mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
m : ℕ
hm : Fintype.card α = n.gcd (Fintype.card α) * m
hm0 : 0 < m
⊢ #(↑(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α))))).toFinset ≤ n

Tactic:
simp only [Set.toFinset_card, SetLike.coe_sort_coe]

Goal state after:
α : Type u_1
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
m : ℕ
hm : Fintype.card α = n.gcd (Fintype.card α) * m
hm0 : 0 < m
⊢ Fintype.card ↥(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α)))) ≤ n
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
⊢ IsPGroup p G ↔ ∃ k, ∀ (g : G), g ^ p ^ k = 1

Tactic:
isPGroup_iff_exists_orderOf_dvd_pow,

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
⊢ (∃ k, ∀ (g : G), orderOf g ∣ p ^ k) ↔ ∃ k, ∀ (g : G), g ^ p ^ k = 1
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
⊢ IsPGroup p G ↔ ∃ n, Monoid.exponent G ∣ p ^ n

Tactic:
isPGroup_iff_exists_orderOf_dvd_pow,

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
⊢ (∃ k, ∀ (g : G), orderOf g ∣ p ^ k) ↔ ∃ n, Monoid.exponent G ∣ p ^ n
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
this : 1 = (orderOf (t ^ p ^ k)).factorization p + 1
⊢ ¬p ^ 1 ∣ orderOf (t ^ p ^ k)

Tactic:
rw [this]

Goal state after:
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
this : 1 = (orderOf (t ^ p ^ k)).factorization p + 1
⊢ ¬p ^ ((orderOf (t ^ p ^ k)).factorization p + 1) ∣ orderOf (t ^ p ^ k)
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: case h
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
hcoprime : (orderOf (t ^ p ^ k)).Coprime (orderOf g)
⊢ orderOf t < orderOf t / p ^ (orderOf t).factorization p * p ^ ((orderOf t).factorization p + a + 1)

Tactic:
pow_add,

Goal state after:
case h
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
hcoprime : (orderOf (t ^ p ^ k)).Coprime (orderOf g)
⊢ orderOf t < orderOf t / p ^ (orderOf t).factorization p * (p ^ ((orderOf t).factorization p + a) * p ^ 1)
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: case h
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
hcoprime : (orderOf (t ^ p ^ k)).Coprime (orderOf g)
⊢ orderOf t < orderOf t / p ^ (orderOf t).factorization p * (p ^ ((orderOf t).factorization p + a) * p ^ 1)

Tactic:
pow_add,

Goal state after:
case h
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
hcoprime : (orderOf (t ^ p ^ k)).Coprime (orderOf g)
⊢ orderOf t < orderOf t / p ^ (orderOf t).factorization p * (p ^ (orderOf t).factorization p * p ^ a * p ^ 1)
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: case h
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
hcoprime : (orderOf (t ^ p ^ k)).Coprime (orderOf g)
⊢ p ^ (orderOf t).factorization p ∣ orderOf t

Tactic:
· exact hpk

Goal state after:
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
⊢ exponent (M₁ × M₂) = GCDMonoid.lcm (exponent M₁) (exponent M₂)

Tactic:
refine dvd_antisymm ?_ (lcm_dvd ?_ ?_)

Goal state after:
case refine_1
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
⊢ exponent (M₁ × M₂) ∣ GCDMonoid.lcm (exponent M₁) (exponent M₂)

case refine_2
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
⊢ exponent M₁ ∣ exponent (M₁ × M₂)

case refine_3
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
⊢ exponent M₂ ∣ exponent (M₁ × M₂)
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
h : IsClosedMap (comap (mapRingHom f))
algInst✝ : Algebra R S := f.toAlgebra
a✝¹ : Nontrivial R
a✝ : Nontrivial S
r : S
p : S[X] := C r * X - 1
H : zeroLocus ↑(Ideal.comap (mapRingHom f) (Ideal.span {p})) = comap (mapRingHom f) '' zeroLocus ↑(Ideal.span {p})
⊢ ∀ (a : PrimeSpectrum S[X]), p ∈ a.asIdeal → X ∉ a.asIdeal

Tactic:
intro q

Goal state after:
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
h : IsClosedMap (comap (mapRingHom f))
algInst✝ : Algebra R S := f.toAlgebra
a✝¹ : Nontrivial R
a✝ : Nontrivial S
r : S
p : S[X] := C r * X - 1
H : zeroLocus ↑(Ideal.comap (mapRingHom f) (Ideal.span {p})) = comap (mapRingHom f) '' zeroLocus ↑(Ideal.span {p})
q : PrimeSpectrum S[X]
⊢ p ∈ q.asIdeal → X ∉ q.asIdeal
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
this✝ : fixedPoints (↥↑P) (Sylow p G) = {P}
this : Nat.card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1
⊢ Nat.card ↑(fixedPoints (↥↑P) (Sylow p G)) ≡ 1 [MOD p]

Tactic:
rw [this]

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: case h1
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : IsSimpleGroup α
g : α
hg : ∀ (x : α), x ∈ Subgroup.zpowers g
hα : orderOf g ≠ 1
h : orderOf g = 0
⊢ ∃ n, ¬orderOf g ∣ n ∧ n.gcd (orderOf g) ≠ 1

Tactic:
exact ⟨37, by simp [h]⟩

Goal state after:
Module: Mathlib.GroupTheory.Torsion

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : CommMonoid G
inst✝ : CommMonoid H
p : ℕ
a b : G
x✝¹ : a ∈ {g | ∃ k, g ^ p ^ k = 1}
x✝ : b ∈ {g | ∃ k, g ^ p ^ k = 1}
m : ℕ
hm : a ^ p ^ m = 1
n : ℕ
hn : b ^ p ^ n = 1
⊢ a ^ p ^ (m + n) * b ^ p ^ (m + n) = 1

Tactic:
pow_add,

Goal state after:
G : Type u_1
H : Type u_2
inst✝¹ : CommMonoid G
inst✝ : CommMonoid H
p : ℕ
a b : G
x✝¹ : a ∈ {g | ∃ k, g ^ p ^ k = 1}
x✝ : b ∈ {g | ∃ k, g ^ p ^ k = 1}
m : ℕ
hm : a ^ p ^ m = 1
n : ℕ
hn : b ^ p ^ n = 1
⊢ a ^ (p ^ m * p ^ n) * b ^ (p ^ m * p ^ n) = 1
Module: Mathlib.GroupTheory.Torsion

Goal state before:
pp: G : Type u_1
inst✝ : CommMonoid G
p : ℕ
hp : Fact (Nat.Prime p)
p' : ℕ
hp' : Fact (Nat.Prime p')
hne : p ≠ p'
g : G
hg : g ∈ primaryComponent G p
hg' : g ∈ primaryComponent G p'
⊢ g = 1

Tactic:
mem_primaryComponent_iff_orderOf

Goal state after:
G : Type u_1
inst✝ : CommMonoid G
p : ℕ
hp : Fact (Nat.Prime p)
p' : ℕ
hp' : Fact (Nat.Prime p')
hne : p ≠ p'
g : G
hg : ∃ n, orderOf g = p ^ n
hg' : ∃ n, orderOf g = p' ^ n
⊢ g = 1
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: G : Type u_2
inst✝¹ : CommGroup G
hG : IsCyclic G
inst✝ : Finite G
d : ℕ
g : G
h : Subgroup.zpowers g = ⊤
⊢ orderOf (g ^ d) = Nat.card G / (Nat.card G).gcd d

Tactic:
orderOf_pow,

Goal state after:
G : Type u_2
inst✝¹ : CommGroup G
hG : IsCyclic G
inst✝ : Finite G
d : ℕ
g : G
h : Subgroup.zpowers g = ⊤
⊢ orderOf g / (orderOf g).gcd d = Nat.card G / (Nat.card G).gcd d
Module: Mathlib.RingTheory.Coprime.Ideal

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
t : Finset ι
h : t.Nonempty
I : ι → Ideal R
this : DecidableEq ι
⊢ ⨆ i ∈ t, ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j = ⊤ ↔ (↑t).Pairwise fun i j ↦ I i ⊔ I j = ⊤

Tactic:
eq_top_iff_one,

Goal state after:
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
t : Finset ι
h : t.Nonempty
I : ι → Ideal R
this : DecidableEq ι
⊢ 1 ∈ ⨆ i ∈ t, ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j ↔ (↑t).Pairwise fun i j ↦ I i ⊔ I j = ⊤
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x✝ y✝ : ↥(normalizer ↑H)
⊢ (leftRel (comap (normalizer H.carrier).subtype H)) x✝ y✝ ↔ (leftRel H) ↑x✝ ↑y✝

Tactic:
rw [leftRel_apply (α := normalizer (H : Set G)), leftRel_apply]

Goal state after:
G : Type u
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x✝ y✝ : ↥(normalizer ↑H)
⊢ x✝⁻¹ * y✝ ∈ comap (normalizer H.carrier).subtype H ↔ (↑x✝)⁻¹ * ↑y✝ ∈ H
Module: Mathlib.RingTheory.Coprime.Ideal

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
I : ι → Ideal R
this✝ : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j ↦ I i ⊔ I j = ⊤
this : Std.Symm fun x1 x2 ↦ I x1 ⊔ I x2 = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ↑(μ a) + ∑ x ∈ t, ↑(μ x) = 1
b : ι
hb : b ∈ ↑t
ab : a ≠ b
⊢ I a ⊔ I b = ⊤

Tactic:
eq_top_iff_one,

Goal state after:
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
I : ι → Ideal R
this✝ : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j ↦ I i ⊔ I j = ⊤
this : Std.Symm fun x1 x2 ↦ I x1 ⊔ I x2 = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ↑(μ a) + ∑ x ∈ t, ↑(μ x) = 1
b : ι
hb : b ∈ ↑t
ab : a ≠ b
⊢ 1 ∈ I a ⊔ I b
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
⊢ IsMaximal (s.toFun ⟨s.length.pred, ⋯⟩) (last s)

Tactic:
last

Goal state after:
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
⊢ IsMaximal (s.toFun ⟨s.length.pred, ⋯⟩) (s.toFun (Fin.last s.length))
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
x₁ x₂ : X
hsat₁ : IsMaximal (last s₁) x₁
hsat₂ : IsMaximal (last s₂) x₂
hequiv : s₁.Equivalent s₂
hlast : Iso (last s₁, x₁) (last s₂, x₂)
e : Fin s₁.length.succ ≃ Fin s₂.length.succ :=
  Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm
⊢ e (Fin.last s₁.length) = Fin.last s₂.length

Tactic:
simp [e, finSuccEquivLast_last, Functor.mapEquiv_apply]

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

Goal state before:
pp: case refine_2.mpr
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
I : ι → Ideal R
this✝¹ : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j ↦ I i ⊔ I j = ⊤
this✝ : Std.Symm fun x1 x2 ↦ I x1 ⊔ I x2 = ⊤
hs : (↑t).Pairwise fun i j ↦ I i ⊔ I j = ⊤
Hb : ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤
μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)
hμ : ∑ i ∈ t, ↑(μ i) = 1
this : I a ⊔ ⨅ i ∈ t, I i = ⊤
⊢ ∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1

Tactic:
eq_top_iff_one,

Goal state after:
case refine_2.mpr
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
I : ι → Ideal R
this✝¹ : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j ↦ I i ⊔ I j = ⊤
this✝ : Std.Symm fun x1 x2 ↦ I x1 ⊔ I x2 = ⊤
hs : (↑t).Pairwise fun i j ↦ I i ⊔ I j = ⊤
Hb : ∀ b ∈ ↑t, a ≠ b → I a ⊔ I b = ⊤
μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)
hμ : ∑ i ∈ t, ↑(μ i) = 1
this : 1 ∈ I a ⊔ ⨅ i ∈ t, I i
⊢ ∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : head s₁ = head s₂
ht : last s₁ = last s₂
x✝ : ℕ
hle : s₁.length = x✝
⊢ s₁.Equivalent s₂

Tactic:
induction hle : s₁.length generalizing s₁ s₂ with

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
a : R
x✝¹ : Submodule (R ⧸ R ∙ a) ↥(torsionBy R M a)
x✝ : ↥(torsionBy R M a)
⊢ x✝ ∈
      (fun p ↦ { carrier := ↑p, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ })
        ((restrictScalarsEmbedding R (R ⧸ R ∙ a) ↥(torsionBy R M a)).toFun x✝¹) ↔
    x✝ ∈ x✝¹

Tactic:
simp [restrictScalarsEmbedding]

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
a : R
x✝¹ : Submodule R ↥(torsionBy R M a)
x✝ : ↥(torsionBy R M a)
⊢ x✝ ∈
      (restrictScalarsEmbedding R (R ⧸ R ∙ a) ↥(torsionBy R M a)).toFun
        ((fun p ↦ { carrier := ↑p, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }) x✝¹) ↔
    x✝ ∈ x✝¹

Tactic:
simp [restrictScalarsEmbedding]

