mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
γ : Type u_3
R : Type u_4
f : σ → τ
g : τ → γ
inst✝² : TendstoCofinite f
inst✝¹ : CommSemiring R
inst✝ : TendstoCofinite g
p : MvPowerSeries σ R
y : γ →₀ ℕ
⊢ Set.InjOn (fun u ↦ (mapDomain f u, u)) ↑⋯.toFinset

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
x✝ : MvPowerSeries σ R
y : σ →₀ ℕ
⊢ ∀ b ∈ ⋯.toFinset, b ≠ y → (coeff b) x✝ = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
x✝ : MvPowerSeries σ R
y : σ →₀ ℕ
⊢ y ∉ ⋯.toFinset → (coeff y) x✝ = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
p : MvPowerSeries σ R
⊢ (rename id) p = p

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst✝¹ : TendstoCofinite f
inst✝ : CommSemiring R
p : MvPowerSeries σ R
⊢ 0 ∉ ⋯.toFinset → (coeff 0) p = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: case C
σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst✝¹ : TendstoCofinite f
inst✝ : CommSemiring R
a : R
⊢ (rename f) ↑(MvPolynomial.C a) = ↑((MvPolynomial.rename f) (MvPolynomial.C a))

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
γ : Type u_3
R : Type u_4
S : Type u_5
f : σ → τ
g : τ → γ
inst✝² : TendstoCofinite f
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
e : σ ≃ τ
x✝ : MvPowerSeries σ R
⊢ (rename ⇑e.symm) ((↑↑(rename ⇑e).toRingHom).toFun x✝) = x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
γ : Type u_3
R : Type u_4
S : Type u_5
f : σ → τ
g : τ → γ
inst✝² : TendstoCofinite f
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
e : σ ≃ τ
x✝ : MvPowerSeries τ R
⊢ (↑↑(rename ⇑e).toRingHom).toFun ((rename ⇑e.symm) x✝) = x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
⊢ ∀ (a : MvPowerSeries σ R), (renameEquiv R (Equiv.refl σ)) a = AlgEquiv.refl a

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
e : σ ↪ τ
t : τ
h : t ∉ Set.range ⇑e
⊢ 1 ≠ 0

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
f : σ → τ
inst✝¹ : TendstoCofinite f
R : Type u_6
inst✝ : CommRing R
⊢ ∀ (s : σ), IsNilpotent (constantCoeff ((X ∘ f) s))

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G A : Type u
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup A
inst✝² : Module k A
inst✝¹ : DistribMulAction G A
inst✝ : SMulCommClass G k A
x : A
hx : x ∈ Coinvariants.ker (Representation.ofDistribMulAction k G A)
⊢ (sum 0 fun g z ↦ g⁻¹ • z - z) = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
⊢ (ComplexShape.down ℕ).next 0 = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
⊢ (ComplexShape.down ℕ).prev 0 = 1

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x : ↑A
⊢ (ComplexShape.down ℕ).next 0 = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x : ↑A
⊢ (ConcreteCategory.hom (chainsIso₀ A).inv) x =
    (ConcreteCategory.hom (iCycles A 0)) ((ConcreteCategory.hom (cyclesIso₀ A).inv) x)

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
⊢ (ComplexShape.down ℕ).prev 1 = 2

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
⊢ (ComplexShape.down ℕ).next 1 = 0

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R⟦X⟧
hXI : X ∈ I
f : R⟦X⟧
⊢ f ∈ I → C (constantCoeff f) ∈ I

Tactic:
hf

Goal state after:
R : Type u_1
inst✝ : CommRing R
I : Ideal R⟦X⟧
hXI : X ∈ I
f : R⟦X⟧
hf : f ∈ I
⊢ C (constantCoeff f) ∈ I
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
ha₂ : ¬↑(span {a}) ∩ ↑(closure {r | Prime r}).toSubsemigroup = ∅
x : R
hx : x ∈ ↑(span {a})
b : R
hx₂ : a * b ∈ ↑(closure {r | Prime r}).toSubsemigroup
hb : b * a = x
hsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}
this : a ∈ closure {r | IsUnit r ∨ Prime r}
⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated f.prod a

Tactic:
clear ha ha₂ hx hb hx₂

Goal state after:
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}
this : a ∈ closure {r | IsUnit r ∨ Prime r}
⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated f.prod a
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x : ↥(cycles₁ A)
⊢ (ComplexShape.down ℕ).next 1 = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x : ↥(cycles₁ A)
⊢ (ModuleCat.Hom.hom (d₁₀ A ≫ (chainsIso₀ A).inv)) ↑x = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
⊢ (ComplexShape.down ℕ).prev 2 = 3

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
⊢ (ComplexShape.down ℕ).next 2 = 1

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}
⊢ (∀ b ∈ ∅, Prime b) ∧ Associated ∅.prod 1

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x : ↥(cycles₂ A)
⊢ (ComplexShape.down ℕ).next 2 = 1

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x : ↥(cycles₂ A)
⊢ (ModuleCat.Hom.hom (d₂₁ A ≫ (chainsIso₁ A).inv)) ↑x = 0

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
inst✝ : A.IsTrivial
⊢ (ComplexShape.down ℕ).prev 0 = 1

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R⟦X⟧
inst✝ : I.IsPrime
hI : X ∉ I
S : Set R
hSI : span S = Ideal.map constantCoeff I
hS : S.Finite
this : SurjOn (⇑constantCoeff) (↑I) S
T : Set R⟦X⟧
hTI : T ⊆ ↑I
hinj : InjOn (⇑constantCoeff) T
hT : ⇑constantCoeff '' T = S
f : R⟦X⟧
⊢ f ∈ I → f ∈ comap constantCoeff (Ideal.map constantCoeff (span T))

Tactic:
hf

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R⟦X⟧
inst✝ : I.IsPrime
hI : X ∉ I
S : Set R
hSI : span S = Ideal.map constantCoeff I
hS : S.Finite
this : SurjOn (⇑constantCoeff) (↑I) S
T : Set R⟦X⟧
hTI : T ⊆ ↑I
hinj : InjOn (⇑constantCoeff) T
hT : ⇑constantCoeff '' T = S
f : R⟦X⟧
hf : f ∈ I
⊢ f ∈ comap constantCoeff (Ideal.map constantCoeff (span T))
Module: Mathlib.RingTheory.PowerSeries.Ideal

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R⟦X⟧
inst✝ : I.IsPrime
hI : X ∉ I
S : Set R
hSI : span S = Ideal.map constantCoeff I
hS : S.Finite
this : SurjOn (⇑constantCoeff) (↑I) S
T : Set R⟦X⟧
hTI : T ⊆ ↑I
hinj : InjOn (⇑constantCoeff) T
hT : ⇑constantCoeff '' T = S
f : R⟦X⟧
hf : constantCoeff f ∈ span S
⊢ constantCoeff 0 = 0

Tactic:
simp

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

Goal state before:
pp: case refine_1
R : Type u_1
inst✝¹ : CommRing R
this : RingHomSurjective constantCoeff
S : Set R⟦X⟧
inst✝ : (Ideal.span S).IsPrime
hI : X ∉ Ideal.span S
hfg : (Ideal.span S).FG
hS : S.Finite
hScard : S.ncard = (generators (Ideal.map constantCoeff (Ideal.span S))).ncard
⊢ S.ncard ≤ spanFinrank (Ideal.map constantCoeff (Ideal.span S))

Tactic:
hScard,

Goal state after:
case refine_1
R : Type u_1
inst✝¹ : CommRing R
this : RingHomSurjective constantCoeff
S : Set R⟦X⟧
inst✝ : (Ideal.span S).IsPrime
hI : X ∉ Ideal.span S
hfg : (Ideal.span S).FG
hS : S.Finite
hScard : S.ncard = (generators (Ideal.map constantCoeff (Ideal.span S))).ncard
⊢ (generators (Ideal.map constantCoeff (Ideal.span S))).ncard ≤ spanFinrank (Ideal.map constantCoeff (Ideal.span S))
Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
U : Set R
i : ℕ
h : ↑(I ^ i) ⊆ U
⊢ ↑(I ^ i) = ↑((fun i ↦ toAddSubgroup (I ^ i • ⊤)) i)

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology

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
m : M
i : ℕ
⊢ ↑(I ^ i • ⊤) = ↑((fun i ↦ toAddSubgroup (I ^ i • ⊤)) i)

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology

Goal state before:
pp: case e'_3
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
this : TopologicalSpace R := I.adicTopology
⊢ ↑(I ^ n) = ↑(I ^ n • ⊤)

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : UniformSpace R
inst✝ : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this✝ : (𝓝 0).IsCountablyGenerated
this : (𝓤 R).IsCountablyGenerated
⊢ IsPrecomplete I R ↔ CompleteSpace R

Tactic:
simp only [isPrecomplete_iff, smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : UniformSpace R
inst✝ : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this✝ : (𝓝 0).IsCountablyGenerated
this : (𝓤 R).IsCountablyGenerated
⊢ (∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n) ↔ CompleteSpace R
Module: Mathlib.RingTheory.AdicCompletion.Topology

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : UniformSpace R
inst✝ : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this✝ : (𝓝 0).IsCountablyGenerated
this : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
N : ℕ → ℕ
hN : ∀ (i m : ℕ), N i ≤ m → ∀ (n : ℕ), N i ≤ n → u n - u m ∈ I ^ i
m✝ n✝ : ℕ
x✝ : m✝ ≤ n✝
⊢ m✝ ∈ Finset.Iic m✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : UniformSpace R
inst✝ : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this✝ : (𝓝 0).IsCountablyGenerated
this : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
N : ℕ → ℕ
hN : ∀ (i m : ℕ), N i ≤ m → ∀ (n : ℕ), N i ≤ n → u n - u m ∈ I ^ i
L : R
hL : ∀ (n : ℕ), u ((Finset.Iic n).sup N) - L ∈ I ^ n
i n : ℕ
hn : (Finset.Iic i).sup N ≤ n
⊢ i ∈ Finset.Iic i

Tactic:
simp

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

Goal state before:
pp: case mpr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : UniformSpace R
inst✝ : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this✝ : (𝓝 0).IsCountablyGenerated
this : (𝓤 R).IsCountablyGenerated
H : CompleteSpace R
f : ℕ → R
⊢ (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n

Tactic:
hf

Goal state after:
case mpr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : UniformSpace R
inst✝ : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this✝ : (𝓝 0).IsCountablyGenerated
this : (𝓤 R).IsCountablyGenerated
H : CompleteSpace R
f : ℕ → R
hf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m
⊢ ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n
Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology

Goal state before:
pp: case mpr.left
A : Type u_2
inst✝² : CommRing A
inst✝¹ : TopologicalSpace A
inst✝ : IsTopologicalRing A
a✝ : DiscreteTopology A
⊢ ∀ (n : ℕ), IsOpen[inst✝¹] ↑(⊥ ^ n)

Tactic:
simp

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

Goal state before:
pp: case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
fg : (maximalIdeal R).FG
this : IsLocalRing (AdicCompletion (maximalIdeal R) R)
x : IsLocalRing.ResidueField (AdicCompletion (maximalIdeal R) R)
y : AdicCompletion (maximalIdeal R) R
hy : (residue (AdicCompletion (maximalIdeal R) R)) y = x
z : R
hz : (Ideal.Quotient.mk (maximalIdeal R ^ 1 • ⊤)) z = ↑y 1
⊢ (residue (AdicCompletion (maximalIdeal R) R)) ((algebraMap R (AdicCompletion (maximalIdeal R) R)) z) =
    (residue (AdicCompletion (maximalIdeal R) R)) y

Tactic:
apply (Ideal.Quotient.mk_eq_mk_iff_sub_mem _ _).mpr

Goal state after:
case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
fg : (maximalIdeal R).FG
this : IsLocalRing (AdicCompletion (maximalIdeal R) R)
x : IsLocalRing.ResidueField (AdicCompletion (maximalIdeal R) R)
y : AdicCompletion (maximalIdeal R) R
hy : (residue (AdicCompletion (maximalIdeal R) R)) y = x
z : R
hz : (Ideal.Quotient.mk (maximalIdeal R ^ 1 • ⊤)) z = ↑y 1
⊢ (algebraMap R (AdicCompletion (maximalIdeal R) R)) z - y ∈ maximalIdeal (AdicCompletion (maximalIdeal R) R)
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: σ✝ : Type u_1
R✝ : Type u_2
n✝ : ℕ
inst✝³ : CommRing R✝
inst✝² : Finite σ✝
σ : Type u_3
R : Type u_4
inst✝¹ : Finite σ
inst✝ : CommRing R
n : ℕ
x✝¹ x✝ : MvPowerSeries σ R
⊢ (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n)) ((truncTotal n) (x✝¹ + x✝)) =
    (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n)) ((truncTotal n) x✝¹) +
      (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n)) ((truncTotal n) x✝)

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A B : Rep k G
f : A ⟶ B
x✝ : ↑(ModuleCat.of k ↑A)
⊢ (ModuleCat.Hom.hom (H0π A ≫ map (MonoidHom.id G) f 0 ≫ (H0Iso B).hom)) x✝ =
    (ModuleCat.Hom.hom (H0π A ≫ (H0Iso A).hom ≫ (coinvariantsFunctor k G).map f)) x✝

Tactic:
simp

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

Goal state before:
pp: case neg
R : Type u_1
inst✝⁷ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : IsNoetherianRing R
inst✝³ : Module.Finite R M
A : Type u_3
inst✝² : CommRing A
inst✝¹ : IsArtinianRing A
inst✝ : IsLocalRing A
f : ℕ → A
n : ℕ
hn : maximalIdeal A ^ n = 0
m : ℕ
hf : f n = f m
h : ¬m ≤ n
⊢ f m ≡ f n [SMOD maximalIdeal A ^ m • ⊤]

Tactic:
hf

Goal state after:
case neg
R : Type u_1
inst✝⁷ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : IsNoetherianRing R
inst✝³ : Module.Finite R M
A : Type u_3
inst✝² : CommRing A
inst✝¹ : IsArtinianRing A
inst✝ : IsLocalRing A
f : ℕ → A
n : ℕ
hn : maximalIdeal A ^ n = 0
m : ℕ
hf : f n = f m
h : ¬m ≤ n
⊢ f m ≡ f m [SMOD maximalIdeal A ^ m • ⊤]
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
inst✝ : Finite σ
n : ℕ
f : AdicCompletion (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)
x : σ →₀ ℕ
h : degree x < n
⊢ (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n • ⊤)) (Quotient.out (↑f n)) = ↑f n

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: case a
σ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
inst✝ : Finite σ
n : ℕ
f : AdicCompletion (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)
x : σ →₀ ℕ
h : degree x < n
⊢ degree x < degree x + 1

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CommRing S
I : Ideal S
inst✝ : IsAdicComplete I S
f : (n : ℕ) → R →+* S ⧸ I ^ n
hf : ∀ {m n : ℕ} (hle : m ≤ n), (factorPow I hle).comp (f n) = f m
x✝ : R
n✝ : ℕ
⊢ ↑(((↑(ofAlgEquiv I)).comp (liftRingHom I f ⋯)) x✝) n✝ = ↑((AdicCompletion.liftRingHom I f ⋯) x✝) n✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CommRing S
I : Ideal S
inst✝ : IsAdicComplete I S
f : (n : ℕ) → R →+* S ⧸ I ^ n
hf : ∀ {m n : ℕ} (hle : m ≤ n), (factorPow I hle).comp (f n) = f m
n : ℕ
x : R
⊢ (evalₐ I n) ((of I S) ((ofAlgEquiv I).symm ((AdicCompletion.liftRingHom I f ⋯) x))) = (f n) x

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CommRing S
I : Ideal S
inst✝ : IsAdicComplete I S
f : (n : ℕ) → R →+* S ⧸ I ^ n
hf : ∀ {m n : ℕ} (hle : m ≤ n), (factorPow I hle).comp (f n) = f m
n : ℕ
x✝ : R
⊢ ((Ideal.Quotient.mk (I ^ n)).comp (liftRingHom I f ⋯)) x✝ = (f n) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: case refine_1
k G H : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Group H
A : Rep k G
B : Rep k H
f : G →* H
a✝¹ : G × G
x✝ : ↑A
a✝ : H × H
⊢ ((ModuleCat.Hom.hom (mapShortComplexH1 f 0).τ₁ ∘ₗ lsingle a✝¹) x✝) a✝ =
    ((ModuleCat.Hom.hom (Hom.τ₁ 0) ∘ₗ lsingle a✝¹) x✝) a✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: case refine_2
k G H : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Group H
A : Rep k G
B : Rep k H
f : G →* H
a✝¹ : G
x✝ : ↑A
a✝ : H
⊢ ((ModuleCat.Hom.hom (mapShortComplexH1 f 0).τ₂ ∘ₗ lsingle a✝¹) x✝) a✝ =
    ((ModuleCat.Hom.hom (Hom.τ₂ 0) ∘ₗ lsingle a✝¹) x✝) a✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CommRing S
I : Ideal S
inst✝ : IsAdicComplete I S
f : (n : ℕ) → R →+* S ⧸ I ^ n
hf : ∀ {m n : ℕ} (hle : m ≤ n), (factorPow I hle).comp (f n) = f m
F : R →+* S
hF : ∀ (n : ℕ), (Ideal.Quotient.mk (I ^ n)).comp F = f n
⊢ F = liftRingHom I f ⋯

Tactic:
by
  apply DFunLike.coe_injective
  apply IsHausdorff.funext' I
  intro n m
  simp [← hF n]

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: k G H : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
⊢ mapCycles₁ f φ ≫ (shortComplexH1 B).moduleCatLeftHomologyData.i =
    (shortComplexH1 A).moduleCatLeftHomologyData.i ≫ chainsMap₁ f φ

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → R →+* S ⧸ I ^ a n
hf : ∀ {m : ℕ}, (factorPow I ⋯).comp (f (m + 1)) = f m
inst✝ : IsAdicComplete I S
n : ℕ
x : R
⊢ (Ideal.Quotient.mk (I ^ a n)) ((liftRingHom I ha f ⋯) x) = (f n) x

Tactic:
simp [liftRingHom, IsAdicComplete.liftRingHom,
      factorPow_comp_eq_of_factorPow_comp_succ_eq' ha f hf ha.le_apply]

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → R →+* S ⧸ I ^ a n
hf : ∀ {m : ℕ}, (factorPow I ⋯).comp (f (m + 1)) = f m
inst✝ : IsAdicComplete I S
n : ℕ
x✝ : R
⊢ ((Ideal.Quotient.mk (I ^ a n)).comp (liftRingHom I ha f ⋯)) x✝ = (f n) x✝

Tactic:
simp

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

Goal state before:
pp: case h
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
x : AdicCompletion I M
hx : ↑x n = 0
⊢ (ofPowSMul I M n) (ofValEqZero I hx) = x

Tactic:
simp

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

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
n : ℕ
s : Finset R
hs : Ideal.span ↑s = I ^ n
⊢ (↑(finsuppLEquivDirectSum (AdicCompletion I R) (AdicCompletion I M) ↥s).symm).range = ⊤

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: case convert_3
k G H : Type u
inst✝⁴ : CommRing k
inst✝³ : Group G
inst✝² : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
⊢ Epi (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv ≫ H1π (A.ofQuotient S))

Tactic:
exact @epi_comp _ _ _ _ _ _ (mapCycles₁_quotientGroupMk'_epi A S) (H1π _) inferInstance

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

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
n : ℕ
s : Finset R
hs : Ideal.span ↑s = I ^ n
x : AdicCompletion I ↥(I ^ n • ⊤)
⊢ comap ((lsum R) fun i ↦ ↑i • LinearMap.id).range.subtype ((lsum R) fun i ↦ ↑i • LinearMap.id).range = ⊤

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
fg : I.FG
x : AdicCompletion I R
⊢ I ^ 1 * ⊤ = I

Tactic:
simp

Goal state after: