mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
H : Subgroup G
A : Type u_2
inst✝² : CommGroup A
ϕ : ↥H →* A
inst✝¹ : H.FiniteIndex
g : G
inst✝ : Fintype (Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
⊢ ϕ
      ⟨(g ^ ZMod.cast 0 * Quotient.out (Quotient.out q))⁻¹ *
          if 0 = 0 then g ^ Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
          else g ^ ZMod.cast 0 * Quotient.out (Quotient.out q),
        ⋯⟩ =
    ϕ
      ⟨(Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q) *
          Quotient.out (Quotient.out q),
        ⋯⟩

pp: G : Type u_1
inst✝³ : Group G
H : Subgroup G
A : Type u_2
inst✝² : CommGroup A
ϕ : ↥H →* A
inst✝¹ : H.FiniteIndex
g : G
inst✝ : Fintype (Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
⊢ ∀ (x : ZMod (Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q))),
    x ≠ 0 →
      ϕ
          ⟨(g ^ x.cast * Quotient.out (Quotient.out q))⁻¹ *
              if x = 0 then g ^ Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
              else g ^ x.cast * Quotient.out (Quotient.out q),
            ⋯⟩ =
        1

Tactic:
·

Goal state after:
G : Type u_1
inst✝³ : Group G
H : Subgroup G
A : Type u_2
inst✝² : CommGroup A
ϕ : ↥H →* A
inst✝¹ : H.FiniteIndex
g : G
inst✝ : Fintype (Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
⊢ ϕ
      ⟨(g ^ ZMod.cast 0 * Quotient.out (Quotient.out q))⁻¹ *
          if 0 = 0 then g ^ Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
          else g ^ ZMod.cast 0 * Quotient.out (Quotient.out q),
        ⋯⟩ =
    ϕ
      ⟨(Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q) *
          Quotient.out (Quotient.out q),
        ⋯⟩
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
H : Subgroup G
A : Type u_2
inst✝² : CommGroup A
ϕ : ↥H →* A
inst✝¹ : H.FiniteIndex
g : G
inst✝ : Fintype (Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
⊢ ∀ (x : ZMod (Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q))),
    x ≠ 0 →
      ϕ
          ⟨(g ^ x.cast * Quotient.out (Quotient.out q))⁻¹ *
              if x = 0 then g ^ Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
              else g ^ x.cast * Quotient.out (Quotient.out q),
            ⋯⟩ =
        1

Tactic:
·

Goal state after:
G : Type u_1
inst✝³ : Group G
H : Subgroup G
A : Type u_2
inst✝² : CommGroup A
ϕ : ↥H →* A
inst✝¹ : H.FiniteIndex
g : G
inst✝ : Fintype (Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
⊢ ∀ (x : ZMod (Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q))),
    x ≠ 0 →
      ϕ
          ⟨(g ^ x.cast * Quotient.out (Quotient.out q))⁻¹ *
              if x = 0 then g ^ Function.minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
              else g ^ x.cast * Quotient.out (Quotient.out q),
            ⋯⟩ =
        1
Module: Mathlib.GroupTheory.Schreier

Goal state before:
pp: case py₁
G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
hU : closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ ↑(closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)))
r : G
hr : r ∈ R
hh : (fun x1 x2 ↦ x1 * x2) g r ∈ H
⊢ r * (↑⟨r, hr⟩)⁻¹ ∈ ↑H

pp: case py₂
G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
hU : closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ ↑(closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)))
r : G
hr : r ∈ R
hh : (fun x1 x2 ↦ x1 * x2) g r ∈ H
⊢ r * (↑⟨1, hR1⟩)⁻¹ ∈ ↑H

Tactic:
·

Goal state after:
case py₁
G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
hU : closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ ↑(closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)))
r : G
hr : r ∈ R
hh : (fun x1 x2 ↦ x1 * x2) g r ∈ H
⊢ r * (↑⟨r, hr⟩)⁻¹ ∈ ↑H
Module: Mathlib.GroupTheory.Schreier

Goal state before:
pp: case py₂
G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
hU : closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ ↑(closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)))
r : G
hr : r ∈ R
hh : (fun x1 x2 ↦ x1 * x2) g r ∈ H
⊢ r * (↑⟨1, hR1⟩)⁻¹ ∈ ↑H

Tactic:
·

Goal state after:
case py₂
G : Type u_1
inst✝ : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H) R
hR1 : 1 ∈ R
hS : closure S = ⊤
hU : closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ ↑(closure ((fun g ↦ g * (↑(hR.toRightFun g))⁻¹) '' (R * S)))
r : G
hr : r ∈ R
hh : (fun x1 x2 ↦ x1 * x2) g r ∈ H
⊢ r * (↑⟨1, hR1⟩)⁻¹ ∈ ↑H
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: case pos
G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
hH : H.index = 0
⊢ g ^ H.index ∈ H

pp: case neg
G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
hH : ¬H.index = 0
⊢ g ^ H.index ∈ H

Tactic:
·

Goal state after:
case pos
G : Type u_1
inst✝ : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
hH : H.index = 0
⊢ g ^ H.index ∈ H
Module: Mathlib.GroupTheory.Schreier

Goal state before:
pp: case pos
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
hG : (center G).index = 0
⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)

pp: case neg
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)

Tactic:
·

Goal state after:
case pos
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
hG : (center G).index = 0
⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)
Module: Mathlib.GroupTheory.FinitelyPresentedGroup

Goal state before:
pp: G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
inst✝² : Group G
inst✝¹ : Group H
inst✝ : Finite α
n : ℕ
toFun✝ : α → Fin n
f : Fin n → α
hf_surj : Function.LeftInverse f toFun✝
hf_inj : Function.RightInverse f toFun✝
⊢ (FreeGroup.map f).ker.IsFinitelyNormallyGenerated

Tactic:
·

Goal state after:
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
inst✝² : Group G
inst✝¹ : Group H
inst✝ : Finite α
n : ℕ
toFun✝ : α → Fin n
f : Fin n → α
hf_surj : Function.LeftInverse f toFun✝
hf_inj : Function.RightInverse f toFun✝
⊢ (FreeGroup.map f).ker.IsFinitelyNormallyGenerated
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: case pos
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : Nat.card G = 1
⊢ normalizer ↑P ≤ centralizer ↑P

pp: case neg
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
⊢ normalizer ↑P ≤ centralizer ↑P

Tactic:
·

Goal state after:
case pos
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : Nat.card G = 1
⊢ normalizer ↑P ≤ centralizer ↑P
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: case neg.inl
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relIndex (normalizer ↑↑P) ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : k = 0
⊢ ((centralizer ↑P).relIndex (normalizer ↑P)).Coprime (Nat.card (MulAut ↥↑P))

pp: case neg.inr
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relIndex (normalizer ↑↑P) ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : 0 < k
⊢ ((centralizer ↑P).relIndex (normalizer ↑P)).Coprime (Nat.card (MulAut ↥↑P))

Tactic:
·

Goal state after:
case neg.inl
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relIndex (normalizer ↑↑P) ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : k = 0
⊢ ((centralizer ↑P).relIndex (normalizer ↑P)).Coprime (Nat.card (MulAut ↥↑P))
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: case neg.inr.refine_1
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relIndex (normalizer ↑↑P) ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : 0 < k
⊢ ((centralizer ↑P).relIndex (normalizer ↑P)).Coprime (Nat.card G).minFac

pp: case neg.inr.refine_2
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relIndex (normalizer ↑↑P) ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : 0 < k
⊢ ((centralizer ↑P).relIndex (normalizer ↑P)).Coprime ((Nat.card G).minFac - 1)

Tactic:
·

Goal state after:
case neg.inr.refine_1
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relIndex (normalizer ↑↑P) ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : 0 < k
⊢ ((centralizer ↑P).relIndex (normalizer ↑P)).Coprime (Nat.card G).minFac
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: case neg.inr.refine_2
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relIndex (normalizer ↑↑P) ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : 0 < k
⊢ ((centralizer ↑P).relIndex (normalizer ↑P)).Coprime ((Nat.card G).minFac - 1)

Tactic:
·

Goal state after:
case neg.inr.refine_2
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relIndex (normalizer ↑↑P) ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : 0 < k
⊢ ((centralizer ↑P).relIndex (normalizer ↑P)).Coprime ((Nat.card G).minFac - 1)
Module: Mathlib.GroupTheory.Focal

Goal state before:
pp: case a
G : Type u_1
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : (↑P).FiniteIndex
⊢ _root_.commutator G ⊓ ↑P ≤ (↑P).focalSubgroup

pp: case a
G : Type u_1
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : (↑P).FiniteIndex
⊢ (↑P).focalSubgroup ≤ _root_.commutator G ⊓ ↑P

Tactic:
·

Goal state after:
case a
G : Type u_1
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : (↑P).FiniteIndex
⊢ _root_.commutator G ⊓ ↑P ≤ (↑P).focalSubgroup
Module: Mathlib.GroupTheory.Focal

Goal state before:
pp: case a
G : Type u_1
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : (↑P).FiniteIndex
⊢ (↑P).focalSubgroup ≤ _root_.commutator G ⊓ ↑P

Tactic:
·

Goal state after:
case a
G : Type u_1
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : (↑P).FiniteIndex
⊢ (↑P).focalSubgroup ≤ _root_.commutator G ⊓ ↑P
Module: Mathlib.GroupTheory.Transfer

Goal state before:
pp: case pos
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : Nat.card G = 1
⊢ (MonoidHom.transferSylow P ⋯).ker.IsComplement' ↑P

pp: case neg
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
⊢ (MonoidHom.transferSylow P ⋯).ker.IsComplement' ↑P

Tactic:
·

Goal state after:
case pos
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : Nat.card G = 1
⊢ (MonoidHom.transferSylow P ⋯).ker.IsComplement' ↑P
Module: Mathlib.Order.Radical

Goal state before:
pp: case e
α : Type u_1
inst✝¹ : CompleteLattice α
β : Type u_2
inst✝ : CompleteLattice β
f : α ≃o β
⊢ α ≃ β

pp: case h
α : Type u_1
inst✝¹ : CompleteLattice α
β : Type u_2
inst✝ : CompleteLattice β
f : α ≃o β
⊢ ∀ (x : α), ⨅ (_ : ?e x ∈ {H | IsCoatom H}), ?e x = ⨅ (_ : x ∈ {H | IsCoatom H}), f x

Tactic:
·

Goal state after:
case e
α : Type u_1
inst✝¹ : CompleteLattice α
β : Type u_2
inst✝ : CompleteLattice β
f : α ≃o β
⊢ α ≃ β
Module: Mathlib.Order.Radical

Goal state before:
pp: case h
α : Type u_1
inst✝¹ : CompleteLattice α
β : Type u_2
inst✝ : CompleteLattice β
f : α ≃o β
⊢ ∀ (x : α), ⨅ (_ : f.toEquiv x ∈ {H | IsCoatom H}), f.toEquiv x = ⨅ (_ : x ∈ {H | IsCoatom H}), f x

Tactic:
·

Goal state after:
case h
α : Type u_1
inst✝¹ : CompleteLattice α
β : Type u_2
inst✝ : CompleteLattice β
f : α ≃o β
⊢ ∀ (x : α), ⨅ (_ : f.toEquiv x ∈ {H | IsCoatom H}), f.toEquiv x = ⨅ (_ : x ∈ {H | IsCoatom H}), f x
Module: Mathlib.Order.Radical

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : IsCoatomic α
h : ⊤ ⊔ radical α = ⊤
⊢ ⊤ = ⊤

pp: case inr
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : IsCoatomic α
a : α
h : a ⊔ radical α = ⊤
w : ∃ a_1, IsCoatom a_1 ∧ a ≤ a_1
⊢ a = ⊤

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : IsCoatomic α
h : ⊤ ⊔ radical α = ⊤
⊢ ⊤ = ⊤
Module: Mathlib.Order.Radical

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : IsCoatomic α
a : α
h : a ⊔ radical α = ⊤
w : ∃ a_1, IsCoatom a_1 ∧ a ≤ a_1
⊢ a = ⊤

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : IsCoatomic α
a : α
h : a ⊔ radical α = ⊤
w : ∃ a_1, IsCoatom a_1 ∧ a ≤ a_1
⊢ a = ⊤
Module: Mathlib.GroupTheory.FreeGroup.CyclicallyReduced

Goal state before:
pp: case refine_1
α : Type u
L₁ L₂ L₃ : List (α × Bool)
h₁ : IsCyclicallyReduced L₂
h₂ : IsReduced (L₁ ++ L₂ ++ L₃)
n✝ n : ℕ
hn : n + 1 ≠ 0
h : ¬L₂ = []
h' : (replicate (n + 1) L₂).flatten ≠ []
⊢ IsReduced (L₁ ++ (replicate (n + 1) L₂).flatten)

pp: case refine_2
α : Type u
L₁ L₂ L₃ : List (α × Bool)
h₁ : IsCyclicallyReduced L₂
h₂ : IsReduced (L₁ ++ L₂ ++ L₃)
n✝ n : ℕ
hn : n + 1 ≠ 0
h : ¬L₂ = []
h' : (replicate (n + 1) L₂).flatten ≠ []
⊢ IsReduced ((replicate (n + 1) L₂).flatten ++ L₃)

Tactic:
·

Goal state after:
case refine_1
α : Type u
L₁ L₂ L₃ : List (α × Bool)
h₁ : IsCyclicallyReduced L₂
h₂ : IsReduced (L₁ ++ L₂ ++ L₃)
n✝ n : ℕ
hn : n + 1 ≠ 0
h : ¬L₂ = []
h' : (replicate (n + 1) L₂).flatten ≠ []
⊢ IsReduced (L₁ ++ (replicate (n + 1) L₂).flatten)
Module: Mathlib.GroupTheory.FreeGroup.CyclicallyReduced

Goal state before:
pp: case refine_2
α : Type u
L₁ L₂ L₃ : List (α × Bool)
h₁ : IsCyclicallyReduced L₂
h₂ : IsReduced (L₁ ++ L₂ ++ L₃)
n✝ n : ℕ
hn : n + 1 ≠ 0
h : ¬L₂ = []
h' : (replicate (n + 1) L₂).flatten ≠ []
⊢ IsReduced ((replicate (n + 1) L₂).flatten ++ L₃)

Tactic:
·

Goal state after:
case refine_2
α : Type u
L₁ L₂ L₃ : List (α × Bool)
h₁ : IsCyclicallyReduced L₂
h₂ : IsReduced (L₁ ++ L₂ ++ L₃)
n✝ n : ℕ
hn : n + 1 ≠ 0
h : ¬L₂ = []
h' : (replicate (n + 1) L₂).flatten ≠ []
⊢ IsReduced ((replicate (n + 1) L₂).flatten ++ L₃)
Module: Mathlib.GroupTheory.FreeGroup.NielsenSchreier

Goal state before:
pp: case refine_1
G A : Type u
inst✝³ : Group G
inst✝² : IsFreeGroup G
inst✝¹ : MulAction G A
X : Type u
inst✝ : Group X
f : Labelling (Generators (ActionCategory G A)) X
f' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e ↦ ⟨fun b ↦ f ⟨e, ⋯⟩, IsFreeGroup.of e⟩
F' : G →* (A → X) ⋊[mulAutArrow] G
hF' : ∀ (a : IsFreeGroup.Generators G), F' (IsFreeGroup.of a) = f' a
uF' :
  ∀ (y : G →* (A → X) ⋊[mulAutArrow] G),
    (fun F ↦ ∀ (a : IsFreeGroup.Generators G), F (IsFreeGroup.of a) = f' a) y → y = F'
⊢ ∀ (g : G), (F' g).right = g

pp: case refine_2
G A : Type u
inst✝³ : Group G
inst✝² : IsFreeGroup G
inst✝¹ : MulAction G A
X : Type u
inst✝ : Group X
f : Labelling (Generators (ActionCategory G A)) X
f' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e ↦ ⟨fun b ↦ f ⟨e, ⋯⟩, IsFreeGroup.of e⟩
F' : G →* (A → X) ⋊[mulAutArrow] G
hF' : ∀ (a : IsFreeGroup.Generators G), F' (IsFreeGroup.of a) = f' a
uF' :
  ∀ (y : G →* (A → X) ⋊[mulAutArrow] G),
    (fun F ↦ ∀ (a : IsFreeGroup.Generators G), F (IsFreeGroup.of a) = f' a) y → y = F'
⊢ (fun F ↦ ∀ (a b : Generators (ActionCategory G A)) (g : a ⟶ b), F.map ⟨IsFreeGroup.of ↑g, ⋯⟩ = f g)
    (uncurry F' ?refine_1)

pp: case refine_3
G A : Type u
inst✝³ : Group G
inst✝² : IsFreeGroup G
inst✝¹ : MulAction G A
X : Type u
inst✝ : Group X
f : Labelling (Generators (ActionCategory G A)) X
f' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e ↦ ⟨fun b ↦ f ⟨e, ⋯⟩, IsFreeGroup.of e⟩
F' : G →* (A → X) ⋊[mulAutArrow] G
hF' : ∀ (a : IsFreeGroup.Generators G), F' (IsFreeGroup.of a) = f' a
uF' :
  ∀ (y : G →* (A → X) ⋊[mulAutArrow] G),
    (fun F ↦ ∀ (a : IsFreeGroup.Generators G), F (IsFreeGroup.of a) = f' a) y → y = F'
⊢ ∀ (y : ActionCategory G A ⥤ CategoryTheory.SingleObj X),
    (fun F ↦ ∀ (a b : Generators (ActionCategory G A)) (g : a ⟶ b), F.map ⟨IsFreeGroup.of ↑g, ⋯⟩ = f g) y →
      y = uncurry F' ?refine_1

Tactic:
·

Goal state after:
case refine_1
G A : Type u
inst✝³ : Group G
inst✝² : IsFreeGroup G
inst✝¹ : MulAction G A
X : Type u
inst✝ : Group X
f : Labelling (Generators (ActionCategory G A)) X
f' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e ↦ ⟨fun b ↦ f ⟨e, ⋯⟩, IsFreeGroup.of e⟩
F' : G →* (A → X) ⋊[mulAutArrow] G
hF' : ∀ (a : IsFreeGroup.Generators G), F' (IsFreeGroup.of a) = f' a
uF' :
  ∀ (y : G →* (A → X) ⋊[mulAutArrow] G),
    (fun F ↦ ∀ (a : IsFreeGroup.Generators G), F (IsFreeGroup.of a) = f' a) y → y = F'
⊢ ∀ (g : G), (F' g).right = g
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case refine_1
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
e : G ≃* H
n : ℕ
h : upperCentralSeries G n = ⊤
⊢ upperCentralSeries H n = ⊤

pp: case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
e : G ≃* H
n : ℕ
h : upperCentralSeries H n = ⊤
⊢ upperCentralSeries G n = ⊤

Tactic:
·

Goal state after:
case refine_1
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
e : G ≃* H
n : ℕ
h : upperCentralSeries G n = ⊤
⊢ upperCentralSeries H n = ⊤
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
e : G ≃* H
n : ℕ
h : upperCentralSeries H n = ⊤
⊢ upperCentralSeries G n = ⊤

Tactic:
·

Goal state after:
case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
e : G ≃* H
n : ℕ
h : upperCentralSeries H n = ⊤
⊢ upperCentralSeries G n = ⊤
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
h'δ : 1 + δ * C < r' ^ 2
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this✝ : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ (trivializationAt F E x).baseSet
this : ‖g' y - g' x‖ ≤ δ
v : E x
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x) v
hgx : ((g x) v) v = ((g' x) w) w
hgy : ((g y) ((trivializationAt F E x).symm y w)) ((trivializationAt F E x).symm y w) = ((g' y) w) w
⊢ ((g' y - g' x) w) w + ((g' x) w) w ≤ ‖((g' y - g' x) w) w‖ + ((g' x) w) w

Tactic:
← Real.le_norm_self

Goal state after:
B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
h'δ : 1 + δ * C < r' ^ 2
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this✝ : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ (trivializationAt F E x).baseSet
this : ‖g' y - g' x‖ ≤ δ
v : E x
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x) v
hgx : ((g x) v) v = ((g' x) w) w
hgy : ((g y) ((trivializationAt F E x).symm y w)) ((trivializationAt F E x).symm y w) = ((g' y) w) w
⊢ ((g' y - g' x) w) w + ((g' x) w) w ≤ ((g' y - g' x) w) w + ((g' x) w) w
Module: Mathlib.GroupTheory.FreeGroup.CyclicallyReduced

Goal state before:
pp: case h₁
α : Type u
L : List (α × Bool)
inst✝ : DecidableEq α
h : IsReduced L
n : ℕ
ih :
  reduce (replicate (n + 1) L).flatten =
    conjugator L ++ (replicate (n + 1) (reduceCyclically L)).flatten ++ invRev (conjugator L)
this :
  ∀ {L₁ L₂ L₃ L₄ L₅ : List (α × Bool)},
    reduce (L₁ ++ L₂ ++ invRev L₃ ++ (L₃ ++ L₄ ++ L₅)) = reduce (L₁ ++ (L₂ ++ L₄) ++ L₅)
⊢ IsCyclicallyReduced (reduceCyclically L)

pp: case h₂
α : Type u
L : List (α × Bool)
inst✝ : DecidableEq α
h : IsReduced L
n : ℕ
ih :
  reduce (replicate (n + 1) L).flatten =
    conjugator L ++ (replicate (n + 1) (reduceCyclically L)).flatten ++ invRev (conjugator L)
this :
  ∀ {L₁ L₂ L₃ L₄ L₅ : List (α × Bool)},
    reduce (L₁ ++ L₂ ++ invRev L₃ ++ (L₃ ++ L₄ ++ L₅)) = reduce (L₁ ++ (L₂ ++ L₄) ++ L₅)
⊢ IsReduced (conjugator L ++ reduceCyclically L ++ invRev (conjugator L))

Tactic:
·

Goal state after:
case h₁
α : Type u
L : List (α × Bool)
inst✝ : DecidableEq α
h : IsReduced L
n : ℕ
ih :
  reduce (replicate (n + 1) L).flatten =
    conjugator L ++ (replicate (n + 1) (reduceCyclically L)).flatten ++ invRev (conjugator L)
this :
  ∀ {L₁ L₂ L₃ L₄ L₅ : List (α × Bool)},
    reduce (L₁ ++ L₂ ++ invRev L₃ ++ (L₃ ++ L₄ ++ L₅)) = reduce (L₁ ++ (L₂ ++ L₄) ++ L₅)
⊢ IsCyclicallyReduced (reduceCyclically L)