ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HeytingHom.comp_assoc
@[simp]
theorem comp_assoc (f : HeytingHom γ δ) (g : HeytingHom β γ) (h : HeytingHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] [HeytingAlgebra δ] (f : HeytingHom γ δ) (g : HeytingHom β γ) (h : HeytingHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- HeytingHom.comp_id
@[simp]
theorem comp_id (f : HeytingHom α β) : f.comp (HeytingHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] (f : HeytingHom α β)
Goal: f.comp (HeytingHom.id α) = f
-- HeytingHom.id_comp
@[simp]
theorem id_comp (f : HeytingHom α β) : (HeytingHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] (f : HeytingHom α β)
Goal: (HeytingHom.id β).comp f = f
-- HeytingHom.cancel_right
@[simp]
theorem cancel_right (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] {f : HeytingHom α β} {g₁ g₂ : HeytingHom β γ} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- HeytingHom.cancel_left
@[simp]
theorem cancel_left (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] {f₁ f₂ : HeytingHom α β} {g : HeytingHom β γ} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- CoheytingHom.toFun_eq_coe
theorem toFun_eq_coe {f : CoheytingHom α β} : f.toFun = (f : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] {f : CoheytingHom α β}
Goal: f.toFun = ⇑f
-- CoheytingHom.toFun_eq_coe_aux
@[simp]
theorem toFun_eq_coe_aux {f : CoheytingHom α β} : (↑f.toLatticeHom) = ⇑f 
Arguments: {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] {f : CoheytingHom α β}
Goal: ⇑f.toLatticeHom = ⇑f
-- CoheytingHom.ext
@[ext]
theorem ext {f g : CoheytingHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] {f g : CoheytingHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- CoheytingHom.coe_copy
@[simp]
theorem coe_copy (f : CoheytingHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] (f : CoheytingHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- CoheytingHom.copy_eq
theorem copy_eq (f : CoheytingHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] (f : CoheytingHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- CoheytingHom.coe_id
@[simp]
theorem coe_id : ⇑(CoheytingHom.id α) = id 
Arguments: (α : Type u_2) [CoheytingAlgebra α]
Goal: ⇑(CoheytingHom.id α) = id
-- CoheytingHom.id_apply
@[simp]
theorem id_apply (a : α) : CoheytingHom.id α a = a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: (CoheytingHom.id α) a = a
-- CoheytingHom.coe_comp
@[simp]
theorem coe_comp (f : CoheytingHom β γ) (g : CoheytingHom α β) : ⇑(f.comp g) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] (f : CoheytingHom β γ) (g : CoheytingHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- CoheytingHom.comp_apply
@[simp]
theorem comp_apply (f : CoheytingHom β γ) (g : CoheytingHom α β) (a : α) : f.comp g a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] (f : CoheytingHom β γ) (g : CoheytingHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- CoheytingHom.comp_assoc
@[simp]
theorem comp_assoc (f : CoheytingHom γ δ) (g : CoheytingHom β γ) (h : CoheytingHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] [CoheytingAlgebra δ] (f : CoheytingHom γ δ) (g : CoheytingHom β γ) (h : CoheytingHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- CoheytingHom.comp_id
@[simp]
theorem comp_id (f : CoheytingHom α β) : f.comp (CoheytingHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] (f : CoheytingHom α β)
Goal: f.comp (CoheytingHom.id α) = f
-- CoheytingHom.id_comp
@[simp]
theorem id_comp (f : CoheytingHom α β) : (CoheytingHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] (f : CoheytingHom α β)
Goal: (CoheytingHom.id β).comp f = f
-- CoheytingHom.cancel_right
@[simp]
theorem cancel_right (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] {f : CoheytingHom α β} {g₁ g₂ : CoheytingHom β γ} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- CoheytingHom.cancel_left
@[simp]
theorem cancel_left (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] {f₁ f₂ : CoheytingHom α β} {g : CoheytingHom β γ} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- BiheytingHom.toFun_eq_coe
theorem toFun_eq_coe {f : BiheytingHom α β} : f.toFun = (f : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] {f : BiheytingHom α β}
Goal: f.toFun = ⇑f
-- BiheytingHom.toFun_eq_coe_aux
@[simp]
theorem toFun_eq_coe_aux {f : BiheytingHom α β} : (↑f.toLatticeHom) = ⇑f 
Arguments: {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] {f : BiheytingHom α β}
Goal: ⇑f.toLatticeHom = ⇑f
-- BiheytingHom.ext
@[ext]
theorem ext {f g : BiheytingHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] {f g : BiheytingHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- BiheytingHom.coe_copy
@[simp]
theorem coe_copy (f : BiheytingHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] (f : BiheytingHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- BiheytingHom.copy_eq
theorem copy_eq (f : BiheytingHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] (f : BiheytingHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- BiheytingHom.coe_id
@[simp]
theorem coe_id : ⇑(BiheytingHom.id α) = id 
Arguments: (α : Type u_2) [BiheytingAlgebra α]
Goal: ⇑(BiheytingHom.id α) = id
-- BiheytingHom.id_apply
@[simp]
theorem id_apply (a : α) : BiheytingHom.id α a = a 
Arguments: {α : Type u_2} [BiheytingAlgebra α] (a : α)
Goal: (BiheytingHom.id α) a = a
-- BiheytingHom.coe_comp
@[simp]
theorem coe_comp (f : BiheytingHom β γ) (g : BiheytingHom α β) : ⇑(f.comp g) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] (f : BiheytingHom β γ) (g : BiheytingHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- BiheytingHom.comp_apply
@[simp]
theorem comp_apply (f : BiheytingHom β γ) (g : BiheytingHom α β) (a : α) : f.comp g a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] (f : BiheytingHom β γ) (g : BiheytingHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- BiheytingHom.comp_assoc
@[simp]
theorem comp_assoc (f : BiheytingHom γ δ) (g : BiheytingHom β γ) (h : BiheytingHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] [BiheytingAlgebra δ] (f : BiheytingHom γ δ) (g : BiheytingHom β γ) (h : BiheytingHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- BiheytingHom.comp_id
@[simp]
theorem comp_id (f : BiheytingHom α β) : f.comp (BiheytingHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] (f : BiheytingHom α β)
Goal: f.comp (BiheytingHom.id α) = f
-- BiheytingHom.id_comp
@[simp]
theorem id_comp (f : BiheytingHom α β) : (BiheytingHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] (f : BiheytingHom α β)
Goal: (BiheytingHom.id β).comp f = f
-- BiheytingHom.cancel_right
@[simp]
theorem cancel_right (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] {f : BiheytingHom α β} {g₁ g₂ : BiheytingHom β γ} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- BiheytingHom.cancel_left
@[simp]
theorem cancel_left (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] {f₁ f₂ : BiheytingHom α β} {g : BiheytingHom β γ} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- Subgroup.exists_leftTransversal_of_FiniteIndex
@[to_additive]
theorem exists_leftTransversal_of_FiniteIndex
    {D H : Subgroup G} [D.FiniteIndex] (hD_le_H : D ≤ H) :
    ∃ t : Finset H,
      (t : Set H) ∈ leftTransversals (D.subgroupOf H) ∧
        ⋃ g ∈ t, (g : G) • (D : Set G) = H 
Arguments: {G : Type u_1} [Group G] {D H : Subgroup G} [D.FiniteIndex] (hD_le_H : D ≤ H)
Goal: ∃ t, ↑t ∈ Subgroup.leftTransversals ↑(D.subgroupOf H) ∧ ⋃ g ∈ t, ↑g • ↑D = ↑H
-- AddSubgroup.exists_leftTransversal_of_FiniteIndex
@[to_additive]
theorem exists_leftTransversal_of_FiniteIndex
    {D H : Subgroup G} [D.FiniteIndex] (hD_le_H : D ≤ H) :
    ∃ t : Finset H,
      (t : Set H) ∈ leftTransversals (D.subgroupOf H) ∧
        ⋃ g ∈ t, (g : G) • (D : Set G) = H 
Arguments: {G : Type u_1} [AddGroup G] {D H : AddSubgroup G} [D.FiniteIndex] (hD_le_H : D ≤ H)
Goal: ∃ t, ↑t ∈ AddSubgroup.leftTransversals ↑(D.addSubgroupOf H) ∧ ⋃ g ∈ t, ↑g +ᵥ ↑D = ↑H
-- AddSubgroup.leftCoset_cover_const_iff_surjOn
@[to_additive]
theorem leftCoset_cover_const_iff_surjOn :
    ⋃ i ∈ s, g i • (H : Set G) = Set.univ ↔ Set.SurjOn (g · : ι → G ⧸ H) s Set.univ 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {s : Finset ι} {H : AddSubgroup G} {g : ι → G}
Goal: ⋃ i ∈ s, g i +ᵥ ↑H = Set.univ ↔ Set.SurjOn (fun x => ↑(g x)) (↑s) Set.univ
-- Subgroup.leftCoset_cover_const_iff_surjOn
@[to_additive]
theorem leftCoset_cover_const_iff_surjOn :
    ⋃ i ∈ s, g i • (H : Set G) = Set.univ ↔ Set.SurjOn (g · : ι → G ⧸ H) s Set.univ 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {s : Finset ι} {H : Subgroup G} {g : ι → G}
Goal: ⋃ i ∈ s, g i • ↑H = Set.univ ↔ Set.SurjOn (fun x => ↑(g x)) (↑s) Set.univ
-- AddSubgroup.finiteIndex_of_leftCoset_cover_const
@[to_additive]
theorem finiteIndex_of_leftCoset_cover_const : H.FiniteIndex 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {s : Finset ι} {H : AddSubgroup G} {g : ι → G} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑H = Set.univ)
Goal: H.FiniteIndex
-- Subgroup.finiteIndex_of_leftCoset_cover_const
@[to_additive]
theorem finiteIndex_of_leftCoset_cover_const : H.FiniteIndex 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {s : Finset ι} {H : Subgroup G} {g : ι → G} (hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ)
Goal: H.FiniteIndex
-- Subgroup.index_le_of_leftCoset_cover_const
@[to_additive]
theorem index_le_of_leftCoset_cover_const : H.index ≤ s.card 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {s : Finset ι} {H : Subgroup G} {g : ι → G} (hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ)
Goal: H.index ≤ s.card
-- AddSubgroup.index_le_of_leftCoset_cover_const
@[to_additive]
theorem index_le_of_leftCoset_cover_const : H.index ≤ s.card 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {s : Finset ι} {H : AddSubgroup G} {g : ι → G} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑H = Set.univ)
Goal: H.index ≤ s.card
-- Subgroup.pairwiseDisjoint_leftCoset_cover_const_of_index_eq
@[to_additive]
theorem pairwiseDisjoint_leftCoset_cover_const_of_index_eq (hind : H.index = s.card) :
    Set.PairwiseDisjoint s (g · • (H : Set G)) 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {s : Finset ι} {H : Subgroup G} {g : ι → G} (hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ) (hind : H.index = s.card)
Goal: (↑s).PairwiseDisjoint fun x => g x • ↑H
-- AddSubgroup.pairwiseDisjoint_leftCoset_cover_const_of_index_eq
@[to_additive]
theorem pairwiseDisjoint_leftCoset_cover_const_of_index_eq (hind : H.index = s.card) :
    Set.PairwiseDisjoint s (g · • (H : Set G)) 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {s : Finset ι} {H : AddSubgroup G} {g : ι → G} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑H = Set.univ) (hind : H.index = s.card)
Goal: (↑s).PairwiseDisjoint fun x => g x +ᵥ ↑H
-- AddSubgroup.exists_finiteIndex_of_leftCoset_cover_aux
@[to_additive]
theorem exists_finiteIndex_of_leftCoset_cover_aux [DecidableEq (Subgroup G)]
    (j : ι) (hj : j ∈ s) (hcovers' : ⋃ i ∈ s.filter (H · = H j), g i • (H i : Set G) ≠ Set.univ) :
    ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ) [DecidableEq (AddSubgroup G)] (j : ι) (hj : j ∈ s) (hcovers' : ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i +ᵥ ↑(H i) ≠ Set.univ)
Goal: ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
-- Subgroup.exists_finiteIndex_of_leftCoset_cover_aux
@[to_additive]
theorem exists_finiteIndex_of_leftCoset_cover_aux [DecidableEq (Subgroup G)]
    (j : ι) (hj : j ∈ s) (hcovers' : ⋃ i ∈ s.filter (H · = H j), g i • (H i : Set G) ≠ Set.univ) :
    ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ) [DecidableEq (Subgroup G)] (j : ι) (hj : j ∈ s) (hcovers' : ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ)
Goal: ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
-- Subgroup.exists_finiteIndex_of_leftCoset_cover
@[to_additive]
theorem exists_finiteIndex_of_leftCoset_cover : ∃ k ∈ s, (H k).FiniteIndex 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ)
Goal: ∃ k ∈ s, (H k).FiniteIndex
-- AddSubgroup.exists_finiteIndex_of_leftCoset_cover
@[to_additive]
theorem exists_finiteIndex_of_leftCoset_cover : ∃ k ∈ s, (H k).FiniteIndex 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ)
Goal: ∃ k ∈ s, (H k).FiniteIndex
-- AddSubgroup.leftCoset_cover_filter_FiniteIndex_aux
@[to_additive]
theorem leftCoset_cover_filter_FiniteIndex_aux
    [DecidablePred (FiniteIndex : Subgroup G → Prop)] :
    (⋃ k ∈ s.filter (fun i => (H i).FiniteIndex), g k • (H k : Set G) = Set.univ) ∧
      (1 ≤ ∑ i ∈ s, ((H i).index : ℚ)⁻¹) ∧
      (∑ i ∈ s, ((H i).index : ℚ)⁻¹ = 1 → Set.PairwiseDisjoint
        (s.filter (fun i => (H i).FiniteIndex)) (fun i ↦ g i • (H i : Set G))) 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ) [DecidablePred AddSubgroup.FiniteIndex]
Goal: ⋃ k ∈ Finset.filter (fun i => (H i).FiniteIndex) s, g k +ᵥ ↑(H k) = Set.univ ∧
  1 ≤ ∑ i ∈ s, (↑(H i).index)⁻¹ ∧
    (∑ i ∈ s, (↑(H i).index)⁻¹ = 1 →
      (↑(Finset.filter (fun i => (H i).FiniteIndex) s)).PairwiseDisjoint fun i => g i +ᵥ ↑(H i))
-- Subgroup.leftCoset_cover_filter_FiniteIndex_aux
@[to_additive]
theorem leftCoset_cover_filter_FiniteIndex_aux
    [DecidablePred (FiniteIndex : Subgroup G → Prop)] :
    (⋃ k ∈ s.filter (fun i => (H i).FiniteIndex), g k • (H k : Set G) = Set.univ) ∧
      (1 ≤ ∑ i ∈ s, ((H i).index : ℚ)⁻¹) ∧
      (∑ i ∈ s, ((H i).index : ℚ)⁻¹ = 1 → Set.PairwiseDisjoint
        (s.filter (fun i => (H i).FiniteIndex)) (fun i ↦ g i • (H i : Set G))) 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ) [DecidablePred Subgroup.FiniteIndex]
Goal: ⋃ k ∈ Finset.filter (fun i => (H i).FiniteIndex) s, g k • ↑(H k) = Set.univ ∧
  1 ≤ ∑ i ∈ s, (↑(H i).index)⁻¹ ∧
    (∑ i ∈ s, (↑(H i).index)⁻¹ = 1 →
      (↑(Finset.filter (fun i => (H i).FiniteIndex) s)).PairwiseDisjoint fun i => g i • ↑(H i))
-- Subgroup.leftCoset_cover_filter_FiniteIndex
@[to_additive]
theorem leftCoset_cover_filter_FiniteIndex
    [DecidablePred (FiniteIndex : Subgroup G → Prop)] :
    ⋃ k ∈ s.filter (fun i => (H i).FiniteIndex), g k • (H k : Set G) = Set.univ 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ) [DecidablePred Subgroup.FiniteIndex]
Goal: ⋃ k ∈ Finset.filter (fun i => (H i).FiniteIndex) s, g k • ↑(H k) = Set.univ
-- AddSubgroup.leftCoset_cover_filter_FiniteIndex
@[to_additive]
theorem leftCoset_cover_filter_FiniteIndex
    [DecidablePred (FiniteIndex : Subgroup G → Prop)] :
    ⋃ k ∈ s.filter (fun i => (H i).FiniteIndex), g k • (H k : Set G) = Set.univ 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ) [DecidablePred AddSubgroup.FiniteIndex]
Goal: ⋃ k ∈ Finset.filter (fun i => (H i).FiniteIndex) s, g k +ᵥ ↑(H k) = Set.univ
-- Subgroup.one_le_sum_inv_index_of_leftCoset_cover
@[to_additive one_le_sum_inv_index_of_leftCoset_cover]
theorem one_le_sum_inv_index_of_leftCoset_cover :
    1 ≤ ∑ i ∈ s, ((H i).index : ℚ)⁻¹ 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ)
Goal: 1 ≤ ∑ i ∈ s, (↑(H i).index)⁻¹
-- AddSubgroup.one_le_sum_inv_index_of_leftCoset_cover
@[to_additive one_le_sum_inv_index_of_leftCoset_cover]
theorem one_le_sum_inv_index_of_leftCoset_cover :
    1 ≤ ∑ i ∈ s, ((H i).index : ℚ)⁻¹ 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ)
Goal: 1 ≤ ∑ i ∈ s, (↑(H i).index)⁻¹
-- AddSubgroup.pairwiseDisjoint_leftCoset_cover_of_sum_neg_index_eq_zero
@[to_additive]
theorem pairwiseDisjoint_leftCoset_cover_of_sum_inv_index_eq_one
    [DecidablePred (FiniteIndex : Subgroup G → Prop)] :
    ∑ i ∈ s, ((H i).index : ℚ)⁻¹ = 1 →
      Set.PairwiseDisjoint (s.filter (fun i => (H i).FiniteIndex))
        (fun i ↦ g i • (H i : Set G)) 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ) [DecidablePred AddSubgroup.FiniteIndex]
Goal: ∑ i ∈ s, (↑(H i).index)⁻¹ = 1 →
  (↑(Finset.filter (fun i => (H i).FiniteIndex) s)).PairwiseDisjoint fun i => g i +ᵥ ↑(H i)
-- Subgroup.pairwiseDisjoint_leftCoset_cover_of_sum_inv_index_eq_one
@[to_additive]
theorem pairwiseDisjoint_leftCoset_cover_of_sum_inv_index_eq_one
    [DecidablePred (FiniteIndex : Subgroup G → Prop)] :
    ∑ i ∈ s, ((H i).index : ℚ)⁻¹ = 1 →
      Set.PairwiseDisjoint (s.filter (fun i => (H i).FiniteIndex))
        (fun i ↦ g i • (H i : Set G)) 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ) [DecidablePred Subgroup.FiniteIndex]
Goal: ∑ i ∈ s, (↑(H i).index)⁻¹ = 1 → (↑(Finset.filter (fun i => (H i).FiniteIndex) s)).PairwiseDisjoint fun i => g i • ↑(H i)
-- Subgroup.exists_index_le_card_of_leftCoset_cover
@[to_additive]
theorem exists_index_le_card_of_leftCoset_cover :
    ∃ i ∈ s, (H i).FiniteIndex ∧ (H i).index ≤ s.card 
Arguments: {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ)
Goal: ∃ i ∈ s, (H i).FiniteIndex ∧ (H i).index ≤ s.card
-- AddSubgroup.exists_index_le_card_of_leftCoset_cover
@[to_additive]
theorem exists_index_le_card_of_leftCoset_cover :
    ∃ i ∈ s, (H i).FiniteIndex ∧ (H i).index ≤ s.card 
Arguments: {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ)
Goal: ∃ i ∈ s, (H i).FiniteIndex ∧ (H i).index ≤ s.card
-- Submodule.exists_finiteIndex_of_cover
theorem Submodule.exists_finiteIndex_of_cover :
    ∃ k ∈ s, (p k).toAddSubgroup.FiniteIndex 
Arguments: {R : Type u_1} {M : Type u_2} {ι : Type u_3} [Ring R] [AddCommGroup M] [Module R M] {p : ι → Submodule R M} {s : Finset ι} (hcovers : ⋃ i ∈ s, ↑(p i) = Set.univ)
Goal: ∃ k ∈ s, (p k).toAddSubgroup.FiniteIndex
-- Subspace.biUnion_ne_univ_of_ne_top
theorem Subspace.biUnion_ne_univ_of_ne_top (hs : ∀ p ∈ s, p ≠ ⊤) :
    ⋃ p ∈ s, (p : Set E) ≠ Set.univ 
Arguments: {k : Type u_1} {E : Type u_2} [DivisionRing k] [Infinite k] [AddCommGroup E] [Module k E] {s : Finset (Subspace k E)} (hs : ∀ p ∈ s, p ≠ ⊤)
Goal: ⋃ p ∈ s, ↑p ≠ Set.univ
-- Subspace.exists_eq_top_of_biUnion_eq_univ
theorem Subspace.exists_eq_top_of_biUnion_eq_univ (hcovers : ⋃ p ∈ s, (p : Set E) = Set.univ) :
    ∃ p ∈ s, p = ⊤ 
Arguments: {k : Type u_1} {E : Type u_2} [DivisionRing k] [Infinite k] [AddCommGroup E] [Module k E] {s : Finset (Subspace k E)} (hcovers : ⋃ p ∈ s, ↑p = Set.univ)
Goal: ∃ p ∈ s, p = ⊤
-- Order.IsPFilter.of_def
theorem IsPFilter.of_def [Preorder P] {F : Set P} (nonempty : F.Nonempty)
    (directed : DirectedOn (· ≥ ·) F) (mem_of_le : ∀ {x y : P}, x ≤ y → x ∈ F → y ∈ F) :
    IsPFilter F 
Arguments: {P : Type u_1} [Preorder P] {F : Set P} (nonempty : F.Nonempty) (directed : DirectedOn (fun x x_1 => x ≥ x_1) F) (mem_of_le : ∀ {x y : P}, x ≤ y → x ∈ F → y ∈ F)
Goal: Order.IsPFilter F
-- Order.PFilter.isPFilter
theorem isPFilter : IsPFilter (F : Set P) 
Arguments: {P : Type u_1} [Preorder P] (F : Order.PFilter P)
Goal: Order.IsPFilter ↑F
-- Order.PFilter.nonempty
protected theorem nonempty : (F : Set P).Nonempty 
Arguments: {P : Type u_1} [Preorder P] (F : Order.PFilter P)
Goal: (↑F).Nonempty
-- Order.PFilter.directed
theorem directed : DirectedOn (· ≥ ·) (F : Set P) 
Arguments: {P : Type u_1} [Preorder P] (F : Order.PFilter P)
Goal: DirectedOn (fun x x_1 => x ≥ x_1) ↑F
-- Order.PFilter.mem_of_le
theorem mem_of_le {F : PFilter P} : x ≤ y → x ∈ F → y ∈ F 
Arguments: {P : Type u_1} [Preorder P] {x y : P} {F : Order.PFilter P}
Goal: x ≤ y → x ∈ F → y ∈ F
-- Order.PFilter.ext
@[ext]
theorem ext (h : (s : Set P) = t) : s = t 
Arguments: {P : Type u_1} [Preorder P] (s t : Order.PFilter P) (h : ↑s = ↑t)
Goal: s = t
-- Order.PFilter.mem_of_mem_of_le
@[trans]
theorem mem_of_mem_of_le {F G : PFilter P} (hx : x ∈ F) (hle : F ≤ G) : x ∈ G 
Arguments: {P : Type u_1} [Preorder P] {x : P} {F G : Order.PFilter P} (hx : x ∈ F) (hle : F ≤ G)
Goal: x ∈ G
-- Order.PFilter.mem_mk
@[simp]
theorem mem_mk (x : P) (I : Ideal Pᵒᵈ) : x ∈ (⟨I⟩ : PFilter P) ↔ toDual x ∈ I 
Arguments: {P : Type u_1} [Preorder P] (x : P) (I : Order.Ideal Pᵒᵈ)
Goal: x ∈ { dual := I } ↔ OrderDual.toDual x ∈ I
-- Order.PFilter.principal_le_iff
@[simp]
theorem principal_le_iff {F : PFilter P} : principal x ≤ F ↔ x ∈ F 
Arguments: {P : Type u_1} [Preorder P] {x : P} {F : Order.PFilter P}
Goal: Order.PFilter.principal x ≤ F ↔ x ∈ F
-- Order.PFilter.mem_principal
@[simp] theorem mem_principal : x ∈ principal y ↔ y ≤ x 
Arguments: {P : Type u_1} [Preorder P] {x y : P}
Goal: x ∈ Order.PFilter.principal y ↔ y ≤ x
-- Order.PFilter.principal_le_principal_iff
theorem principal_le_principal_iff {p q : P} : principal q ≤ principal p ↔ p ≤ q 
Arguments: {P : Type u_1} [Preorder P] {p q : P}
Goal: Order.PFilter.principal q ≤ Order.PFilter.principal p ↔ p ≤ q
-- Order.PFilter.antitone_principal
theorem antitone_principal : Antitone (principal : P → PFilter P) 
Arguments: {P : Type u_1} [Preorder P]
Goal: Antitone Order.PFilter.principal
-- Order.PFilter.top_mem
@[simp] theorem top_mem : ⊤ ∈ F 
Arguments: {P : Type u_1} [Preorder P] [OrderTop P] {F : Order.PFilter P}
Goal: ⊤ ∈ F
-- Order.PFilter.inf_mem
theorem inf_mem (hx : x ∈ F) (hy : y ∈ F) : x ⊓ y ∈ F 
Arguments: {P : Type u_1} [SemilatticeInf P] {x y : P} {F : Order.PFilter P} (hx : x ∈ F) (hy : y ∈ F)
Goal: x ⊓ y ∈ F