.hom (↟(Finsupp.lapply PUnit.unit) ≫ ↟(Finsupp.lsingle PUnit.unit))) (freeMk PUnit.unit) =
    (ConcreteCategory.hom (𝟙 ((free R).obj (𝟙_ (Type u))))) (freeMk PUnit.unit)
Module: Mathlib.CategoryTheory.Monad.Adjunction

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
adj : L ⊣ R
j : R ⋙ L ≅ 𝟭 D
⊢ ((adj.toComonad.transport j).δ ≫ j.inv) ≫ adj.counit = 𝟙 (𝟭 D)

Tactic:
rw [assoc]

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
adj : L ⊣ R
j : R ⋙ L ≅ 𝟭 D
⊢ (adj.toComonad.transport j).δ ≫ j.inv ≫ adj.counit = 𝟙 (𝟭 D)
Module: Mathlib.CategoryTheory.Limits.Shapes.Reflexive

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : WalkingReflexivePair
⊢ Nonempty (StructuredArrow X inclusionWalkingReflexivePair)

Tactic:
cases X with
  | zero => exact ⟨StructuredArrow.mk (Y := one) (𝟙 _)⟩
  | one => exact ⟨StructuredArrow.mk (Y := zero) (𝟙 _)⟩

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

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : CreatesLimitOfIsCosplitPair F
I : ∀ {A B : C} (f g : A ⟶ B) [F.IsCosplitPair f g], HasLimit (parallelPair f g ⋙ F)
this✝ : HasEqualizerOfIsCosplitPair F
this : PreservesLimitOfIsCosplitPair F
⊢ ComonadicLeftAdjoint F

Tactic:
have : ReflectsLimitOfIsCosplitPair F := ⟨by intros; infer_instance⟩

Goal state after:
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : CreatesLimitOfIsCosplitPair F
I : ∀ {A B : C} (f g : A ⟶ B) [F.IsCosplitPair f g], HasLimit (parallelPair f g ⋙ F)
this✝¹ : HasEqualizerOfIsCosplitPair F
this✝ : PreservesLimitOfIsCosplitPair F
this : ReflectsLimitOfIsCosplitPair F
⊢ ComonadicLeftAdjoint F
Module: Mathlib.RingTheory.Flat.Stability

Goal state before:
pp: R : Type u
S : Type v
M : Type w
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : Flat R S
inst✝ : Flat S M
⊢ ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P),
    Injective ⇑(LinearMap.lTensor M N.subtype)

Tactic:
introv

Goal state after:
R : Type u
S : Type v
M : Type w
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : Module S M
inst✝⁴ : IsScalarTower R S M
inst✝³ : Flat R S
inst✝² : Flat S M
P : Type u
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
N : Submodule R P
⊢ Injective ⇑(LinearMap.lTensor M N.subtype)
Module: Mathlib.RingTheory.Ideal.Over

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
p : Ideal R
P : Ideal S
inst✝² : Algebra R S
inst✝¹ : Algebra (R ⧸ p) (S ⧸ P)
inst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)
h : Function.Injective ⇑(algebraMap (R ⧸ p) (S ⧸ P))
x : R
⊢ (algebraMap R S) x ∈ P ↔ x ∈ p

Tactic:
← Quotient.eq_zero_iff_mem,

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
p : Ideal R
P : Ideal S
inst✝² : Algebra R S
inst✝¹ : Algebra (R ⧸ p) (S ⧸ P)
inst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)
h : Function.Injective ⇑(algebraMap (R ⧸ p) (S ⧸ P))
x : R
⊢ (Quotient.mk P) ((algebraMap R S) x) = 0 ↔ x ∈ p
Module: Mathlib.RingTheory.Ideal.Over

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
p : Ideal R
P : Ideal S
inst✝² : Algebra R S
inst✝¹ : Algebra (R ⧸ p) (S ⧸ P)
inst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)
h : Function.Injective ⇑(algebraMap (R ⧸ p) (S ⧸ P))
x : R
⊢ (Quotient.mk P) ((algebraMap R S) x) = 0 ↔ x ∈ p

Tactic:
← Quotient.eq_zero_iff_mem,

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
p : Ideal R
P : Ideal S
inst✝² : Algebra R S
inst✝¹ : Algebra (R ⧸ p) (S ⧸ P)
inst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)
h : Function.Injective ⇑(algebraMap (R ⧸ p) (S ⧸ P))
x : R
⊢ (Quotient.mk P) ((algebraMap R S) x) = 0 ↔ (Quotient.mk p) x = 0
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 N₁ N₂ : Submodule R M
S S₁ S₂ : Set M
P : Submodule R M
r s : R
hr : r ∈ N.colon ↑P
p : M
hp : p ∈ (r * s) • ↑P
⊢ p ∈ ↑N

Tactic:
obtain ⟨p, hp, rfl⟩ := hp

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N N₁ N₂ : Submodule R M
S S₁ S₂ : Set M
P : Submodule R M
r s : R
hr : r ∈ N.colon ↑P
p : M
hp : p ∈ ↑P
⊢ (fun x ↦ (r * s) • x) p ∈ ↑N
Module: Mathlib.RingTheory.Ideal.Colon

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
⊢ Module.annihilator R (M ⧸ N) = N.colon Set.univ

Tactic:
by
  ext r
  have htop : (⊤ : Submodule R (M ⧸ N)) = (⊤ : Submodule R M).map N.mkQ := by
    simpa [map_top] using (LinearMap.range_eq_top.mpr (mkQ_surjective N)).symm
  rw [← annihilator_top (R := R) (M := M ⧸ N), htop,
    annihilator_map_mkQ_eq_colon (N := N) (P := ⊤), Submodule.top_coe]

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✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid 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
⊢ IsUnit ((algebraMap R (Module.End R ↥(localized₀ p f M'))) ↑x)

Tactic:
by
    simp_rw [Module.End.isUnit_iff]
    constructor
    · exact fun _ _ e ↦ Subtype.ext
        (IsLocalizedModule.smul_injective f x (congr_arg Subtype.val e))
    · rintro ⟨_, m, hm, s, rfl⟩
      refine ⟨⟨IsLocalizedModule.mk' f m (s * x), ⟨_, hm, _, rfl⟩⟩, Subtype.ext ?_⟩
      rw [Module.algebraMap_end_apply, SetLike.val_smul_of_tower,
        ← IsLocalizedModule.mk'_smul, ← Submonoid.smul_def, IsLocalizedModule.mk'_cancel_right]

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

Goal state before:
pp: case refine_1
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
⊢ I ≤ comap (algebraMap R S) P

Tactic:
← Ideal.under_def,

Goal state after:
case refine_1
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
⊢ I ≤ under R P
Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent

Goal state before:
pp: case neg.zero
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I
h₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J
S : Type u_1
inst✝ : CommRing S
I : Ideal S
hI' : ¬I = ⊥
H : ∀ m < 0, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S), I ^ m = ⊥ → P I
hI : I ^ 0 = ⊥
⊢ P I

Tactic:
rw [pow_zero, Ideal.one_eq_top] at hI
    have := subsingleton_of_bot_eq_top hI.symm
    exact (hI' (Subsingleton.elim _ _)).elim

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

Goal state before:
pp: case refine_1
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
⊢ I ≤ p

Tactic:
exact hI.le

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

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
I J : Ideal R
x : S
i : R
hi : i ∈ I
mi : ↥M
j : R
hj : j ∈ J
mj : ↥M
hi' : x = mk' S i mi
hj' : x = mk' S j mj
m : ↥M
hm : i * (↑m * ↑mj) = j * (↑m * ↑mi)
⊢ x = mk' S (↑⟨i * ↑(m * mj), ⋯⟩) (mi * (m * mj))

Tactic:
Subtype.coe_mk,

Goal state after:
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
I J : Ideal R
x : S
i : R
hi : i ∈ I
mi : ↥M
j : R
hj : j ∈ J
mj : ↥M
hi' : x = mk' S i mi
hj' : x = mk' S j mj
m : ↥M
hm : i * (↑m * ↑mj) = j * (↑m * ↑mi)
⊢ x = mk' S (i * ↑(m * mj)) (mi * (m * mj))
Module: Mathlib.RingTheory.Localization.Algebra

Goal state before:
pp: R✝ : Type u_1
S : Type u_2
P : Type u_3
Q : Type u_4
inst✝²⁹ : CommSemiring R✝
inst✝²⁸ : CommSemiring S
inst✝²⁷ : CommSemiring P
inst✝²⁶ : CommSemiring Q
M✝ : Submonoid R✝
T : Submonoid P
inst✝²⁵ : Algebra R✝ S
inst✝²⁴ : Algebra P Q
inst✝²³ : IsLocalization M✝ S
inst✝²² : IsLocalization T Q
g : R✝ →+* P
R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
⊢ IsLocalization (Submonoid.map (↑f) (algebraMapSubmonoid A M)) Bₚ

Tactic:
← Submonoid.map_coe_toMonoidHom,

Goal state after:
R✝ : Type u_1
S : Type u_2
P : Type u_3
Q : Type u_4
inst✝²⁹ : CommSemiring R✝
inst✝²⁸ : CommSemiring S
inst✝²⁷ : CommSemiring P
inst✝²⁶ : CommSemiring Q
M✝ : Submonoid R✝
T : Submonoid P
inst✝²⁵ : Algebra R✝ S
inst✝²⁴ : Algebra P Q
inst✝²³ : IsLocalization M✝ S
inst✝²² : IsLocalization T Q
g : R✝ →+* P
R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
⊢ IsLocalization (Submonoid.map (↑↑f) (algebraMapSubmonoid A M)) Bₚ
Module: Mathlib.RingTheory.Localization.Algebra

Goal state before:
pp: R✝ : Type u_1
S : Type u_2
P : Type u_3
Q : Type u_4
inst✝²⁹ : CommSemiring R✝
inst✝²⁸ : CommSemiring S
inst✝²⁷ : CommSemiring P
inst✝²⁶ : CommSemiring Q
M✝ : Submonoid R✝
T : Submonoid P
inst✝²⁵ : Algebra R✝ S
inst✝²⁴ : Algebra P Q
inst✝²³ : IsLocalization M✝ S
inst✝²² : IsLocalization T Q
g : R✝ →+* P
R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
r : Rₚ
⊢ (↑↑(map Bₚ f.toRingHom ⋯)).toFun ((algebraMap Rₚ Aₚ) r) = (algebraMap Rₚ Bₚ) r

Tactic:
obtain ⟨a, m, rfl⟩ := IsLocalization.exists_mk'_eq M r

Goal state after:
R✝ : Type u_1
S : Type u_2
P : Type u_3
Q : Type u_4
inst✝²⁹ : CommSemiring R✝
inst✝²⁸ : CommSemiring S
inst✝²⁷ : CommSemiring P
inst✝²⁶ : CommSemiring Q
M✝ : Submonoid R✝
T : Submonoid P
inst✝²⁵ : Algebra R✝ S
inst✝²⁴ : Algebra P Q
inst✝²³ : IsLocalization M✝ S
inst✝²² : IsLocalization T Q
g : R✝ →+* P
R : Type u_5
inst✝²¹ : CommSemiring R
M : Submonoid R
A : Type u_6
inst✝²⁰ : CommSemiring A
inst✝¹⁹ : Algebra R A
B : Type u_7
inst✝¹⁸ : CommSemiring B
inst✝¹⁷ : Algebra R B
Rₚ : Type u_8
inst✝¹⁶ : CommSemiring Rₚ
inst✝¹⁵ : Algebra R Rₚ
inst✝¹⁴ : IsLocalization M Rₚ
Aₚ : Type u_9
inst✝¹³ : CommSemiring Aₚ
inst✝¹² : Algebra R Aₚ
inst✝¹¹ : Algebra A Aₚ
inst✝¹⁰ : IsScalarTower R A Aₚ
inst✝⁹ : IsLocalization (algebraMapSubmonoid A M) Aₚ
Bₚ : Type u_10
inst✝⁸ : CommSemiring Bₚ
inst✝⁷ : Algebra R Bₚ
inst✝⁶ : Algebra B Bₚ
inst✝⁵ : IsScalarTower R B Bₚ
inst✝⁴ : IsLocalization (algebraMapSubmonoid B M) Bₚ
inst✝³ : Algebra Rₚ Aₚ
inst✝² : Algebra Rₚ Bₚ
inst✝¹ : IsScalarTower R Rₚ Aₚ
inst✝ : IsScalarTower R Rₚ Bₚ
f : A →ₐ[R] B
a : R
m : ↥M
⊢ (↑↑(map Bₚ f.toRingHom ⋯)).toFun ((algebraMap Rₚ Aₚ) (mk' Rₚ a m)) = (algebraMap Rₚ Bₚ) (mk' Rₚ a m)
Module: Mathlib.RingTheory.Localization.LocalizationLocalization

Goal state before:
pp: case h
R : Type u_1
inst✝⁸ : CommRing R
M : Submonoid R
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsFractionRing R T
hM : M ≤ nonZeroDivisors R
this : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T
x : S
left✝ : x ∈ ↑(nonZeroDivisorsLeft S)
hx : x ∈ ↑(nonZeroDivisorsRight S)
y : R
s : ↥M
e : x * (algebraMap R S) ↑(y, s).2 = (algebraMap R S) (y, s).1
⊢ ↑⟨x, ⋯⟩ * (algebraMap R S) ↑s ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)

Tactic:
Subtype.coe_mk,

Goal state after:
case h
R : Type u_1
inst✝⁸ : CommRing R
M : Submonoid R
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsFractionRing R T
hM : M ≤ nonZeroDivisors R
this : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T
x : S
left✝ : x ∈ ↑(nonZeroDivisorsLeft S)
hx : x ∈ ↑(nonZeroDivisorsRight S)
y : R
s : ↥M
e : x * (algebraMap R S) ↑(y, s).2 = (algebraMap R S) (y, s).1
⊢ x * (algebraMap R S) ↑s ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)
Module: Mathlib.RingTheory.Localization.LocalizationLocalization

Goal state before:
pp: case h
R : Type u_1
inst✝⁸ : CommRing R
M : Submonoid R
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsFractionRing R T
hM : M ≤ nonZeroDivisors R
this : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T
x : S
left✝ : x ∈ ↑(nonZeroDivisorsLeft S)
hx : x ∈ ↑(nonZeroDivisorsRight S)
y : R
s : ↥M
e : x * (algebraMap R S) ↑(y, s).2 = (algebraMap R S) (y, s).1
z : R
⊢ z * (y, s).1 = 0 → z = 0

Tactic:
hz

Goal state after:
case h
R : Type u_1
inst✝⁸ : CommRing R
M : Submonoid R
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsFractionRing R T
hM : M ≤ nonZeroDivisors R
this : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T
x : S
left✝ : x ∈ ↑(nonZeroDivisorsLeft S)
hx : x ∈ ↑(nonZeroDivisorsRight S)
y : R
s : ↥M
e : x * (algebraMap R S) ↑(y, s).2 = (algebraMap R S) (y, s).1
z : R
hz : z * (y, s).1 = 0
⊢ z = 0
Module: Mathlib.RingTheory.LocalProperties.Exactness

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
s : Set R
hs : Ideal.span s = ⊤
Rᵣ : ↑s → Type u_3
inst✝⁵ : (r : ↑s) → CommSemiring (Rᵣ r)
inst✝⁴ : (r : ↑s) → Algebra R (Rᵣ r)
Sᵣ : ↑s → Type u_4
inst✝³ : (r : ↑s) → CommSemiring (Sᵣ r)
inst✝² : (r : ↑s) → Algebra S (Sᵣ r)
f : R →+* S
inst✝¹ : ∀ (r : ↑s), IsLocalization.Away (↑r) (Rᵣ r)
inst✝ : ∀ (r : ↑s), IsLocalization.Away (f ↑r) (Sᵣ r)
h : ∀ (r : ↑s), Function.Injective ⇑(IsLocalization.Away.map (Rᵣ r) (Sᵣ r) f ↑r)
algInst✝ : Algebra R S := ⋯
this✝³ : (r : ↑s) → Algebra R (Sᵣ r) := ⋯
this✝² : ∀ (r : ↑s), IsScalarTower R S (Sᵣ r)
this✝¹ : ∀ (r : ↑s), IsLocalization.Away ((algebraMap R S) ↑r) (Sᵣ r)
this✝ : (r : ↑s) → Algebra (Rᵣ r) (Sᵣ r) := ⋯
this : ∀ (r : ↑s), IsScalarTower R (Rᵣ r) (Sᵣ r)
⊢ ∀ (r : ↑s),
    Function.Injective
      ⇑((IsLocalizedModule.map (Submonoid.powers ↑r) (Algebra.linearMap R (Rᵣ r))
            (IsScalarTower.toAlgHom R S (Sᵣ r)).toLinearMap)
          (Algebra.linearMap R S))

Tactic:
simpa [IsLocalization.map_linearMap_eq_toLinearMap_mapₐ] using! h

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Exactness

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
s : Set R
hs : Ideal.span s = ⊤
Rᵣ : ↑s → Type u_3
inst✝⁵ : (r : ↑s) → CommSemiring (Rᵣ r)
inst✝⁴ : (r : ↑s) → Algebra R (Rᵣ r)
Sᵣ : ↑s → Type u_4
inst✝³ : (r : ↑s) → CommSemiring (Sᵣ r)
inst✝² : (r : ↑s) → Algebra S (Sᵣ r)
f : R →+* S
inst✝¹ : ∀ (r : ↑s), IsLocalization.Away (↑r) (Rᵣ r)
inst✝ : ∀ (r : ↑s), IsLocalization.Away (f ↑r) (Sᵣ r)
h : ∀ (r : ↑s), Function.Surjective ⇑(IsLocalization.Away.map (Rᵣ r) (Sᵣ r) f ↑r)
algInst✝ : Algebra R S := ⋯
this✝³ : (r : ↑s) → Algebra R (Sᵣ r) := ⋯
this✝² : ∀ (r : ↑s), IsScalarTower R S (Sᵣ r)
this✝¹ : ∀ (r : ↑s), IsLocalization.Away ((algebraMap R S) ↑r) (Sᵣ r)
this✝ : (r : ↑s) → Algebra (Rᵣ r) (Sᵣ r) := ⋯
this : ∀ (r : ↑s), IsScalarTower R (Rᵣ r) (Sᵣ r)
⊢ ∀ (r : ↑s),
    Function.Surjective
      ⇑((IsLocalizedModule.map (Submonoid.powers ↑r) (Algebra.linearMap R (Rᵣ r))
            (IsScalarTower.toAlgHom R S (Sᵣ r)).toLinearMap)
          (Algebra.linearMap R S))

Tactic:
simpa [IsLocalization.map_linearMap_eq_toLinearMap_mapₐ] using! h

Goal state after: