mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
this : A ≃ₐ[A] Localization.Away 1
⊢ smoothLocus R A = Set.univ ↔ ↑(PrimeSpectrum.basicOpen 1) ⊆ smoothLocus R A

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Smooth.Pi

Goal state before:
pp: case h
R : Type u_1
I : Type u_2
A : I → Type u_3
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : FormallySmooth R ((i : I) → A i)
i : I
x : A i
⊢ Pi.single i x i = x

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
hx' : (Ideal.Quotient.mk I) ((aeval x) (minpoly K pb.gen)) = 0
⊢ ∃ u, (aeval x) (derivative (minpoly K pb.gen)) * u + 1 ∈ I

Tactic:
by
    have := (isUnit_iff_ne_zero.mpr ((Algebra.IsSeparable.isSeparable K
      pb.gen).aeval_derivative_ne_zero (minpoly.aeval K _))).map f
    rw [← aeval_algHom_apply, ← hx, ← helper, aeval_algHom_apply, helper] at this
    obtain ⟨u, hu⟩ := Ideal.Quotient.mk_surjective (-this.unit⁻¹ : B ⧸ I)
    use u
    rw [← Ideal.Quotient.eq_zero_iff_mem, map_add, map_mul, map_one, hu, mul_neg,
      IsUnit.mul_val_inv, neg_add_cancel]

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsSeparable K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h✝ : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
x y : L
h : x ∈ K⟮y⟯
e : K⟮x⟯ ≤ K⟮y⟯
⊢ ∀ (x_1 : ↥K⟮x⟯),
    (f.comp ((IsScalarTower.toAlgHom K (↥K⟮y⟯) L).comp (IntermediateField.inclusion e))) x_1 =
      (f.comp (IsScalarTower.toAlgHom K (↥K⟮x⟯) L)) x_1

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsSeparable K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x y : L
this✝ : FiniteDimensional K ↥K⟮x, y⟯
this : Algebra.IsSeparable K ↥K⟮x, y⟯
α : L
hα : α ∈ K⟮x, y⟯
e : K⟮α⟯ = K⟮x, y⟯
⊢ x ∈ {x, y}

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsSeparable K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x y : L
this✝ : FiniteDimensional K ↥K⟮x, y⟯
this : Algebra.IsSeparable K ↥K⟮x, y⟯
α : L
hα : α ∈ K⟮x, y⟯
e : K⟮α⟯ = K⟮x, y⟯
hx : x ∈ K⟮α⟯
⊢ y ∈ {x, y}

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
⊢ ∀ (a : A), a • 0 = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
⊢ ∀ (a : A) (x y : S ⊗[R] Ω[A⁄R]), a • (x + y) = a • x + a • y

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Smooth.Local

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : IsLocalRing S
inst✝¹¹ : Algebra R S
P : Type u_3
K : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : CommRing P
inst✝⁸ : Algebra R P
inst✝⁷ : Algebra P S
inst✝⁶ : IsScalarTower R P S
inst✝⁵ : Algebra S K
inst✝⁴ : Algebra P K
inst✝³ : IsScalarTower P S K
inst✝² : FormallySmooth R P
inst✝¹ : Module.Free P Ω[P⁄R]
inst✝ : Module.Finite P Ω[P⁄R]
h₁ : Function.Surjective ⇑(algebraMap P S)
h₂ : (RingHom.ker (algebraMap P S)).FG
h₃ : maximalIdeal S ≤ RingHom.ker (algebraMap S K)
f : ResidueField S →ₐ[S] K := Ideal.Quotient.liftₐ (maximalIdeal S) (ofId S K) h₃
this✝¹ : Algebra (ResidueField S) K := f.toAlgebra
this✝ : IsScalarTower S (ResidueField S) K
this : IsScalarTower P (ResidueField S) K
x✝ : ↥(RingHom.ker (algebraMap P S))
⊢ (1 • 1) ⊗ₜ[P] (D R P) ↑x✝ = 1 ⊗ₜ[P] (D R P) ↑x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : IsSepClosed K
x : K
p : PrimeSpectrum K
this : Field (K ⧸ p.asIdeal) := ⋯
⊢ (AlgEquiv.ofBijective (ofId K (K ⧸ p.asIdeal)) ⋯)
      ((AlgEquiv.ofBijective (ofId K (K ⧸ p.asIdeal)) ⋯).symm ((Ideal.Quotient.mk p.asIdeal) x)) =
    (AlgEquiv.ofBijective (ofId K (K ⧸ p.asIdeal)) ⋯) x

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
A : Type u
inst✝⁹ : Field K
inst✝⁸ : CommRing A
inst✝⁷ : Algebra K A
B : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : EssFiniteType K A
inst✝⁴ : FormallyEtale K A
inst✝³ : Algebra K B
inst✝² : EssFiniteType K B
inst✝¹ : FormallyEtale K B
inst✝ : IsSepClosed K
f : A →ₐ[K] B
x : A
p : PrimeSpectrum B
⊢ ofId K (B ⧸ p.asIdeal) = (Ideal.quotientMapₐ p.asIdeal f ⋯).comp (ofId K (A ⧸ Ideal.comap f p.asIdeal))

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
A : Type u
inst✝⁹ : Field K
inst✝⁸ : CommRing A
inst✝⁷ : Algebra K A
B : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : EssFiniteType K A
inst✝⁴ : FormallyEtale K A
inst✝³ : Algebra K B
inst✝² : EssFiniteType K B
inst✝¹ : FormallyEtale K B
inst✝ : IsSepClosed K
f : A →ₐ[K] B
x : A
p : PrimeSpectrum B
heq : ofId K (B ⧸ p.asIdeal) = (Ideal.quotientMapₐ p.asIdeal f ⋯).comp (ofId K (A ⧸ Ideal.comap f p.asIdeal))
⊢ (Ideal.quotientMapₐ p.asIdeal f ⋯) ((Ideal.Quotient.mk (Ideal.comap f p.asIdeal)) x) =
    (Ideal.Quotient.mk p.asIdeal) (f x)

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case zero
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
⊢ (LinearMap.liftBaseChange S (↑R (map R S A B)))
      ((derivationTensorProduct R S A B).liftKaehlerDifferential ((D S B).tensorProductTo 0)) =
    (D S B).tensorProductTo 0

Tactic:
simp

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
u : R
⊢ W'.dblX (u • P) = u ^ 4 * W'.dblX P

Tactic:
by
  simp only [dblX, smul_fin3_ext]
  ring1

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case zero
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
this : Algebra.IsPushout R A S B
e₁ : B ⊗[A] Ω[A⁄R] ≃ₗ[A] Ω[A⁄R] ⊗[R] S :=
  AlgebraTensorModule.congr (↑(Algebra.IsPushout.equiv R A S B).symm) (LinearEquiv.refl A Ω[A⁄R]) ≪≫ₗ
      TensorProduct.comm A (A ⊗[R] S) Ω[A⁄R] ≪≫ₗ
    AlgebraTensorModule.cancelBaseChange R A A Ω[A⁄R] S
e₂ : B ⊗[A] Ω[A⁄R] ≃ₗ[R] Ω[B⁄S] :=
  LinearEquiv.restrictScalars R e₁ ≪≫ₗ TensorProduct.comm R Ω[A⁄R] S ≪≫ₗ
    LinearEquiv.restrictScalars R (tensorKaehlerEquivBase R S A B)
x : B ⊗[A] Ω[A⁄R]
⊢ e₂ ((Algebra.IsPushout.equiv R A S B) 0 • x) = (Algebra.IsPushout.equiv R A S B) 0 • e₂ x

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.zero
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
this : Algebra.IsPushout R A S B
e₁ : B ⊗[A] Ω[A⁄R] ≃ₗ[A] Ω[A⁄R] ⊗[R] S :=
  AlgebraTensorModule.congr (↑(Algebra.IsPushout.equiv R A S B).symm) (LinearEquiv.refl A Ω[A⁄R]) ≪≫ₗ
      TensorProduct.comm A (A ⊗[R] S) Ω[A⁄R] ≪≫ₗ
    AlgebraTensorModule.cancelBaseChange R A A Ω[A⁄R] S
e₂ : B ⊗[A] Ω[A⁄R] ≃ₗ[R] Ω[B⁄S] :=
  LinearEquiv.restrictScalars R e₁ ≪≫ₗ TensorProduct.comm R Ω[A⁄R] S ≪≫ₗ
    LinearEquiv.restrictScalars R (tensorKaehlerEquivBase R S A B)
a : A
b : S
⊢ e₂ ((Algebra.IsPushout.equiv R A S B) (a ⊗ₜ[R] b) • 0) = (Algebra.IsPushout.equiv R A S B) (a ⊗ₜ[R] b) • e₂ 0

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.tmul.zero
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
this : Algebra.IsPushout R A S B
e₁ : B ⊗[A] Ω[A⁄R] ≃ₗ[A] Ω[A⁄R] ⊗[R] S :=
  AlgebraTensorModule.congr (↑(Algebra.IsPushout.equiv R A S B).symm) (LinearEquiv.refl A Ω[A⁄R]) ≪≫ₗ
      TensorProduct.comm A (A ⊗[R] S) Ω[A⁄R] ≪≫ₗ
    AlgebraTensorModule.cancelBaseChange R A A Ω[A⁄R] S
e₂ : B ⊗[A] Ω[A⁄R] ≃ₗ[R] Ω[B⁄S] :=
  LinearEquiv.restrictScalars R e₁ ≪≫ₗ TensorProduct.comm R Ω[A⁄R] S ≪≫ₗ
    LinearEquiv.restrictScalars R (tensorKaehlerEquivBase R S A B)
a : A
b : S
y : Ω[A⁄R]
⊢ e₂ ((Algebra.IsPushout.equiv R A S B) (a ⊗ₜ[R] b) • (Algebra.IsPushout.equiv R A S B) 0 ⊗ₜ[A] y) =
    (Algebra.IsPushout.equiv R A S B) (a ⊗ₜ[R] b) • e₂ ((Algebra.IsPushout.equiv R A S B) 0 ⊗ₜ[A] y)

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case zero
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
a : A
this : Algebra.IsPushout R A S B
⊢ (tensorKaehlerEquiv R S A B) ((Algebra.IsPushout.equiv R A S B) 0 ⊗ₜ[A] (D R A) a) =
    (Algebra.IsPushout.equiv R A S B) 0 • (D S B) ((algebraMap A B) a)

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing A
inst✝ : Algebra R A
s : S
a : A
⊢ s ⊗ₜ[R] 1 = (algebraMap S (S ⊗[R] A)) s

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
m n : ℕ
hmn : m ≤ n
x : M
⊢ ((transitionMap I N hmn ∘ₗ ↑R (reduceModIdeal (I ^ n) f)) ∘ₗ (I ^ n • ⊤).mkQ) x =
    ((↑R (reduceModIdeal (I ^ m) f) ∘ₗ transitionMap I M hmn) ∘ₗ (I ^ n • ⊤).mkQ) x

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_3
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : Type u_4
inst✝³ : AddCommGroup P
inst✝² : Module R P
T : Type u_5
inst✝¹ : AddCommGroup T
inst✝ : Module (AdicCompletion I R) T
f : M →ₗ[R] N
a b : AdicCauchySequence I M
n : ℕ
⊢ ↑⟨fun n ↦ f (↑(a + b) n), ⋯⟩ n = ↑(⟨fun n ↦ f (↑a n), ⋯⟩ + ⟨fun n ↦ f (↑b n), ⋯⟩) n

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_3
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : Type u_4
inst✝³ : AddCommGroup P
inst✝² : Module R P
T : Type u_5
inst✝¹ : AddCommGroup T
inst✝ : Module (AdicCompletion I R) T
f : M →ₗ[R] N
r : R
a : AdicCauchySequence I M
n : ℕ
⊢ ↑⟨fun n ↦ f (↑(r • a) n), ⋯⟩ n = ↑((RingHom.id R) r • ⟨fun n ↦ f (↑a n), ⋯⟩) n

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
a : AdicCauchySequence I M
n : ℕ
⊢ ↑((map I LinearMap.id) ((mk I M) a)) n = ↑(LinearMap.id ((mk I M) a)) n

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : Type u_4
inst✝¹ : AddCommGroup P
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
a✝ : AdicCauchySequence I M
n✝ : ℕ
⊢ ↑((map I g ∘ₗ map I f) ((mk I M) a✝)) n✝ = ↑((map I (g ∘ₗ f)) ((mk I M) a✝)) n✝

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
a✝ : AdicCauchySequence I M
n✝ : ℕ
⊢ ↑((map I 0) ((mk I M) a✝)) n✝ = ↑(0 ((mk I M) a✝)) n✝

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
ι : Type u_6
M : ι → Type u_7
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
j : ι
x : AdicCauchySequence I (M j)
n : ℕ
⊢ Submodule.Quotient.mk (Eq.recOn ⋯ (↑x n)) = ↑(Eq.recOn ⋯ ((mk I (M j)) x)) n

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: case isFalse
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
ι : Type u_6
M : ι → Type u_7
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
j : ι
x : AdicCauchySequence I (M j)
i : ι
n : ℕ
h✝ : ¬j = i
⊢ Submodule.Quotient.mk 0 = ↑0 n

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : AdicCompletion I R
n m✝ n✝ : ℕ
hmn : m✝ ≤ n✝
⊢ (transitionMap I R hmn) ((↑x ^ n) n✝) = (↑x ^ n) m✝

Tactic:
by simp [x.property, transitionMap_map_pow I hmn]

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
x : R
⊢ x ∈ I ^ n • ⊤ ↔ x ∈ I ^ n

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
n : ℕ
x : AdicCompletion I R
h : I ^ n ≤ I ^ n • ⊤
⊢ (Ideal.Quotient.factor h) ((evalₐ I n) x) = (eval I R n) x

Tactic:
by
  simp [evalₐ]

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
n : ℕ
x : AdicCompletion I R
h : I ^ n • ⊤ ≤ I ^ n
⊢ (factor h) ((eval I R n) x) = (evalₐ I n) x

Tactic:
by
  simp [evalₐ]

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
n : ℕ
x : R
⊢ (evalₐ I n) ((of I R) x) = (Ideal.Quotient.mk (I ^ n)) x

Tactic:
by
  simp [evalₐ]

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
n : ℕ
x : AdicCauchySequence I R
⊢ (evalₐ I n) ((mk I R) x) = (Ideal.Quotient.mk (I ^ n)) (↑x n)

Tactic:
by
  simp [evalₐ]

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
x✝ y : AdicCompletion I R
H : ∀ (n : ℕ), (evalₐ I n) x✝ = (evalₐ I n) y
n : ℕ
x : R
⊢ x ∈ I ^ n • ⊤ ↔ x ∈ I ^ n

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ I ^ 1 ≤ I

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : AdicCompletion I R
⊢ I ^ 1 ≤ I

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
⊢ I ^ 1 ≤ I

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
m n : ℕ
x✝ : m ≤ n
⊢ 1 m ≡ 1 n [SMOD I ^ m • ⊤]

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
m n : ℕ
hmn : m ≤ n
r : AdicCauchySequence I R
⊢ I ^ m = I ^ m • ⊤

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: case h.right
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
h : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
x : M
y : N
n : ℕ
hxy : f x ≡ y [SMOD I ^ n • ⊤]
z : M
hz : z ∈ I ^ n • ⊤
hz' : f z ≡ y - f x [SMOD I ^ (n + 1) • ⊤]
⊢ f (x + z) ≡ y [SMOD I ^ (n + 1) • ⊤]

Tactic:
· simpa [SModEq.sub_mem, sub_sub_eq_add_sub, add_comm] using hz'

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
r : R ⧸ I • ⊤
b₁ b₂ : R
h : b₁ ≈ b₂
a✝ : M
⊢ I = I • ⊤

Tactic:
simp

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

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
I : Ideal R
M : Type u
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : Type u
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
g : N →ₗ[R] P
hf : Function.Injective ⇑f
hfg : Function.Exact ⇑f ⇑g
hg : Function.Surjective ⇑g
k : ℕ
hkn : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCauchySequence I N
hker : ∀ (n : ℕ), g (↑x n) ∈ I ^ n • ⊤
n : ℕ
d : N
hdmem : d ∈ I ^ (k + n + 1) • ⊤
y yₙ : M
hd : f y = ↑x (k + n + 1) - d
hyₙ : f yₙ - ↑x (k + n) ∈ I ^ (k + n) • ⊤
⊢ f (y - yₙ) ∈ I ^ (k + n) • ⊤

Tactic:
simp only [map_sub, hd]
    convert_to x (k + n + 1) - x (k + n) - d - (f yₙ - x (k + n)) ∈ I ^ (k + n) • ⊤
    · abel
    · refine Submodule.sub_mem _ (Submodule.sub_mem _ ?_ ?_) hyₙ
      · rw [← Submodule.Quotient.eq]
        exact AdicCauchySequence.mk_eq_mk (by lia) _
      · exact (Submodule.smul_mono_left (Ideal.pow_le_pow_right (by lia))) hdmem

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
h : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
y : AdicCompletion I N
⊢ Submodule.map (↑(LinearEquiv.refl R N)) (I ^ 0 • ⊤) = ⊤

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
h : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
y : AdicCompletion I N
n : ℕ
xn : { m // Submodule.Quotient.mk (f m) = (eval I N n) y }
⊢ (factor ⋯) ((eval I N (n + 1)) y) = (eval I N n) y

Tactic:
simp

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

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
I : Ideal R
M : Type u
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : Type u
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
g : N →ₗ[R] P
hf : Function.Injective ⇑f
hfg : Function.Exact ⇑f ⇑g
hg : Function.Surjective ⇑g
k : ℕ
hkn : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCauchySequence I N
hker : ∀ (n : ℕ), g (↑x n) ∈ I ^ n • ⊤
n : ℕ
d : N
hdmem : d ∈ I ^ (k + n + 1) • ⊤
y yₙ : M
hd : f y = ↑x (k + n + 1) - d
hyₙ : f yₙ - ↑x (k + n) ∈ I ^ (k + n) • ⊤
h : f (y - yₙ) ∈ I ^ (k + n) • ⊤
hincl : I ^ (k + n - k) • (I ^ k • ⊤ ⊓ f.range) ≤ I ^ (k + n - k) • f.range
⊢ I ^ n • ⊤ = I ^ (k + n - k) • ⊤

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
h : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
y : AdicCompletion I N
x : (n : ℕ) → { m // Submodule.Quotient.mk (f m) = (eval I N n) y } :=
  fun n ↦ Nat.recAux ⟨0, ⋯⟩ (fun n xn ↦ (fun z hz ↦ ⟨z, ⋯⟩) (Classical.choose ⋯) ⋯) n
n : ℕ
⊢ (factor ⋯) ((eval I N (n + 1)) y) = (eval I N n) y

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
r : AdicCompletion I R
n : ℕ
⊢ ↑(r • 0) n = ↑0 n

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
r : AdicCompletion I R
x y : AdicCompletion I M
n : ℕ
⊢ ↑(r • (x + y)) n = ↑(r • x + r • y) n

Tactic:
simp

Goal state after: