mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ (Ideal.Quotient.mk (I ^ a n)).comp (liftRingHom I ha f ⋯) = f n

Tactic:
ext; 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
⊢ restrictScalars R (ofPowSMul I M n).range = (eval I M n).ker

Tactic:
refine le_antisymm (fun x hx ↦ ?_) (fun x hx ↦ ?_)
  · rcases hx with ⟨y, rfl⟩
    rw [LinearMap.mem_ker, eval_apply, ofPowSMul_val_apply_eq_zero _ (by rfl)]
  simp only [LinearMap.mem_ker, coe_eval] at hx
  use ofValEqZero I hx; 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
⊢ (ofPowSMul I M n).range ≤ ⇑(algebraMap R (AdicCompletion I R)) '' ↑s • ⊤

Tactic:
image_smul_top_eq_range_lsum

Goal state after:
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
⊢ (ofPowSMul I M n).range ≤
    ((lsum (AdicCompletion I R)) fun i ↦ (algebraMap R (AdicCompletion I R)) ↑i • LinearMap.id).range
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: 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)
⊢ map S.subtype (𝟙 (res S.subtype A)) 1 ≫ map (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv 1 = 0

Tactic:
by rw [← map_comp, congr (QuotientGroup.mk'_comp_subtype S) (map (n := 1)), map₁_one]

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

Goal state before:
pp: k G H : Type u
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst✝ : S.Normal
⊢ map S.subtype (𝟙 (res S.subtype A)) 1 ≫ map (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S) 1 = 0

Tactic:
by rw [← map_comp, congr (QuotientGroup.mk'_comp_subtype S) (map (n := 1)), map₁_one]

Goal state after:
Module: Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
G : Type u_5
H : Type u_6
I : Type u_7
M : Type u_8
N : Type u_9
O : Type u_10
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Bialgebra R A
inst✝³ : Bialgebra R B
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid O
⊢ (LinearMap.mul R A[M]).compr₂ counit = (LinearMap.mul R R).compl₁₂ counit counit

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
G : Type u_5
H : Type u_6
I : Type u_7
M : Type u_8
N : Type u_9
O : Type u_10
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Bialgebra R A
inst✝³ : Bialgebra R B
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid O
f : M →* N
⊢ (counitAlgHom R R[N]).comp (mapDomainAlgHom R R f) = counitAlgHom R R[M]

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
G : Type u_5
H : Type u_6
I : Type u_7
M : Type u_8
N : Type u_9
O : Type u_10
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Bialgebra R A
inst✝³ : Bialgebra R B
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid O
f : M →* N
⊢ (Algebra.TensorProduct.map (mapDomainAlgHom R R f) (mapDomainAlgHom R R f)).comp (comulAlgHom R R[M]) =
    (comulAlgHom R R[N]).comp (mapDomainAlgHom R R f)

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
M : Type u_8
inst✝¹ : CommSemiring R
inst✝ : Monoid M
⊢ mapDomainBialgHom R (MonoidHom.id M) = BialgHom.id R R[M]

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
M : Type u_8
N : Type u_9
O : Type u_10
inst✝³ : CommSemiring R
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid O
f : N →* O
g : M →* N
x : R[M]
⊢ (mapDomainBialgHom R f) ((mapDomainBialgHom R g) x) = (mapDomainBialgHom R (f.comp g)) x

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.SInteger

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
S : Set (HeightOneSpectrum R)
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
⊢ ↑(S.integer K).toSubring = ↑(⨅ v, ⨅ (_ : v ∉ S), (HeightOneSpectrum.valuation K v).valuationSubring.toSubring)

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.SInteger

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
S : Set (HeightOneSpectrum R)
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x : (↥(S.integer K))ˣ
v : HeightOneSpectrum R
hv : v ∉ S
⊢ (HeightOneSpectrum.valuation K v) 1 = 1

