ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Monoid.Coprod.range_swap
@[to_additive (attr := simp)] theorem range_swap : MonoidHom.range (swap G H) = ⊤ 
Arguments: {G : Type u_1} {H : Type u_2} [Group G] [Group H]
Goal: (Monoid.Coprod.swap G H).range = ⊤
-- AddMonoid.Coprod.range_swap
@[to_additive (attr := simp)] theorem range_swap : MonoidHom.range (swap G H) = ⊤ 
Arguments: {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H]
Goal: (AddMonoid.Coprod.swap G H).range = ⊤
-- AddMonoid.Coprod.range_eq
@[to_additive] theorem range_eq (f : G ∗ H →* K) :
    MonoidHom.range f = MonoidHom.range (f.comp inl) ⊔ MonoidHom.range (f.comp inr) 
Arguments: {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {K : Type u_3} [AddGroup K] (f : AddMonoid.Coprod G H →+ K)
Goal: f.range = (f.comp AddMonoid.Coprod.inl).range ⊔ (f.comp AddMonoid.Coprod.inr).range
-- Monoid.Coprod.range_eq
@[to_additive] theorem range_eq (f : G ∗ H →* K) :
    MonoidHom.range f = MonoidHom.range (f.comp inl) ⊔ MonoidHom.range (f.comp inr) 
Arguments: {G : Type u_1} {H : Type u_2} [Group G] [Group H] {K : Type u_3} [Group K] (f : Monoid.Coprod G H →* K)
Goal: f.range = (f.comp Monoid.Coprod.inl).range ⊔ (f.comp Monoid.Coprod.inr).range
-- Monoid.Coprod.range_lift
@[to_additive (attr := simp)] theorem range_lift (f : G →* K) (g : H →* K) :
    MonoidHom.range (lift f g) = MonoidHom.range f ⊔ MonoidHom.range g 
Arguments: {G : Type u_1} {H : Type u_2} [Group G] [Group H] {K : Type u_3} [Group K] (f : G →* K) (g : H →* K)
Goal: (Monoid.Coprod.lift f g).range = f.range ⊔ g.range
-- AddMonoid.Coprod.range_lift
@[to_additive (attr := simp)] theorem range_lift (f : G →* K) (g : H →* K) :
    MonoidHom.range (lift f g) = MonoidHom.range f ⊔ MonoidHom.range g 
Arguments: {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {K : Type u_3} [AddGroup K] (f : G →+ K) (g : H →+ K)
Goal: (AddMonoid.Coprod.lift f g).range = f.range ⊔ g.range
-- AddMonoid.MulEquiv.coprodCongr_apply
@[to_additive (attr := simps! (config := .asFn)) "Lift two additive monoid
equivalences `e : M ≃+ N` and `e' : M' ≃+ N'` to an additive monoid equivalence
`(AddMonoid.Coprod M M') ≃+ (AddMonoid.Coprod N N')`."]
def coprodCongr (e : M ≃* N) (e' : M' ≃* N') : (M ∗ M') ≃* (N ∗ N') 
Arguments: {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [AddZeroClass M] [AddZeroClass N] [AddZeroClass M'] [AddZeroClass N'] (e : M ≃+ N) (e' : M' ≃+ N')
Goal: ⇑(AddMonoid.MulEquiv.coprodCongr e e') = ⇑(AddMonoid.Coprod.map ↑e ↑e')
-- AddMonoid.MulEquiv.coprodCongr_symm_apply
@[to_additive (attr := simps! (config := .asFn)) "Lift two additive monoid
equivalences `e : M ≃+ N` and `e' : M' ≃+ N'` to an additive monoid equivalence
`(AddMonoid.Coprod M M') ≃+ (AddMonoid.Coprod N N')`."]
def coprodCongr (e : M ≃* N) (e' : M' ≃* N') : (M ∗ M') ≃* (N ∗ N') 
Arguments: {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [AddZeroClass M] [AddZeroClass N] [AddZeroClass M'] [AddZeroClass N'] (e : M ≃+ N) (e' : M' ≃+ N')
Goal: ⇑(AddMonoid.MulEquiv.coprodCongr e e').symm = ⇑(AddMonoid.Coprod.map ↑e.symm ↑e'.symm)
-- Monoid.MulEquiv.coprodCongr_symm_apply
@[to_additive (attr := simps! (config := .asFn)) "Lift two additive monoid
equivalences `e : M ≃+ N` and `e' : M' ≃+ N'` to an additive monoid equivalence
`(AddMonoid.Coprod M M') ≃+ (AddMonoid.Coprod N N')`."]
def coprodCongr (e : M ≃* N) (e' : M' ≃* N') : (M ∗ M') ≃* (N ∗ N') 
Arguments: {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (e : M ≃* N) (e' : M' ≃* N')
Goal: ⇑(Monoid.MulEquiv.coprodCongr e e').symm = ⇑(Monoid.Coprod.map ↑e.symm ↑e'.symm)
-- Monoid.MulEquiv.coprodCongr_apply
@[to_additive (attr := simps! (config := .asFn)) "Lift two additive monoid
equivalences `e : M ≃+ N` and `e' : M' ≃+ N'` to an additive monoid equivalence
`(AddMonoid.Coprod M M') ≃+ (AddMonoid.Coprod N N')`."]
def coprodCongr (e : M ≃* N) (e' : M' ≃* N') : (M ∗ M') ≃* (N ∗ N') 
Arguments: {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [MulOneClass M] [MulOneClass N] [MulOneClass M'] [MulOneClass N'] (e : M ≃* N) (e' : M' ≃* N')
Goal: ⇑(Monoid.MulEquiv.coprodCongr e e') = ⇑(Monoid.Coprod.map ↑e ↑e')
-- AddMonoid.MulEquiv.coprodComm_apply
@[to_additive (attr := simps! (config := .asFn))
  "An `AddEquiv` version of `AddMonoid.Coprod.swap`."]
def coprodComm : M ∗ N ≃* N ∗ M 
Arguments: (M : Type u_1) (N : Type u_2) [AddZeroClass M] [AddZeroClass N]
Goal: ⇑(AddMonoid.MulEquiv.coprodComm M N) = ⇑(AddMonoid.Coprod.swap M N)
-- AddMonoid.MulEquiv.coprodComm_symm_apply
@[to_additive (attr := simps! (config := .asFn))
  "An `AddEquiv` version of `AddMonoid.Coprod.swap`."]
def coprodComm : M ∗ N ≃* N ∗ M 
Arguments: (M : Type u_1) (N : Type u_2) [AddZeroClass M] [AddZeroClass N]
Goal: ⇑(AddMonoid.MulEquiv.coprodComm M N).symm = ⇑(AddMonoid.Coprod.swap N M)
-- Monoid.MulEquiv.coprodComm_symm_apply
@[to_additive (attr := simps! (config := .asFn))
  "An `AddEquiv` version of `AddMonoid.Coprod.swap`."]
def coprodComm : M ∗ N ≃* N ∗ M 
Arguments: (M : Type u_1) (N : Type u_2) [MulOneClass M] [MulOneClass N]
Goal: ⇑(Monoid.MulEquiv.coprodComm M N).symm = ⇑(Monoid.Coprod.swap N M)
-- Monoid.MulEquiv.coprodComm_apply
@[to_additive (attr := simps! (config := .asFn))
  "An `AddEquiv` version of `AddMonoid.Coprod.swap`."]
def coprodComm : M ∗ N ≃* N ∗ M 
Arguments: (M : Type u_1) (N : Type u_2) [MulOneClass M] [MulOneClass N]
Goal: ⇑(Monoid.MulEquiv.coprodComm M N) = ⇑(Monoid.Coprod.swap M N)
-- AddMonoid.MulEquiv.coprodAssoc_apply_inl_inl
@[to_additive (attr := simp)]
theorem coprodAssoc_apply_inl_inl (x : M) : coprodAssoc M N P (inl (inl x)) = inl x 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [AddMonoid M] [AddMonoid N] [AddMonoid P] (x : M)
Goal: (AddMonoid.MulEquiv.coprodAssoc M N P) (AddMonoid.Coprod.inl (AddMonoid.Coprod.inl x)) = AddMonoid.Coprod.inl x
-- Monoid.MulEquiv.coprodAssoc_apply_inl_inl
@[to_additive (attr := simp)]
theorem coprodAssoc_apply_inl_inl (x : M) : coprodAssoc M N P (inl (inl x)) = inl x 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [Monoid M] [Monoid N] [Monoid P] (x : M)
Goal: (Monoid.MulEquiv.coprodAssoc M N P) (Monoid.Coprod.inl (Monoid.Coprod.inl x)) = Monoid.Coprod.inl x
-- Monoid.MulEquiv.coprodAssoc_apply_inl_inr
@[to_additive (attr := simp)]
theorem coprodAssoc_apply_inl_inr (x : N) : coprodAssoc M N P (inl (inr x)) = inr (inl x) 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [Monoid M] [Monoid N] [Monoid P] (x : N)
Goal: (Monoid.MulEquiv.coprodAssoc M N P) (Monoid.Coprod.inl (Monoid.Coprod.inr x)) = Monoid.Coprod.inr (Monoid.Coprod.inl x)
-- AddMonoid.MulEquiv.coprodAssoc_apply_inl_inr
@[to_additive (attr := simp)]
theorem coprodAssoc_apply_inl_inr (x : N) : coprodAssoc M N P (inl (inr x)) = inr (inl x) 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [AddMonoid M] [AddMonoid N] [AddMonoid P] (x : N)
Goal: (AddMonoid.MulEquiv.coprodAssoc M N P) (AddMonoid.Coprod.inl (AddMonoid.Coprod.inr x)) =
  AddMonoid.Coprod.inr (AddMonoid.Coprod.inl x)
-- AddMonoid.MulEquiv.coprodAssoc_apply_inr
@[to_additive (attr := simp)]
theorem coprodAssoc_apply_inr (x : P) : coprodAssoc M N P (inr x) = inr (inr x) 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [AddMonoid M] [AddMonoid N] [AddMonoid P] (x : P)
Goal: (AddMonoid.MulEquiv.coprodAssoc M N P) (AddMonoid.Coprod.inr x) = AddMonoid.Coprod.inr (AddMonoid.Coprod.inr x)
-- Monoid.MulEquiv.coprodAssoc_apply_inr
@[to_additive (attr := simp)]
theorem coprodAssoc_apply_inr (x : P) : coprodAssoc M N P (inr x) = inr (inr x) 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [Monoid M] [Monoid N] [Monoid P] (x : P)
Goal: (Monoid.MulEquiv.coprodAssoc M N P) (Monoid.Coprod.inr x) = Monoid.Coprod.inr (Monoid.Coprod.inr x)
-- Monoid.MulEquiv.coprodAssoc_symm_apply_inl
@[to_additive (attr := simp)]
theorem coprodAssoc_symm_apply_inl (x : M) : (coprodAssoc M N P).symm (inl x) = inl (inl x) 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [Monoid M] [Monoid N] [Monoid P] (x : M)
Goal: (Monoid.MulEquiv.coprodAssoc M N P).symm (Monoid.Coprod.inl x) = Monoid.Coprod.inl (Monoid.Coprod.inl x)
-- AddMonoid.MulEquiv.coprodAssoc_symm_apply_inl
@[to_additive (attr := simp)]
theorem coprodAssoc_symm_apply_inl (x : M) : (coprodAssoc M N P).symm (inl x) = inl (inl x) 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [AddMonoid M] [AddMonoid N] [AddMonoid P] (x : M)
Goal: (AddMonoid.MulEquiv.coprodAssoc M N P).symm (AddMonoid.Coprod.inl x) = AddMonoid.Coprod.inl (AddMonoid.Coprod.inl x)
-- Monoid.MulEquiv.coprodAssoc_symm_apply_inr_inl
@[to_additive (attr := simp)]
theorem coprodAssoc_symm_apply_inr_inl (x : N) :
    (coprodAssoc M N P).symm (inr (inl x)) = inl (inr x) 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [Monoid M] [Monoid N] [Monoid P] (x : N)
Goal: (Monoid.MulEquiv.coprodAssoc M N P).symm (Monoid.Coprod.inr (Monoid.Coprod.inl x)) =
  Monoid.Coprod.inl (Monoid.Coprod.inr x)
-- AddMonoid.MulEquiv.coprodAssoc_symm_apply_inr_inl
@[to_additive (attr := simp)]
theorem coprodAssoc_symm_apply_inr_inl (x : N) :
    (coprodAssoc M N P).symm (inr (inl x)) = inl (inr x) 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [AddMonoid M] [AddMonoid N] [AddMonoid P] (x : N)
Goal: (AddMonoid.MulEquiv.coprodAssoc M N P).symm (AddMonoid.Coprod.inr (AddMonoid.Coprod.inl x)) =
  AddMonoid.Coprod.inl (AddMonoid.Coprod.inr x)
-- AddMonoid.MulEquiv.coprodAssoc_symm_apply_inr_inr
@[to_additive (attr := simp)]
theorem coprodAssoc_symm_apply_inr_inr (x : P) :
    (coprodAssoc M N P).symm (inr (inr x)) = inr x 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [AddMonoid M] [AddMonoid N] [AddMonoid P] (x : P)
Goal: (AddMonoid.MulEquiv.coprodAssoc M N P).symm (AddMonoid.Coprod.inr (AddMonoid.Coprod.inr x)) = AddMonoid.Coprod.inr x
-- Monoid.MulEquiv.coprodAssoc_symm_apply_inr_inr
@[to_additive (attr := simp)]
theorem coprodAssoc_symm_apply_inr_inr (x : P) :
    (coprodAssoc M N P).symm (inr (inr x)) = inr x 
Arguments: {M : Type u_1} {N : Type u_2} {P : Type u_3} [Monoid M] [Monoid N] [Monoid P] (x : P)
Goal: (Monoid.MulEquiv.coprodAssoc M N P).symm (Monoid.Coprod.inr (Monoid.Coprod.inr x)) = Monoid.Coprod.inr x
-- Monoid.MulEquiv.coprodPUnit_apply
@[simps! (config := .asFn)]
def coprodPUnit : M ∗ PUnit ≃* M 
Arguments: (M : Type u_1) [Monoid M]
Goal: ⇑(Monoid.MulEquiv.coprodPUnit M) = ⇑Monoid.Coprod.fst
-- Monoid.MulEquiv.coprodPUnit_symm_apply
@[simps! (config := .asFn)]
def coprodPUnit : M ∗ PUnit ≃* M 
Arguments: (M : Type u_1) [Monoid M]
Goal: ⇑(Monoid.MulEquiv.coprodPUnit M).symm = ⇑Monoid.Coprod.inl
-- Monoid.MulEquiv.punitCoprod_apply
@[simps! (config := .asFn)]
def punitCoprod : PUnit ∗ M ≃* M 
Arguments: (M : Type u_1) [Monoid M]
Goal: ⇑(Monoid.MulEquiv.punitCoprod M) = ⇑Monoid.Coprod.snd
-- Monoid.MulEquiv.punitCoprod_symm_apply
@[simps! (config := .asFn)]
def punitCoprod : PUnit ∗ M ≃* M 
Arguments: (M : Type u_1) [Monoid M]
Goal: ⇑(Monoid.MulEquiv.punitCoprod M).symm = ⇑Monoid.Coprod.inr
-- Monoid.AddEquiv.coprodUnit_apply
@[simps! (config := .asFn)]
def coprodUnit : AddMonoid.Coprod M PUnit ≃+ M 
Arguments: {M : Type u_1} [AddMonoid M]
Goal: ⇑Monoid.AddEquiv.coprodUnit = ⇑AddMonoid.Coprod.fst
-- Monoid.AddEquiv.coprodUnit_symm_apply
@[simps! (config := .asFn)]
def coprodUnit : AddMonoid.Coprod M PUnit ≃+ M 
Arguments: {M : Type u_1} [AddMonoid M]
Goal: ⇑Monoid.AddEquiv.coprodUnit.symm = ⇑AddMonoid.Coprod.inl
-- Monoid.AddEquiv.punitCoprod_symm_apply
@[simps! (config := .asFn)]
def punitCoprod : AddMonoid.Coprod PUnit M ≃+ M 
Arguments: {M : Type u_1} [AddMonoid M]
Goal: ⇑Monoid.AddEquiv.punitCoprod.symm = ⇑AddMonoid.Coprod.inr
-- Monoid.AddEquiv.punitCoprod_apply
@[simps! (config := .asFn)]
def punitCoprod : AddMonoid.Coprod PUnit M ≃+ M 
Arguments: {M : Type u_1} [AddMonoid M]
Goal: ⇑Monoid.AddEquiv.punitCoprod = ⇑AddMonoid.Coprod.snd
-- mem_leftCoset
@[to_additive mem_leftAddCoset]
theorem mem_leftCoset {s : Set α} {x : α} (a : α) (hxS : x ∈ s) : a * x ∈ a • s 
Arguments: {α : Type u_1} [Mul α] {s : Set α} {x : α} (a : α) (hxS : x ∈ s)
Goal: a * x ∈ a • s
-- mem_leftAddCoset
@[to_additive mem_leftAddCoset]
theorem mem_leftCoset {s : Set α} {x : α} (a : α) (hxS : x ∈ s) : a * x ∈ a • s 
Arguments: {α : Type u_1} [Add α] {s : Set α} {x : α} (a : α) (hxS : x ∈ s)
Goal: a + x ∈ a +ᵥ s
-- mem_rightCoset
@[to_additive mem_rightAddCoset]
theorem mem_rightCoset {s : Set α} {x : α} (a : α) (hxS : x ∈ s) : x * a ∈ op a • s 
Arguments: {α : Type u_1} [Mul α] {s : Set α} {x : α} (a : α) (hxS : x ∈ s)
Goal: x * a ∈ MulOpposite.op a • s
-- mem_rightAddCoset
@[to_additive mem_rightAddCoset]
theorem mem_rightCoset {s : Set α} {x : α} (a : α) (hxS : x ∈ s) : x * a ∈ op a • s 
Arguments: {α : Type u_1} [Add α] {s : Set α} {x : α} (a : α) (hxS : x ∈ s)
Goal: x + a ∈ AddOpposite.op a +ᵥ s
-- leftAddCosetEquivalence_rel
@[to_additive leftAddCosetEquivalence_rel]
theorem leftCosetEquivalence_rel (s : Set α) : Equivalence (LeftCosetEquivalence s) 
Arguments: {α : Type u_1} [Add α] (s : Set α)
Goal: Equivalence (LeftAddCosetEquivalence s)
-- leftCosetEquivalence_rel
@[to_additive leftAddCosetEquivalence_rel]
theorem leftCosetEquivalence_rel (s : Set α) : Equivalence (LeftCosetEquivalence s) 
Arguments: {α : Type u_1} [Mul α] (s : Set α)
Goal: Equivalence (LeftCosetEquivalence s)
-- rightCosetEquivalence_rel
@[to_additive rightAddCosetEquivalence_rel]
theorem rightCosetEquivalence_rel (s : Set α) : Equivalence (RightCosetEquivalence s) 
Arguments: {α : Type u_1} [Mul α] (s : Set α)
Goal: Equivalence (RightCosetEquivalence s)
-- rightAddCosetEquivalence_rel
@[to_additive rightAddCosetEquivalence_rel]
theorem rightCosetEquivalence_rel (s : Set α) : Equivalence (RightCosetEquivalence s) 
Arguments: {α : Type u_1} [Add α] (s : Set α)
Goal: Equivalence (RightAddCosetEquivalence s)
-- leftCoset_assoc
@[to_additive leftAddCoset_assoc]
theorem leftCoset_assoc (s : Set α) (a b : α) : a • (b • s) = (a * b) • s 
Arguments: {α : Type u_1} [Semigroup α] (s : Set α) (a b : α)
Goal: a • b • s = (a * b) • s
-- leftAddCoset_assoc
@[to_additive leftAddCoset_assoc]
theorem leftCoset_assoc (s : Set α) (a b : α) : a • (b • s) = (a * b) • s 
Arguments: {α : Type u_1} [AddSemigroup α] (s : Set α) (a b : α)
Goal: a +ᵥ (b +ᵥ s) = a + b +ᵥ s
-- rightCoset_assoc
@[to_additive rightAddCoset_assoc]
theorem rightCoset_assoc (s : Set α) (a b : α) : op b • op a • s = op (a * b) • s 
Arguments: {α : Type u_1} [Semigroup α] (s : Set α) (a b : α)
Goal: MulOpposite.op b • MulOpposite.op a • s = MulOpposite.op (a * b) • s
-- rightAddCoset_assoc
@[to_additive rightAddCoset_assoc]
theorem rightCoset_assoc (s : Set α) (a b : α) : op b • op a • s = op (a * b) • s 
Arguments: {α : Type u_1} [AddSemigroup α] (s : Set α) (a b : α)
Goal: AddOpposite.op b +ᵥ (AddOpposite.op a +ᵥ s) = AddOpposite.op (a + b) +ᵥ s
-- leftAddCoset_rightAddCoset
@[to_additive leftAddCoset_rightAddCoset]
theorem leftCoset_rightCoset (s : Set α) (a b : α) : op b • a • s = a • (op b • s) 
Arguments: {α : Type u_1} [AddSemigroup α] (s : Set α) (a b : α)
Goal: AddOpposite.op b +ᵥ (a +ᵥ s) = a +ᵥ (AddOpposite.op b +ᵥ s)
-- leftCoset_rightCoset
@[to_additive leftAddCoset_rightAddCoset]
theorem leftCoset_rightCoset (s : Set α) (a b : α) : op b • a • s = a • (op b • s) 
Arguments: {α : Type u_1} [Semigroup α] (s : Set α) (a b : α)
Goal: MulOpposite.op b • a • s = a • MulOpposite.op b • s
-- zero_leftAddCoset
@[to_additive zero_leftAddCoset]
theorem one_leftCoset : (1 : α) • s = s 
Arguments: {α : Type u_1} [AddMonoid α] (s : Set α)
Goal: 0 +ᵥ s = s
-- one_leftCoset
@[to_additive zero_leftAddCoset]
theorem one_leftCoset : (1 : α) • s = s 
Arguments: {α : Type u_1} [Monoid α] (s : Set α)
Goal: 1 • s = s
-- rightCoset_one
@[to_additive rightAddCoset_zero]
theorem rightCoset_one : op (1 : α) • s = s 
Arguments: {α : Type u_1} [Monoid α] (s : Set α)
Goal: MulOpposite.op 1 • s = s
-- rightAddCoset_zero
@[to_additive rightAddCoset_zero]
theorem rightCoset_one : op (1 : α) • s = s 
Arguments: {α : Type u_1} [AddMonoid α] (s : Set α)
Goal: AddOpposite.op 0 +ᵥ s = s
-- mem_own_leftCoset
@[to_additive mem_own_leftAddCoset]
theorem mem_own_leftCoset (a : α) : a ∈ a • (s : Set α) 
Arguments: {α : Type u_1} [Monoid α] (s : Submonoid α) (a : α)
Goal: a ∈ a • ↑s
-- mem_own_leftAddCoset
@[to_additive mem_own_leftAddCoset]
theorem mem_own_leftCoset (a : α) : a ∈ a • (s : Set α) 
Arguments: {α : Type u_1} [AddMonoid α] (s : AddSubmonoid α) (a : α)
Goal: a ∈ a +ᵥ ↑s
-- mem_own_rightCoset
@[to_additive mem_own_rightAddCoset]
theorem mem_own_rightCoset (a : α) : a ∈ op a • (s : Set α) 
Arguments: {α : Type u_1} [Monoid α] (s : Submonoid α) (a : α)
Goal: a ∈ MulOpposite.op a • ↑s
-- mem_own_rightAddCoset
@[to_additive mem_own_rightAddCoset]
theorem mem_own_rightCoset (a : α) : a ∈ op a • (s : Set α) 
Arguments: {α : Type u_1} [AddMonoid α] (s : AddSubmonoid α) (a : α)
Goal: a ∈ AddOpposite.op a +ᵥ ↑s
-- mem_leftAddCoset_leftAddCoset
@[to_additive mem_leftAddCoset_leftAddCoset]
theorem mem_leftCoset_leftCoset {a : α} (ha : a • (s : Set α) = s) : a ∈ s 
Arguments: {α : Type u_1} [AddMonoid α] (s : AddSubmonoid α) {a : α} (ha : a +ᵥ ↑s = ↑s)
Goal: a ∈ s
-- mem_leftCoset_leftCoset
@[to_additive mem_leftAddCoset_leftAddCoset]
theorem mem_leftCoset_leftCoset {a : α} (ha : a • (s : Set α) = s) : a ∈ s 
Arguments: {α : Type u_1} [Monoid α] (s : Submonoid α) {a : α} (ha : a • ↑s = ↑s)
Goal: a ∈ s
-- mem_rightAddCoset_rightAddCoset
@[to_additive mem_rightAddCoset_rightAddCoset]
theorem mem_rightCoset_rightCoset {a : α} (ha : op a • (s : Set α) = s) : a ∈ s 
Arguments: {α : Type u_1} [AddMonoid α] (s : AddSubmonoid α) {a : α} (ha : AddOpposite.op a +ᵥ ↑s = ↑s)
Goal: a ∈ s
-- mem_rightCoset_rightCoset
@[to_additive mem_rightAddCoset_rightAddCoset]
theorem mem_rightCoset_rightCoset {a : α} (ha : op a • (s : Set α) = s) : a ∈ s 
Arguments: {α : Type u_1} [Monoid α] (s : Submonoid α) {a : α} (ha : MulOpposite.op a • ↑s = ↑s)
Goal: a ∈ s
-- mem_leftAddCoset_iff
@[to_additive mem_leftAddCoset_iff]
theorem mem_leftCoset_iff (a : α) : x ∈ a • s ↔ a⁻¹ * x ∈ s 
Arguments: {α : Type u_1} [AddGroup α] {s : Set α} {x : α} (a : α)
Goal: x ∈ a +ᵥ s ↔ -a + x ∈ s
-- mem_leftCoset_iff
@[to_additive mem_leftAddCoset_iff]
theorem mem_leftCoset_iff (a : α) : x ∈ a • s ↔ a⁻¹ * x ∈ s 
Arguments: {α : Type u_1} [Group α] {s : Set α} {x : α} (a : α)
Goal: x ∈ a • s ↔ a⁻¹ * x ∈ s
-- mem_rightCoset_iff
@[to_additive mem_rightAddCoset_iff]
theorem mem_rightCoset_iff (a : α) : x ∈ op a • s ↔ x * a⁻¹ ∈ s 
Arguments: {α : Type u_1} [Group α] {s : Set α} {x : α} (a : α)
Goal: x ∈ MulOpposite.op a • s ↔ x * a⁻¹ ∈ s
-- mem_rightAddCoset_iff
@[to_additive mem_rightAddCoset_iff]
theorem mem_rightCoset_iff (a : α) : x ∈ op a • s ↔ x * a⁻¹ ∈ s 
Arguments: {α : Type u_1} [AddGroup α] {s : Set α} {x : α} (a : α)
Goal: x ∈ AddOpposite.op a +ᵥ s ↔ x + -a ∈ s
-- leftAddCoset_mem_leftAddCoset
@[to_additive leftAddCoset_mem_leftAddCoset]
theorem leftCoset_mem_leftCoset {a : α} (ha : a ∈ s) : a • (s : Set α) = s 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α) {a : α} (ha : a ∈ s)
Goal: a +ᵥ ↑s = ↑s
-- leftCoset_mem_leftCoset
@[to_additive leftAddCoset_mem_leftAddCoset]
theorem leftCoset_mem_leftCoset {a : α} (ha : a ∈ s) : a • (s : Set α) = s 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α) {a : α} (ha : a ∈ s)
Goal: a • ↑s = ↑s
-- rightAddCoset_mem_rightAddCoset
@[to_additive rightAddCoset_mem_rightAddCoset]
theorem rightCoset_mem_rightCoset {a : α} (ha : a ∈ s) : op a • (s : Set α) = s 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α) {a : α} (ha : a ∈ s)
Goal: AddOpposite.op a +ᵥ ↑s = ↑s
-- rightCoset_mem_rightCoset
@[to_additive rightAddCoset_mem_rightAddCoset]
theorem rightCoset_mem_rightCoset {a : α} (ha : a ∈ s) : op a • (s : Set α) = s 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α) {a : α} (ha : a ∈ s)
Goal: MulOpposite.op a • ↑s = ↑s
-- orbit_addSubgroup_eq_rightCoset
@[to_additive]
theorem orbit_subgroup_eq_rightCoset (a : α) : MulAction.orbit s a = op a • s 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α) (a : α)
Goal: AddAction.orbit (↥s) a = AddOpposite.op a +ᵥ ↑s
-- orbit_subgroup_eq_rightCoset
@[to_additive]
theorem orbit_subgroup_eq_rightCoset (a : α) : MulAction.orbit s a = op a • s 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α) (a : α)
Goal: MulAction.orbit (↥s) a = MulOpposite.op a • ↑s
-- orbit_addSubgroup_eq_self_of_mem
@[to_additive]
theorem orbit_subgroup_eq_self_of_mem {a : α} (ha : a ∈ s) : MulAction.orbit s a = s 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α) {a : α} (ha : a ∈ s)
Goal: AddAction.orbit (↥s) a = ↑s
-- orbit_subgroup_eq_self_of_mem
@[to_additive]
theorem orbit_subgroup_eq_self_of_mem {a : α} (ha : a ∈ s) : MulAction.orbit s a = s 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α) {a : α} (ha : a ∈ s)
Goal: MulAction.orbit (↥s) a = ↑s
-- orbit_subgroup_one_eq_self
@[to_additive]
theorem orbit_subgroup_one_eq_self : MulAction.orbit s (1 : α) = s 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α)
Goal: MulAction.orbit (↥s) 1 = ↑s
-- orbit_addSubgroup_zero_eq_self
@[to_additive]
theorem orbit_subgroup_one_eq_self : MulAction.orbit s (1 : α) = s 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α)
Goal: AddAction.orbit (↥s) 0 = ↑s
-- eq_cosets_of_normal
@[to_additive eq_addCosets_of_normal]
theorem eq_cosets_of_normal (N : s.Normal) (g : α) : g • (s : Set α) = op g • s 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α) (N : s.Normal) (g : α)
Goal: g • ↑s = MulOpposite.op g • ↑s
-- eq_addCosets_of_normal
@[to_additive eq_addCosets_of_normal]
theorem eq_cosets_of_normal (N : s.Normal) (g : α) : g • (s : Set α) = op g • s 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α) (N : s.Normal) (g : α)
Goal: g +ᵥ ↑s = AddOpposite.op g +ᵥ ↑s
-- normal_of_eq_addCosets
@[to_additive normal_of_eq_addCosets]
theorem normal_of_eq_cosets (h : ∀ g : α, g • (s : Set α) = op g • s) : s.Normal 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α) (h : ∀ (g : α), g +ᵥ ↑s = AddOpposite.op g +ᵥ ↑s)
Goal: s.Normal
-- normal_of_eq_cosets
@[to_additive normal_of_eq_addCosets]
theorem normal_of_eq_cosets (h : ∀ g : α, g • (s : Set α) = op g • s) : s.Normal 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α) (h : ∀ (g : α), g • ↑s = MulOpposite.op g • ↑s)
Goal: s.Normal
-- normal_iff_eq_cosets
@[to_additive normal_iff_eq_addCosets]
theorem normal_iff_eq_cosets : s.Normal ↔ ∀ g : α, g • (s : Set α) = op g • s 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α)
Goal: s.Normal ↔ ∀ (g : α), g • ↑s = MulOpposite.op g • ↑s
-- normal_iff_eq_addCosets
@[to_additive normal_iff_eq_addCosets]
theorem normal_iff_eq_cosets : s.Normal ↔ ∀ g : α, g • (s : Set α) = op g • s 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α)
Goal: s.Normal ↔ ∀ (g : α), g +ᵥ ↑s = AddOpposite.op g +ᵥ ↑s
-- leftCoset_eq_iff
@[to_additive leftAddCoset_eq_iff]
theorem leftCoset_eq_iff {x y : α} : x • (s : Set α) = y • s ↔ x⁻¹ * y ∈ s 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α) {x y : α}
Goal: x • ↑s = y • ↑s ↔ x⁻¹ * y ∈ s
-- leftAddCoset_eq_iff
@[to_additive leftAddCoset_eq_iff]
theorem leftCoset_eq_iff {x y : α} : x • (s : Set α) = y • s ↔ x⁻¹ * y ∈ s 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α) {x y : α}
Goal: x +ᵥ ↑s = y +ᵥ ↑s ↔ -x + y ∈ s
-- rightCoset_eq_iff
@[to_additive rightAddCoset_eq_iff]
theorem rightCoset_eq_iff {x y : α} : op x • (s : Set α) = op y • s ↔ y * x⁻¹ ∈ s 
Arguments: {α : Type u_1} [Group α] (s : Subgroup α) {x y : α}
Goal: MulOpposite.op x • ↑s = MulOpposite.op y • ↑s ↔ y * x⁻¹ ∈ s
-- rightAddCoset_eq_iff
@[to_additive rightAddCoset_eq_iff]
theorem rightCoset_eq_iff {x y : α} : op x • (s : Set α) = op y • s ↔ y * x⁻¹ ∈ s 
Arguments: {α : Type u_1} [AddGroup α] (s : AddSubgroup α) {x y : α}
Goal: AddOpposite.op x +ᵥ ↑s = AddOpposite.op y +ᵥ ↑s ↔ y + -x ∈ s
-- QuotientAddGroup.leftRel_apply
@[to_additive]
theorem leftRel_apply {x y : α} : @Setoid.r _ (leftRel s) x y ↔ x⁻¹ * y ∈ s 
Arguments: {α : Type u_1} [AddGroup α] {s : AddSubgroup α} {x y : α}
Goal: Setoid.r x y ↔ -x + y ∈ s
-- QuotientGroup.leftRel_apply
@[to_additive]
theorem leftRel_apply {x y : α} : @Setoid.r _ (leftRel s) x y ↔ x⁻¹ * y ∈ s 
Arguments: {α : Type u_1} [Group α] {s : Subgroup α} {x y : α}
Goal: Setoid.r x y ↔ x⁻¹ * y ∈ s