mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
inst✝² : IsInvariant A B G
P Q : Ideal B
inst✝¹ : P.IsPrime
inst✝ : Q.IsPrime
hPQ : Ideal.under A P = Ideal.under A Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
F : OpenNormalSubgroup G ⥤ Type u :=
  { obj := fun N ↦ { g // Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) Q = g • Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) P },
    map := fun {N N'} f ↦
      ↾fun x ↦ ⟨(QuotientGroup.map (↑N.toOpenSubgroup) N'.toOpenSubgroup.1 (MonoidHom.id G) ⋯) ↑x, ⋯⟩,
    map_id := ⋯, map_comp := ⋯ }
this✝ : ∀ (N : OpenNormalSubgroup G), Nonempty (F.obj N)
s : (j : OpenNormalSubgroup G) → F.obj j
hs : s ∈ F.sections
a : ↑(ProfiniteGrp.of G).toProfinite.toTop :=
  (ProfiniteGrp.Hom.hom (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv) ⟨fun N ↦ ↑(s N), ⋯⟩
this : ∀ (N : OpenNormalSubgroup G), ↑a = ↑(s N)
⊢ Q = a • P

Tactic:
ext x

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
inst✝² : IsInvariant A B G
P Q : Ideal B
inst✝¹ : P.IsPrime
inst✝ : Q.IsPrime
hPQ : Ideal.under A P = Ideal.under A Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
F : OpenNormalSubgroup G ⥤ Type u :=
  { obj := fun N ↦ { g // Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) Q = g • Ideal.under (↥(B' ↥↑N.toOpenSubgroup)) P },
    map := fun {N N'} f ↦
      ↾fun x ↦ ⟨(QuotientGroup.map (↑N.toOpenSubgroup) N'.toOpenSubgroup.1 (MonoidHom.id G) ⋯) ↑x, ⋯⟩,
    map_id := ⋯, map_comp := ⋯ }
this✝ : ∀ (N : OpenNormalSubgroup G), Nonempty (F.obj N)
s : (j : OpenNormalSubgroup G) → F.obj j
hs : s ∈ F.sections
a : ↑(ProfiniteGrp.of G).toProfinite.toTop :=
  (ProfiniteGrp.Hom.hom (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv) ⟨fun N ↦ ↑(s N), ⋯⟩
this : ∀ (N : OpenNormalSubgroup G), ↑a = ↑(s N)
x : B
⊢ x ∈ Q ↔ x ∈ a • P
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalIso R S
inst✝ : IsLocalIso S T
s : Set S := {g | IsStandardOpenImmersion R (Localization.Away g)}
T' : S → Type u_3 := fun g ↦ Localization.Away ((algebraMap S T) g)
this✝³ : (g : S) → Algebra (Localization.Away g) (T' g) := fun g ↦ localizationAlgebra (Submonoid.powers g) T
T'' : S → T → Type u_3 := fun g x ↦ Localization.Away ((algebraMap T (T' g)) x)
t : S → Set T := fun g ↦ {x | IsStandardOpenImmersion (Localization.Away g) (T'' g x)}
ι : Type (max u_3 u_2) := (i : ↑s) × ↑(t ↑i)
this✝² : ∀ (i : ι), IsStandardOpenImmersion (Localization.Away ↑i.fst) (T'' ↑i.fst ↑i.snd)
h1 : Ideal.span (⇑(algebraMap S T) '' {g | IsStandardOpenImmersion R (Localization.Away g)}) = ⊤
g : S
hg : IsStandardOpenImmersion R (Localization.Away g)
x : T
hx : IsStandardOpenImmersion S (Localization.Away x)
this✝¹ : Algebra (Localization.Away x) (T'' g x) := localizationAlgebra (Submonoid.powers x) (T' g)
this✝ : IsScalarTower S (Localization.Away x) (T'' g x)
this :
  algebraMapSubmonoid (Localization.Away x) (Submonoid.powers g) =
    algebraMapSubmonoid (Localization.Away x) (Submonoid.powers ((algebraMap S T) g))
⊢ IsLocalization (algebraMapSubmonoid (Localization.Away x) (Submonoid.powers ((algebraMap S T) g))) (T'' g x)

Tactic:
exact .commutes _ (T' g) _ (.powers x) (.powers (algebraMap S T g))

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

Goal state before:
pp: case succ
R : Type u
inst✝² : CommRing R
inst✝¹ : Small.{v, u} R
S : Submonoid R
this✝¹ : Small.{v, u} (Localization S)
n : ℕ
ih : ∀ (M : ModuleCat R) [HasProjectiveDimensionLE M n], HasProjectiveDimensionLE (M.localizedModule S) n
M : ModuleCat R
inst✝ : HasProjectiveDimensionLE M (n + 1)
P : ModuleCat R
projective✝ : Projective P
f : P ⟶ M
epi✝ : Epi f
T : ShortComplex (ModuleCat R) := { X₁ := kernel f, X₂ := P, X₃ := M, f := kernel.ι f, g := f, zero := ⋯ }
T_exact : T.ShortExact
TS : ShortComplex (ModuleCat (Localization S)) := T.map (localizedModuleFunctor S)
TS_exact : TS.ShortExact
this✝ : Projective TS.X₂
this : HasProjectiveDimensionLT T.X₁ (n + 1)
⊢ HasProjectiveDimensionLE (M.localizedModule S) (n + 1)

Tactic:
exact (TS_exact.hasProjectiveDimensionLT_X₃_iff n ‹_›).mpr (ih (kernel f))

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : FaithfulSMul R S
inst✝ : Etale R S
β : S
hadj : R[β] = ⊤
⊢ map (residue R) (minpoly R β) = minpoly (ResidueField R) ((residue S) β)

Tactic:
have h := minpoly.monic <| Algebra.IsIntegral.isIntegral (R := R) β

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : FaithfulSMul R S
inst✝ : Etale R S
β : S
hadj : R[β] = ⊤
h : (minpoly R β).Monic
⊢ map (residue R) (minpoly R β) = minpoly (ResidueField R) ((residue S) β)
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: case refine_1
A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
P : Ideal A
Q : Ideal B
inst✝² : Q.IsPrime
inst✝¹ : Q.LiesOver P
inst✝ : Algebra.IsInvariant A B G
σ : (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q
B' : (G : Type ?u.58) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
s : (j : OpenNormalSubgroup G) → (stabilizerHomSurjectiveAuxFunctor P Q σ).obj j
hs : s ∈ (stabilizerHomSurjectiveAuxFunctor P Q σ).sections
a : ↑(ProfiniteGrp.of G).toProfinite.toTop :=
  (ProfiniteGrp.Hom.hom (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv) ⟨fun N ↦ ↑↑(s N), ⋯⟩
this : ∀ (N : OpenNormalSubgroup G), ↑a = ↑↑(s N)
⊢ a ∈ MulAction.stabilizer G Q

Tactic:
ext x

Goal state after:
case refine_1
A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
P : Ideal A
Q : Ideal B
inst✝² : Q.IsPrime
inst✝¹ : Q.LiesOver P
inst✝ : Algebra.IsInvariant A B G
σ : (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q
B' : (G : Type ?u.58) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
s : (j : OpenNormalSubgroup G) → (stabilizerHomSurjectiveAuxFunctor P Q σ).obj j
hs : s ∈ (stabilizerHomSurjectiveAuxFunctor P Q σ).sections
a : ↑(ProfiniteGrp.of G).toProfinite.toTop :=
  (ProfiniteGrp.Hom.hom (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv) ⟨fun N ↦ ↑↑(s N), ⋯⟩
this : ∀ (N : OpenNormalSubgroup G), ↑a = ↑↑(s N)
x : B
⊢ x ∈ a • Q ↔ x ∈ Q
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: case refine_2
A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
P : Ideal A
Q : Ideal B
inst✝² : Q.IsPrime
inst✝¹ : Q.LiesOver P
inst✝ : Algebra.IsInvariant A B G
σ : (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
s : (j : OpenNormalSubgroup G) → (stabilizerHomSurjectiveAuxFunctor P Q σ).obj j
hs : s ∈ (stabilizerHomSurjectiveAuxFunctor P Q σ).sections
a : ↑(ProfiniteGrp.of G).toProfinite.toTop :=
  (ProfiniteGrp.Hom.hom (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv) ⟨fun N ↦ ↑↑(s N), ⋯⟩
this : ∀ (N : OpenNormalSubgroup G), ↑a = ↑↑(s N)
⊢ (stabilizerHom Q P G) ⟨a, ⋯⟩ = σ

Tactic:
ext x

Goal state after:
case refine_2
A : Type u_1
B : Type u_2
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
G : Type u
inst✝¹² : Group G
inst✝¹¹ : MulSemiringAction G B
inst✝¹⁰ : SMulCommClass G A B
inst✝⁹ : TopologicalSpace G
inst✝⁸ : CompactSpace G
inst✝⁷ : TotallyDisconnectedSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : TopologicalSpace B
inst✝⁴ : DiscreteTopology B
inst✝³ : ContinuousSMul G B
P : Ideal A
Q : Ideal B
inst✝² : Q.IsPrime
inst✝¹ : Q.LiesOver P
inst✝ : Algebra.IsInvariant A B G
σ : (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q
B' : (G : Type u) → [inst : Monoid G] → [inst_1 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
s : (j : OpenNormalSubgroup G) → (stabilizerHomSurjectiveAuxFunctor P Q σ).obj j
hs : s ∈ (stabilizerHomSurjectiveAuxFunctor P Q σ).sections
a : ↑(ProfiniteGrp.of G).toProfinite.toTop :=
  (ProfiniteGrp.Hom.hom (ProfiniteGrp.of G).isoLimittoFiniteQuotientFunctor.inv) ⟨fun N ↦ ↑↑(s N), ⋯⟩
this : ∀ (N : OpenNormalSubgroup G), ↑a = ↑↑(s N)
x : B ⧸ Q
⊢ ((stabilizerHom Q P G) ⟨a, ⋯⟩) x = σ x
Module: Mathlib.RingTheory.MvPolynomial.EulerIdentity

Goal state before:
pp: case refine_1
R : Type u_1
σ : Type u_2
M : Type u_3
inst✝¹ : CommSemiring R
φ : MvPolynomial σ R
inst✝ : AddCancelCommMonoid M
w : σ → M
n n' : M
i : σ
h : φ ∈ Submodule.span R ((fun m ↦ AddMonoidAlgebra.single m 1) '' {d | (weight w) d = n})
h' : n' + w i = n
m : σ →₀ ℕ
hm : m ∈ {d | (weight w) d = n}
⊢ IsWeightedHomogeneous w ((monomial (m - single i 1)) ↑(m i)) n'

Tactic:
by_cases hi : m i = 0

Goal state after:
case pos
R : Type u_1
σ : Type u_2
M : Type u_3
inst✝¹ : CommSemiring R
φ : MvPolynomial σ R
inst✝ : AddCancelCommMonoid M
w : σ → M
n n' : M
i : σ
h : φ ∈ Submodule.span R ((fun m ↦ AddMonoidAlgebra.single m 1) '' {d | (weight w) d = n})
h' : n' + w i = n
m : σ →₀ ℕ
hm : m ∈ {d | (weight w) d = n}
hi : m i = 0
⊢ IsWeightedHomogeneous w ((monomial (m - single i 1)) ↑(m i)) n'

case neg
R : Type u_1
σ : Type u_2
M : Type u_3
inst✝¹ : CommSemiring R
φ : MvPolynomial σ R
inst✝ : AddCancelCommMonoid M
w : σ → M
n n' : M
i : σ
h : φ ∈ Submodule.span R ((fun m ↦ AddMonoidAlgebra.single m 1) '' {d | (weight w) d = n})
h' : n' + w i = n
m : σ →₀ ℕ
hm : m ∈ {d | (weight w) d = n}
hi : ¬m i = 0
⊢ IsWeightedHomogeneous w ((monomial (m - single i 1)) ↑(m i)) n'
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case neg
K : Type u_2
inst✝ : Field K
d n : ℤ
f g : K⸨X⸩
H : Valued.v (g - f) ≤ exp (-d)
triv : ¬g = f
hn : n < d
⊢ (g - f).coeff n = 0

Tactic:
apply coeff_zero_of_lt_valuation K H hn

Goal state after:
Module: Mathlib.RingTheory.Localization.AtPrime.Extension

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁷ : CommRing R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : Algebra R S
p : Ideal R
inst✝¹⁴ : p.IsPrime
Rₚ : Type u_3
inst✝¹³ : CommRing Rₚ
inst✝¹² : Algebra R Rₚ
inst✝¹¹ : IsLocalization.AtPrime Rₚ p
inst✝¹⁰ : IsLocalRing Rₚ
Sₚ : Type u_4
inst✝⁹ : CommRing Sₚ
inst✝⁸ : Algebra S Sₚ
inst✝⁷ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝⁶ : Algebra Rₚ Sₚ
P : Ideal S
hPp : P.LiesOver p
inst✝⁵ : Algebra R Sₚ
inst✝⁴ : IsScalarTower R S Sₚ
inst✝³ : IsScalarTower R Rₚ Sₚ
inst✝² : p.IsMaximal
inst✝¹ : P.IsMaximal
inst✝ : (Ideal.map (algebraMap S Sₚ) P).LiesOver (maximalIdeal Rₚ)
⊢ (algebraMap (R ⧸ p) (S ⧸ P)).comp (equivQuotMaximalIdeal p Rₚ).symm.toRingHom =
    (equivQuotientMapOfIsMaximal p Sₚ P).symm.toRingHom.comp
      (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ Ideal.map (algebraMap S Sₚ) P))

Tactic:
ext x

Goal state after:
R : Type u_1
S : Type u_2
inst✝¹⁷ : CommRing R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : Algebra R S
p : Ideal R
inst✝¹⁴ : p.IsPrime
Rₚ : Type u_3
inst✝¹³ : CommRing Rₚ
inst✝¹² : Algebra R Rₚ
inst✝¹¹ : IsLocalization.AtPrime Rₚ p
inst✝¹⁰ : IsLocalRing Rₚ
Sₚ : Type u_4
inst✝⁹ : CommRing Sₚ
inst✝⁸ : Algebra S Sₚ
inst✝⁷ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝⁶ : Algebra Rₚ Sₚ
P : Ideal S
hPp : P.LiesOver p
inst✝⁵ : Algebra R Sₚ
inst✝⁴ : IsScalarTower R S Sₚ
inst✝³ : IsScalarTower R Rₚ Sₚ
inst✝² : p.IsMaximal
inst✝¹ : P.IsMaximal
inst✝ : (Ideal.map (algebraMap S Sₚ) P).LiesOver (maximalIdeal Rₚ)
x : Rₚ
⊢ (((algebraMap (R ⧸ p) (S ⧸ P)).comp (equivQuotMaximalIdeal p Rₚ).symm.toRingHom).comp
        (Ideal.Quotient.mk (maximalIdeal Rₚ)))
      x =
    (((equivQuotientMapOfIsMaximal p Sₚ P).symm.toRingHom.comp
            (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ Ideal.map (algebraMap S Sₚ) P))).comp
        (Ideal.Quotient.mk (maximalIdeal Rₚ)))
      x
Module: Mathlib.RingTheory.MvPolynomial.Expand

Goal state before:
pp: case succ
σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
f : MvPolynomial σ R
k : ℕ
n_ih : (map (iterateFrobenius R p k)) ((expand (p ^ k)) f) = f ^ p ^ k
⊢ (map (iterateFrobenius R p k)) ((expand (p ^ k)) f) ^ p =
    (map (iterateFrobenius R p (k + 1))) ((expand (p ^ (k + 1))) f)

Tactic:
← map_frobenius_expand p,

Goal state after:
case succ
σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
f : MvPolynomial σ R
k : ℕ
n_ih : (map (iterateFrobenius R p k)) ((expand (p ^ k)) f) = f ^ p ^ k
⊢ (map (frobenius R p)) ((expand p) ((map (iterateFrobenius R p k)) ((expand (p ^ k)) f))) =
    (map (iterateFrobenius R p (k + 1))) ((expand (p ^ (k + 1))) f)
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: K : Type u_2
inst✝ : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
x✝ : UniformSpace K := ⊥
N : ℤ
hN : ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0
⊢ ∃ N, ∀ n < N, coeff hℱ n = 0

Tactic:
refine ⟨N, fun n hn ↦ Ultrafilter.eq_of_le_pure (hℱ.map (uniformContinuous_coeff n)).1
      ((principal_singleton _).symm ▸ coeff_tendsto _ _) ?_⟩

Goal state after:
K : Type u_2
inst✝ : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
x✝ : UniformSpace K := ⊥
N : ℤ
hN : ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0
n : ℤ
hn : n < N
⊢ Filter.map (fun f ↦ f.coeff n) ℱ ≤ pure 0
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: case inr
σ : Type u_1
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h1 : t.2 ∉ t.1
⊢ #(pairMap σ t).1 ≤ k ∧ (#(pairMap σ t).1 = k → (pairMap σ t).2 ∈ (pairMap σ t).1)

Tactic:
rw [pairMap_of_snd_notMem_fst σ h1]

Goal state after:
case inr
σ : Type u_1
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h1 : t.2 ∉ t.1
⊢ #(cons t.2 t.1 h1, t.2).1 ≤ k ∧ (#(cons t.2 t.1 h1, t.2).1 = k → (cons t.2 t.1 h1, t.2).2 ∈ (cons t.2 t.1 h1, t.2).1)
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: case inr
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (-1) ^ #t.1 * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - #t.1)) +
      (-1) ^ #(pairMap σ t).1 * ((∏ a ∈ (pairMap σ t).1, X a) * X (pairMap σ t).2 ^ (k - #(pairMap σ t).1)) =
    0

Tactic:
rw [pairMap_of_snd_notMem_fst σ h1]

Goal state after:
case inr
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (-1) ^ #t.1 * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - #t.1)) +
      (-1) ^ #(cons t.2 t.1 h1, t.2).1 *
        ((∏ a ∈ (cons t.2 t.1 h1, t.2).1, X a) * X (cons t.2 t.1 h1, t.2).2 ^ (k - #(cons t.2 t.1 h1, t.2).1)) =
    0
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
⊢ ↑k * esymm σ R k = (-1) ^ (k + 1) * ∑ a ∈ antidiagonal k with a.1 < k, (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2

Tactic:
rw [NewtonIdentities.esymm_to_weight σ R k, NewtonIdentities.esymm_mul_psum_to_weight σ R k,
    eq_comm, ← sub_eq_zero, sub_eq_add_neg, neg_mul_eq_neg_mul,
    neg_eq_neg_one_mul ((-1 : MvPolynomial σ R) ^ k)]

Goal state after:
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
⊢ (-1) ^ (k + 1) * ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 < k, NewtonIdentities.weight σ R k t +
      -1 * (-1) ^ k * ∑ t ∈ NewtonIdentities.pairs σ k with #t.1 = k, NewtonIdentities.weight σ R k t =
    0
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: K : Type u_2
inst✝ : Field K
a : K
⊢ (algebraMap K K⸨X⸩) a = HahnSeries.C a

Tactic:
by
  simp [RingHom.algebraMap_toAlgebra]

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPowerSeries σ R
⊢ ((expand p hp) φ).order = p • φ.order

Tactic:
by_cases! hφ : φ = 0

Goal state after:
case pos
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPowerSeries σ R
hφ : φ = 0
⊢ ((expand p hp) φ).order = p • φ.order

case neg
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPowerSeries σ R
hφ : φ ≠ 0
⊢ ((expand p hp) φ).order = p • φ.order
Module: Mathlib.RingTheory.MvPowerSeries.Expand

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPolynomial σ R
n : σ →₀ ℕ
h : ∀ (i : σ), p ∣ n i
⊢ (coeff n) ((expand p hp) ↑φ) = MvPolynomial.coeff n ((MvPolynomial.expand p) φ)

pp: case neg
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPolynomial σ R
n : σ →₀ ℕ
h : ∃ i, ¬p ∣ n i
⊢ (coeff n) ((expand p hp) ↑φ) = MvPolynomial.coeff n ((MvPolynomial.expand p) φ)

Tactic:
· obtain ⟨m, hm⟩ : ∃ m, p • m = n := ⟨n.mapRange (fun a ↦ a / p) (by simp),
      by ext i; simp [(Nat.mul_div_cancel' (h i))]⟩
    rw [← hm, coeff_expand_smul p hp _ _, φ.coeff_expand_smul _ hp, φ.coeff_coe]

Goal state after:
case neg
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPolynomial σ R
n : σ →₀ ℕ
h : ∃ i, ¬p ∣ n i
⊢ (coeff n) ((expand p hp) ↑φ) = MvPolynomial.coeff n ((MvPolynomial.expand p) φ)
Module: Mathlib.RingTheory.Morita.Matrix

Goal state before:
pp: R : Type u
ι : Type v
inst✝² : Ring R
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
X : ModuleCat R
⊢ ↑(toModuleCatFromModuleCatLinearEquiv R ((toMatrixModCat R ι).obj X) i).symm ∘ₗ
      LinearMap.mapMatrixModule ι (Hom.hom ((unitIso R i).inv.app X)) =
    LinearMap.id

Tactic:
ext x

Goal state after:
R : Type u
ι : Type v
inst✝² : Ring R
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
X : ModuleCat R
x : ι → ↑((𝟭 (ModuleCat R)).obj X)
⊢ (↑(toModuleCatFromModuleCatLinearEquiv R ((toMatrixModCat R ι).obj X) i).symm ∘ₗ
        LinearMap.mapMatrixModule ι (Hom.hom ((unitIso R i).inv.app X)))
      x =
    LinearMap.id x
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: K : Type u_2
inst✝ : Field K
⊢ Subring.map (LaurentSeriesRingEquiv K).toRingHom (powerSeries_as_subring K) =
    (adicCompletionIntegers K⟮X⟯ (idealX K)).toSubring

Tactic:
ext x

Goal state after:
K : Type u_2
inst✝ : Field K
x : RatFuncAdicCompl K
⊢ x ∈ Subring.map (LaurentSeriesRingEquiv K).toRingHom (powerSeries_as_subring K) ↔
    x ∈ (adicCompletionIntegers K⟮X⟯ (idealX K)).toSubring
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Invertible ↑p
n : ℕ
⊢ (X n - ∑ i, C (↑p ^ ↑i) * xInTermsOfW p R ↑i ^ p ^ (n - ↑i)) * C (⅟↑p ^ n) =
    (X n - ∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p R i ^ p ^ (n - i)) * C (⅟↑p ^ n)

Tactic:
← Fin.sum_univ_eq_sum_range

Goal state after:
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Invertible ↑p
n : ℕ
⊢ (X n - ∑ i, C (↑p ^ ↑i) * xInTermsOfW p R ↑i ^ p ^ (n - ↑i)) * C (⅟↑p ^ n) =
    (X n - ∑ i, C (↑p ^ ↑i) * xInTermsOfW p R ↑i ^ p ^ (n - ↑i)) * C (⅟↑p ^ n)
Module: Mathlib.RingTheory.WittVector.Basic

Goal state before:
pp: case hmn
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (Nat.Prime p)
x : 𝕎 R
n : ℕ
hx : ∀ i ≤ n, ↑p ∣ x.coeff i
i : ℕ
hi : i < n + 1
this : (aeval x.coeff) ((monomial (Finsupp.single i (p ^ (n - i)))) (↑p ^ i)) = ↑p ^ i * x.coeff i ^ p ^ (n - i)
⊢ n - i + 1 ≤ p ^ (n - i)

Tactic:
exact ((n - i).lt_two_pow_self).succ_le.trans
        (pow_left_mono (n - i) (Nat.Prime.two_le Fact.out))

Goal state after:
Module: Mathlib.RingTheory.WittVector.Basic

Goal state before:
pp: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (Nat.Prime p)
x : 𝕎 R
n : ℕ
hx : ∀ i ≤ n, ↑p ∣ x.coeff i
⊢ ↑p ^ (n + 1) ∣ (ghostComponent n) x

Tactic:
by
  rw [WittVector.ghostComponent_apply, wittPolynomial, MvPolynomial.aeval_sum]
  apply Finset.dvd_sum
  intro i hi
  simp only [Finset.mem_range] at hi
  have : (MvPolynomial.aeval x.coeff) ((MvPolynomial.monomial (R := ℤ)
      (Finsupp.single i (p ^ (n - i)))) (p ^ i)) = ((p : R) ^ i) * (x.coeff i) ^ (p ^ (n - i)) := by
    simp [MvPolynomial.aeval_monomial, map_pow]
  rw [this, show n + 1 = (n - i) + 1 + i by lia, pow_add, mul_comm]
  gcongr
  · exact hx i (Nat.le_of_lt_succ hi)
  · exact ((n - i).lt_two_pow_self).succ_le.trans
        (pow_left_mono (n - i) (Nat.Prime.two_le Fact.out))

Goal state after:
Module: Mathlib.RingTheory.WittVector.StructurePolynomial

Goal state before:
pp: case h
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH : ∀ m < n, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ ∀ (d : idx × ℕ →₀ ℕ),
    (Int.castRingHom ℚ) (coeff d (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) n)).num =
      coeff d (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) n)

Tactic:
Int.coe_castRingHom

Goal state after:
case h
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH : ∀ m < n, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ ∀ (d : idx × ℕ →₀ ℕ),
    (fun x ↦ ↑x) (coeff d (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) n)).num =
      coeff d (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) n)
Module: Mathlib.RingTheory.WittVector.Frobenius

Goal state before:
pp: p n : ℕ
⊢ (X (n + 1) -
      ∑ i,
        have x := ⋯;
        ∑ j ∈ range (p ^ (n - ↑i)),
          (X ↑i ^ p) ^ (p ^ (n - ↑i) - (j + 1)) * frobeniusPolyAux p ↑i ^ (j + 1) *
            C ↑((p ^ (n - ↑i)).choose (j + 1) / p ^ (n - ↑i - v p (j + 1)) * p ^ (j - v p (j + 1)))) =
    X (n + 1) -
      ∑ i ∈ range n,
        ∑ j ∈ range (p ^ (n - i)),
          (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *
            C ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p (j + 1)) * p ^ (j - v p (j + 1)))

Tactic:
← Fin.sum_univ_eq_sum_range

Goal state after:
p n : ℕ
⊢ (X (n + 1) -
      ∑ i,
        have x := ⋯;
        ∑ j ∈ range (p ^ (n - ↑i)),
          (X ↑i ^ p) ^ (p ^ (n - ↑i) - (j + 1)) * frobeniusPolyAux p ↑i ^ (j + 1) *
            C ↑((p ^ (n - ↑i)).choose (j + 1) / p ^ (n - ↑i - v p (j + 1)) * p ^ (j - v p (j + 1)))) =
    X (n + 1) -
      ∑ i,
        ∑ j ∈ range (p ^ (n - ↑i)),
          (X ↑i ^ p) ^ (p ^ (n - ↑i) - (j + 1)) * frobeniusPolyAux p ↑i ^ (j + 1) *
            C ↑((p ^ (n - ↑i)).choose (j + 1) / p ^ (n - ↑i - v p (j + 1)) * p ^ (j - v p (j + 1)))