mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := { toSubgroup := M, isOpen' := hMOpen, isNormal' := ⋯ }
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
a : OpenNormalSubgroup ↑P.toProfinite.toTop
a_in_J : a ∈ ↑J
M_to_Na : m ⟶ a := ⋯.hom
⊢ (Hom.hom (P.diagram.map M_to_Na)) (↑((Hom.hom P.toLimit) origin) m) ∈ fJ a

Tactic:
change (P.toFiniteQuotientFunctor.map M_to_Na) (QuotientGroup.mk' M origin) ∈ _

Goal state after:
case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := { toSubgroup := M, isOpen' := hMOpen, isNormal' := ⋯ }
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
a : OpenNormalSubgroup ↑P.toProfinite.toTop
a_in_J : a ∈ ↑J
M_to_Na : m ⟶ a := ⋯.hom
⊢ (ConcreteCategory.hom (P.toFiniteQuotientFunctor.map M_to_Na)) ((QuotientGroup.mk' M) origin) ∈ fJ a
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
s : Set R
p : Ideal R
inst✝ : p.IsPrime
hrm : s ⊆ ↑p
⊢ (map (Quotient.mk (span s)) p).height + ↑(Submodule.spanFinrank (span s)) ≤
    (map (Quotient.mk (span s)) p).height + s.encard

Tactic:
gcongr
    exact Submodule.spanFinrank_span_le_encard _

Goal state after:
Module: Mathlib.RingTheory.LittleWedderburn

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this✝¹ : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] := cyclotomic n ℤ
key : ↑q - 1 + ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card) = ↑q ^ n - 1
i✝ : ConjClasses Dˣ
x : Dˣ
hx : ConjClasses.mk x ∈ ConjClasses.noncenter Dˣ
Zx : Subring D := Subring.centralizer {↑x}
hZx : Zx ≠ ⊤
this✝ : Field ↥Zx := hD.field ⋯
this : Algebra ↥Z ↥Zx := (Subring.inclusion ⋯).toAlgebra
d : ℕ := finrank ↥Z ↥Zx
card_Zx : card ↥Zx = q ^ d
⊢ 1 ≤ q ^ d

Tactic:
← card_Zx

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this✝¹ : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] := cyclotomic n ℤ
key : ↑q - 1 + ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card) = ↑q ^ n - 1
i✝ : ConjClasses Dˣ
x : Dˣ
hx : ConjClasses.mk x ∈ ConjClasses.noncenter Dˣ
Zx : Subring D := Subring.centralizer {↑x}
hZx : Zx ≠ ⊤
this✝ : Field ↥Zx := hD.field ⋯
this : Algebra ↥Z ↥Zx := (Subring.inclusion ⋯).toAlgebra
d : ℕ := finrank ↥Z ↥Zx
card_Zx : card ↥Zx = q ^ d
⊢ 1 ≤ card ↥Zx
Module: Mathlib.RingTheory.LittleWedderburn

Goal state before:
pp: case intro.mk
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this✝² : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] := cyclotomic n ℤ
key : ↑q - 1 + ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card) = ↑q ^ n - 1
i✝ : ConjClasses Dˣ
x : Dˣ
hx : ConjClasses.mk x ∈ ConjClasses.noncenter Dˣ
Zx : Subring D := Subring.centralizer {↑x}
hZx : Zx ≠ ⊤
this✝¹ : Field ↥Zx := hD.field ⋯
this✝ : Algebra ↥Z ↥Zx := (Subring.inclusion ⋯).toAlgebra
d : ℕ := finrank ↥Z ↥Zx
card_Zx : card ↥Zx = q ^ d
h1qd : 1 ≤ q ^ d
this : IsScalarTower (↥Z) (↥Zx) D
aux : ∀ {k : ℕ}, eval (↑q) (X ^ k - 1) = ↑q ^ k - 1
⊢ q ^ d < card D

Tactic:
← card_Zx

Goal state after:
case intro.mk
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this✝² : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] := cyclotomic n ℤ
key : ↑q - 1 + ↑(∑ x ∈ (ConjClasses.noncenter Dˣ).toFinset, x.carrier.toFinset.card) = ↑q ^ n - 1
i✝ : ConjClasses Dˣ
x : Dˣ
hx : ConjClasses.mk x ∈ ConjClasses.noncenter Dˣ
Zx : Subring D := Subring.centralizer {↑x}
hZx : Zx ≠ ⊤
this✝¹ : Field ↥Zx := hD.field ⋯
this✝ : Algebra ↥Z ↥Zx := (Subring.inclusion ⋯).toAlgebra
d : ℕ := finrank ↥Z ↥Zx
card_Zx : card ↥Zx = q ^ d
h1qd : 1 ≤ q ^ d
this : IsScalarTower (↥Z) (↥Zx) D
aux : ∀ {k : ℕ}, eval (↑q) (X ^ k - 1) = ↑q ^ k - 1
⊢ card ↥Zx < card D
Module: Mathlib.RingTheory.LittleWedderburn

Goal state before:
pp: n : ℕ
IH : ∀ m < n, ∀ (D : Type u_1) [inst : DivisionRing D] [Finite D] (val : Fintype D), card D = m → Subring.center D = ⊤
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
val✝ : Fintype D
hn : card D = n
R : Subring D
hR : R < ⊤
x y : D
hx : x ∈ R
hy : y ∈ R
R_dr : DivisionRing ↥R := divisionRingOfIsDomain ↥R
⊢ card ↥R < n

Tactic:
← hn,

Goal state after:
n : ℕ
IH : ∀ m < n, ∀ (D : Type u_1) [inst : DivisionRing D] [Finite D] (val : Fintype D), card D = m → Subring.center D = ⊤
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
val✝ : Fintype D
hn : card D = n
R : Subring D
hR : R < ⊤
x y : D
hx : x ∈ R
hy : y ∈ R
R_dr : DivisionRing ↥R := divisionRingOfIsDomain ↥R
⊢ card ↥R < card D
Module: Mathlib.RingTheory.LocalIso

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
ι : Type u_3
f : ι → S
h✝ : Ideal.span (Set.range f) = ⊤
T : ι → Type u_4
inst✝⁵ : (i : ι) → CommSemiring (T i)
inst✝⁴ : (i : ι) → Algebra R (T i)
inst✝³ : (i : ι) → Algebra S (T i)
inst✝² : ∀ (i : ι), IsScalarTower R S (T i)
inst✝¹ : ∀ (i : ι), IsLocalization.Away (f i) (T i)
inst✝ : ∀ (i : ι), IsLocalIso R (T i)
q : Ideal S
hq : q.IsPrime
i : ι
hi : f i ∉ q
q' : PrimeSpectrum (T i)
g' : T i
hg' : g' ∉ q'.asIdeal
h : IsStandardOpenImmersion R (Localization.Away g')
n : ℕ
g : S
hg : g' * (algebraMap S (T i)) (f i) ^ n = (algebraMap S (T i)) g
hq' : Ideal.comap (algebraMap S (T i)) q'.asIdeal = q
⊢ g ∉ q

Tactic:
← hq',

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
ι : Type u_3
f : ι → S
h✝ : Ideal.span (Set.range f) = ⊤
T : ι → Type u_4
inst✝⁵ : (i : ι) → CommSemiring (T i)
inst✝⁴ : (i : ι) → Algebra R (T i)
inst✝³ : (i : ι) → Algebra S (T i)
inst✝² : ∀ (i : ι), IsScalarTower R S (T i)
inst✝¹ : ∀ (i : ι), IsLocalization.Away (f i) (T i)
inst✝ : ∀ (i : ι), IsLocalIso R (T i)
q : Ideal S
hq : q.IsPrime
i : ι
hi : f i ∉ q
q' : PrimeSpectrum (T i)
g' : T i
hg' : g' ∉ q'.asIdeal
h : IsStandardOpenImmersion R (Localization.Away g')
n : ℕ
g : S
hg : g' * (algebraMap S (T i)) (f i) ^ n = (algebraMap S (T i)) g
hq' : Ideal.comap (algebraMap S (T i)) q'.asIdeal = q
⊢ g ∉ Ideal.comap (algebraMap S (T i)) q'.asIdeal
Module: Mathlib.RingTheory.LocalProperties.Injective

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
H : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)
I : Ideal R
x✝¹ : FinitePresentation R ↥I := finitePresentation_of_finite R ↥I
m : Ideal R
x✝ : m.IsMaximal
Rₘ : Type u := Localization.AtPrime m
Mₘ : Type (max u v) := LocalizedModule m.primeCompl M
⊢ Function.Surjective ⇑((LocalizedModule.map m.primeCompl) (LinearMap.lcomp R M (Submodule.subtype I)))

Tactic:
let f := LocalizedModule.mkLinearMap m.primeCompl M

Goal state after:
R : Type u
inst✝⁴ : CommRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Small.{v, u} R
inst✝ : IsNoetherianRing R
H : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)
I : Ideal R
x✝¹ : FinitePresentation R ↥I := finitePresentation_of_finite R ↥I
m : Ideal R
x✝ : m.IsMaximal
Rₘ : Type u := Localization.AtPrime m
Mₘ : Type (max u v) := LocalizedModule m.primeCompl M
f : M →ₗ[R] LocalizedModule m.primeCompl M := LocalizedModule.mkLinearMap m.primeCompl M
⊢ Function.Surjective ⇑((LocalizedModule.map m.primeCompl) (LinearMap.lcomp R M (Submodule.subtype I)))
Module: Mathlib.RingTheory.Invariant.Profinite

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
G : Type u
inst✝³ : Group G
inst✝² : MulSemiringAction G B
inst✝¹ : SMulCommClass G A B
inst✝ : TopologicalSpace G
Q : Ideal B
N N' : OpenNormalSubgroup G
e : N ≤ N'
h : FixedPoints.subalgebra A B ↥↑N'.toOpenSubgroup ≤ FixedPoints.subalgebra A B ↥↑N.toOpenSubgroup
x : G
hx : ↑x ∈ MulAction.stabilizer (G ⧸ ↑N.toOpenSubgroup) (under (↥(FixedPoints.subalgebra A B ↥↑N.toOpenSubgroup)) Q)
⊢ (QuotientGroup.map (↑N.toOpenSubgroup) N'.toOpenSubgroup.1 (MonoidHom.id G) e) ↑x •
      under (↥(FixedPoints.subalgebra A B ↥↑N'.toOpenSubgroup)) Q =
    under (↥(FixedPoints.subalgebra A B ↥↑N'.toOpenSubgroup)) Q

Tactic:
replace hx := congr(Ideal.comap (Subalgebra.inclusion h) $hx)

Goal state after:
A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
G : Type u
inst✝³ : Group G
inst✝² : MulSemiringAction G B
inst✝¹ : SMulCommClass G A B
inst✝ : TopologicalSpace G
Q : Ideal B
N N' : OpenNormalSubgroup G
e : N ≤ N'
h : FixedPoints.subalgebra A B ↥↑N'.toOpenSubgroup ≤ FixedPoints.subalgebra A B ↥↑N.toOpenSubgroup
x : G
hx :
  comap (Subalgebra.inclusion h) (↑x • under (↥(FixedPoints.subalgebra A B ↥↑N.toOpenSubgroup)) Q) =
    comap (Subalgebra.inclusion h) (under (↥(FixedPoints.subalgebra A B ↥↑N.toOpenSubgroup)) Q)
⊢ (QuotientGroup.map (↑N.toOpenSubgroup) N'.toOpenSubgroup.1 (MonoidHom.id G) e) ↑x •
      under (↥(FixedPoints.subalgebra A B ↥↑N'.toOpenSubgroup)) Q =
    under (↥(FixedPoints.subalgebra A B ↥↑N'.toOpenSubgroup)) Q
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
S : Type u_3
inst✝¹ : Semiring S
inst✝ : Module R S
r : R
x : S⟦X⟧
⊢ (ofPowerSeries ℤ S) (r • x) = r • (ofPowerSeries ℤ S) x

Tactic:
by
  ext
  simp [coeff_coe, coeff_smul, smul_ite]

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

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : IsLocalRing S
f : R →+* S
hf : Function.Injective ⇑f
h : ∀ a ∈ R⁰, IsUnit a
⊢ IsLocalRing R

Tactic:
have : Nontrivial R := f.domain_nontrivial

Goal state after:
R : Type u_2
S : Type u_1
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : IsLocalRing S
f : R →+* S
hf : Function.Injective ⇑f
h : ∀ a ∈ R⁰, IsUnit a
this : Nontrivial R
⊢ IsLocalRing R
Module: Mathlib.RingTheory.MvPolynomial.EulerIdentity

Goal state before:
pp: case refine_3
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
p q : MvPolynomial σ R
x✝¹ : p ∈ Submodule.span R ((fun m ↦ AddMonoidAlgebra.single m 1) '' {d | (weight w) d = n})
x✝ : q ∈ Submodule.span R ((fun m ↦ AddMonoidAlgebra.single m 1) '' {d | (weight w) d = n})
hp : IsWeightedHomogeneous w ((pderiv i) p) n'
hq : IsWeightedHomogeneous w ((pderiv i) q) n'
⊢ IsWeightedHomogeneous w ((pderiv i) p + (pderiv i) q) n'

Tactic:
exact hp.add hq

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: i n m : ℕ
hin : i < n
him : i + 1 < m
t : Fin n → ℕ
⊢ (accumulate n m) t ⟨i, ⋯⟩ = t ⟨i, hin⟩ + (accumulate n m) t ⟨i + 1, him⟩

Tactic:
accumulate_apply

Goal state after:
i n m : ℕ
hin : i < n
him : i + 1 < m
t : Fin n → ℕ
⊢ ∑ x with i ≤ ↑x, t x = t ⟨i, hin⟩ + ∑ x with i + 1 ≤ ↑x, t x
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: i n m : ℕ
hin : i < n
hmi : m = i + 1
t : Fin n → ℕ
ht : ∀ (j : Fin n), m ≤ ↑j → t j = 0
⊢ (accumulate n m) t ⟨i, ⋯⟩ = t ⟨i, hin⟩

Tactic:
accumulate_apply

Goal state after:
i n m : ℕ
hin : i < n
hmi : m = i + 1
t : Fin n → ℕ
ht : ∀ (j : Fin n), m ≤ ↑j → t j = 0
⊢ ∑ i with ↑⟨i, ⋯⟩ ≤ ↑i, t i = t ⟨i, hin⟩
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case refine_1
n m : ℕ
hmn : m ≤ n
s : Fin m → ℕ
hs : Antitone s
x✝¹ : Fin m
i✝ : ℕ
this : i✝ ≤ m - 1
i : ℕ
hi : i + 1 < m
x✝ : i✝ ≤ i
ih : ∀ (hi : i + 1 < m), (accumulate n m) (invAccumulate n m s) ⟨i + 1, hi⟩ = s ⟨i + 1, hi⟩
him : i < m
⊢ (accumulate n m) (invAccumulate n m s) ⟨i, him⟩ = s ⟨i, him⟩

Tactic:
accumulate_rec (him.trans_le hmn) hi,

Goal state after:
case refine_1
n m : ℕ
hmn : m ≤ n
s : Fin m → ℕ
hs : Antitone s
x✝¹ : Fin m
i✝ : ℕ
this : i✝ ≤ m - 1
i : ℕ
hi : i + 1 < m
x✝ : i✝ ≤ i
ih : ∀ (hi : i + 1 < m), (accumulate n m) (invAccumulate n m s) ⟨i + 1, hi⟩ = s ⟨i + 1, hi⟩
him : i < m
⊢ invAccumulate n m s ⟨i, ⋯⟩ + (accumulate n m) (invAccumulate n m s) ⟨i + 1, hi⟩ = s ⟨i, him⟩
Module: Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝ : CommRing R
c : n →₀ R
i : n
⊢ ∀ b ∈ c.support, b ≠ i → coeff (Finsupp.single i 1) (c b • X b) = 0

Tactic:
intro j hj hji

Goal state after:
n : Type u_1
R : Type u_2
inst✝ : CommRing R
c : n →₀ R
i j : n
hj : j ∈ c.support
hji : j ≠ i
⊢ coeff (Finsupp.single i 1) (c j • X j) = 0
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case mul_X
R : Type u_3
n m : ℕ
inst✝ : CommRing R
h : m ≤ n
p : MvPolynomial (Fin m) R
i : Fin m
hp : (esymmAlgHom (Fin m) R m) p ∈ (esymmAlgHom (Fin m) R n).range
⊢ ∃ x, (esymmAlgHom (Fin m) R n) x = (esymmAlgHom (Fin m) R m) (X i)

Tactic:
refine ⟨X ⟨i, i.2.trans_le h⟩, ?_⟩

Goal state after:
case mul_X
R : Type u_3
n m : ℕ
inst✝ : CommRing R
h : m ≤ n
p : MvPolynomial (Fin m) R
i : Fin m
hp : (esymmAlgHom (Fin m) R m) p ∈ (esymmAlgHom (Fin m) R n).range
⊢ (esymmAlgHom (Fin m) R n) (X ⟨↑i, ⋯⟩) = (esymmAlgHom (Fin m) R m) (X i)
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
⊢ (-1) ^ (k + 1) *
      ∑ x ∈ {t ∈ NewtonIdentities.pairs σ k | #t.1 < k}.disjUnion ({t ∈ NewtonIdentities.pairs σ k | #t.1 = k}) ⋯,
        NewtonIdentities.weight σ R k x =
    0

Tactic:
NewtonIdentities.disjUnion_filter_pairs_eq_pairs σ k,

Goal state after:
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
⊢ (-1) ^ (k + 1) * ∑ x ∈ NewtonIdentities.pairs σ k, NewtonIdentities.weight σ R k x = 0
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
m : σ →₀ ℕ
this : ∀ {d : σ →₀ ℕ}, (d.prod fun s e ↦ (X s ^ p) ^ e) = (monomial (p • d)) 1
⊢ (coeff m) φ * (coeff (p • m)) (m.prod fun s e ↦ (X s ^ p) ^ e) = (coeff m) φ

pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPowerSeries σ R
m : σ →₀ ℕ
this : ∀ {d : σ →₀ ℕ}, (d.prod fun s e ↦ (X s ^ p) ^ e) = (monomial (p • d)) 1
⊢ ∀ (x : σ →₀ ℕ), x ≠ m → (coeff x) φ * (coeff (p • m)) (x.prod fun s e ↦ (X s ^ p) ^ e) = 0

Tactic:
· rw [this, coeff_monomial, if_pos rfl, mul_one]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPowerSeries σ R
m : σ →₀ ℕ
this : ∀ {d : σ →₀ ℕ}, (d.prod fun s e ↦ (X s ^ p) ^ e) = (monomial (p • d)) 1
⊢ ∀ (x : σ →₀ ℕ), x ≠ m → (coeff x) φ * (coeff (p • m)) (x.prod fun s e ↦ (X s ^ p) ^ e) = 0
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
m : σ →₀ ℕ
i : σ
h : ∑ᶠ (d : σ →₀ ℕ), (coeff d) φ • (coeff m) (d.prod fun s e ↦ (X s ^ p) ^ e) ≠ 0
⊢ p ∣ m i

Tactic:
obtain ⟨d, hd⟩ : ∃ (d : σ →₀ ℕ), (coeff m) (d.prod fun s e ↦ ((X s (R := R)) ^ p) ^ e) ≠ 0 := by
    by_contra! hc
    rw [finsum_eq_zero_of_forall_eq_zero fun d => by simp [hc d]] at h
    contradiction

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPowerSeries σ R
m : σ →₀ ℕ
i : σ
h : ∑ᶠ (d : σ →₀ ℕ), (coeff d) φ • (coeff m) (d.prod fun s e ↦ (X s ^ p) ^ e) ≠ 0
d : σ →₀ ℕ
hd : (coeff m) (d.prod fun s e ↦ (X s ^ p) ^ e) ≠ 0
⊢ p ∣ m i
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
d : σ →₀ ℕ
this : ∀ (i : σ), p ∣ d i
m : σ →₀ ℕ := Finsupp.mapRange (fun n ↦ n / p) ⋯ d
hd : (expand p hp) φ (p • m) ≠ 0
eq_aux : p • m = d
⊢ d ∈ (fun x ↦ p • x) '' Function.support φ

Tactic:
← coeff_apply (expand p hp φ),

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
p : ℕ
hp : p ≠ 0
φ : MvPowerSeries σ R
d : σ →₀ ℕ
this : ∀ (i : σ), p ∣ d i
m : σ →₀ ℕ := Finsupp.mapRange (fun n ↦ n / p) ⋯ d
hd : (coeff (p • m)) ((expand p hp) φ) ≠ 0
eq_aux : p • m = d
⊢ d ∈ (fun x ↦ p • x) '' Function.support φ
Module: Mathlib.RingTheory.Morita.Matrix

Goal state before:
pp: R : Type u
ι : Type v
inst✝⁴ : Ring R
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
M : Type u_1
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ι
x✝ : ↥(toModuleCatObj R (ι → M) i)
x : ι → M
hx : x ∈ toModuleCatObj R (ι → M) i
y : ι → M
hy : single i i 1 • y = x
⊢ (fun x ↦ ⟨Pi.single i x, ⋯⟩) ((fun x ↦ ∑ i_1, ↑x i_1) ⟨x, hx⟩) = ⟨x, hx⟩

Tactic:
single_smul

Goal state after:
R : Type u
ι : Type v
inst✝⁴ : Ring R
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
M : Type u_1
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ι
x✝ : ↥(toModuleCatObj R (ι → M) i)
x : ι → M
hx : x ∈ toModuleCatObj R (ι → M) i
y : ι → M
hy : Pi.single i (1 • y i) = x
⊢ (fun x ↦ ⟨Pi.single i x, ⋯⟩) ((fun x ↦ ∑ i_1, ↑x i_1) ⟨x, hx⟩) = ⟨x, hx⟩
Module: Mathlib.RingTheory.WittVector.WittPolynomial

Goal state before:
pp: case ind
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
⊢ m ∈ range n → constantCoeff (C (↑p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0

Tactic:
H

Goal state after:
case ind
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
hp : Fact (Nat.Prime p)
inst✝ : Invertible ↑p
n : ℕ
IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0
m : ℕ
H : m ∈ range n
⊢ constantCoeff (C (↑p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0
Module: Mathlib.RingTheory.Teichmuller

Goal state before:
pp: p : ℕ
inst✝³ : Fact (Nat.Prime p)
R : Type u_1
inst✝² : CommRing R
I : Ideal R
inst✝¹ : CharP (R ⧸ I) p
inst✝ : IsAdicComplete I R
x y : Perfection (R ⧸ I) p
n : ℕ
⊢ Quotient.out ((coeff (R ⧸ I) p n) x) ^ p ^ n * Quotient.out ((coeff (R ⧸ I) p n) y) ^ p ^ n ≡
    x.teichmullerFun * y.teichmullerFun [SMOD I ^ (n + 1)]

Tactic:
refine (teichmullerFun_sModEq ?_).symm.mul (teichmullerFun_sModEq ?_).symm <;> simp

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

Goal state before:
pp: case refine_2
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℚ
φ : ℕ → MvPolynomial (idx × ℕ) ℚ
⊢ (fun φ ↦ ∀ (n : ℕ), (bind₁ φ) (W_ ℚ n) = (bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℚ n)) Φ) φ → φ = wittStructureRat p Φ

Tactic:
H

Goal state after:
case refine_2
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℚ
φ : ℕ → MvPolynomial (idx × ℕ) ℚ
H : ∀ (n : ℕ), (bind₁ φ) (W_ ℚ n) = (bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℚ n)) Φ
⊢ φ = wittStructureRat p Φ
Module: Mathlib.RingTheory.NoetherNormalization

Goal state before:
pp: k : Type u_1
inst✝ : Field k
n d : ℕ
hd : ∀ (I : Ideal (MvPolynomial (Fin d) k)), I ≠ ⊤ → ∃ s ≤ d, ∃ g, Function.Injective ⇑g ∧ g.IsIntegral
I : Ideal (MvPolynomial (Fin (d + 1)) k)
hi : I ≠ ⊤
eqi : ¬I = 0
f : MvPolynomial (Fin (d + 1)) k
fi : f ∈ I
fne : f ≠ 0
ϕ : MvPolynomial (Fin d) k ⧸ ker (hom2 f I) →ₐ[k] MvPolynomial (Fin (d + 1)) k ⧸ I := kerLiftAlg (hom2 f I)
this : Nontrivial (MvPolynomial (Fin (d + 1)) k ⧸ I)
s : ℕ
left✝ : s ≤ d
g : MvPolynomial (Fin s) k →ₐ[k] MvPolynomial (Fin d) k ⧸ ker (hom2 f I)
injg : Function.Injective ⇑g
intg : g.IsIntegral
a : MvPolynomial (Fin d) k
⊢ ((kerLiftAlg (hom2 f I)).comp (Quotient.mkₐ k (ker (hom2 f I)))) a = (hom2 f I) a

Tactic:
simp only [AlgHom.coe_comp, Quotient.mkₐ_eq_mk, Function.comp_apply, kerLiftAlg_mk]

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

Goal state before:
pp: case h.h₁
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
IH : ∀ m < n, 0 < m → (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ m) = 0
hn : 0 < n
⊢ 0 ∉ Finset.range n → C ↑p ^ 0 * (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ 0) ^ p ^ (n - 0) = 0

Tactic:
· grind

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

Goal state before:
pp: p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
h : constantCoeff Φ = 0
n : ℕ
inj : Function.Injective ⇑(Int.castRingHom ℚ)
⊢ (Int.castRingHom ℚ) (constantCoeff (wittStructureInt p Φ n)) = (Int.castRingHom ℚ) 0

Tactic:
rw [← constantCoeff_map, map_wittStructureInt, constantCoeff_wittStructureRat, map_zero]

Goal state after:
case h
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
h : constantCoeff Φ = 0
n : ℕ
inj : Function.Injective ⇑(Int.castRingHom ℚ)
⊢ constantCoeff ((map (Int.castRingHom ℚ)) Φ) = 0
Module: Mathlib.RingTheory.WittVector.Verschiebung

Goal state before:
pp: p : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x : 𝕎 R
⊢ ↑p ^ 0 * x.verschiebungFun.coeff 0 ^ p ^ (0 - 0) = 0

Tactic:
verschiebungFun_coeff_zero,

Goal state after:
p : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x : 𝕎 R
⊢ ↑p ^ 0 * 0 ^ p ^ (0 - 0) = 0