ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AddSubgroup.mem_sSup_of_directedOn
@[to_additive]
theorem mem_sSup_of_directedOn {K : Set (Subgroup G)} (Kne : K.Nonempty) (hK : DirectedOn (· ≤ ·) K)
    {x : G} : x ∈ sSup K ↔ ∃ s ∈ K, x ∈ s 
Arguments: {G : Type u_1} [AddGroup G] {K : Set (AddSubgroup G)} (Kne : K.Nonempty) (hK : DirectedOn (fun x x_1 => x ≤ x_1) K) {x : G}
Goal: x ∈ sSup K ↔ ∃ s ∈ K, x ∈ s
-- Subgroup.mem_sSup_of_directedOn
@[to_additive]
theorem mem_sSup_of_directedOn {K : Set (Subgroup G)} (Kne : K.Nonempty) (hK : DirectedOn (· ≤ ·) K)
    {x : G} : x ∈ sSup K ↔ ∃ s ∈ K, x ∈ s 
Arguments: {G : Type u_1} [Group G] {K : Set (Subgroup G)} (Kne : K.Nonempty) (hK : DirectedOn (fun x x_1 => x ≤ x_1) K) {x : G}
Goal: x ∈ sSup K ↔ ∃ s ∈ K, x ∈ s
-- Subgroup.coe_comap
@[to_additive (attr := simp)]
theorem coe_comap (K : Subgroup N) (f : G →* N) : (K.comap f : Set G) = f ⁻¹' K 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (K : Subgroup N) (f : G →* N)
Goal: ↑(Subgroup.comap f K) = ⇑f ⁻¹' ↑K
-- AddSubgroup.coe_comap
@[to_additive (attr := simp)]
theorem coe_comap (K : Subgroup N) (f : G →* N) : (K.comap f : Set G) = f ⁻¹' K 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (K : AddSubgroup N) (f : G →+ N)
Goal: ↑(AddSubgroup.comap f K) = ⇑f ⁻¹' ↑K
-- Subgroup.toAddSubgroup_comap
@[simp]
theorem toAddSubgroup_comap {G₂ : Type*} [Group G₂] (f : G →* G₂) (s : Subgroup G₂) :
    s.toAddSubgroup.comap (MonoidHom.toAdditive f) = Subgroup.toAddSubgroup (s.comap f) 
Arguments: {G : Type u_1} [Group G] {G₂ : Type u_7} [Group G₂] (f : G →* G₂) (s : Subgroup G₂)
Goal: AddSubgroup.comap (MonoidHom.toAdditive f) (Subgroup.toAddSubgroup s) = Subgroup.toAddSubgroup (Subgroup.comap f s)
-- AddSubgroup.toSubgroup_comap
@[simp]
theorem _root_.AddSubgroup.toSubgroup_comap {A A₂ : Type*} [AddGroup A] [AddGroup A₂]
    (f : A →+ A₂)  (s : AddSubgroup A₂) :
    s.toSubgroup.comap (AddMonoidHom.toMultiplicative f) = AddSubgroup.toSubgroup (s.comap f) 
Arguments: {A : Type u_7} {A₂ : Type u_8} [AddGroup A] [AddGroup A₂] (f : A →+ A₂) (s : AddSubgroup A₂)
Goal: Subgroup.comap (AddMonoidHom.toMultiplicative f) (AddSubgroup.toSubgroup s) =
  AddSubgroup.toSubgroup (AddSubgroup.comap f s)
-- Subgroup.mem_comap
@[to_additive (attr := simp)]
theorem mem_comap {K : Subgroup N} {f : G →* N} {x : G} : x ∈ K.comap f ↔ f x ∈ K 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {K : Subgroup N} {f : G →* N} {x : G}
Goal: x ∈ Subgroup.comap f K ↔ f x ∈ K
-- AddSubgroup.mem_comap
@[to_additive (attr := simp)]
theorem mem_comap {K : Subgroup N} {f : G →* N} {x : G} : x ∈ K.comap f ↔ f x ∈ K 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {K : AddSubgroup N} {f : G →+ N} {x : G}
Goal: x ∈ AddSubgroup.comap f K ↔ f x ∈ K
-- AddSubgroup.comap_mono
@[to_additive]
theorem comap_mono {f : G →* N} {K K' : Subgroup N} : K ≤ K' → comap f K ≤ comap f K' 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K K' : AddSubgroup N}
Goal: K ≤ K' → AddSubgroup.comap f K ≤ AddSubgroup.comap f K'
-- Subgroup.comap_mono
@[to_additive]
theorem comap_mono {f : G →* N} {K K' : Subgroup N} : K ≤ K' → comap f K ≤ comap f K' 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K K' : Subgroup N}
Goal: K ≤ K' → Subgroup.comap f K ≤ Subgroup.comap f K'
-- Subgroup.comap_comap
@[to_additive]
theorem comap_comap (K : Subgroup P) (g : N →* P) (f : G →* N) :
    (K.comap g).comap f = K.comap (g.comp f) 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {P : Type u_6} [Group P] (K : Subgroup P) (g : N →* P) (f : G →* N)
Goal: Subgroup.comap f (Subgroup.comap g K) = Subgroup.comap (g.comp f) K
-- AddSubgroup.comap_comap
@[to_additive]
theorem comap_comap (K : Subgroup P) (g : N →* P) (f : G →* N) :
    (K.comap g).comap f = K.comap (g.comp f) 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {P : Type u_6} [AddGroup P] (K : AddSubgroup P) (g : N →+ P) (f : G →+ N)
Goal: AddSubgroup.comap f (AddSubgroup.comap g K) = AddSubgroup.comap (g.comp f) K
-- Subgroup.comap_id
@[to_additive (attr := simp)]
theorem comap_id (K : Subgroup N) : K.comap (MonoidHom.id _) = K 
Arguments: {N : Type u_5} [Group N] (K : Subgroup N)
Goal: Subgroup.comap (MonoidHom.id N) K = K
-- AddSubgroup.comap_id
@[to_additive (attr := simp)]
theorem comap_id (K : Subgroup N) : K.comap (MonoidHom.id _) = K 
Arguments: {N : Type u_5} [AddGroup N] (K : AddSubgroup N)
Goal: AddSubgroup.comap (AddMonoidHom.id N) K = K
-- Subgroup.coe_map
@[to_additive (attr := simp)]
theorem coe_map (f : G →* N) (K : Subgroup G) : (K.map f : Set N) = f '' K 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (K : Subgroup G)
Goal: ↑(Subgroup.map f K) = ⇑f '' ↑K
-- AddSubgroup.coe_map
@[to_additive (attr := simp)]
theorem coe_map (f : G →* N) (K : Subgroup G) : (K.map f : Set N) = f '' K 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (K : AddSubgroup G)
Goal: ↑(AddSubgroup.map f K) = ⇑f '' ↑K
-- Subgroup.mem_map
@[to_additive (attr := simp)]
theorem mem_map {f : G →* N} {K : Subgroup G} {y : N} : y ∈ K.map f ↔ ∃ x ∈ K, f x = y 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K : Subgroup G} {y : N}
Goal: y ∈ Subgroup.map f K ↔ ∃ x ∈ K, f x = y
-- AddSubgroup.mem_map
@[to_additive (attr := simp)]
theorem mem_map {f : G →* N} {K : Subgroup G} {y : N} : y ∈ K.map f ↔ ∃ x ∈ K, f x = y 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K : AddSubgroup G} {y : N}
Goal: y ∈ AddSubgroup.map f K ↔ ∃ x ∈ K, f x = y
-- AddSubgroup.mem_map_of_mem
@[to_additive]
theorem mem_map_of_mem (f : G →* N) {K : Subgroup G} {x : G} (hx : x ∈ K) : f x ∈ K.map f 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) {K : AddSubgroup G} {x : G} (hx : x ∈ K)
Goal: f x ∈ AddSubgroup.map f K
-- Subgroup.mem_map_of_mem
@[to_additive]
theorem mem_map_of_mem (f : G →* N) {K : Subgroup G} {x : G} (hx : x ∈ K) : f x ∈ K.map f 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) {K : Subgroup G} {x : G} (hx : x ∈ K)
Goal: f x ∈ Subgroup.map f K
-- AddSubgroup.apply_coe_mem_map
@[to_additive]
theorem apply_coe_mem_map (f : G →* N) (K : Subgroup G) (x : K) : f x ∈ K.map f 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (K : AddSubgroup G) (x : ↥K)
Goal: f ↑x ∈ AddSubgroup.map f K
-- Subgroup.apply_coe_mem_map
@[to_additive]
theorem apply_coe_mem_map (f : G →* N) (K : Subgroup G) (x : K) : f x ∈ K.map f 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (K : Subgroup G) (x : ↥K)
Goal: f ↑x ∈ Subgroup.map f K
-- AddSubgroup.map_mono
@[to_additive]
theorem map_mono {f : G →* N} {K K' : Subgroup G} : K ≤ K' → map f K ≤ map f K' 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K K' : AddSubgroup G}
Goal: K ≤ K' → AddSubgroup.map f K ≤ AddSubgroup.map f K'
-- Subgroup.map_mono
@[to_additive]
theorem map_mono {f : G →* N} {K K' : Subgroup G} : K ≤ K' → map f K ≤ map f K' 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K K' : Subgroup G}
Goal: K ≤ K' → Subgroup.map f K ≤ Subgroup.map f K'
-- AddSubgroup.map_id
@[to_additive (attr := simp)]
theorem map_id : K.map (MonoidHom.id G) = K 
Arguments: {G : Type u_1} [AddGroup G] (K : AddSubgroup G)
Goal: AddSubgroup.map (AddMonoidHom.id G) K = K
-- Subgroup.map_id
@[to_additive (attr := simp)]
theorem map_id : K.map (MonoidHom.id G) = K 
Arguments: {G : Type u_1} [Group G] (K : Subgroup G)
Goal: Subgroup.map (MonoidHom.id G) K = K
-- Subgroup.map_map
@[to_additive]
theorem map_map (g : N →* P) (f : G →* N) : (K.map f).map g = K.map (g.comp f) 
Arguments: {G : Type u_1} [Group G] (K : Subgroup G) {N : Type u_5} [Group N] {P : Type u_6} [Group P] (g : N →* P) (f : G →* N)
Goal: Subgroup.map g (Subgroup.map f K) = Subgroup.map (g.comp f) K
-- AddSubgroup.map_map
@[to_additive]
theorem map_map (g : N →* P) (f : G →* N) : (K.map f).map g = K.map (g.comp f) 
Arguments: {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {N : Type u_5} [AddGroup N] {P : Type u_6} [AddGroup P] (g : N →+ P) (f : G →+ N)
Goal: AddSubgroup.map g (AddSubgroup.map f K) = AddSubgroup.map (g.comp f) K
-- Subgroup.map_one_eq_bot
@[to_additive (attr := simp)]
theorem map_one_eq_bot : K.map (1 : G →* N) = ⊥ 
Arguments: {G : Type u_1} [Group G] (K : Subgroup G) {N : Type u_5} [Group N]
Goal: Subgroup.map 1 K = ⊥
-- AddSubgroup.map_zero_eq_bot
@[to_additive (attr := simp)]
theorem map_one_eq_bot : K.map (1 : G →* N) = ⊥ 
Arguments: {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {N : Type u_5} [AddGroup N]
Goal: AddSubgroup.map 0 K = ⊥
-- Subgroup.mem_map_equiv
@[to_additive]
theorem mem_map_equiv {f : G ≃* N} {K : Subgroup G} {x : N} :
    x ∈ K.map f.toMonoidHom ↔ f.symm x ∈ K 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G ≃* N} {K : Subgroup G} {x : N}
Goal: x ∈ Subgroup.map f.toMonoidHom K ↔ f.symm x ∈ K
-- AddSubgroup.mem_map_equiv
@[to_additive]
theorem mem_map_equiv {f : G ≃* N} {K : Subgroup G} {x : N} :
    x ∈ K.map f.toMonoidHom ↔ f.symm x ∈ K 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G ≃+ N} {K : AddSubgroup G} {x : N}
Goal: x ∈ AddSubgroup.map f.toAddMonoidHom K ↔ f.symm x ∈ K
-- AddSubgroup.mem_map_iff_mem
@[to_additive (attr := simp 1100, nolint simpNF)]
theorem mem_map_iff_mem {f : G →* N} (hf : Function.Injective f) {K : Subgroup G} {x : G} :
    f x ∈ K.map f ↔ x ∈ K 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (hf : Function.Injective ⇑f) {K : AddSubgroup G} {x : G}
Goal: f x ∈ AddSubgroup.map f K ↔ x ∈ K
-- Subgroup.mem_map_iff_mem
@[to_additive (attr := simp 1100, nolint simpNF)]
theorem mem_map_iff_mem {f : G →* N} (hf : Function.Injective f) {K : Subgroup G} {x : G} :
    f x ∈ K.map f ↔ x ∈ K 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (hf : Function.Injective ⇑f) {K : Subgroup G} {x : G}
Goal: f x ∈ Subgroup.map f K ↔ x ∈ K
-- Subgroup.map_equiv_eq_comap_symm'
@[to_additive]
theorem map_equiv_eq_comap_symm' (f : G ≃* N) (K : Subgroup G) :
    K.map f.toMonoidHom = K.comap f.symm.toMonoidHom 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G ≃* N) (K : Subgroup G)
Goal: Subgroup.map f.toMonoidHom K = Subgroup.comap f.symm.toMonoidHom K
-- AddSubgroup.map_equiv_eq_comap_symm'
@[to_additive]
theorem map_equiv_eq_comap_symm' (f : G ≃* N) (K : Subgroup G) :
    K.map f.toMonoidHom = K.comap f.symm.toMonoidHom 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G ≃+ N) (K : AddSubgroup G)
Goal: AddSubgroup.map f.toAddMonoidHom K = AddSubgroup.comap f.symm.toAddMonoidHom K
-- AddSubgroup.map_equiv_eq_comap_symm
@[to_additive]
theorem map_equiv_eq_comap_symm (f : G ≃* N) (K : Subgroup G) :
    K.map f = K.comap (G := N) f.symm 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G ≃+ N) (K : AddSubgroup G)
Goal: AddSubgroup.map (↑f) K = AddSubgroup.comap (↑f.symm) K
-- Subgroup.map_equiv_eq_comap_symm
@[to_additive]
theorem map_equiv_eq_comap_symm (f : G ≃* N) (K : Subgroup G) :
    K.map f = K.comap (G := N) f.symm 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G ≃* N) (K : Subgroup G)
Goal: Subgroup.map (↑f) K = Subgroup.comap (↑f.symm) K
-- Subgroup.comap_equiv_eq_map_symm
@[to_additive]
theorem comap_equiv_eq_map_symm (f : N ≃* G) (K : Subgroup G) :
    K.comap (G := N) f = K.map f.symm 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : N ≃* G) (K : Subgroup G)
Goal: Subgroup.comap (↑f) K = Subgroup.map (↑f.symm) K
-- AddSubgroup.comap_equiv_eq_map_symm
@[to_additive]
theorem comap_equiv_eq_map_symm (f : N ≃* G) (K : Subgroup G) :
    K.comap (G := N) f = K.map f.symm 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : N ≃+ G) (K : AddSubgroup G)
Goal: AddSubgroup.comap (↑f) K = AddSubgroup.map (↑f.symm) K
-- AddSubgroup.comap_equiv_eq_map_symm'
@[to_additive]
theorem comap_equiv_eq_map_symm' (f : N ≃* G) (K : Subgroup G) :
    K.comap f.toMonoidHom = K.map f.symm.toMonoidHom 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : N ≃+ G) (K : AddSubgroup G)
Goal: AddSubgroup.comap f.toAddMonoidHom K = AddSubgroup.map f.symm.toAddMonoidHom K
-- Subgroup.comap_equiv_eq_map_symm'
@[to_additive]
theorem comap_equiv_eq_map_symm' (f : N ≃* G) (K : Subgroup G) :
    K.comap f.toMonoidHom = K.map f.symm.toMonoidHom 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : N ≃* G) (K : Subgroup G)
Goal: Subgroup.comap f.toMonoidHom K = Subgroup.map f.symm.toMonoidHom K
-- AddSubgroup.map_symm_eq_iff_map_eq
@[to_additive]
theorem map_symm_eq_iff_map_eq {H : Subgroup N} {e : G ≃* N} :
    H.map ↑e.symm = K ↔ K.map ↑e = H 
Arguments: {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {N : Type u_5} [AddGroup N] {H : AddSubgroup N} {e : G ≃+ N}
Goal: AddSubgroup.map (↑e.symm) H = K ↔ AddSubgroup.map (↑e) K = H
-- Subgroup.map_symm_eq_iff_map_eq
@[to_additive]
theorem map_symm_eq_iff_map_eq {H : Subgroup N} {e : G ≃* N} :
    H.map ↑e.symm = K ↔ K.map ↑e = H 
Arguments: {G : Type u_1} [Group G] (K : Subgroup G) {N : Type u_5} [Group N] {H : Subgroup N} {e : G ≃* N}
Goal: Subgroup.map (↑e.symm) H = K ↔ Subgroup.map (↑e) K = H
-- AddSubgroup.map_le_iff_le_comap
@[to_additive]
theorem map_le_iff_le_comap {f : G →* N} {K : Subgroup G} {H : Subgroup N} :
    K.map f ≤ H ↔ K ≤ H.comap f 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K : AddSubgroup G} {H : AddSubgroup N}
Goal: AddSubgroup.map f K ≤ H ↔ K ≤ AddSubgroup.comap f H
-- Subgroup.map_le_iff_le_comap
@[to_additive]
theorem map_le_iff_le_comap {f : G →* N} {K : Subgroup G} {H : Subgroup N} :
    K.map f ≤ H ↔ K ≤ H.comap f 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K : Subgroup G} {H : Subgroup N}
Goal: Subgroup.map f K ≤ H ↔ K ≤ Subgroup.comap f H
-- AddSubgroup.gc_map_comap
@[to_additive]
theorem gc_map_comap (f : G →* N) : GaloisConnection (map f) (comap f) 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N)
Goal: GaloisConnection (AddSubgroup.map f) (AddSubgroup.comap f)
-- Subgroup.gc_map_comap
@[to_additive]
theorem gc_map_comap (f : G →* N) : GaloisConnection (map f) (comap f) 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N)
Goal: GaloisConnection (Subgroup.map f) (Subgroup.comap f)
-- AddSubgroup.map_sup
@[to_additive]
theorem map_sup (H K : Subgroup G) (f : G →* N) : (H ⊔ K).map f = H.map f ⊔ K.map f 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N)
Goal: AddSubgroup.map f (H ⊔ K) = AddSubgroup.map f H ⊔ AddSubgroup.map f K
-- Subgroup.map_sup
@[to_additive]
theorem map_sup (H K : Subgroup G) (f : G →* N) : (H ⊔ K).map f = H.map f ⊔ K.map f 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup G) (f : G →* N)
Goal: Subgroup.map f (H ⊔ K) = Subgroup.map f H ⊔ Subgroup.map f K
-- AddSubgroup.map_iSup
@[to_additive]
theorem map_iSup {ι : Sort*} (f : G →* N) (s : ι → Subgroup G) :
    (iSup s).map f = ⨆ i, (s i).map f 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {ι : Sort u_7} (f : G →+ N) (s : ι → AddSubgroup G)
Goal: AddSubgroup.map f (iSup s) = ⨆ i, AddSubgroup.map f (s i)
-- Subgroup.map_iSup
@[to_additive]
theorem map_iSup {ι : Sort*} (f : G →* N) (s : ι → Subgroup G) :
    (iSup s).map f = ⨆ i, (s i).map f 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {ι : Sort u_7} (f : G →* N) (s : ι → Subgroup G)
Goal: Subgroup.map f (iSup s) = ⨆ i, Subgroup.map f (s i)
-- AddSubgroup.comap_sup_comap_le
@[to_additive]
theorem comap_sup_comap_le (H K : Subgroup N) (f : G →* N) :
    comap f H ⊔ comap f K ≤ comap f (H ⊔ K) 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup N) (f : G →+ N)
Goal: AddSubgroup.comap f H ⊔ AddSubgroup.comap f K ≤ AddSubgroup.comap f (H ⊔ K)
-- Subgroup.comap_sup_comap_le
@[to_additive]
theorem comap_sup_comap_le (H K : Subgroup N) (f : G →* N) :
    comap f H ⊔ comap f K ≤ comap f (H ⊔ K) 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup N) (f : G →* N)
Goal: Subgroup.comap f H ⊔ Subgroup.comap f K ≤ Subgroup.comap f (H ⊔ K)
-- AddSubgroup.iSup_comap_le
@[to_additive]
theorem iSup_comap_le {ι : Sort*} (f : G →* N) (s : ι → Subgroup N) :
    ⨆ i, (s i).comap f ≤ (iSup s).comap f 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {ι : Sort u_7} (f : G →+ N) (s : ι → AddSubgroup N)
Goal: ⨆ i, AddSubgroup.comap f (s i) ≤ AddSubgroup.comap f (iSup s)
-- Subgroup.iSup_comap_le
@[to_additive]
theorem iSup_comap_le {ι : Sort*} (f : G →* N) (s : ι → Subgroup N) :
    ⨆ i, (s i).comap f ≤ (iSup s).comap f 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {ι : Sort u_7} (f : G →* N) (s : ι → Subgroup N)
Goal: ⨆ i, Subgroup.comap f (s i) ≤ Subgroup.comap f (iSup s)
-- Subgroup.comap_inf
@[to_additive]
theorem comap_inf (H K : Subgroup N) (f : G →* N) : (H ⊓ K).comap f = H.comap f ⊓ K.comap f 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup N) (f : G →* N)
Goal: Subgroup.comap f (H ⊓ K) = Subgroup.comap f H ⊓ Subgroup.comap f K
-- AddSubgroup.comap_inf
@[to_additive]
theorem comap_inf (H K : Subgroup N) (f : G →* N) : (H ⊓ K).comap f = H.comap f ⊓ K.comap f 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup N) (f : G →+ N)
Goal: AddSubgroup.comap f (H ⊓ K) = AddSubgroup.comap f H ⊓ AddSubgroup.comap f K
-- Subgroup.comap_iInf
@[to_additive]
theorem comap_iInf {ι : Sort*} (f : G →* N) (s : ι → Subgroup N) :
    (iInf s).comap f = ⨅ i, (s i).comap f 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] {ι : Sort u_7} (f : G →* N) (s : ι → Subgroup N)
Goal: Subgroup.comap f (iInf s) = ⨅ i, Subgroup.comap f (s i)
-- AddSubgroup.comap_iInf
@[to_additive]
theorem comap_iInf {ι : Sort*} (f : G →* N) (s : ι → Subgroup N) :
    (iInf s).comap f = ⨅ i, (s i).comap f 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {ι : Sort u_7} (f : G →+ N) (s : ι → AddSubgroup N)
Goal: AddSubgroup.comap f (iInf s) = ⨅ i, AddSubgroup.comap f (s i)
-- Subgroup.map_inf_le
@[to_additive]
theorem map_inf_le (H K : Subgroup G) (f : G →* N) : map f (H ⊓ K) ≤ map f H ⊓ map f K 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup G) (f : G →* N)
Goal: Subgroup.map f (H ⊓ K) ≤ Subgroup.map f H ⊓ Subgroup.map f K
-- AddSubgroup.map_inf_le
@[to_additive]
theorem map_inf_le (H K : Subgroup G) (f : G →* N) : map f (H ⊓ K) ≤ map f H ⊓ map f K 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N)
Goal: AddSubgroup.map f (H ⊓ K) ≤ AddSubgroup.map f H ⊓ AddSubgroup.map f K
-- AddSubgroup.map_inf_eq
@[to_additive]
theorem map_inf_eq (H K : Subgroup G) (f : G →* N) (hf : Function.Injective f) :
    map f (H ⊓ K) = map f H ⊓ map f K 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N) (hf : Function.Injective ⇑f)
Goal: AddSubgroup.map f (H ⊓ K) = AddSubgroup.map f H ⊓ AddSubgroup.map f K
-- Subgroup.map_inf_eq
@[to_additive]
theorem map_inf_eq (H K : Subgroup G) (f : G →* N) (hf : Function.Injective f) :
    map f (H ⊓ K) = map f H ⊓ map f K 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup G) (f : G →* N) (hf : Function.Injective ⇑f)
Goal: Subgroup.map f (H ⊓ K) = Subgroup.map f H ⊓ Subgroup.map f K
-- Subgroup.map_bot
@[to_additive (attr := simp)]
theorem map_bot (f : G →* N) : (⊥ : Subgroup G).map f = ⊥ 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N)
Goal: Subgroup.map f ⊥ = ⊥
-- AddSubgroup.map_bot
@[to_additive (attr := simp)]
theorem map_bot (f : G →* N) : (⊥ : Subgroup G).map f = ⊥ 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N)
Goal: AddSubgroup.map f ⊥ = ⊥
-- Subgroup.map_top_of_surjective
@[to_additive (attr := simp)]
theorem map_top_of_surjective (f : G →* N) (h : Function.Surjective f) : Subgroup.map f ⊤ = ⊤ 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (h : Function.Surjective ⇑f)
Goal: Subgroup.map f ⊤ = ⊤
-- AddSubgroup.map_top_of_surjective
@[to_additive (attr := simp)]
theorem map_top_of_surjective (f : G →* N) (h : Function.Surjective f) : Subgroup.map f ⊤ = ⊤ 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (h : Function.Surjective ⇑f)
Goal: AddSubgroup.map f ⊤ = ⊤
-- AddSubgroup.comap_top
@[to_additive (attr := simp)]
theorem comap_top (f : G →* N) : (⊤ : Subgroup N).comap f = ⊤ 
Arguments: {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N)
Goal: AddSubgroup.comap f ⊤ = ⊤
-- Subgroup.comap_top
@[to_additive (attr := simp)]
theorem comap_top (f : G →* N) : (⊤ : Subgroup N).comap f = ⊤ 
Arguments: {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N)
Goal: Subgroup.comap f ⊤ = ⊤
-- Subgroup.subgroupOfEquivOfLe_symm_apply_coe_coe
@[to_additive (attr := simps) "If `H ≤ K`, then `H` as a subgroup of `K` is isomorphic to `H`."]
def subgroupOfEquivOfLe {G : Type*} [Group G] {H K : Subgroup G} (h : H ≤ K) :
    H.subgroupOf K ≃* H 
Arguments: {G : Type u_7} [Group G] {H K : Subgroup G} (h : H ≤ K) (g : ↥H)
Goal: ↑↑((Subgroup.subgroupOfEquivOfLe h).symm g) = ↑g
-- Subgroup.subgroupOfEquivOfLe_apply_coe
@[to_additive (attr := simps) "If `H ≤ K`, then `H` as a subgroup of `K` is isomorphic to `H`."]
def subgroupOfEquivOfLe {G : Type*} [Group G] {H K : Subgroup G} (h : H ≤ K) :
    H.subgroupOf K ≃* H 
Arguments: {G : Type u_7} [Group G] {H K : Subgroup G} (h : H ≤ K) (g : ↥(H.subgroupOf K))
Goal: ↑((Subgroup.subgroupOfEquivOfLe h) g) = ↑↑g
-- AddSubgroup.addSubgroupOfEquivOfLe_symm_apply_coe_coe
@[to_additive (attr := simps) "If `H ≤ K`, then `H` as a subgroup of `K` is isomorphic to `H`."]
def subgroupOfEquivOfLe {G : Type*} [Group G] {H K : Subgroup G} (h : H ≤ K) :
    H.subgroupOf K ≃* H 
Arguments: {G : Type u_7} [AddGroup G] {H K : AddSubgroup G} (h : H ≤ K) (g : ↥H)
Goal: ↑↑((AddSubgroup.addSubgroupOfEquivOfLe h).symm g) = ↑g
-- AddSubgroup.addSubgroupOfEquivOfLe_apply_coe
@[to_additive (attr := simps) "If `H ≤ K`, then `H` as a subgroup of `K` is isomorphic to `H`."]
def subgroupOfEquivOfLe {G : Type*} [Group G] {H K : Subgroup G} (h : H ≤ K) :
    H.subgroupOf K ≃* H 
Arguments: {G : Type u_7} [AddGroup G] {H K : AddSubgroup G} (h : H ≤ K) (g : ↥(H.addSubgroupOf K))
Goal: ↑((AddSubgroup.addSubgroupOfEquivOfLe h) g) = ↑↑g
-- Subgroup.comap_subtype
@[to_additive (attr := simp)]
theorem comap_subtype (H K : Subgroup G) : H.comap K.subtype = H.subgroupOf K 
Arguments: {G : Type u_1} [Group G] (H K : Subgroup G)
Goal: Subgroup.comap K.subtype H = H.subgroupOf K
-- AddSubgroup.comap_subtype
@[to_additive (attr := simp)]
theorem comap_subtype (H K : Subgroup G) : H.comap K.subtype = H.subgroupOf K 
Arguments: {G : Type u_1} [AddGroup G] (H K : AddSubgroup G)
Goal: AddSubgroup.comap K.subtype H = H.addSubgroupOf K
-- AddSubgroup.comap_inclusion_addSubgroupOf
@[to_additive (attr := simp)]
theorem comap_inclusion_subgroupOf {K₁ K₂ : Subgroup G} (h : K₁ ≤ K₂) (H : Subgroup G) :
    (H.subgroupOf K₂).comap (inclusion h) = H.subgroupOf K₁ 
Arguments: {G : Type u_1} [AddGroup G] {K₁ K₂ : AddSubgroup G} (h : K₁ ≤ K₂) (H : AddSubgroup G)
Goal: AddSubgroup.comap (AddSubgroup.inclusion h) (H.addSubgroupOf K₂) = H.addSubgroupOf K₁
-- Subgroup.comap_inclusion_subgroupOf
@[to_additive (attr := simp)]
theorem comap_inclusion_subgroupOf {K₁ K₂ : Subgroup G} (h : K₁ ≤ K₂) (H : Subgroup G) :
    (H.subgroupOf K₂).comap (inclusion h) = H.subgroupOf K₁ 
Arguments: {G : Type u_1} [Group G] {K₁ K₂ : Subgroup G} (h : K₁ ≤ K₂) (H : Subgroup G)
Goal: Subgroup.comap (Subgroup.inclusion h) (H.subgroupOf K₂) = H.subgroupOf K₁
-- Subgroup.coe_subgroupOf
@[to_additive]
theorem coe_subgroupOf (H K : Subgroup G) : (H.subgroupOf K : Set K) = K.subtype ⁻¹' H 
Arguments: {G : Type u_1} [Group G] (H K : Subgroup G)
Goal: ↑(H.subgroupOf K) = ⇑K.subtype ⁻¹' ↑H
-- AddSubgroup.coe_addSubgroupOf
@[to_additive]
theorem coe_subgroupOf (H K : Subgroup G) : (H.subgroupOf K : Set K) = K.subtype ⁻¹' H 
Arguments: {G : Type u_1} [AddGroup G] (H K : AddSubgroup G)
Goal: ↑(H.addSubgroupOf K) = ⇑K.subtype ⁻¹' ↑H