Goal state after:
Module: Mathlib.RingTheory.Nakayama

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I J : Ideal R
N : Submodule R M
hN : N.FG
hIN : N ≤ I • N
hIjac : I ≤ J.jacobson
n : M
hn : n ∈ N
r : R
hr : r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
s : R
hs : s * r - 1 ∈ J
this : n = -(s * r - 1) • n
⊢ n ∈ J • N

Tactic:
rw [this]

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I J : Ideal R
N : Submodule R M
hN : N.FG
hIN : N ≤ I • N
hIjac : I ≤ J.jacobson
n : M
hn : n ∈ N
r : R
hr : r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
s : R
hs : s * r - 1 ∈ J
this : n = -(s * r - 1) • n
⊢ -(s * r - 1) • n ∈ J • N
Module: Mathlib.RingTheory.Nakayama

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I J : Ideal R
N : Submodule R M
hN : N.FG
hIN : N ≤ I • N
hIjac : I ≤ J.jacobson
n : M
hn : n ∈ N
r : R
hr : r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
s : R
hs : s * r - 1 ∈ J
this : n = -(s * r - 1) • n
⊢ -(s * r - 1) • n ∈ J • N

Tactic:
exact Submodule.smul_mem_smul (Submodule.neg_mem _ hs) hn

Goal state after:
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
i j k : ℕ
⊢ I ^ (k + i) • F.N j ≤ I ^ k • F.N (i + j)

Tactic:
pow_add,

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
i j k : ℕ
⊢ (I ^ k * I ^ i) • F.N j ≤ I ^ k • F.N (i + j)
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F✝ F' : I.Filtration M
S : Set (I.Filtration M)
i : ℕ
F : I.Filtration M
hF : F ∈ S
⊢ ∃ b ∈ Set.range fun f ↦ ↑f i, (fun f ↦ ↑f (i + 1)) ⟨F.N, ⋯⟩ ≤ b

Tactic:
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩

Goal state after:
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F✝ F' : I.Filtration M
S : Set (I.Filtration M)
i : ℕ
F : I.Filtration M
hF : F ∈ S
⊢ ∃ b ∈ Set.range fun f ↦ ↑f (i + 1), b ≤ (fun f ↦ ↑f i) ⟨F.N, ⋯⟩

Tactic:
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: case h
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : (x : M) → Decidable (x = 0)
p : R
hM : IsTorsion' M ↥(Submonoid.powers p)
d : ℕ
hd : d ≠ 0
s : Fin d → M
hs : span R (Set.range s) = ⊤
oj : Option (Fin d) := List.argmax (fun i ↦ pOrder hM (s i)) (List.finRange d)
hoj : oj.isSome = true
i : Fin d
this : pOrder hM (s i) ≤ pOrder hM (s (oj.get hoj))
⊢ p ^ (pOrder hM (s (oj.get hoj)) - pOrder hM (s i) + pOrder hM (s i)) • s i = 0

Tactic:
pow_add,

Goal state after:
case h
R : Type u_1
M : Type u_2
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : (x : M) → Decidable (x = 0)
p : R
hM : IsTorsion' M ↥(Submonoid.powers p)
d : ℕ
hd : d ≠ 0
s : Fin d → M
hs : span R (Set.range s) = ⊤
oj : Option (Fin d) := List.argmax (fun i ↦ pOrder hM (s i)) (List.finRange d)
hoj : oj.isSome = true
i : Fin d
this : pOrder hM (s i) ≤ pOrder hM (s (oj.get hoj))
⊢ (p ^ (pOrder hM (s (oj.get hoj)) - pOrder hM (s i)) * p ^ pOrder hM (s i)) • s i = 0
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I✝ : Ideal R
F F' : I✝.Filtration M
I : Ideal R
N : Submodule R M
i : ℕ
⊢ I ^ (1 + i) • N ≤ I ^ i • N

Tactic:
pow_add,

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I✝ : Ideal R
F F' : I✝.Filtration M
I : Ideal R
N : Submodule R M
i : ℕ
⊢ (I ^ 1 * I ^ i) • N ≤ I ^ i • N
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I✝ : Ideal R
F F' : I✝.Filtration M
I : Ideal R
N : Submodule R M
i : ℕ
⊢ I ^ (i + 1) • N ≤ I ^ i • N

Tactic:
by rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right

Goal state after:
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I✝ : Ideal R
F F' : I✝.Filtration M
I : Ideal R
N : Submodule R M
i : ℕ
⊢ I • I ^ i • N ≤ I ^ (1 + i) • N

Tactic:
pow_add,

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I✝ : Ideal R
F F' : I✝.Filtration M
I : Ideal R
N : Submodule R M
i : ℕ
⊢ I • I ^ i • N ≤ (I ^ 1 * I ^ i) • N
Module: Mathlib.RingTheory.Nakayama

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N N' P : Submodule R M
hN' : N'.FG
hN'le : N' ≤ P
hNN' : P ≤ N ⊔ I • N'
hNN'' : P ≤ N ⊔ N'
h1 : map N.mkQ P = map N.mkQ N'
h2 : map N.mkQ P = map N.mkQ (I • N')
| map N.mkQ P

Tactic:
rw [h2]

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N N' P : Submodule R M
hN' : N'.FG
hN'le : N' ≤ P
hNN' : P ≤ N ⊔ I • N'
hNN'' : P ≤ N ⊔ N'
h1 : map N.mkQ P = map N.mkQ N'
h2 : map N.mkQ P = map N.mkQ (I • N')
| map N.mkQ (I • N')
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: case h
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
n : ℕ
a✝ : n ≥ 0
⊢ I • I ^ n • N = I ^ (1 + n) • N

Tactic:
pow_add,

Goal state after:
case h
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
n : ℕ
a✝ : n ≥ 0
⊢ I • I ^ n • N = (I ^ 1 * I ^ n) • N
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: case h.succ
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
hn : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
n✝ : ℕ
ih : F.N (n₀ + n✝) = I ^ n✝ • F.N n₀
⊢ I • I ^ n✝ • F.N n₀ = I ^ (1 + n✝) • F.N n₀

pp: case h.succ.a
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
hn : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
n✝ : ℕ
ih : F.N (n₀ + n✝) = I ^ n✝ • F.N n₀
⊢ n₀ + n✝ ≥ n₀

Tactic:
pow_add,

Goal state after:
case h.succ
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
hn : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
n✝ : ℕ
ih : F.N (n₀ + n✝) = I ^ n✝ • F.N n₀
⊢ I • I ^ n✝ • F.N n₀ = (I ^ 1 * I ^ n✝) • F.N n₀

case h.succ.a
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
hn : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
n✝ : ℕ
ih : F.N (n₀ + n✝) = I ^ n✝ • F.N n₀
⊢ n₀ + n✝ ≥ n₀
Module: Mathlib.RingTheory.Ideal.Cotangent

Goal state before:
pp: case refine_2
R : Type u
inst✝⁴ : CommRing R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra R S
I : Ideal S
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ↥I →ₗ[R] M
hf : ∀ (x y : ↥I), f (x * y) = 0
h : Function.Surjective ⇑f
⊢ Function.Surjective ⇑(lift f hf)

Tactic:
dsimp [Cotangent.lift]

Goal state after:
case refine_2
R : Type u
inst✝⁴ : CommRing R
S : Type u_3
inst✝³ : CommRing S
inst✝² : Algebra R S
I : Ideal S
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ↥I →ₗ[R] M
hf : ∀ (x y : ↥I), f (x * y) = 0
h : Function.Surjective ⇑f
⊢ Function.Surjective ⇑(QuotientAddGroup.lift (I • ⊤).toAddSubgroup f.toAddMonoidHom ⋯)
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: case mpr.succ
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
x : M
N : Submodule R M := ⨅ i, I ^ i • ⊤
hN : ∀ (k : ℕ), (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N
r : ↥I
eq : ↑r • x = x
i : ℕ
hi : x ∈ I ^ i • ⊤
⊢ x ∈ I ^ (1 + i) • ⊤

Tactic:
pow_add,

Goal state after:
case mpr.succ
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
x : M
N : Submodule R M := ⨅ i, I ^ i • ⊤
hN : ∀ (k : ℕ), (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N
r : ↥I
eq : ↑r • x = x
i : ℕ
hi : x ∈ I ^ i • ⊤
⊢ x ∈ (I ^ 1 * I ^ i) • ⊤
Module: Mathlib.RingTheory.Ideal.Cotangent

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
inst✝ : IsNoetherianRing R
⊢ ⊤.IsPrincipal ↔ Submodule.IsPrincipal (maximalIdeal R)

Tactic:
isPrincipal_iff,

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
inst✝ : IsNoetherianRing R
⊢ (∃ a, ⊤ = ResidueField R ∙ a) ↔ Submodule.IsPrincipal (maximalIdeal R)