Tactic:
map_one

Goal state after:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
S : Set (HeightOneSpectrum R)
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x : (↥(S.integer K))ˣ
v : HeightOneSpectrum R
hv : v ∉ S
⊢ 1 = 1
Module: Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
G : Type u_5
H : Type u_6
I : Type u_7
M : Type u_8
N : Type u_9
O : Type u_10
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Bialgebra R A
inst✝³ : Bialgebra R B
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid O
⊢ (counitAlgHom R A).comp ((lift R A (GroupLike R A)) { toFun := fun g ↦ ↑g, map_one' := ⋯, map_mul' := ⋯ }) =
    counitAlgHom R R[GroupLike R A]

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
G : Type u_5
H : Type u_6
I : Type u_7
M : Type u_8
N : Type u_9
O : Type u_10
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Bialgebra R A
inst✝³ : Bialgebra R B
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid O
⊢ (Algebra.TensorProduct.map ((lift R A (GroupLike R A)) { toFun := fun g ↦ ↑g, map_one' := ⋯, map_mul' := ⋯ })
          ((lift R A (GroupLike R A)) { toFun := fun g ↦ ↑g, map_one' := ⋯, map_mul' := ⋯ })).comp
      (comulAlgHom R R[GroupLike R A]) =
    (comulAlgHom R A).comp ((lift R A (GroupLike R A)) { toFun := fun g ↦ ↑g, map_one' := ⋯, map_mul' := ⋯ })

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
M : Type u_8
N : Type u_9
inst✝² : CommSemiring R
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : M →* N
⊢ mapDomainBialgHom R (f * g) = (toConv (mapDomainBialgHom R f) * toConv (mapDomainBialgHom R g)).ofConv

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
G : Type u_5
H : Type u_6
I : Type u_7
M : Type u_8
N : Type u_9
O : Type u_10
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid N
f : R →+* S
⊢ (mapRingHom M f).comp (algebraMap R R[M]) = (algebraMap S S[M]).comp f

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.SelmerGroup

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
n : ℕ
x : Rˣ
⊢ (AddEquiv.toMultiplicative (Int.quotientZMultiplesNatEquivZMod n)).toMonoidHom 1 = 1

Tactic:
map_one

Goal state after:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
n : ℕ
x : Rˣ
⊢ 1 = 1
Module: Mathlib.RingTheory.DedekindDomain.SelmerGroup

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
S S' : Set (HeightOneSpectrum R)
n : ℕ
x✝¹ : HeightOneSpectrum R
x✝ : x✝¹ ∉ S
⊢ (x✝¹.valuationOfNeZeroMod n) 1 = 1

Tactic:
map_one

Goal state after:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
S S' : Set (HeightOneSpectrum R)
n : ℕ
x✝¹ : HeightOneSpectrum R
x✝ : x✝¹ ∉ S
⊢ 1 = 1
Module: Mathlib.RingTheory.Regular.IsSMulRegular

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
r : R
⊢ IsSMulRegular (M ⧸ N) r → r • ⊤ ⊓ N ≤ r • N

Tactic:
convert! map_mono ∘ (isSMulRegular_on_quot_iff_lsmul_comap_le N r).mp using 2

Goal state after:
case e'_3
R : Type u_1
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
r : R
a✝ : IsSMulRegular (M ⧸ N) r
⊢ r • ⊤ ⊓ N = map (DistribSMul.toLinearMap R M r) (comap ((LinearMap.lsmul R M) r) N)
Module: Mathlib.RingTheory.DividedPowers.DPMorphism

Goal state before:
pp: case pos
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
hn : n = 0
a✝ : A
ha : a✝ ∈ I
⊢ 1 = f 1

Tactic:
map_one

Goal state after:
case pos
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
hn : n = 0
a✝ : A
ha : a✝ ∈ I
⊢ 1 = 1
Module: Mathlib.RingTheory.DividedPowers.DPMorphism

Goal state before:
pp: case zero
A✝ : Type u_1
B✝ : Type u_2
inst✝³ : CommSemiring A✝
inst✝² : CommSemiring B✝
I✝ : Ideal A✝
J✝ : Ideal B✝
hI✝ : DividedPowers I✝
hJ✝ : DividedPowers J✝
A : Type u_3
B : Type u_4
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
⊢ f 1 = 1

Tactic:
map_one

Goal state after:
case zero
A✝ : Type u_1
B✝ : Type u_2
inst✝³ : CommSemiring A✝
inst✝² : CommSemiring B✝
I✝ : Ideal A✝
J✝ : Ideal B✝
hI✝ : DividedPowers I✝
hJ✝ : DividedPowers J✝
A : Type u_3
B : Type u_4
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hf : map f I ≤ J
⊢ 1 = 1
Module: Mathlib.RingTheory.DividedPowerAlgebra.Init

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_4
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Ideal A
hI : DividedPowers I
g : M →ₗ[R] A
hg : ∀ (m : M), g m ∈ I
⊢ (lift hI g hg).toLinearMap ∘ₗ embed R M = g

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.DividedPowerAlgebra.Init

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
S : Type u_4
inst✝⁵ : CommSemiring S
N : Type u_5
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R N
inst✝² : Module S N
f : M →ₗ[R] N
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R S N
⊢ (map S f).toLinearMap ∘ₗ embed R M = ↑R (embed S N) ∘ₗ f

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Regular.RegularSequence

Goal state before:
pp: case cons
R : Type u_1
M : Type u_3
M₂ : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M
inst✝³ : Module R M₂
inst✝² : Module.Flat R M₂
rs : List R
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
r : R
rs' : List R
h1 : IsSMulRegular N r
h2✝ : IsWeaklyRegular (QuotSMulTop r N) rs'
ih : IsWeaklyRegular (QuotSMulTop r N ⊗[R] M₂) rs'
⊢ IsWeaklyRegular (N ⊗[R] M₂) (r :: rs')

Tactic:
let e := quotSMulTopTensorEquivQuotSMulTop r M₂ N

Goal state after:
case cons
R : Type u_1
M : Type u_3
M₂ : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M
inst✝³ : Module R M₂
inst✝² : Module.Flat R M₂
rs : List R
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
r : R
rs' : List R
h1 : IsSMulRegular N r
h2✝ : IsWeaklyRegular (QuotSMulTop r N) rs'
ih : IsWeaklyRegular (QuotSMulTop r N ⊗[R] M₂) rs'
e : QuotSMulTop r N ⊗[R] M₂ ≃ₗ[R] QuotSMulTop r (N ⊗[R] M₂) := quotSMulTopTensorEquivQuotSMulTop r M₂ N
⊢ IsWeaklyRegular (N ⊗[R] M₂) (r :: rs')
Module: Mathlib.RingTheory.DividedPowers.Padic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
f : A →+* B
hf : Injective ⇑f
hJ : DividedPowers J
hIJ : Ideal.map f I = J
hmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)
n : ℕ
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ (if hx : x + y ∈ I then ⋯.choose else 0) =
    ∑ k ∈ Finset.antidiagonal n, (if hx : x ∈ I then ⋯.choose else 0) * if hx : y ∈ I then ⋯.choose else 0

Tactic:
have hxy : x + y ∈ I := Ideal.add_mem _ hx hy

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
f : A →+* B
hf : Injective ⇑f
hJ : DividedPowers J
hIJ : Ideal.map f I = J
hmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)
n : ℕ
x y : A
hx : x ∈ I
hy : y ∈ I
hxy : x + y ∈ I
⊢ (if hx : x + y ∈ I then ⋯.choose else 0) =
    ∑ k ∈ Finset.antidiagonal n, (if hx : x ∈ I then ⋯.choose else 0) * if hx : y ∈ I then ⋯.choose else 0
Module: Mathlib.RingTheory.DividedPowers.RatAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
I : Ideal R
inst✝¹ : DecidablePred fun x ↦ x ∈ I
inst✝ : Algebra ℚ R
hI : DividedPowers I
n : ℕ
x : R
hx : x ∈ I
⊢ hI.dpow n x = (dividedPowers I).dpow n x

Tactic:
rw [dpow_apply, if_pos hx, eq_comm, inverse_mul_eq_iff_eq_mul _ _ _
      (IsUnit.natCast_factorial_of_algebra ℚ n), factorial_mul_dpow_eq_pow _ hx]

Goal state after:
Module: Mathlib.RingTheory.DividedPowers.SubDPIdeal

Goal state before:
pp: A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
J : Ideal B
hf : Function.Surjective ⇑f
hIJ : J = Ideal.map f I
hIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)
a : A
ha : a ∈ I
hx : f a ∈ J
⊢ f 1 = 1

Tactic:
map_one

Goal state after:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
J : Ideal B
hf : Function.Surjective ⇑f
hIJ : J = Ideal.map f I
hIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)
a : A
ha : a ∈ I
hx : f a ∈ J
⊢ 1 = 1
Module: Mathlib.RingTheory.Frobenius

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
φ : S →ₐ[R] S
Q : Ideal S
H : φ.IsArithFrobAt Q
inst✝ : IsDomain S
ζ : S
m : ℕ
hζ✝ : ζ ^ m = 1
hk'✝ : ↑m ∉ Q
q : ℕ := Nat.card (R ⧸ Ideal.under R Q)
hm : m ≠ 0
k : ℕ
hk : k > 0
hζ : IsPrimitiveRoot ζ k
hk' : ↑k ∉ Q
this : NeZero k
⊢ φ 1 = 1

Tactic:
map_one

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
φ : S →ₐ[R] S
Q : Ideal S
H : φ.IsArithFrobAt Q
inst✝ : IsDomain S
ζ : S
m : ℕ
hζ✝ : ζ ^ m = 1
hk'✝ : ↑m ∉ Q
q : ℕ := Nat.card (R ⧸ Ideal.under R Q)
hm : m ≠ 0
k : ℕ
hk : k > 0
hζ : IsPrimitiveRoot ζ k
hk' : ↑k ∉ Q
this : NeZero k
⊢ 1 = 1
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring

Goal state before:
pp: case mul
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : AddMonoid ι
inst✝⁴ : Semiring A
inst✝³ : SetLike σ A
inst✝² : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝¹ : DecidableEq ι
inst✝ : GradedRing 𝒜
s : Set A
h : ∀ (i : ι) ⦃x : A⦄, x ∈ s → ↑(((decompose 𝒜) x) i) ∈ s
x✝ x y : A
hx✝ : x ∈ Subsemiring.closure s
hy✝ : y ∈ Subsemiring.closure s
h₁ : ∀ (i : ι), ↑(((decompose 𝒜) x) i) ∈ Subsemiring.closure s
h₂ : ∀ (i : ι), ↑(((decompose 𝒜) y) i) ∈ Subsemiring.closure s
i : ι
⊢ ∑ i_1 ∈ DFinsupp.support ((decompose 𝒜) x),
      ↑((DFinsupp.sum ((decompose 𝒜) y) fun x_1 aj ↦
            (of (fun i ↦ ↥(𝒜 i)) (i_1 + x_1)) (GradedMonoid.GMul.mul (((decompose 𝒜) x) i_1) aj))
          i) ∈
    Subsemiring.closure s

Tactic:
refine sum_mem fun j _ ↦ ?_

Goal state after:
case mul
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : AddMonoid ι
inst✝⁴ : Semiring A
inst✝³ : SetLike σ A
inst✝² : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝¹ : DecidableEq ι
inst✝ : GradedRing 𝒜
s : Set A
h : ∀ (i : ι) ⦃x : A⦄, x ∈ s → ↑(((decompose 𝒜) x) i) ∈ s
x✝¹ x y : A
hx✝ : x ∈ Subsemiring.closure s
hy✝ : y ∈ Subsemiring.closure s
h₁ : ∀ (i : ι), ↑(((decompose 𝒜) x) i) ∈ Subsemiring.closure s
h₂ : ∀ (i : ι), ↑(((decompose 𝒜) y) i) ∈ Subsemiring.closure s
i j : ι
x✝ : j ∈ DFinsupp.support ((decompose 𝒜) x)
⊢ ↑((DFinsupp.sum ((decompose 𝒜) y) fun x_1 aj ↦
          (of (fun i ↦ ↥(𝒜 i)) (j + x_1)) (GradedMonoid.GMul.mul (((decompose 𝒜) x) j) aj))
        i) ∈
    Subsemiring.closure s
Module: Mathlib.RingTheory.GradedAlgebra.AlgHom

Goal state before:
pp: R : Type u_1
A : Type u_6
ι : Type u_10
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : 𝒜 →ₐᵍ[R] 𝒜
n : ℕ
⊢ ⇑(f ^ Nat.zero) = (⇑f)^[Nat.zero]

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Grassmannian

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
M : Type v
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
k : ℕ
A✝ : Type w
inst✝⁵ : CommRing A✝
inst✝⁴ : Algebra R A✝
B : Type w
inst✝³ : CommRing B
inst✝² : Algebra R B
f : A✝ →ₐ[R] B
C : Type w
inst✝¹ : CommRing C
inst✝ : Algebra R C
g : B →ₐ[R] C
A : CommAlgCat R
⊢ ↾map (CommAlgCat.Hom.hom (𝟙 A)) = 𝟙 G(k, ↑A ⊗[R] M; ↑A)

Tactic:
ext N : 1; exact map_id k A N

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

Goal state before:
pp: case zero
S : Type u_1
inst✝ : CommRing S
d : ℕ
⊢ mk 1 ^ (0 + 1) = mk fun n ↦ ↑((0 + n).choose 0)

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.HEval

Goal state before:
pp: case h
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
u : R⟦X⟧ˣ
⊢ (heval x) 1 = 1

Tactic:
map_one

Goal state after:
case h
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
u : R⟦X⟧ˣ
⊢ 1 = 1
Module: Mathlib.RingTheory.HopfAlgebra.GroupLike

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : HopfAlgebra R A
a✝ b : A
a : GroupLike R A
⊢ a⁻¹ * a = 1

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ c n + -(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ) ≡ a₀ + 0 [SMOD I]

Tactic:
refine ih.add ?_

Goal state after:
case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ -(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ) ≡ 0 [SMOD I]
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : HopfAlgebraStruct R A
I : Ideal A
inst✝¹ : I.IsTwoSided
inst✝ : Ideal.IsHopfIdeal R I
⊢ antipode R ∘ₗ (mkₐ R I).toLinearMap = (mkₐ R I).toLinearMap ∘ₗ antipode R

Tactic:
ext; simp

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
hfcI : ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)
m k : ℕ
ih : c m ≡ c (m + k) [SMOD I ^ m]
⊢ c m + 0 ≡ c (m + k) + -(Polynomial.eval (c (m + k)) f * (Polynomial.eval (c (m + k)) f')⁻¹ʳ) [SMOD I ^ m]

Tactic:
refine ih.add ?_

Goal state after:
case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
hfcI : ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)
m k : ℕ
ih : c m ≡ c (m + k) [SMOD I ^ m]
⊢ 0 ≡ -(Polynomial.eval (c (m + k)) f * (Polynomial.eval (c (m + k)) f')⁻¹ʳ) [SMOD I ^ m]
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
p : Ideal R
inst✝ : p.IsPrime
I : Ideal R
hI : p ∈ I.minimalPrimes
hr : Submodule.spanRank I ≤ ↑p.height
hs : (Submodule.generators I).Finite
⊢ ↑hs.toFinset.card = p.height

Tactic:
← Set.ncard_eq_toFinset_card (hs := hs),

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
p : Ideal R
inst✝ : p.IsPrime
I : Ideal R
hI : p ∈ I.minimalPrimes
hr : Submodule.spanRank I ≤ ↑p.height
hs : (Submodule.generators I).Finite
⊢ ↑(Submodule.generators I).ncard = p.height
Module: Mathlib.RingTheory.KrullDimension.Polynomial

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
ι : Type u_2
inst✝ : Finite ι
⊢ ringKrullDim (MvPolynomial ι R) = ringKrullDim R + ↑(Nat.card ι)

Tactic:
induction ι using Finite.induction_empty_option with

Goal state after: