mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.WhitneyEmbedding

Goal state before:
pp: case convert_2
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
L : (ι → E × ℝ) →L[ℝ] E := ContinuousLinearMap.fst ℝ E ℝ ∘SL ContinuousLinearMap.proj (f.ind x hx)
this : HasMFDerivAt% (⇑L ∘ ⇑f.embeddingPiTangent) x (L ∘SL mfderiv% ⇑f.embeddingPiTangent x)
y : M
hy : ↑(f.toFun (f.ind x hx)) y = 1 y
⊢ ↑(extChartAt I (f.c (f.ind x hx))) y = ↑(extChartAt I (f.c (f.ind x hx))) y

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Commensurable

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
g : ConjAct G
a b : ↥K
⊢ ↑(a⁻¹ * b) ∈ H ↔ g • ↑(a⁻¹ * b) ∈ g • H

Tactic:
]

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
g : ConjAct G
a b : ↥K
⊢ ↑(a⁻¹ * b) ∈ H ↔ g • ↑(a⁻¹ * b) ∈ g • H
Module: Mathlib.GroupTheory.Commensurable

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
g : ConjAct G
⊢ H.Commensurable (g⁻¹ • H) ↔ H.Commensurable (g⁻¹ • H)

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Commensurable

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
a✝ b✝ : ConjAct G
ha : a✝ ∈ {g | (g • H).Commensurable H}
hb : b✝ ∈ {g | (g • H).Commensurable H}
⊢ (a✝ • b✝ • H).Commensurable H

Tactic:
]

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
a✝ b✝ : ConjAct G
ha : a✝ ∈ {g | (g • H).Commensurable H}
hb : b✝ ∈ {g | (g • H).Commensurable H}
⊢ (a✝ • b✝ • H).Commensurable H
Module: Mathlib.GroupTheory.Commensurable

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ H.Commensurable H

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Commensurable

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
x✝¹ : ConjAct G
x✝ : x✝¹ ∈ {g | (g • H).Commensurable H}
⊢ (x✝¹ • H).Commensurable H

Tactic:
]

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
x✝¹ : ConjAct G
x✝ : x✝¹ ∈ {g | (g • H).Commensurable H}
⊢ (x✝¹ • H).Commensurable H
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
⊢ ∀ (i : η), map (MonoidHom.mulSingle Gs i) ⁅H i, K i⁆ ≤ ⁅pi Set.univ H, pi Set.univ K⁆

Tactic:
]

Goal state after:
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
⊢ ∀ (i : η), map (MonoidHom.mulSingle Gs i) ⁅H i, K i⁆ ≤ ⁅pi Set.univ H, pi Set.univ K⁆
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
⊢ hi ∈ ⁅map (MonoidHom.mulSingle Gs i) (H i), map (MonoidHom.mulSingle Gs i) (K i)⁆ →
    hi ∈ ⁅pi Set.univ H, pi Set.univ K⁆

Tactic:
]

Goal state after:
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
⊢ hi ∈ ⁅map (MonoidHom.mulSingle Gs i) (H i), map (MonoidHom.mulSingle Gs i) (K i)⁆ →
    hi ∈ ⁅pi Set.univ H, pi Set.univ K⁆
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: case h₁
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
⊢ ∀ i_1 ∈ Set.univ, map (MonoidHom.mulSingle Gs i) (H i) ≤ comap (Pi.evalMonoidHom Gs i_1) (H i_1)

Tactic:
]

Goal state after:
case h₁
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
⊢ ∀ i_1 ∈ Set.univ, map (MonoidHom.mulSingle Gs i) (H i) ≤ comap (Pi.evalMonoidHom Gs i_1) (H i_1)
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: case h₂
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
⊢ ∀ i_1 ∈ Set.univ, map (MonoidHom.mulSingle Gs i) (K i) ≤ comap (Pi.evalMonoidHom Gs i_1) (K i_1)

Tactic:
]

Goal state after:
case h₂
η : Type u_2
inst✝¹ : Finite η
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
H K : (i : η) → Subgroup (Gs i)
i : η
hi : (i : η) → Gs i
⊢ ∀ i_1 ∈ Set.univ, map (MonoidHom.mulSingle Gs i) (K i) ≤ comap (Pi.evalMonoidHom Gs i_1) (K i_1)
Module: Mathlib.Geometry.Manifold.WhitneyEmbedding

Goal state before:
pp: ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
⊢ (↑(mfderiv% ⇑f.embeddingPiTangent x)).ker ≤
    (↑((ContinuousLinearMap.fst ℝ E ℝ ∘SL ContinuousLinearMap.proj (f.ind x hx)) ∘SL
          mfderiv% ⇑f.embeddingPiTangent x)).ker

Tactic:
]

Goal state after:
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
⊢ (↑(mfderiv% ⇑f.embeddingPiTangent x)).ker ≤
    (↑((ContinuousLinearMap.fst ℝ E ℝ ∘SL ContinuousLinearMap.proj (f.ind x hx)) ∘SL
          mfderiv% ⇑f.embeddingPiTangent x)).ker
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Group.FG ↥(closure (commutatorSet G))

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Group.FG ↥(closure (commutatorSet G))
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Group.rank ↥(closure (commutatorSet G)) ≤ Nat.card ↑(commutatorSet G)

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Group.rank ↥(closure (commutatorSet G)) ≤ Nat.card ↑(commutatorSet G)
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : Finite ↑(commutatorSet G)
inst✝ : Group.FG G
S : Finset G
hS1 : S.card = Group.rank G
hS2 : closure ↑S = ⊤
⊢ (center G).index ≤ Nat.card (↑↑S → ↑(commutatorSet G))

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : Finite ↑(commutatorSet G)
inst✝ : Group.FG G
S : Finset G
hS1 : S.card = Group.rank G
hS2 : closure ↑S = ⊤
⊢ (center G).index ≤ Nat.card (↑↑S → ↑(commutatorSet G))
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝ : Group G
⊢ Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) =
    Nat.card ↑(⇑(closureCommutatorRepresentatives G).subtype '' commutatorSet ↥(closureCommutatorRepresentatives G))

Tactic:
]

Goal state after:
G : Type u_1
inst✝ : Group G
⊢ Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) =
    Nat.card ↑(⇑(closureCommutatorRepresentatives G).subtype '' commutatorSet ↥(closureCommutatorRepresentatives G))
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝ : Group G
⊢ Nat.card ↥(_root_.commutator ↥(closureCommutatorRepresentatives G)) =
    Nat.card
      ↥(map (closureCommutatorRepresentatives G).subtype (_root_.commutator ↥(closureCommutatorRepresentatives G)))

Tactic:
]

Goal state after:
G : Type u_1
inst✝ : Group G
⊢ Nat.card ↥(_root_.commutator ↥(closureCommutatorRepresentatives G)) =
    Nat.card
      ↥(map (closureCommutatorRepresentatives G).subtype (_root_.commutator ↥(closureCommutatorRepresentatives G)))
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Group.rank ↥(closureCommutatorRepresentatives G) ≤ Nat.card ↑(commutatorSet G) + Nat.card ↑(commutatorSet G)

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Group.rank ↥(closureCommutatorRepresentatives G) ≤ Nat.card ↑(commutatorSet G) + Nat.card ↑(commutatorSet G)
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Nat.card ↑(commutatorSet G) ≠ 0

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite ↑(commutatorSet G)
⊢ Nat.card ↑(commutatorSet G) ≠ 0
Module: Mathlib.Geometry.Manifold.WhitneyEmbedding

Goal state before:
pp: case intro
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
ι : Type u_1
inst✝ : Finite ι
f : SmoothBumpCovering ι I M
val✝ : Fintype ι
F : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))
this✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance
this : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance
eEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯
x : M
⊢ Injective ⇑(↑eEF ∘SL mfderiv% ⇑f.embeddingPiTangent x)

Tactic:
]

Goal state after:
case intro
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
ι : Type u_1
inst✝ : Finite ι
f : SmoothBumpCovering ι I M
val✝ : Fintype ι
F : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))
this✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance
this : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance
eEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯
x : M
⊢ Injective ⇑(↑eEF ∘SL mfderiv% ⇑f.embeddingPiTangent x)
Module: Mathlib.GroupTheory.GroupAction.CardCommute

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : Group α
inst✝⁴ : MulAction α β
inst✝³ : Fintype α
inst✝² : Fintype β
inst✝¹ : Fintype (Quotient (orbitRel α β))
inst✝ : (b : β) → Fintype ↥(stabilizer α b)
φ : Quotient (orbitRel α β) → β
hφ : LeftInverse Quotient.mk'' φ
ω : Quotient (orbitRel α β)
⊢ Fintype.card (α ⧸ stabilizer α (φ ω)) = Fintype.card (α ⧸ stabilizer α (φ ω))

pp: case H
α : Type u_1
β : Type u_2
inst✝⁵ : Group α
inst✝⁴ : MulAction α β
inst✝³ : Fintype α
inst✝² : Fintype β
inst✝¹ : Fintype (Quotient (orbitRel α β))
inst✝ : (b : β) → Fintype ↥(stabilizer α b)
φ : Quotient (orbitRel α β) → β
hφ : LeftInverse Quotient.mk'' φ
ω : Quotient (orbitRel α β)
⊢ 0 < Fintype.card ↥(stabilizer α (φ ω))

Tactic:
]

Goal state after:
case H
α : Type u_1
β : Type u_2
inst✝⁵ : Group α
inst✝⁴ : MulAction α β
inst✝³ : Fintype α
inst✝² : Fintype β
inst✝¹ : Fintype (Quotient (orbitRel α β))
inst✝ : (b : β) → Fintype ↥(stabilizer α b)
φ : Quotient (orbitRel α β) → β
hφ : LeftInverse Quotient.mk'' φ
ω : Quotient (orbitRel α β)
⊢ 0 < Fintype.card ↥(stabilizer α (φ ω))
Module: Mathlib.GroupTheory.GroupAction.CardCommute

Goal state before:
pp: case inr
G : Type u_3
inst✝ : Group G
val✝ : Infinite G
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
⊢ Nat.card (Additive G) = Nat.card G

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
⊢ Nat.card (Additive G) = Nat.card G
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: G : Type u_1
inst✝¹ : AddGroup G
inst✝ : IsAddKleinFour G
⊢ Nat.card (Multiplicative G) = Nat.card G

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : AddGroup G
inst✝ : IsAddKleinFour G
⊢ Nat.card (Multiplicative G) = Nat.card G
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
x : G
⊢ x = x

Tactic:
]

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

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ #{x * y, x, y, 1} = 4

Tactic:
]

Goal state after:
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ #{x * y, x, y, 1} = 4
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ #{x, y, 1} + 1 = 4

pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}

Tactic:
]

Goal state after:
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ #{x, y, 1} + 1 = 4

G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ #{y, 1} + 1 + 1 = 4

pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}

pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}

Tactic:
]

Goal state after:
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ #{y, 1} + 1 + 1 = 4

G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}

G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ #{1} + 1 + 1 + 1 = 4

pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ y ∉ {1}

pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}

pp: G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}

Tactic:
]

Goal state after:
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ #{1} + 1 + 1 + 1 = 4

G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ y ∉ {1}

G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}

G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: case neg
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
x y : G₁
hx : e x ≠ e 1
hy : e y ≠ e 1
hxy : e x ≠ e y
univ₂ : {e (x * y), e x, e y, 1} = univ
⊢ e (x * y) = e x * e y

Tactic:
]

Goal state after:
case neg
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
x y : G₁
hx : e x ≠ e 1
hy : e y ≠ e 1
hxy : e x ≠ e y
univ₂ : {e (x * y), e x, e y, 1} = univ
⊢ e (x * y) = e x * e y
Module: Mathlib.GroupTheory.SpecificGroups.KleinFour

Goal state before:
pp: case neg
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
x y : G₁
hx : e x ≠ 1
hy : e y ≠ 1
hxy : e x ≠ e y
univ₂ : {e (x * y), e x, e y, 1} = univ
⊢ e (x * y) = e x * e y

Tactic:
]

Goal state after:
case neg
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
x y : G₁
hx : e x ≠ 1
hy : e y ≠ 1
hxy : e x ≠ e y
univ₂ : {e (x * y), e x, e y, 1} = univ
⊢ e (x * y) = e x * e y
Module: Mathlib.GroupTheory.Congruence.Star

Goal state before:
pp: M : Type u_1
inst✝¹ : Mul M
inst✝ : StarMul M
r : M → M → Prop
hr : ∀ (a b : M), r a b → r (Star.star a) (Star.star b)
a b w✝ x✝ y✝ z✝ : M
h1 : Rel r w✝ x✝
h2 : Rel r y✝ z✝
⊢ Rel r (Star.star y✝ * Star.star w✝) (Star.star z✝ * Star.star x✝)

Tactic:
]

Goal state after:
M : Type u_1
inst✝¹ : Mul M
inst✝ : StarMul M
r : M → M → Prop
hr : ∀ (a b : M), r a b → r (Star.star a) (Star.star b)
a b w✝ x✝ y✝ z✝ : M
h1 : Rel r w✝ x✝
h2 : Rel r y✝ z✝
⊢ Rel r (Star.star y✝ * Star.star w✝) (Star.star z✝ * Star.star x✝)
Module: Mathlib.GroupTheory.Congruence.Star

Goal state before:
pp: A : Type u_1
inst✝¹ : AddMonoid A
inst✝ : StarAddMonoid A
r : A → A → Prop
hr : ∀ (a b : A), r a b → r (Star.star a) (Star.star b)
a b w✝ x✝ y✝ z✝ : A
h1 : Rel r w✝ x✝
h2 : Rel r y✝ z✝
⊢ Rel r (Star.star w✝ + Star.star y✝) (Star.star x✝ + Star.star z✝)

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹ : AddMonoid A
inst✝ : StarAddMonoid A
r : A → A → Prop
hr : ∀ (a b : A), r a b → r (Star.star a) (Star.star b)
a b w✝ x✝ y✝ z✝ : A
h1 : Rel r w✝ x✝
h2 : Rel r y✝ z✝
⊢ Rel r (Star.star w✝ + Star.star y✝) (Star.star x✝ + Star.star z✝)
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case succ
n : ℕ
i : ZMod n
k : ℕ
IH : r i ^ k = r (i * ↑k)
⊢ i * ↑k + i = i * ↑k + i

Tactic:
]

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

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ n + n = n + n

Tactic:
]

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

Goal state before:
pp: case zero
⊢ 0 = 2 * 0

Tactic:
]

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

Goal state before:
pp: case succ
n✝ : ℕ
⊢ 2 * (n✝ + 1) = 2 * (n✝ + 1)

Tactic:
]

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

Goal state before:
pp: case succ
n✝ : ℕ
⊢ Nat.card (DihedralGroup (n✝ + 1)) = 2 * (n✝ + 1)

Tactic:
rw [Nat.card_eq_fintype_card, card]

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

Goal state before:
pp: case hg
n : ℕ
i : ZMod n
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: case inl
⊢ ∀ (n : ℕ), 0 < n → r 1 ^ n ≠ 1

Tactic:
]

Goal state after:
case inl
⊢ ∀ (n : ℕ), 0 < n → r 1 ^ n ≠ 1
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case inl
n : ℕ
hn : 0 < n
⊢ r ↑n ≠ r 0

Tactic:
]

Goal state after:
case inl
n : ℕ
hn : 0 < n
⊢ r ↑n ≠ r 0
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case inr
n : ℕ
hn : NeZero n
h : orderOf (r 1) < n
h1 : r ↑(orderOf (r 1)) = 1
⊢ False

Tactic:
]

Goal state after:
case inr
n : ℕ
hn : NeZero n
h : orderOf (r 1) < n
h1 : r ↑(orderOf (r 1)) = 1
⊢ False
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case inr
n : ℕ
hn : NeZero n
h : orderOf (r 1) < n
h2 : orderOf (r 1) = 0
⊢ False

Tactic:
]

Goal state after:
case inr
n : ℕ
hn : NeZero n
h : orderOf (r 1) < n
h2 : orderOf (r 1) = 0
⊢ False
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
i : ZMod n
⊢ n / n.gcd i.val = n / n.gcd i.val

Tactic:
]

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

Goal state before:
pp: case inr.a.r
n : ℕ
hn : NeZero n
m : ZMod n
⊢ n / n.gcd m.val ∣ lcm n 2

Tactic:
]

Goal state after:
case inr.a.r
n : ℕ
hn : NeZero n
m : ZMod n
⊢ n / n.gcd m.val ∣ lcm n 2
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case inr.a.sr
n : ℕ
hn : NeZero n
m : ZMod n
⊢ 2 ∣ lcm n 2

Tactic:
]

Goal state after:
case inr.a.sr
n : ℕ
hn : NeZero n
m : ZMod n
⊢ 2 ∣ lcm n 2
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: n : ℕ
x✝¹ : n + 3 ≠ 1
x✝ : n + 3 ≠ 2
h' : IsMulCommutative (DihedralGroup (n + 3))
this : 2 % (n + 3) = 0
⊢ False

Tactic:
]

Goal state after:
n : ℕ
x✝¹ : n + 3 ≠ 1
x✝ : n + 3 ≠ 2
h' : IsMulCommutative (DihedralGroup (n + 3))
this : 2 % (n + 3) = 0
⊢ False
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Mul β
⊢ commProb β ^ card α = commProb β ^ card α

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
⊢ Nat.card { p // Commute p.1 p.2 } ≤ Nat.card (M × M)

Tactic:
]

Goal state after:
M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
⊢ Nat.card { p // Commute p.1 p.2 } ≤ Nat.card (M × M)
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
h : Nonempty M
this : Fintype M
⊢ ↑(card ↑{x | Commute x.1 x.2}) / ↑(card M) ^ 2 = 1 ↔ IsMulCommutative M

Tactic:
]

Goal state after:
M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
h : Nonempty M
this : Fintype M
⊢ ↑(card ↑{x | Commute x.1 x.2}) / ↑(card M) ^ 2 = 1 ↔ IsMulCommutative M
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
h : Nonempty M
this : Fintype M
⊢ (∀ (x : M × M), x ∈ {x | Commute x.1 x.2}) ↔ IsMulCommutative M

pp: M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
h : Nonempty M
this : Fintype M
⊢ ↑(card M) ^ 2 ≠ 0

Tactic:
]

Goal state after:
M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
h : Nonempty M
this : Fintype M
⊢ (∀ (x : M × M), x ∈ {x | Commute x.1 x.2}) ↔ IsMulCommutative M

M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
h : Nonempty M
this : Fintype M
⊢ ↑(card M) ^ 2 ≠ 0
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: G : Type u_2
inst✝ : Group G
⊢ ↑(Nat.card (ConjClasses G)) * ↑(Nat.card G) / (↑(Nat.card G) * ↑(Nat.card G)) =
    ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)

Tactic:
]

Goal state after:
G : Type u_2
inst✝ : Group G
⊢ ↑(Nat.card (ConjClasses G)) * ↑(Nat.card G) / (↑(Nat.card G) * ↑(Nat.card G)) =
    ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: case pos
G : Type u_2
inst✝ : Group G
h : ↑(Nat.card G) = 0
⊢ 0 = 0

Tactic:
]

Goal state after: