ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FreeAbelianGroup.ofMulHom_coe
@[simp]
theorem ofMulHom_coe : (ofMulHom : α → FreeAbelianGroup α) = of 
Arguments: {α : Type u} [Monoid α]
Goal: ⇑FreeAbelianGroup.ofMulHom = FreeAbelianGroup.of
-- FreeAbelianGroup.liftMonoid_coe_addMonoidHom
@[simp]
theorem liftMonoid_coe_addMonoidHom (f : α →* R) : ↑(liftMonoid f) = lift f 
Arguments: {α : Type u} {R : Type u_1} [Monoid α] [Ring R] (f : α →* R)
Goal: ↑(FreeAbelianGroup.liftMonoid f) = FreeAbelianGroup.lift ⇑f
-- FreeAbelianGroup.liftMonoid_coe
@[simp]
theorem liftMonoid_coe (f : α →* R) : ⇑(liftMonoid f) = lift f 
Arguments: {α : Type u} {R : Type u_1} [Monoid α] [Ring R] (f : α →* R)
Goal: ⇑(FreeAbelianGroup.liftMonoid f) = ⇑(FreeAbelianGroup.lift ⇑f)
-- FreeAbelianGroup.liftMonoid_symm_coe
@[simp]
-- Porting note: Added a type to `↑f`.
theorem liftMonoid_symm_coe (f : FreeAbelianGroup α →+* R) :
    ⇑(liftMonoid.symm f) = lift.symm (↑f : FreeAbelianGroup α →+ R) 
Arguments: {α : Type u} {R : Type u_1} [Monoid α] [Ring R] (f : FreeAbelianGroup α →+* R)
Goal: ⇑(FreeAbelianGroup.liftMonoid.symm f) = FreeAbelianGroup.lift.symm ↑f
-- FreeAbelianGroup.one_def
theorem one_def : (1 : FreeAbelianGroup α) = of 1 
Arguments: {α : Type u} [Monoid α]
Goal: 1 = FreeAbelianGroup.of 1
-- FreeAbelianGroup.of_one
theorem of_one : (of 1 : FreeAbelianGroup α) = 1 
Arguments: {α : Type u} [Monoid α]
Goal: FreeAbelianGroup.of 1 = 1
-- AddAction.mem_orbit_iff
@[to_additive]
theorem mem_orbit_iff {a₁ a₂ : α} : a₂ ∈ orbit M a₁ ↔ ∃ x : M, x • a₁ = a₂ 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {a₁ a₂ : α}
Goal: a₂ ∈ AddAction.orbit M a₁ ↔ ∃ x, x +ᵥ a₁ = a₂
-- MulAction.mem_orbit_iff
@[to_additive]
theorem mem_orbit_iff {a₁ a₂ : α} : a₂ ∈ orbit M a₁ ↔ ∃ x : M, x • a₁ = a₂ 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {a₁ a₂ : α}
Goal: a₂ ∈ MulAction.orbit M a₁ ↔ ∃ x, x • a₁ = a₂
-- MulAction.mem_orbit
@[to_additive (attr := simp)]
theorem mem_orbit (a : α) (m : M) : m • a ∈ orbit M a 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] (a : α) (m : M)
Goal: m • a ∈ MulAction.orbit M a
-- AddAction.mem_orbit
@[to_additive (attr := simp)]
theorem mem_orbit (a : α) (m : M) : m • a ∈ orbit M a 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] (a : α) (m : M)
Goal: m +ᵥ a ∈ AddAction.orbit M a
-- MulAction.mem_orbit_self
@[to_additive (attr := simp)]
theorem mem_orbit_self (a : α) : a ∈ orbit M a 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] (a : α)
Goal: a ∈ MulAction.orbit M a
-- AddAction.mem_orbit_self
@[to_additive (attr := simp)]
theorem mem_orbit_self (a : α) : a ∈ orbit M a 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] (a : α)
Goal: a ∈ AddAction.orbit M a
-- AddAction.orbit_nonempty
@[to_additive]
theorem orbit_nonempty (a : α) : Set.Nonempty (orbit M a) 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] (a : α)
Goal: (AddAction.orbit M a).Nonempty
-- MulAction.orbit_nonempty
@[to_additive]
theorem orbit_nonempty (a : α) : Set.Nonempty (orbit M a) 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] (a : α)
Goal: (MulAction.orbit M a).Nonempty
-- MulAction.mapsTo_smul_orbit
@[to_additive]
theorem mapsTo_smul_orbit (m : M) (a : α) : Set.MapsTo (m • ·) (orbit M a) (orbit M a) 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] (m : M) (a : α)
Goal: Set.MapsTo (fun x => m • x) (MulAction.orbit M a) (MulAction.orbit M a)
-- AddAction.mapsTo_vadd_orbit
@[to_additive]
theorem mapsTo_smul_orbit (m : M) (a : α) : Set.MapsTo (m • ·) (orbit M a) (orbit M a) 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] (m : M) (a : α)
Goal: Set.MapsTo (fun x => m +ᵥ x) (AddAction.orbit M a) (AddAction.orbit M a)
-- MulAction.smul_orbit_subset
@[to_additive]
theorem smul_orbit_subset (m : M) (a : α) : m • orbit M a ⊆ orbit M a 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] (m : M) (a : α)
Goal: m • MulAction.orbit M a ⊆ MulAction.orbit M a
-- AddAction.vadd_orbit_subset
@[to_additive]
theorem smul_orbit_subset (m : M) (a : α) : m • orbit M a ⊆ orbit M a 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] (m : M) (a : α)
Goal: m +ᵥ AddAction.orbit M a ⊆ AddAction.orbit M a
-- AddAction.orbit_vadd_subset
@[to_additive]
theorem orbit_smul_subset (m : M) (a : α) : orbit M (m • a) ⊆ orbit M a 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] (m : M) (a : α)
Goal: AddAction.orbit M (m +ᵥ a) ⊆ AddAction.orbit M a
-- MulAction.orbit_smul_subset
@[to_additive]
theorem orbit_smul_subset (m : M) (a : α) : orbit M (m • a) ⊆ orbit M a 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] (m : M) (a : α)
Goal: MulAction.orbit M (m • a) ⊆ MulAction.orbit M a
-- AddAction.orbit.coe_vadd
@[to_additive (attr := simp)]
theorem orbit.coe_smul {a : α} {m : M} {a' : orbit M a} : ↑(m • a') = m • (a' : α) 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {a : α} {m : M} {a' : ↑(AddAction.orbit M a)}
Goal: ↑(m +ᵥ a') = m +ᵥ ↑a'
-- MulAction.orbit.coe_smul
@[to_additive (attr := simp)]
theorem orbit.coe_smul {a : α} {m : M} {a' : orbit M a} : ↑(m • a') = m • (a' : α) 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {a : α} {m : M} {a' : ↑(MulAction.orbit M a)}
Goal: ↑(m • a') = m • ↑a'
-- AddAction.orbit_addSubmonoid_subset
@[to_additive]
lemma orbit_submonoid_subset (S : Submonoid M) (a : α) : orbit S a ⊆ orbit M a 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] (S : AddSubmonoid M) (a : α)
Goal: AddAction.orbit (↥S) a ⊆ AddAction.orbit M a
-- MulAction.orbit_submonoid_subset
@[to_additive]
lemma orbit_submonoid_subset (S : Submonoid M) (a : α) : orbit S a ⊆ orbit M a 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] (S : Submonoid M) (a : α)
Goal: MulAction.orbit (↥S) a ⊆ MulAction.orbit M a
-- MulAction.mem_orbit_of_mem_orbit_submonoid
@[to_additive]
lemma mem_orbit_of_mem_orbit_submonoid {S : Submonoid M} {a b : α} (h : a ∈ orbit S b) :
    a ∈ orbit M b 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {S : Submonoid M} {a b : α} (h : a ∈ MulAction.orbit (↥S) b)
Goal: a ∈ MulAction.orbit M b
-- AddAction.mem_orbit_of_mem_orbit_addSubmonoid
@[to_additive]
lemma mem_orbit_of_mem_orbit_submonoid {S : Submonoid M} {a b : α} (h : a ∈ orbit S b) :
    a ∈ orbit M b 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {S : AddSubmonoid M} {a b : α} (h : a ∈ AddAction.orbit (↥S) b)
Goal: a ∈ AddAction.orbit M b
-- MulAction.fst_mem_orbit_of_mem_orbit
@[to_additive]
lemma fst_mem_orbit_of_mem_orbit {x y : α × β} (h : x ∈ MulAction.orbit M y) :
    x.1 ∈ MulAction.orbit M y.1 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {β : Type u_1} [MulAction M β] {x y : α × β} (h : x ∈ MulAction.orbit M y)
Goal: x.1 ∈ MulAction.orbit M y.1
-- AddAction.fst_mem_orbit_of_mem_orbit
@[to_additive]
lemma fst_mem_orbit_of_mem_orbit {x y : α × β} (h : x ∈ MulAction.orbit M y) :
    x.1 ∈ MulAction.orbit M y.1 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {β : Type u_1} [AddAction M β] {x y : α × β} (h : x ∈ AddAction.orbit M y)
Goal: x.1 ∈ AddAction.orbit M y.1
-- AddAction.snd_mem_orbit_of_mem_orbit
@[to_additive]
lemma snd_mem_orbit_of_mem_orbit {x y : α × β} (h : x ∈ MulAction.orbit M y) :
    x.2 ∈ MulAction.orbit M y.2 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {β : Type u_1} [AddAction M β] {x y : α × β} (h : x ∈ AddAction.orbit M y)
Goal: x.2 ∈ AddAction.orbit M y.2
-- MulAction.snd_mem_orbit_of_mem_orbit
@[to_additive]
lemma snd_mem_orbit_of_mem_orbit {x y : α × β} (h : x ∈ MulAction.orbit M y) :
    x.2 ∈ MulAction.orbit M y.2 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {β : Type u_1} [MulAction M β] {x y : α × β} (h : x ∈ MulAction.orbit M y)
Goal: x.2 ∈ MulAction.orbit M y.2
-- AddAction.orbit_eq_univ
@[to_additive]
theorem orbit_eq_univ [IsPretransitive M α] (a : α) : orbit M a = Set.univ 
Arguments: (M : Type u) [AddMonoid M] {α : Type v} [AddAction M α] [AddAction.IsPretransitive M α] (a : α)
Goal: AddAction.orbit M a = Set.univ
-- MulAction.orbit_eq_univ
@[to_additive]
theorem orbit_eq_univ [IsPretransitive M α] (a : α) : orbit M a = Set.univ 
Arguments: (M : Type u) [Monoid M] {α : Type v} [MulAction M α] [MulAction.IsPretransitive M α] (a : α)
Goal: MulAction.orbit M a = Set.univ
-- MulAction.fixed_eq_iInter_fixedBy
@[to_additive]
theorem fixed_eq_iInter_fixedBy : fixedPoints M α = ⋂ m : M, fixedBy α m 
Arguments: (M : Type u) [Monoid M] (α : Type v) [MulAction M α]
Goal: MulAction.fixedPoints M α = ⋂ m, MulAction.fixedBy α m
-- AddAction.fixed_eq_iInter_fixedBy
@[to_additive]
theorem fixed_eq_iInter_fixedBy : fixedPoints M α = ⋂ m : M, fixedBy α m 
Arguments: (M : Type u) [AddMonoid M] (α : Type v) [AddAction M α]
Goal: AddAction.fixedPoints M α = ⋂ m, AddAction.fixedBy α m
-- MulAction.mem_fixedPoints
@[to_additive (attr := simp)]
theorem mem_fixedPoints {a : α} : a ∈ fixedPoints M α ↔ ∀ m : M, m • a = a 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {a : α}
Goal: a ∈ MulAction.fixedPoints M α ↔ ∀ (m : M), m • a = a
-- AddAction.mem_fixedPoints
@[to_additive (attr := simp)]
theorem mem_fixedPoints {a : α} : a ∈ fixedPoints M α ↔ ∀ m : M, m • a = a 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {a : α}
Goal: a ∈ AddAction.fixedPoints M α ↔ ∀ (m : M), m +ᵥ a = a
-- MulAction.mem_fixedBy
@[to_additive (attr := simp)]
theorem mem_fixedBy {m : M} {a : α} : a ∈ fixedBy α m ↔ m • a = a 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {m : M} {a : α}
Goal: a ∈ MulAction.fixedBy α m ↔ m • a = a
-- AddAction.mem_fixedBy
@[to_additive (attr := simp)]
theorem mem_fixedBy {m : M} {a : α} : a ∈ fixedBy α m ↔ m • a = a 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {m : M} {a : α}
Goal: a ∈ AddAction.fixedBy α m ↔ m +ᵥ a = a
-- AddAction.mem_fixedPoints'
@[to_additive]
theorem mem_fixedPoints' {a : α} : a ∈ fixedPoints M α ↔ ∀ a', a' ∈ orbit M a → a' = a 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {a : α}
Goal: a ∈ AddAction.fixedPoints M α ↔ ∀ a' ∈ AddAction.orbit M a, a' = a
-- MulAction.mem_fixedPoints'
@[to_additive]
theorem mem_fixedPoints' {a : α} : a ∈ fixedPoints M α ↔ ∀ a', a' ∈ orbit M a → a' = a 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {a : α}
Goal: a ∈ MulAction.fixedPoints M α ↔ ∀ a' ∈ MulAction.orbit M a, a' = a
-- MulAction.mem_fixedPoints_iff_card_orbit_eq_one
@[to_additive mem_fixedPoints_iff_card_orbit_eq_one]
theorem mem_fixedPoints_iff_card_orbit_eq_one {a : α} [Fintype (orbit M a)] :
    a ∈ fixedPoints M α ↔ Fintype.card (orbit M a) = 1 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {a : α} [Fintype ↑(MulAction.orbit M a)]
Goal: a ∈ MulAction.fixedPoints M α ↔ Fintype.card ↑(MulAction.orbit M a) = 1
-- AddAction.mem_fixedPoints_iff_card_orbit_eq_one
@[to_additive mem_fixedPoints_iff_card_orbit_eq_one]
theorem mem_fixedPoints_iff_card_orbit_eq_one {a : α} [Fintype (orbit M a)] :
    a ∈ fixedPoints M α ↔ Fintype.card (orbit M a) = 1 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {a : α} [Fintype ↑(AddAction.orbit M a)]
Goal: a ∈ AddAction.fixedPoints M α ↔ Fintype.card ↑(AddAction.orbit M a) = 1
-- MulAction.mem_stabilizerSubmonoid_iff
@[to_additive (attr := simp)]
theorem mem_stabilizerSubmonoid_iff {a : α} {m : M} : m ∈ stabilizerSubmonoid M a ↔ m • a = a 
Arguments: {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {a : α} {m : M}
Goal: m ∈ MulAction.stabilizerSubmonoid M a ↔ m • a = a
-- AddAction.mem_stabilizerAddSubmonoid_iff
@[to_additive (attr := simp)]
theorem mem_stabilizerSubmonoid_iff {a : α} {m : M} : m ∈ stabilizerSubmonoid M a ↔ m • a = a 
Arguments: {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {a : α} {m : M}
Goal: m ∈ AddAction.stabilizerAddSubmonoid M a ↔ m +ᵥ a = a
-- FixedPoints.mem_submonoid
@[simp]
lemma FixedPoints.mem_submonoid (a : α) : a ∈ submonoid M α ↔ ∀ m : M, m • a = a 
Arguments: (M : Type u) (α : Type v) [Monoid M] [Monoid α] [MulDistribMulAction M α] (a : α)
Goal: a ∈ FixedPoints.submonoid M α ↔ ∀ (m : M), m • a = a
-- FixedPoints.mem_subgroup
@[simp]
lemma mem_subgroup (a : α) : a ∈ α^*M ↔ ∀ m : M, m • a = a 
Arguments: (M : Type u) (α : Type v) [Monoid M] [Group α] [MulDistribMulAction M α] (a : α)
Goal: a ∈ FixedPoints.subgroup M α ↔ ∀ (m : M), m • a = a
-- FixedPoints.subgroup_toSubmonoid
@[simp]
lemma subgroup_toSubmonoid : (α^*M).toSubmonoid = submonoid M α 
Arguments: (M : Type u) (α : Type v) [Monoid M] [Group α] [MulDistribMulAction M α]
Goal: (FixedPoints.subgroup M α).toSubmonoid = FixedPoints.submonoid M α
-- FixedPoints.mem_addSubmonoid
@[simp]
lemma FixedPoints.mem_addSubmonoid (a : α) : a ∈ addSubmonoid M α ↔ ∀ m : M, m • a = a 
Arguments: (M : Type u) (α : Type v) [Monoid M] [AddMonoid α] [DistribMulAction M α] (a : α)
Goal: a ∈ FixedPoints.addSubmonoid M α ↔ ∀ (m : M), m • a = a
-- FixedPoints.mem_addSubgroup
@[simp]
lemma FixedPoints.mem_addSubgroup (a : α) : a ∈ α^+M ↔ ∀ m : M, m • a = a 
Arguments: (M : Type u) (α : Type v) [Monoid M] [AddGroup α] [DistribMulAction M α] (a : α)
Goal: a ∈ α^+M ↔ ∀ (m : M), m • a = a
-- FixedPoints.addSubgroup_toAddSubmonoid
@[simp]
lemma FixedPoints.addSubgroup_toAddSubmonoid : (α^+M).toAddSubmonoid = addSubmonoid M α 
Arguments: (M : Type u) (α : Type v) [Monoid M] [AddGroup α] [DistribMulAction M α]
Goal: (α^+M).toAddSubmonoid = FixedPoints.addSubmonoid M α
-- smul_cancel_of_non_zero_divisor
theorem smul_cancel_of_non_zero_divisor {M R : Type*} [Monoid M] [NonUnitalNonAssocRing R]
    [DistribMulAction M R] (k : M) (h : ∀ x : R, k • x = 0 → x = 0) {a b : R} (h' : k • a = k • b) :
    a = b 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [NonUnitalNonAssocRing R] [DistribMulAction M R] (k : M) (h : ∀ (x : R), k • x = 0 → x = 0) {a b : R} (h' : k • a = k • b)
Goal: a = b
-- MulAction.smul_orbit
@[to_additive (attr := simp)]
theorem smul_orbit (g : G) (a : α) : g • orbit G a = orbit G a 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (g : G) (a : α)
Goal: g • MulAction.orbit G a = MulAction.orbit G a
-- AddAction.vadd_orbit
@[to_additive (attr := simp)]
theorem smul_orbit (g : G) (a : α) : g • orbit G a = orbit G a 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (g : G) (a : α)
Goal: g +ᵥ AddAction.orbit G a = AddAction.orbit G a
-- AddAction.orbit_vadd
@[to_additive (attr := simp)]
theorem orbit_smul (g : G) (a : α) : orbit G (g • a) = orbit G a 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (g : G) (a : α)
Goal: AddAction.orbit G (g +ᵥ a) = AddAction.orbit G a
-- MulAction.orbit_smul
@[to_additive (attr := simp)]
theorem orbit_smul (g : G) (a : α) : orbit G (g • a) = orbit G a 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (g : G) (a : α)
Goal: MulAction.orbit G (g • a) = MulAction.orbit G a
-- AddAction.orbit_eq_iff
@[to_additive]
theorem orbit_eq_iff {a b : α} : orbit G a = orbit G b ↔ a ∈ orbit G b 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {a b : α}
Goal: AddAction.orbit G a = AddAction.orbit G b ↔ a ∈ AddAction.orbit G b
-- MulAction.orbit_eq_iff
@[to_additive]
theorem orbit_eq_iff {a b : α} : orbit G a = orbit G b ↔ a ∈ orbit G b 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {a b : α}
Goal: MulAction.orbit G a = MulAction.orbit G b ↔ a ∈ MulAction.orbit G b
-- AddAction.mem_orbit_vadd
@[to_additive]
theorem mem_orbit_smul (g : G) (a : α) : a ∈ orbit G (g • a) 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (g : G) (a : α)
Goal: a ∈ AddAction.orbit G (g +ᵥ a)
-- MulAction.mem_orbit_smul
@[to_additive]
theorem mem_orbit_smul (g : G) (a : α) : a ∈ orbit G (g • a) 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (g : G) (a : α)
Goal: a ∈ MulAction.orbit G (g • a)
-- MulAction.smul_mem_orbit_smul
@[to_additive]
theorem smul_mem_orbit_smul (g h : G) (a : α) : g • a ∈ orbit G (h • a) 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (g h : G) (a : α)
Goal: g • a ∈ MulAction.orbit G (h • a)
-- AddAction.vadd_mem_orbit_vadd
@[to_additive]
theorem smul_mem_orbit_smul (g h : G) (a : α) : g • a ∈ orbit G (h • a) 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (g h : G) (a : α)
Goal: g +ᵥ a ∈ AddAction.orbit G (h +ᵥ a)
-- AddAction.orbit_addSubgroup_subset
@[to_additive]
lemma orbit_subgroup_subset (H : Subgroup G) (a : α) : orbit H a ⊆ orbit G a 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (H : AddSubgroup G) (a : α)
Goal: AddAction.orbit (↥H) a ⊆ AddAction.orbit G a
-- MulAction.orbit_subgroup_subset
@[to_additive]
lemma orbit_subgroup_subset (H : Subgroup G) (a : α) : orbit H a ⊆ orbit G a 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (H : Subgroup G) (a : α)
Goal: MulAction.orbit (↥H) a ⊆ MulAction.orbit G a
-- AddAction.mem_orbit_of_mem_orbit_addSubgroup
@[to_additive]
lemma mem_orbit_of_mem_orbit_subgroup {H : Subgroup G} {a b : α} (h : a ∈ orbit H b) :
    a ∈ orbit G b 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {H : AddSubgroup G} {a b : α} (h : a ∈ AddAction.orbit (↥H) b)
Goal: a ∈ AddAction.orbit G b
-- MulAction.mem_orbit_of_mem_orbit_subgroup
@[to_additive]
lemma mem_orbit_of_mem_orbit_subgroup {H : Subgroup G} {a b : α} (h : a ∈ orbit H b) :
    a ∈ orbit G b 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {H : Subgroup G} {a b : α} (h : a ∈ MulAction.orbit (↥H) b)
Goal: a ∈ MulAction.orbit G b
-- MulAction.mem_orbit_symm
@[to_additive]
lemma mem_orbit_symm {a₁ a₂ : α} : a₁ ∈ orbit G a₂ ↔ a₂ ∈ orbit G a₁ 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {a₁ a₂ : α}
Goal: a₁ ∈ MulAction.orbit G a₂ ↔ a₂ ∈ MulAction.orbit G a₁
-- AddAction.mem_orbit_symm
@[to_additive]
lemma mem_orbit_symm {a₁ a₂ : α} : a₁ ∈ orbit G a₂ ↔ a₂ ∈ orbit G a₁ 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {a₁ a₂ : α}
Goal: a₁ ∈ AddAction.orbit G a₂ ↔ a₂ ∈ AddAction.orbit G a₁
-- AddAction.mem_addSubgroup_orbit_iff
@[to_additive]
lemma mem_subgroup_orbit_iff {H : Subgroup G} {x : α} {a b : orbit G x} :
    a ∈ MulAction.orbit H b ↔ (a : α) ∈ MulAction.orbit H (b : α) 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {H : AddSubgroup G} {x : α} {a b : ↑(AddAction.orbit G x)}
Goal: a ∈ AddAction.orbit (↥H) b ↔ ↑a ∈ AddAction.orbit ↥H ↑b
-- MulAction.mem_subgroup_orbit_iff
@[to_additive]
lemma mem_subgroup_orbit_iff {H : Subgroup G} {x : α} {a b : orbit G x} :
    a ∈ MulAction.orbit H b ↔ (a : α) ∈ MulAction.orbit H (b : α) 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {H : Subgroup G} {x : α} {a b : ↑(MulAction.orbit G x)}
Goal: a ∈ MulAction.orbit (↥H) b ↔ ↑a ∈ MulAction.orbit ↥H ↑b
-- AddAction.orbitRel_apply
@[to_additive]
theorem orbitRel_apply {a b : α} : (orbitRel G α).Rel a b ↔ a ∈ orbit G b 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {a b : α}
Goal: (AddAction.orbitRel G α).Rel a b ↔ a ∈ AddAction.orbit G b
-- MulAction.orbitRel_apply
@[to_additive]
theorem orbitRel_apply {a b : α} : (orbitRel G α).Rel a b ↔ a ∈ orbit G b 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {a b : α}
Goal: (MulAction.orbitRel G α).Rel a b ↔ a ∈ MulAction.orbit G b
-- MulAction.orbitRel_r_apply
@[to_additive]
lemma orbitRel_r_apply {a b : α} : (orbitRel G _).r a b ↔ a ∈ orbit G b 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {a b : α}
Goal: Setoid.r a b ↔ a ∈ MulAction.orbit G b
-- AddAction.orbitRel_r_apply
@[to_additive]
lemma orbitRel_r_apply {a b : α} : (orbitRel G _).r a b ↔ a ∈ orbit G b 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {a b : α}
Goal: Setoid.r a b ↔ a ∈ AddAction.orbit G b
-- MulAction.orbitRel_subgroup_le
@[to_additive]
lemma orbitRel_subgroup_le (H : Subgroup G) : orbitRel H α ≤ orbitRel G α 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (H : Subgroup G)
Goal: MulAction.orbitRel (↥H) α ≤ MulAction.orbitRel G α
-- AddAction.orbitRel_addSubgroup_le
@[to_additive]
lemma orbitRel_subgroup_le (H : Subgroup G) : orbitRel H α ≤ orbitRel G α 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (H : AddSubgroup G)
Goal: AddAction.orbitRel (↥H) α ≤ AddAction.orbitRel G α
-- AddAction.quotient_preimage_image_eq_union_add
@[to_additive
      "When you take a set `U` in `α`, push it down to the quotient, and pull back, you get the
      union of the orbit of `U` under `G`."]
theorem quotient_preimage_image_eq_union_mul (U : Set α) :
    letI := orbitRel G α
    Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ g : G, (g • ·) '' U 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (U : Set α)
Goal: Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ g, (fun x => g +ᵥ x) '' U
-- MulAction.quotient_preimage_image_eq_union_mul
@[to_additive
      "When you take a set `U` in `α`, push it down to the quotient, and pull back, you get the
      union of the orbit of `U` under `G`."]
theorem quotient_preimage_image_eq_union_mul (U : Set α) :
    letI := orbitRel G α
    Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ g : G, (g • ·) '' U 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (U : Set α)
Goal: Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ g, (fun x => g • x) '' U
-- MulAction.disjoint_image_image_iff
@[to_additive]
theorem disjoint_image_image_iff {U V : Set α} :
    letI := orbitRel G α
    Disjoint (Quotient.mk' '' U) (Quotient.mk' '' V) ↔ ∀ x ∈ U, ∀ g : G, g • x ∉ V 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {U V : Set α}
Goal: Disjoint (Quotient.mk' '' U) (Quotient.mk' '' V) ↔ ∀ x ∈ U, ∀ (g : G), g • x ∉ V
-- AddAction.disjoint_image_image_iff
@[to_additive]
theorem disjoint_image_image_iff {U V : Set α} :
    letI := orbitRel G α
    Disjoint (Quotient.mk' '' U) (Quotient.mk' '' V) ↔ ∀ x ∈ U, ∀ g : G, g • x ∉ V 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {U V : Set α}
Goal: Disjoint (Quotient.mk' '' U) (Quotient.mk' '' V) ↔ ∀ x ∈ U, ∀ (g : G), g +ᵥ x ∉ V
-- MulAction.image_inter_image_iff
@[to_additive]
theorem image_inter_image_iff (U V : Set α) :
    letI := orbitRel G α
    Quotient.mk' '' U ∩ Quotient.mk' '' V = ∅ ↔ ∀ x ∈ U, ∀ g : G, g • x ∉ V 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (U V : Set α)
Goal: Quotient.mk' '' U ∩ Quotient.mk' '' V = ∅ ↔ ∀ x ∈ U, ∀ (g : G), g • x ∉ V
-- AddAction.image_inter_image_iff
@[to_additive]
theorem image_inter_image_iff (U V : Set α) :
    letI := orbitRel G α
    Quotient.mk' '' U ∩ Quotient.mk' '' V = ∅ ↔ ∀ x ∈ U, ∀ g : G, g • x ∉ V 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (U V : Set α)
Goal: Quotient.mk' '' U ∩ Quotient.mk' '' V = ∅ ↔ ∀ x ∈ U, ∀ (g : G), g +ᵥ x ∉ V
-- AddAction.pretransitive_iff_subsingleton_quotient
@[to_additive "An additive action is pretransitive if and only if the quotient by
`AddAction.orbitRel` is a subsingleton."]
theorem pretransitive_iff_subsingleton_quotient :
    IsPretransitive G α ↔ Subsingleton (orbitRel.Quotient G α) 
Arguments: (G : Type u_1) (α : Type u_2) [AddGroup G] [AddAction G α]
Goal: AddAction.IsPretransitive G α ↔ Subsingleton (AddAction.orbitRel.Quotient G α)
-- MulAction.pretransitive_iff_subsingleton_quotient
@[to_additive "An additive action is pretransitive if and only if the quotient by
`AddAction.orbitRel` is a subsingleton."]
theorem pretransitive_iff_subsingleton_quotient :
    IsPretransitive G α ↔ Subsingleton (orbitRel.Quotient G α) 
Arguments: (G : Type u_1) (α : Type u_2) [Group G] [MulAction G α]
Goal: MulAction.IsPretransitive G α ↔ Subsingleton (MulAction.orbitRel.Quotient G α)
-- MulAction.pretransitive_iff_unique_quotient_of_nonempty
@[to_additive "If `α` is non-empty, an additive action is pretransitive if and only if the
quotient has exactly one element."]
theorem pretransitive_iff_unique_quotient_of_nonempty [Nonempty α] :
    IsPretransitive G α ↔ Nonempty (Unique <| orbitRel.Quotient G α) 
Arguments: (G : Type u_1) (α : Type u_2) [Group G] [MulAction G α] [Nonempty α]
Goal: MulAction.IsPretransitive G α ↔ Nonempty (Unique (MulAction.orbitRel.Quotient G α))