ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Equiv.Perm.support_swap_mul_swap
theorem support_swap_mul_swap {x y z : α} (h : List.Nodup [x, y, z]) :
    support (swap x y * swap y z) = {x, y, z} 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {x y z : α} (h : [x, y, z].Nodup)
Goal: (Equiv.swap x y * Equiv.swap y z).support = {x, y, z}
-- Equiv.Perm.support_swap_mul_ge_support_diff
theorem support_swap_mul_ge_support_diff (f : Perm α) (x y : α) :
    f.support \ {x, y} ≤ (swap x y * f).support 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) (x y : α)
Goal: f.support \ {x, y} ≤ (Equiv.swap x y * f).support
-- Equiv.Perm.support_swap_mul_eq
theorem support_swap_mul_eq (f : Perm α) (x : α) (h : f (f x) ≠ x) :
    (swap x (f x) * f).support = f.support \ {x} 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] (f : Equiv.Perm α) (x : α) (h : f (f x) ≠ x)
Goal: (Equiv.swap x (f x) * f).support = f.support \ {x}
-- Equiv.Perm.mem_support_swap_mul_imp_mem_support_ne
theorem mem_support_swap_mul_imp_mem_support_ne {x y : α} (hy : y ∈ support (swap x (f x) * f)) :
    y ∈ support f ∧ y ≠ x 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} {x y : α} (hy : y ∈ (Equiv.swap x (f x) * f).support)
Goal: y ∈ f.support ∧ y ≠ x
-- Equiv.Perm.Disjoint.mem_imp
theorem Disjoint.mem_imp (h : Disjoint f g) {x : α} (hx : x ∈ f.support) : x ∉ g.support 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f g : Equiv.Perm α} (h : f.Disjoint g) {x : α} (hx : x ∈ f.support)
Goal: x ∉ g.support
-- Equiv.Perm.eq_on_support_mem_disjoint
theorem eq_on_support_mem_disjoint {l : List (Perm α)} (h : f ∈ l) (hl : l.Pairwise Disjoint) :
    ∀ x ∈ f.support, f x = l.prod x 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} {l : List (Equiv.Perm α)} (h : f ∈ l) (hl : List.Pairwise Equiv.Perm.Disjoint l) (x : α)
Goal: x ∈ f.support → f x = l.prod x
-- Equiv.Perm.Disjoint.mono
theorem Disjoint.mono {x y : Perm α} (h : Disjoint f g) (hf : x.support ≤ f.support)
    (hg : y.support ≤ g.support) : Disjoint x y 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f g x y : Equiv.Perm α} (h : f.Disjoint g) (hf : x.support ≤ f.support) (hg : y.support ≤ g.support)
Goal: x.Disjoint y
-- Equiv.Perm.support_le_prod_of_mem
theorem support_le_prod_of_mem {l : List (Perm α)} (h : f ∈ l) (hl : l.Pairwise Disjoint) :
    f.support ≤ l.prod.support 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} {l : List (Equiv.Perm α)} (h : f ∈ l) (hl : List.Pairwise Equiv.Perm.Disjoint l)
Goal: f.support ≤ l.prod.support
-- Equiv.Perm.support_extend_domain
@[simp]
theorem support_extend_domain (f : α ≃ Subtype p) {g : Perm α} :
    support (g.extendDomain f) = g.support.map f.asEmbedding 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {β : Type u_2} [DecidableEq β] [Fintype β] {p : β → Prop} [DecidablePred p] (f : α ≃ Subtype p) {g : Equiv.Perm α}
Goal: (g.extendDomain f).support = Finset.map f.asEmbedding g.support
-- Equiv.Perm.card_support_extend_domain
theorem card_support_extend_domain (f : α ≃ Subtype p) {g : Perm α} :
    (g.extendDomain f).support.card = g.support.card 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {β : Type u_2} [DecidableEq β] [Fintype β] {p : β → Prop} [DecidablePred p] (f : α ≃ Subtype p) {g : Equiv.Perm α}
Goal: (g.extendDomain f).support.card = g.support.card
-- Equiv.Perm.card_support_eq_zero
theorem card_support_eq_zero {f : Perm α} : f.support.card = 0 ↔ f = 1 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f : Equiv.Perm α}
Goal: f.support.card = 0 ↔ f = 1
-- Equiv.Perm.one_lt_card_support_of_ne_one
theorem one_lt_card_support_of_ne_one {f : Perm α} (h : f ≠ 1) : 1 < f.support.card 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} (h : f ≠ 1)
Goal: 1 < f.support.card
-- Equiv.Perm.card_support_ne_one
theorem card_support_ne_one (f : Perm α) : f.support.card ≠ 1 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] (f : Equiv.Perm α)
Goal: f.support.card ≠ 1
-- Equiv.Perm.card_support_le_one
@[simp]
theorem card_support_le_one {f : Perm α} : f.support.card ≤ 1 ↔ f = 1 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f : Equiv.Perm α}
Goal: f.support.card ≤ 1 ↔ f = 1
-- Equiv.Perm.two_le_card_support_of_ne_one
theorem two_le_card_support_of_ne_one {f : Perm α} (h : f ≠ 1) : 2 ≤ f.support.card 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} (h : f ≠ 1)
Goal: 2 ≤ f.support.card
-- Equiv.Perm.card_support_swap_mul
theorem card_support_swap_mul {f : Perm α} {x : α} (hx : f x ≠ x) :
    (swap x (f x) * f).support.card < f.support.card 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f : Equiv.Perm α} {x : α} (hx : f x ≠ x)
Goal: (Equiv.swap x (f x) * f).support.card < f.support.card
-- Equiv.Perm.card_support_swap
theorem card_support_swap {x y : α} (hxy : x ≠ y) : (swap x y).support.card = 2 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {x y : α} (hxy : x ≠ y)
Goal: (Equiv.swap x y).support.card = 2
-- Equiv.Perm.card_support_eq_two
@[simp]
theorem card_support_eq_two {f : Perm α} : f.support.card = 2 ↔ IsSwap f 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f : Equiv.Perm α}
Goal: f.support.card = 2 ↔ f.IsSwap
-- Equiv.Perm.Disjoint.card_support_mul
theorem Disjoint.card_support_mul (h : Disjoint f g) :
    (f * g).support.card = f.support.card + g.support.card 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {f g : Equiv.Perm α} (h : f.Disjoint g)
Goal: (f * g).support.card = f.support.card + g.support.card
-- Equiv.Perm.card_support_prod_list_of_pairwise_disjoint
theorem card_support_prod_list_of_pairwise_disjoint {l : List (Perm α)} (h : l.Pairwise Disjoint) :
    l.prod.support.card = (l.map (Finset.card ∘ support)).sum 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {l : List (Equiv.Perm α)} (h : List.Pairwise Equiv.Perm.Disjoint l)
Goal: l.prod.support.card = (List.map (Finset.card ∘ Equiv.Perm.support) l).sum
-- Equiv.Perm.support_subtype_perm
@[simp]
theorem support_subtype_perm [DecidableEq α] {s : Finset α} (f : Perm α) (h) :
    ((f.subtypePerm h : Perm { x // x ∈ s }).support) =
    (s.attach.filter ((fun x => decide (f x ≠ x))) : Finset { x // x ∈ s }) 
Arguments: {α : Type u_1} [DecidableEq α] {s : Finset α} (f : Equiv.Perm α) (h : ∀ (x : α), x ∈ s ↔ f x ∈ s)
Goal: (f.subtypePerm h).support = Finset.filter (fun x => decide (f ↑x ≠ ↑x) = true) s.attach
-- Equiv.Perm.fixed_point_card_lt_of_ne_one
theorem fixed_point_card_lt_of_ne_one [DecidableEq α] [Fintype α] {σ : Perm α} (h : σ ≠ 1) :
    (filter (fun x => σ x = x) univ).card < Fintype.card α - 1 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {σ : Equiv.Perm α} (h : σ ≠ 1)
Goal: (Finset.filter (fun x => σ x = x) Finset.univ).card < Fintype.card α - 1
-- Equiv.Perm.support_conj
@[simp]
theorem support_conj : (σ * τ * σ⁻¹).support = τ.support.map σ.toEmbedding 
Arguments: {α : Type u_1} [Fintype α] [DecidableEq α] {σ τ : Equiv.Perm α}
Goal: (σ * τ * σ⁻¹).support = Finset.map (Equiv.toEmbedding σ) τ.support
-- Equiv.Perm.card_support_conj
theorem card_support_conj : (σ * τ * σ⁻¹).support.card = τ.support.card 
Arguments: {α : Type u_1} [Fintype α] [DecidableEq α] {σ τ : Equiv.Perm α}
Goal: (σ * τ * σ⁻¹).support.card = τ.support.card
-- IsFreeGroup.nonempty_basis
class IsFreeGroup (G : Type u) [Group G] : Prop where
  nonempty_basis : ∃ (ι : Type u), Nonempty (FreeGroupBasis ι G)
Arguments: {G : Type u} [Group G] [self : IsFreeGroup G]
Goal: ∃ ι, Nonempty (FreeGroupBasis ι G)
-- FreeGroupBasis.repr_apply_coe
@[simp] lemma repr_apply_coe (b : FreeGroupBasis ι G) (i : ι) : b.repr (b i) = FreeGroup.of i 
Arguments: {ι : Type u_1} {G : Type u_3} [Group G] (b : FreeGroupBasis ι G) (i : ι)
Goal: b.repr (b i) = FreeGroup.of i
-- FreeGroupBasis.ofFreeGroup_apply
@[simp] lemma ofFreeGroup_apply {X : Type*} (x : X) :
    FreeGroupBasis.ofFreeGroup X x = FreeGroup.of x 
Arguments: {X : Type u_5} (x : X)
Goal: (FreeGroupBasis.ofFreeGroup X) x = FreeGroup.of x
-- FreeGroupBasis.reindex_apply
@[simp] lemma reindex_apply (b : FreeGroupBasis ι G) (e : ι ≃ ι') (x : ι') :
    b.reindex e x = b (e.symm x) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {G : Type u_3} [Group G] (b : FreeGroupBasis ι G) (e : ι ≃ ι') (x : ι')
Goal: (b.reindex e) x = b (e.symm x)
-- FreeGroupBasis.map_apply
@[simp] lemma map_apply (b : FreeGroupBasis ι G) (e : G ≃* H) (x : ι) :
    b.map e x = e (b x) 
Arguments: {ι : Type u_1} {G : Type u_3} {H : Type u_4} [Group G] [Group H] (b : FreeGroupBasis ι G) (e : G ≃* H) (x : ι)
Goal: (b.map e) x = e (b x)
-- FreeGroupBasis.injective
protected lemma injective (b : FreeGroupBasis ι G) : Injective b 
Arguments: {ι : Type u_1} {G : Type u_3} [Group G] (b : FreeGroupBasis ι G)
Goal: Function.Injective ⇑b
-- FreeGroupBasis.isFreeGroup
lemma isFreeGroup (b : FreeGroupBasis ι G) : IsFreeGroup G 
Arguments: {ι : Type u_1} {G : Type u_3} [Group G] (b : FreeGroupBasis ι G)
Goal: IsFreeGroup G
-- FreeGroupBasis.lift_apply_apply
@[simps!]
def lift (b : FreeGroupBasis ι G) : (ι → H) ≃ (G →* H) 
Arguments: {ι : Type u_1} {G : Type u_3} {H : Type u_4} [Group G] [Group H] (b : FreeGroupBasis ι G)
Goal: ∀ (a : ι → H) (a_1 : G), (b.lift a) a_1 = (FreeGroup.lift a) (b.repr a_1)
-- FreeGroupBasis.lift_symm_apply
@[simps!]
def lift (b : FreeGroupBasis ι G) : (ι → H) ≃ (G →* H) 
Arguments: {ι : Type u_1} {G : Type u_3} {H : Type u_4} [Group G] [Group H] (b : FreeGroupBasis ι G)
Goal: ∀ (a : G →* H) (a_1 : ι), b.lift.symm a a_1 = a (b.repr.symm (FreeGroup.of a_1))
-- FreeGroupBasis.ext_hom
lemma ext_hom (b : FreeGroupBasis ι G) (f g : G →* H) (h : ∀ i, f (b i) = g (b i)) : f = g 
Arguments: {ι : Type u_1} {G : Type u_3} {H : Type u_4} [Group G] [Group H] (b : FreeGroupBasis ι G) (f g : G →* H) (h : ∀ (i : ι), f (b i) = g (b i))
Goal: f = g
-- IsFreeGroup.mulEquiv_def
irreducible_def mulEquiv : FreeGroup (Generators G) ≃* G 
Arguments: (G : Type u_2) [Group G] [IsFreeGroup G]
Goal: IsFreeGroup.mulEquiv G = ⋯.some.repr.symm
-- IsFreeGroup.toFreeGroup_symm_apply
@[simps!]
def toFreeGroup : G ≃* FreeGroup (Generators G) 
Arguments: (G : Type u_1) [Group G] [IsFreeGroup G]
Goal: ∀ (a : FreeGroup (IsFreeGroup.Generators G)), (IsFreeGroup.toFreeGroup G).symm a = (IsFreeGroup.mulEquiv G) a
-- IsFreeGroup.toFreeGroup_apply
@[simps!]
def toFreeGroup : G ≃* FreeGroup (Generators G) 
Arguments: (G : Type u_1) [Group G] [IsFreeGroup G]
Goal: ∀ (a : G), (IsFreeGroup.toFreeGroup G) a = (IsFreeGroup.mulEquiv G).symm a
-- IsFreeGroup.lift_of
@[simp]
theorem lift_of (f : Generators G → H) (a : Generators G) : lift f (of a) = f a 
Arguments: {G : Type u_1} [Group G] [IsFreeGroup G] {H : Type u_2} [Group H] (f : IsFreeGroup.Generators G → H) (a : IsFreeGroup.Generators G)
Goal: (IsFreeGroup.lift f) (IsFreeGroup.of a) = f a
-- IsFreeGroup.lift_symm_apply
@[simp]
theorem lift_symm_apply (f : G →* H) (a : Generators G) : (lift.symm f) a = f (of a) 
Arguments: {G : Type u_1} [Group G] [IsFreeGroup G] {H : Type u_2} [Group H] (f : G →* H) (a : IsFreeGroup.Generators G)
Goal: IsFreeGroup.lift.symm f a = f (IsFreeGroup.of a)
-- IsFreeGroup.ext_hom
theorem ext_hom ⦃f g : G →* H⦄ (h : ∀ a : Generators G, f (of a) = g (of a)) : f = g 
Arguments: {G : Type u_1} [Group G] [IsFreeGroup G] {H : Type u_2} [Group H] ⦃f g : G →* H⦄ (h : ∀ (a : IsFreeGroup.Generators G), f (IsFreeGroup.of a) = g (IsFreeGroup.of a))
Goal: f = g
-- IsFreeGroup.unique_lift
theorem unique_lift (f : Generators G → H) : ∃! F : G →* H, ∀ a, F (of a) = f a 
Arguments: {G : Type u_1} [Group G] [IsFreeGroup G] {H : Type u_2} [Group H] (f : IsFreeGroup.Generators G → H)
Goal: ∃! F, ∀ (a : IsFreeGroup.Generators G), F (IsFreeGroup.of a) = f a
-- IsFreeGroup.ofLift
lemma ofLift {G : Type u} [Group G] (X : Type u) (of : X → G)
    (lift : ∀ {H : Type u} [Group H], (X → H) ≃ (G →* H))
    (lift_of : ∀ {H : Type u} [Group H], ∀ (f : X → H) (a), lift f (of a) = f a) :
    IsFreeGroup G 
Arguments: {G : Type u} [Group G] (X : Type u) (of : X → G) (lift : {H : Type u} → [inst : Group H] → (X → H) ≃ (G →* H)) (lift_of : ∀ {H : Type u} [inst : Group H] (f : X → H) (a : X), (lift f) (of a) = f a)
Goal: IsFreeGroup G
-- IsFreeGroup.ofUniqueLift
lemma ofUniqueLift {G : Type u} [Group G] (X : Type u) (of : X → G)
    (h : ∀ {H : Type u} [Group H] (f : X → H), ∃! F : G →* H, ∀ a, F (of a) = f a) :
    IsFreeGroup G 
Arguments: {G : Type u} [Group G] (X : Type u) (of : X → G) (h : ∀ {H : Type u} [inst : Group H] (f : X → H), ∃! F, ∀ (a : X), F (of a) = f a)
Goal: IsFreeGroup G
-- IsFreeGroup.ofMulEquiv
lemma ofMulEquiv (e : G ≃* H) : IsFreeGroup H 
Arguments: {G : Type u_1} [Group G] [IsFreeGroup G] {H : Type u_2} [Group H] (e : G ≃* H)
Goal: IsFreeGroup H
-- Sylow.isPGroup'
structure Sylow extends Subgroup G where
  isPGroup' : IsPGroup p toSubgroup
  is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
Arguments: {p : ℕ} {G : Type u_1} [Group G] (self : Sylow p G)
Goal: IsPGroup p ↥↑self
-- Sylow.is_maximal'
structure Sylow extends Subgroup G where
  isPGroup' : IsPGroup p toSubgroup
  is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
Arguments: {p : ℕ} {G : Type u_1} [Group G] (self : Sylow p G) {Q : Subgroup G}
Goal: IsPGroup p ↥Q → ↑self ≤ Q → Q = ↑self
-- Sylow.ext
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {P Q : Sylow p G} (h : ↑P = ↑Q)
Goal: P = Q
-- Sylow.ext_iff
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {P Q : Sylow p G}
Goal: P = Q ↔ ↑P = ↑Q
-- Sylow.coe_comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
    (P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P 
Arguments: {p : ℕ} {G : Type u_1} [Group G] (P : Sylow p G) {K : Type u_2} [Group K] (ϕ : K →* G) (hϕ : IsPGroup p ↥ϕ.ker) (h : ↑P ≤ ϕ.range)
Goal: ↑(P.comapOfKerIsPGroup ϕ hϕ h) = Subgroup.comap ϕ ↑P
-- Sylow.coe_comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
    ↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P 
Arguments: {p : ℕ} {G : Type u_1} [Group G] (P : Sylow p G) {K : Type u_2} [Group K] (ϕ : K →* G) (hϕ : Function.Injective ⇑ϕ) (h : ↑P ≤ ϕ.range)
Goal: ↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P
-- Sylow.coe_subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N 
Arguments: {p : ℕ} {G : Type u_1} [Group G] (P : Sylow p G) {N : Subgroup G} (h : ↑P ≤ N)
Goal: ↑(P.subtype h) = (↑P).subgroupOf N
-- Sylow.subtype_injective
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
    (h : P.subtype hP = Q.subtype hQ) : P = Q 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {N : Subgroup G} {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N} (h : P.subtype hP = Q.subtype hQ)
Goal: P = Q
-- IsPGroup.exists_le_sylow
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {P : Subgroup G} (hP : IsPGroup p ↥P)
Goal: ∃ Q, P ≤ ↑Q
-- Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
    (hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {H : Type u_2} [Group H] (P : Sylow p H) {f : H →* G} (hf : IsPGroup p ↥f.ker)
Goal: ∃ Q, Subgroup.comap f ↑Q = ↑P
-- Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
    (hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {H : Type u_2} [Group H] (P : Sylow p H) {f : H →* G} (hf : Function.Injective ⇑f)
Goal: ∃ Q, Subgroup.comap f ↑Q = ↑P
-- Sylow.exists_comap_subtype_eq
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
    ∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} (P : Sylow p ↥H)
Goal: ∃ Q, Subgroup.comap H.subtype ↑Q = ↑P
-- Sylow.finite_of_ker_is_pGroup
theorem Sylow.finite_of_ker_is_pGroup {H : Type*} [Group H] {f : H →* G}
    (hf : IsPGroup p f.ker) [Finite (Sylow p G)] : Finite (Sylow p H) 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {H : Type u_2} [Group H] {f : H →* G} (hf : IsPGroup p ↥f.ker) [Finite (Sylow p G)]
Goal: Finite (Sylow p H)
-- Sylow.finite_of_injective
theorem Sylow.finite_of_injective {H : Type*} [Group H] {f : H →* G}
    (hf : Function.Injective f) [Finite (Sylow p G)] : Finite (Sylow p H) 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {H : Type u_2} [Group H] {f : H →* G} (hf : Function.Injective ⇑f) [Finite (Sylow p G)]
Goal: Finite (Sylow p H)
-- Sylow.pointwise_smul_def
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
    {P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {α : Type u_2} [Group α] [MulDistribMulAction α G] {g : α} {P : Sylow p G}
Goal: ↑(g • P) = g • ↑P
-- Sylow.smul_def
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {g : G} {P : Sylow p G}
Goal: g • P = MulAut.conj g • P
-- Sylow.coe_subgroup_smul
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
    ↑(g • P) = MulAut.conj g • (P : Subgroup G) 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {g : G} {P : Sylow p G}
Goal: ↑(g • P) = MulAut.conj g • ↑P
-- Sylow.coe_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {g : G} {P : Sylow p G}
Goal: ↑(g • P) = MulAut.conj g • ↑P
-- Sylow.smul_le
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : ↥H)
Goal: ↑(h • P) ≤ H
-- Sylow.smul_subtype
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
    h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : ↥H)
Goal: h • P.subtype hP = (h • P).subtype ⋯
-- Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
    g • P = P ↔ g ∈ (P : Subgroup G).normalizer 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {g : G} {P : Sylow p G}
Goal: g • P = P ↔ g ∈ (↑P).normalizer
-- Sylow.smul_eq_of_normal
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] :
    g • P = P 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {g : G} {P : Sylow p G} [h : (↑P).Normal]
Goal: g • P = P
-- Subgroup.sylow_mem_fixedPoints_iff
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
    P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer 
Arguments: {p : ℕ} {G : Type u_1} [Group G] (H : Subgroup G) {P : Sylow p G}
Goal: P ∈ MulAction.fixedPoints (↥H) (Sylow p G) ↔ H ≤ (↑P).normalizer
-- IsPGroup.inf_normalizer_sylow
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
    P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {P : Subgroup G} (hP : IsPGroup p ↥P) (Q : Sylow p G)
Goal: P ⊓ (↑Q).normalizer = P ⊓ ↑Q
-- IsPGroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
    Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q 
Arguments: {p : ℕ} {G : Type u_1} [Group G] {P : Subgroup G} (hP : IsPGroup p ↥P) {Q : Sylow p G}
Goal: Q ∈ MulAction.fixedPoints (↥P) (Sylow p G) ↔ P ≤ ↑Q
-- card_sylow_modEq_one
theorem card_sylow_modEq_one [Fact p.Prime] [Finite (Sylow p G)] :
    Nat.card (Sylow p G) ≡ 1 [MOD p] 
Arguments: (p : ℕ) (G : Type u_1) [Group G] [Fact (Nat.Prime p)] [Finite (Sylow p G)]
Goal: Nat.card (Sylow p G) ≡ 1 [MOD p]
-- not_dvd_card_sylow
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] : ¬p ∣ Nat.card (Sylow p G) 
Arguments: (p : ℕ) (G : Type u_1) [Group G] [hp : Fact (Nat.Prime p)] [Finite (Sylow p G)]
Goal: ¬p ∣ Nat.card (Sylow p G)
-- Sylow.orbit_eq_top
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ 
Arguments: {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] [Finite (Sylow p G)] (P : Sylow p G)
Goal: MulAction.orbit G P = ⊤
-- Sylow.stabilizer_eq_normalizer
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
    stabilizer G P = (P : Subgroup G).normalizer 
Arguments: {p : ℕ} {G : Type u_1} [Group G] (P : Sylow p G)
Goal: MulAction.stabilizer G P = (↑P).normalizer
-- Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n 
Arguments: {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] [Finite (Sylow p G)] (P : Sylow p G) (x g : G) (hx : x ∈ Subgroup.centralizer ↑P) (hy : g⁻¹ * x * g ∈ Subgroup.centralizer ↑P)
Goal: ∃ n ∈ (↑P).normalizer, g⁻¹ * x * g = n⁻¹ * x * n
-- Sylow.conj_eq_normalizer_conj_of_mem
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
    [_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n 
Arguments: {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] [Finite (Sylow p G)] (P : Sylow p G) [_hP : (↑P).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P)
Goal: ∃ n ∈ (↑P).normalizer, g⁻¹ * x * g = n⁻¹ * x * n
-- card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) : Nat.card (Sylow p G) = Nat.card (G ⧸ (P : Subgroup G).normalizer) 
Arguments: {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] [Finite (Sylow p G)] (P : Sylow p G)
Goal: Nat.card (Sylow p G) = Nat.card (G ⧸ (↑P).normalizer)
-- card_sylow_eq_index_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
    Nat.card (Sylow p G) = (P : Subgroup G).normalizer.index 
Arguments: {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] [Finite (Sylow p G)] (P : Sylow p G)
Goal: Nat.card (Sylow p G) = (↑P).normalizer.index
-- card_sylow_dvd_index
theorem card_sylow_dvd_index [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
    Nat.card (Sylow p G) ∣ (P : Subgroup G).index 
Arguments: {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] [Finite (Sylow p G)] (P : Sylow p G)
Goal: Nat.card (Sylow p G) ∣ (↑P).index
-- not_dvd_index_sylow'
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
    [fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index 
Arguments: {p : ℕ} {G : Type u_1} [Group G] [hp : Fact (Nat.Prime p)] (P : Sylow p G) [(↑P).Normal] [fP : (↑P).FiniteIndex]
Goal: ¬p ∣ (↑P).index
-- not_dvd_index_sylow
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
    (hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index 
Arguments: {p : ℕ} {G : Type u_1} [Group G] [hp : Fact (Nat.Prime p)] [Finite (Sylow p G)] (P : Sylow p G) (hP : (↑P).relindex (↑P).normalizer ≠ 0)
Goal: ¬p ∣ (↑P).index
-- Sylow.normalizer_sup_eq_top
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
    [Finite (Sylow p N)] (P : Sylow p N) :
    ((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ 
Arguments: {G : Type u_1} [Group G] {p : ℕ} [Fact (Nat.Prime p)] {N : Subgroup G} [N.Normal] [Finite (Sylow p ↥N)] (P : Sylow p ↥N)
Goal: (Subgroup.map N.subtype ↑P).normalizer ⊔ N = ⊤
-- Sylow.normalizer_sup_eq_top'
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
    [Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ 
Arguments: {G : Type u_1} [Group G] {p : ℕ} [Fact (Nat.Prime p)] {N : Subgroup G} [N.Normal] [Finite (Sylow p ↥N)] (P : Sylow p G) (hP : ↑P ≤ N)
Goal: (↑P).normalizer ⊔ N = ⊤