Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M	input stringlengths 73 2.09M
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	simp only [Set.mem_setOf]	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i ⊢ x ∈ {x \| ∃ x_1 ∈ n125Set X, List.prod x_1 = x}	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i ⊢ ∃ x_1 ∈ n125Set X, List.prod x_1 = x	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i ⊢ x ∈ {x \| ∃ x_1 ∈ n125Set X, List.prod x_1 = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	specialize hx (n125 X) _	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i ⊢ ∃ x_1 ∈ n125Set X, List.prod x_1 = x	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i ⊢ n125 X ∈ {K \| X ⊆ ↑K} case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : x ∈ ↑(n125 X) ⊢ ∃ x_1 ∈ n125Set X, List.prod x_1 = x	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i ⊢ ∃ x_1 ∈ n125Set X, List.prod x_1 = x TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	simpa [n125] using hx	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : x ∈ ↑(n125 X) ⊢ ∃ x_1 ∈ n125Set X, List.prod x_1 = x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : x ∈ ↑(n125 X) ⊢ ∃ x_1 ∈ n125Set X, List.prod x_1 = x TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	intro y hy	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i ⊢ n125 X ∈ {K \| X ⊆ ↑K}	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i y : G hy : y ∈ X ⊢ y ∈ ↑(n125 X)	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i ⊢ n125 X ∈ {K \| X ⊆ ↑K} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	simp only [n125, Set.mem_setOf, Subgroup.coe_set_mk]	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i y : G hy : y ∈ X ⊢ y ∈ ↑(n125 X)	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i y : G hy : y ∈ X ⊢ ∃ x ∈ n125Set X, List.prod x = y	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i y : G hy : y ∈ X ⊢ y ∈ ↑(n125 X) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	refine' ⟨[y], _, List.prod_singleton⟩	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i y : G hy : y ∈ X ⊢ ∃ x ∈ n125Set X, List.prod x = y	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i y : G hy : y ∈ X ⊢ [y] ∈ n125Set X	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i y : G hy : y ∈ X ⊢ ∃ x ∈ n125Set X, List.prod x = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	simp only [hy, n125Set, forall_eq, true_or_iff, Set.mem_setOf, List.mem_singleton, Set.mem_union]	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i y : G hy : y ∈ X ⊢ [y] ∈ n125Set X	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ ↑i y : G hy : y ∈ X ⊢ [y] ∈ n125Set X TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	intro x hx	case a G : Type u_1 inst✝ : Group G X : Set G ⊢ {x \| ∃ x_1 ∈ n125Set X, List.prod x_1 = x} ⊆ ⋂ s ∈ {K \| X ⊆ ↑K}, ↑s	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : x ∈ {x \| ∃ x_1 ∈ n125Set X, List.prod x_1 = x} ⊢ x ∈ ⋂ s ∈ {K \| X ⊆ ↑K}, ↑s	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G ⊢ {x \| ∃ x_1 ∈ n125Set X, List.prod x_1 = x} ⊆ ⋂ s ∈ {K \| X ⊆ ↑K}, ↑s TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	simp only [Set.mem_setOf, n125Set] at hx	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : x ∈ {x \| ∃ x_1 ∈ n125Set X, List.prod x_1 = x} ⊢ x ∈ ⋂ s ∈ {K \| X ⊆ ↑K}, ↑s	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x ⊢ x ∈ ⋂ s ∈ {K \| X ⊆ ↑K}, ↑s	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : x ∈ {x \| ∃ x_1 ∈ n125Set X, List.prod x_1 = x} ⊢ x ∈ ⋂ s ∈ {K \| X ⊆ ↑K}, ↑s TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	simp only [Set.mem_iInter, SetLike.mem_coe]	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x ⊢ x ∈ ⋂ s ∈ {K \| X ⊆ ↑K}, ↑s	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x ⊢ ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ i	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x ⊢ x ∈ ⋂ s ∈ {K \| X ⊆ ↑K}, ↑s TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	intro i hi	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x ⊢ ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ i	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} ⊢ x ∈ i	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x ⊢ ∀ i ∈ {K \| X ⊆ ↑K}, x ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	obtain ⟨y, h1, rfl⟩ := hx	case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} ⊢ x ∈ i	case a.intro.intro G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y : List G h1 : ∀ x ∈ y, x ∈ X ∪ X⁻¹ ⊢ List.prod y ∈ i	Please generate a tactic in lean4 to solve the state. STATE: case a G : Type u_1 inst✝ : Group G X : Set G x : G hx : ∃ x_1, (∀ x ∈ x_1, x ∈ X ∪ X⁻¹) ∧ List.prod x_1 = x i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} ⊢ x ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	simp only [List.prod_nil, one_mem]	case a.intro.intro.nil G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} h1 : ∀ x ∈ [], x ∈ X ∪ X⁻¹ ⊢ List.prod [] ∈ i	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.intro.intro.nil G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} h1 : ∀ x ∈ [], x ∈ X ∪ X⁻¹ ⊢ List.prod [] ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	simp only [List.prod_cons]	case a.intro.intro.cons G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ List.prod (y_hd :: tail) ∈ i	case a.intro.intro.cons G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ y_hd * List.prod tail ∈ i	Please generate a tactic in lean4 to solve the state. STATE: case a.intro.intro.cons G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ List.prod (y_hd :: tail) ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	apply mul_mem	case a.intro.intro.cons G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ y_hd * List.prod tail ∈ i	case a.intro.intro.cons.a G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ y_hd ∈ i case a.intro.intro.cons.a G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ List.prod tail ∈ i	Please generate a tactic in lean4 to solve the state. STATE: case a.intro.intro.cons G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ y_hd * List.prod tail ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	specialize h1 y_hd (List.mem_cons_self _ _)	case a.intro.intro.cons.a G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ y_hd ∈ i	case a.intro.intro.cons.a G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : y_hd ∈ X ∪ X⁻¹ ⊢ y_hd ∈ i	Please generate a tactic in lean4 to solve the state. STATE: case a.intro.intro.cons.a G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ y_hd ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	obtain h1 \| h1 := (Set.mem_union _ _ _).mp h1	case a.intro.intro.cons.a G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : y_hd ∈ X ∪ X⁻¹ ⊢ y_hd ∈ i	case a.intro.intro.cons.a.inl G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1✝ : y_hd ∈ X ∪ X⁻¹ h1 : y_hd ∈ X ⊢ y_hd ∈ i case a.intro.intro.cons.a.inr G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1✝ : y_hd ∈ X ∪ X⁻¹ h1 : y_hd ∈ X⁻¹ ⊢ y_hd ∈ i	Please generate a tactic in lean4 to solve the state. STATE: case a.intro.intro.cons.a G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : y_hd ∈ X ∪ X⁻¹ ⊢ y_hd ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	exact hi h1	case a.intro.intro.cons.a.inl G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1✝ : y_hd ∈ X ∪ X⁻¹ h1 : y_hd ∈ X ⊢ y_hd ∈ i	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.intro.intro.cons.a.inl G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1✝ : y_hd ∈ X ∪ X⁻¹ h1 : y_hd ∈ X ⊢ y_hd ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	rw [← inv_inv y_hd]	case a.intro.intro.cons.a.inr G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1✝ : y_hd ∈ X ∪ X⁻¹ h1 : y_hd ∈ X⁻¹ ⊢ y_hd ∈ i	case a.intro.intro.cons.a.inr G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1✝ : y_hd ∈ X ∪ X⁻¹ h1 : y_hd ∈ X⁻¹ ⊢ y_hd⁻¹⁻¹ ∈ i	Please generate a tactic in lean4 to solve the state. STATE: case a.intro.intro.cons.a.inr G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1✝ : y_hd ∈ X ∪ X⁻¹ h1 : y_hd ∈ X⁻¹ ⊢ y_hd ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	exact Subgroup.inv_mem _ (hi h1)	case a.intro.intro.cons.a.inr G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1✝ : y_hd ∈ X ∪ X⁻¹ h1 : y_hd ∈ X⁻¹ ⊢ y_hd⁻¹⁻¹ ∈ i	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.intro.intro.cons.a.inr G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1✝ : y_hd ∈ X ∪ X⁻¹ h1 : y_hd ∈ X⁻¹ ⊢ y_hd⁻¹⁻¹ ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n1_2_5'	[142, 1]	[171, 64]	exact y_ih fun y hy => h1 _ (List.mem_cons_of_mem _ hy)	case a.intro.intro.cons.a G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ List.prod tail ∈ i	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a.intro.intro.cons.a G : Type u_1 inst✝ : Group G X : Set G i : Subgroup G hi : i ∈ {K \| X ⊆ ↑K} y_hd : G tail : List G y_ih : (∀ x ∈ tail, x ∈ X ∪ X⁻¹) → List.prod tail ∈ i h1 : ∀ x ∈ y_hd :: tail, x ∈ X ∪ X⁻¹ ⊢ List.prod tail ∈ i TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	d1_2_10'	[189, 1]	[195, 83]	have : Fintype H := by classical infer_instance	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G ⊢ Nat.card (G ⧸ H) = Nat.card G / Nat.card ↥H	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this : Fintype ↥H ⊢ Nat.card (G ⧸ H) = Nat.card G / Nat.card ↥H	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G ⊢ Nat.card (G ⧸ H) = Nat.card G / Nat.card ↥H TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	d1_2_10'	[189, 1]	[195, 83]	have : 0 < Nat.card H := by rw [Nat.card_eq_fintype_card, Fintype.card_pos_iff] exact ⟨⟨1, one_mem _⟩⟩	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this : Fintype ↥H ⊢ Nat.card (G ⧸ H) = Nat.card G / Nat.card ↥H	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this✝ : Fintype ↥H this : 0 < Nat.card ↥H ⊢ Nat.card (G ⧸ H) = Nat.card G / Nat.card ↥H	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this : Fintype ↥H ⊢ Nat.card (G ⧸ H) = Nat.card G / Nat.card ↥H TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	d1_2_10'	[189, 1]	[195, 83]	rw [← Subgroup.card_mul_index H, Nat.mul_div_cancel_left _ this, Subgroup.index]	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this✝ : Fintype ↥H this : 0 < Nat.card ↥H ⊢ Nat.card (G ⧸ H) = Nat.card G / Nat.card ↥H	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this✝ : Fintype ↥H this : 0 < Nat.card ↥H ⊢ Nat.card (G ⧸ H) = Nat.card G / Nat.card ↥H TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	d1_2_10'	[189, 1]	[195, 83]	classical infer_instance	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G ⊢ Fintype ↥H	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G ⊢ Fintype ↥H TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	d1_2_10'	[189, 1]	[195, 83]	infer_instance	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G ⊢ Fintype ↥H	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G ⊢ Fintype ↥H TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	d1_2_10'	[189, 1]	[195, 83]	rw [Nat.card_eq_fintype_card, Fintype.card_pos_iff]	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this : Fintype ↥H ⊢ 0 < Nat.card ↥H	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this : Fintype ↥H ⊢ Nonempty ↥H	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this : Fintype ↥H ⊢ 0 < Nat.card ↥H TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	d1_2_10'	[189, 1]	[195, 83]	exact ⟨⟨1, one_mem _⟩⟩	G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this : Fintype ↥H ⊢ Nonempty ↥H	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G H : Subgroup G this : Fintype ↥H ⊢ Nonempty ↥H TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	xmp_1_2_11	[197, 1]	[199, 41]	simp [ex1232, AddSubgroup.index, Nat.card_congr (Int.quotientZMultiplesNatEquivZMod n).toEquiv, Nat.card_eq_fintype_card, ZMod.card]	n : ℕ+ ⊢ AddSubgroup.index (ex1232 ↑↑n) = ↑n	no goals	Please generate a tactic in lean4 to solve the state. STATE: n : ℕ+ ⊢ AddSubgroup.index (ex1232 ↑↑n) = ↑n TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n_1_2_13	[201, 1]	[202, 102]	rintro rfl	case mpr G : Type u_1 inst✝ : Group G A B : Set G g : G ⊢ {x \| ∃ a ∈ A, g⁻¹ * a = x} = B → A = {x \| ∃ b ∈ B, g * b = x}	case mpr G : Type u_1 inst✝ : Group G A : Set G g : G ⊢ A = {x \| ∃ b ∈ {x \| ∃ a ∈ A, g⁻¹ * a = x}, g * b = x}	Please generate a tactic in lean4 to solve the state. STATE: case mpr G : Type u_1 inst✝ : Group G A B : Set G g : G ⊢ {x \| ∃ a ∈ A, g⁻¹ * a = x} = B → A = {x \| ∃ b ∈ B, g * b = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n_1_2_13	[201, 1]	[202, 102]	ext	case mpr G : Type u_1 inst✝ : Group G A : Set G g : G ⊢ A = {x \| ∃ b ∈ {x \| ∃ a ∈ A, g⁻¹ * a = x}, g * b = x}	case mpr.h G : Type u_1 inst✝ : Group G A : Set G g x✝ : G ⊢ x✝ ∈ A ↔ x✝ ∈ {x \| ∃ b ∈ {x \| ∃ a ∈ A, g⁻¹ * a = x}, g * b = x}	Please generate a tactic in lean4 to solve the state. STATE: case mpr G : Type u_1 inst✝ : Group G A : Set G g : G ⊢ A = {x \| ∃ b ∈ {x \| ∃ a ∈ A, g⁻¹ * a = x}, g * b = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	n_1_2_13	[201, 1]	[202, 102]	simp	case mpr.h G : Type u_1 inst✝ : Group G A : Set G g x✝ : G ⊢ x✝ ∈ A ↔ x✝ ∈ {x \| ∃ b ∈ {x \| ∃ a ∈ A, g⁻¹ * a = x}, g * b = x}	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mpr.h G : Type u_1 inst✝ : Group G A : Set G g x✝ : G ⊢ x✝ ∈ A ↔ x✝ ∈ {x \| ∃ b ∈ {x \| ∃ a ∈ A, g⁻¹ * a = x}, g * b = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_4	[207, 1]	[222, 66]	simp only [ex124', Set.mem_prod, exists_prop, Prod.exists]	G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑{x \| ∃ x_1 ∈ A ×ˢ B, ex124' x_1 = x} ≤ #↑A * #↑B	G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑{x \| ∃ a b, (a ∈ A ∧ b ∈ B) ∧ a * b = x} ≤ #↑A * #↑B	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑{x \| ∃ x_1 ∈ A ×ˢ B, ex124' x_1 = x} ≤ #↑A * #↑B TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_4	[207, 1]	[222, 66]	calc (#{x \| ∃ a b : G, (a ∈ A ∧ b ∈ B) ∧ a * b = x}) = (#(⋃ (a ∈ A) (b ∈ B), ({(a : G) * (b : G)} : Set G))) := ?_ _ = (#(⋃ (a : A) (b : B), ({(a : G) * (b : G)} : Set G))) := ?_ _ ≤ Cardinal.sum fun a : A => Cardinal.sum fun b : B => #({(a : G) * (b : G)} : Set G) := ?_ _ = (#A) * (#B) := ?_	G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑{x \| ∃ a b, (a ∈ A ∧ b ∈ B) ∧ a * b = x} ≤ #↑A * #↑B	case calc_1 G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑{x \| ∃ a b, (a ∈ A ∧ b ∈ B) ∧ a * b = x} = #↑(⋃ a ∈ A, ⋃ b ∈ B, {a * b}) case calc_2 G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑(⋃ a ∈ A, ⋃ b ∈ B, {a * b}) = #↑(⋃ a, ⋃ b, {↑a * ↑b}) case calc_3 G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑(⋃ a, ⋃ b, {↑a * ↑b}) ≤ Cardinal.sum fun a => Cardinal.sum fun b => #↑{↑a * ↑b} case calc_4 G : Type u_1 inst✝ : Group G A B : Set G ⊢ (Cardinal.sum fun a => Cardinal.sum fun b => #↑{↑a * ↑b}) = #↑A * #↑B	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑{x \| ∃ a b, (a ∈ A ∧ b ∈ B) ∧ a * b = x} ≤ #↑A * #↑B TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_4	[207, 1]	[222, 66]	congr 2 with x	case calc_1 G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑{x \| ∃ a b, (a ∈ A ∧ b ∈ B) ∧ a * b = x} = #↑(⋃ a ∈ A, ⋃ b ∈ B, {a * b})	case calc_1.e_a.e_s.h G : Type u_1 inst✝ : Group G A B : Set G x : G ⊢ x ∈ {x \| ∃ a b, (a ∈ A ∧ b ∈ B) ∧ a * b = x} ↔ x ∈ ⋃ a ∈ A, ⋃ b ∈ B, {a * b}	Please generate a tactic in lean4 to solve the state. STATE: case calc_1 G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑{x \| ∃ a b, (a ∈ A ∧ b ∈ B) ∧ a * b = x} = #↑(⋃ a ∈ A, ⋃ b ∈ B, {a * b}) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_4	[207, 1]	[222, 66]	simp [Set.setOf_exists, Set.setOf_and, and_assoc]	case calc_1.e_a.e_s.h G : Type u_1 inst✝ : Group G A B : Set G x : G ⊢ x ∈ {x \| ∃ a b, (a ∈ A ∧ b ∈ B) ∧ a * b = x} ↔ x ∈ ⋃ a ∈ A, ⋃ b ∈ B, {a * b}	no goals	Please generate a tactic in lean4 to solve the state. STATE: case calc_1.e_a.e_s.h G : Type u_1 inst✝ : Group G A B : Set G x : G ⊢ x ∈ {x \| ∃ a b, (a ∈ A ∧ b ∈ B) ∧ a * b = x} ↔ x ∈ ⋃ a ∈ A, ⋃ b ∈ B, {a * b} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_4	[207, 1]	[222, 66]	simp only [Set.iUnion_coe_set]	case calc_2 G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑(⋃ a ∈ A, ⋃ b ∈ B, {a * b}) = #↑(⋃ a, ⋃ b, {↑a * ↑b})	no goals	Please generate a tactic in lean4 to solve the state. STATE: case calc_2 G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑(⋃ a ∈ A, ⋃ b ∈ B, {a * b}) = #↑(⋃ a, ⋃ b, {↑a * ↑b}) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_4	[207, 1]	[222, 66]	exact Cardinal.mk_iUnion_le_sum_mk.trans (Cardinal.sum_le_sum _ _ fun a => Cardinal.mk_iUnion_le_sum_mk)	case calc_3 G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑(⋃ a, ⋃ b, {↑a * ↑b}) ≤ Cardinal.sum fun a => Cardinal.sum fun b => #↑{↑a * ↑b}	no goals	Please generate a tactic in lean4 to solve the state. STATE: case calc_3 G : Type u_1 inst✝ : Group G A B : Set G ⊢ #↑(⋃ a, ⋃ b, {↑a * ↑b}) ≤ Cardinal.sum fun a => Cardinal.sum fun b => #↑{↑a * ↑b} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_4	[207, 1]	[222, 66]	simp_rw [Cardinal.mk_singleton, Cardinal.sum_const', mul_one]	case calc_4 G : Type u_1 inst✝ : Group G A B : Set G ⊢ (Cardinal.sum fun a => Cardinal.sum fun b => #↑{↑a * ↑b}) = #↑A * #↑B	no goals	Please generate a tactic in lean4 to solve the state. STATE: case calc_4 G : Type u_1 inst✝ : Group G A B : Set G ⊢ (Cardinal.sum fun a => Cardinal.sum fun b => #↑{↑a * ↑b}) = #↑A * #↑B TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_5	[224, 1]	[232, 28]	ext y	G : Type u_1 inst✝ : Group G A B : Set G g : G ⊢ {x \| ∃ x_1 ∈ A ∩ B, g * x_1 = x} = {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x}	case h G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ y ∈ {x \| ∃ x_1 ∈ A ∩ B, g * x_1 = x} ↔ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G A B : Set G g : G ⊢ {x \| ∃ x_1 ∈ A ∩ B, g * x_1 = x} = {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_5	[224, 1]	[232, 28]	simp only [Set.mem_inter, Set.mem_setOf_eq, exists_prop]	case h G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ y ∈ {x \| ∃ x_1 ∈ A ∩ B, g * x_1 = x} ↔ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x}	case h G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ (∃ x ∈ A ∩ B, g * x = y) ↔ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x}	Please generate a tactic in lean4 to solve the state. STATE: case h G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ y ∈ {x \| ∃ x_1 ∈ A ∩ B, g * x_1 = x} ↔ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_5	[224, 1]	[232, 28]	constructor	case h G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ (∃ x ∈ A ∩ B, g * x = y) ↔ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x}	case h.mp G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ (∃ x ∈ A ∩ B, g * x = y) → y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x} case h.mpr G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x} → ∃ x ∈ A ∩ B, g * x = y	Please generate a tactic in lean4 to solve the state. STATE: case h G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ (∃ x ∈ A ∩ B, g * x = y) ↔ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_5	[224, 1]	[232, 28]	rintro ⟨x, ⟨ha, hb⟩, hx⟩	case h.mp G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ (∃ x ∈ A ∩ B, g * x = y) → y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x}	case h.mp.intro.intro.intro G : Type u_1 inst✝ : Group G A B : Set G g y x : G hx : g * x = y ha : x ∈ A hb : x ∈ B ⊢ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x}	Please generate a tactic in lean4 to solve the state. STATE: case h.mp G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ (∃ x ∈ A ∩ B, g * x = y) → y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_5	[224, 1]	[232, 28]	exact ⟨⟨x, ha, hx⟩, ⟨x, hb, hx⟩⟩	case h.mp.intro.intro.intro G : Type u_1 inst✝ : Group G A B : Set G g y x : G hx : g * x = y ha : x ∈ A hb : x ∈ B ⊢ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x}	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.mp.intro.intro.intro G : Type u_1 inst✝ : Group G A B : Set G g y x : G hx : g * x = y ha : x ∈ A hb : x ∈ B ⊢ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_5	[224, 1]	[232, 28]	rintro ⟨⟨x, ha, hx⟩, ⟨x', hb, hx'⟩⟩	case h.mpr G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x} → ∃ x ∈ A ∩ B, g * x = y	case h.mpr.intro.intro.intro.intro.intro G : Type u_1 inst✝ : Group G A B : Set G g y x : G ha : x ∈ A hx : g * x = y x' : G hb : x' ∈ B hx' : g * x' = y ⊢ ∃ x ∈ A ∩ B, g * x = y	Please generate a tactic in lean4 to solve the state. STATE: case h.mpr G : Type u_1 inst✝ : Group G A B : Set G g y : G ⊢ y ∈ {x \| ∃ x_1 ∈ A, g * x_1 = x} ∩ {x \| ∃ x_1 ∈ B, g * x_1 = x} → ∃ x ∈ A ∩ B, g * x = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_5	[224, 1]	[232, 28]	obtain rfl : x = x' := by rw [← mul_right_inj g, hx, hx']	case h.mpr.intro.intro.intro.intro.intro G : Type u_1 inst✝ : Group G A B : Set G g y x : G ha : x ∈ A hx : g * x = y x' : G hb : x' ∈ B hx' : g * x' = y ⊢ ∃ x ∈ A ∩ B, g * x = y	case h.mpr.intro.intro.intro.intro.intro G : Type u_1 inst✝ : Group G A B : Set G g y x : G ha : x ∈ A hx : g * x = y hb : x ∈ B hx' : g * x = y ⊢ ∃ x ∈ A ∩ B, g * x = y	Please generate a tactic in lean4 to solve the state. STATE: case h.mpr.intro.intro.intro.intro.intro G : Type u_1 inst✝ : Group G A B : Set G g y x : G ha : x ∈ A hx : g * x = y x' : G hb : x' ∈ B hx' : g * x' = y ⊢ ∃ x ∈ A ∩ B, g * x = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_5	[224, 1]	[232, 28]	exact ⟨x, ⟨ha, hb⟩, hx⟩	case h.mpr.intro.intro.intro.intro.intro G : Type u_1 inst✝ : Group G A B : Set G g y x : G ha : x ∈ A hx : g * x = y hb : x ∈ B hx' : g * x = y ⊢ ∃ x ∈ A ∩ B, g * x = y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.mpr.intro.intro.intro.intro.intro G : Type u_1 inst✝ : Group G A B : Set G g y x : G ha : x ∈ A hx : g * x = y hb : x ∈ B hx' : g * x = y ⊢ ∃ x ∈ A ∩ B, g * x = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	ex_1_2_5	[224, 1]	[232, 28]	rw [← mul_right_inj g, hx, hx']	G : Type u_1 inst✝ : Group G A B : Set G g y x : G ha : x ∈ A hx : g * x = y x' : G hb : x' ∈ B hx' : g * x' = y ⊢ x = x'	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G A B : Set G g y x : G ha : x ∈ A hx : g * x = y x' : G hb : x' ∈ B hx' : g * x' = y ⊢ x = x' TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_15_ii	[251, 1]	[254, 80]	ext	G : Type u_1 inst✝ : Group G g : G ⊢ ↑(Subgroup.closure {g}) = {x \| ∃ k, g ^ k = x}	case h G : Type u_1 inst✝ : Group G g x✝ : G ⊢ x✝ ∈ ↑(Subgroup.closure {g}) ↔ x✝ ∈ {x \| ∃ k, g ^ k = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G ⊢ ↑(Subgroup.closure {g}) = {x \| ∃ k, g ^ k = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_15_ii	[251, 1]	[254, 80]	simp only [SetLike.mem_coe, Set.mem_setOf_eq, Subgroup.mem_closure_singleton]	case h G : Type u_1 inst✝ : Group G g x✝ : G ⊢ x✝ ∈ ↑(Subgroup.closure {g}) ↔ x✝ ∈ {x \| ∃ k, g ^ k = x}	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h G : Type u_1 inst✝ : Group G g x✝ : G ⊢ x✝ ∈ ↑(Subgroup.closure {g}) ↔ x✝ ∈ {x \| ∃ k, g ^ k = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_17	[256, 1]	[262, 40]	rw [Subgroup.mem_closure_singleton] at ha hb	G : Type u_1 inst✝ : Group G g a b : G ha : a ∈ Subgroup.closure {g} hb : b ∈ Subgroup.closure {g} ⊢ a * b = b * a	G : Type u_1 inst✝ : Group G g a b : G ha : ∃ n, g ^ n = a hb : ∃ n, g ^ n = b ⊢ a * b = b * a	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g a b : G ha : a ∈ Subgroup.closure {g} hb : b ∈ Subgroup.closure {g} ⊢ a * b = b * a TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_17	[256, 1]	[262, 40]	obtain ⟨n, rfl⟩ := ha	G : Type u_1 inst✝ : Group G g a b : G ha : ∃ n, g ^ n = a hb : ∃ n, g ^ n = b ⊢ a * b = b * a	case intro G : Type u_1 inst✝ : Group G g b : G hb : ∃ n, g ^ n = b n : ℤ ⊢ g ^ n * b = b * g ^ n	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g a b : G ha : ∃ n, g ^ n = a hb : ∃ n, g ^ n = b ⊢ a * b = b * a TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_17	[256, 1]	[262, 40]	obtain ⟨m, rfl⟩ := hb	case intro G : Type u_1 inst✝ : Group G g b : G hb : ∃ n, g ^ n = b n : ℤ ⊢ g ^ n * b = b * g ^ n	case intro.intro G : Type u_1 inst✝ : Group G g : G n m : ℤ ⊢ g ^ n * g ^ m = g ^ m * g ^ n	Please generate a tactic in lean4 to solve the state. STATE: case intro G : Type u_1 inst✝ : Group G g b : G hb : ∃ n, g ^ n = b n : ℤ ⊢ g ^ n * b = b * g ^ n TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_17	[256, 1]	[262, 40]	rw [← zpow_add, ← zpow_add, add_comm]	case intro.intro G : Type u_1 inst✝ : Group G g : G n m : ℤ ⊢ g ^ n * g ^ m = g ^ m * g ^ n	no goals	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro G : Type u_1 inst✝ : Group G g : G n m : ℤ ⊢ g ^ n * g ^ m = g ^ m * g ^ n TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	simp only [← zpowers_eq_closure, exists_prop]	G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x ⊢ y ∈ closure {x} ↔ ∃ n < orderOf x, x ^ n = y	G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x ⊢ y ∈ zpowers x ↔ ∃ n < orderOf x, x ^ n = y	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x ⊢ y ∈ closure {x} ↔ ∃ n < orderOf x, x ^ n = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	constructor <;> rintro ⟨i, hi⟩	G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x ⊢ y ∈ zpowers x ↔ ∃ n < orderOf x, x ^ n = y	case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ⊢ ∃ n < orderOf x, x ^ n = y case mpr.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℕ hi : i < orderOf x ∧ x ^ i = y ⊢ y ∈ zpowers x	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x ⊢ y ∈ zpowers x ↔ ∃ n < orderOf x, x ^ n = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	have ho : 0 < (orderOf x : ℤ) := by rwa [Int.coe_nat_pos, pos_iff_ne_zero, Ne.def, orderOf_eq_zero_iff, Classical.not_not]	case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ⊢ ∃ n, 0 ≤ n ∧ n < ↑(orderOf x) ∧ x ^ n = y	case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ho : 0 < ↑(orderOf x) ⊢ ∃ n, 0 ≤ n ∧ n < ↑(orderOf x) ∧ x ^ n = y	Please generate a tactic in lean4 to solve the state. STATE: case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ⊢ ∃ n, 0 ≤ n ∧ n < ↑(orderOf x) ∧ x ^ n = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	refine' ⟨i % orderOf x, Int.emod_nonneg _ ho.ne', Int.emod_lt_of_pos _ ho, _⟩	case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ho : 0 < ↑(orderOf x) ⊢ ∃ n, 0 ≤ n ∧ n < ↑(orderOf x) ∧ x ^ n = y	case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ho : 0 < ↑(orderOf x) ⊢ x ^ (i % ↑(orderOf x)) = y	Please generate a tactic in lean4 to solve the state. STATE: case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ho : 0 < ↑(orderOf x) ⊢ ∃ n, 0 ≤ n ∧ n < ↑(orderOf x) ∧ x ^ n = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	rw [← hi, l1_2_20_i _ _ _ _ h]	case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ho : 0 < ↑(orderOf x) ⊢ x ^ (i % ↑(orderOf x)) = y	case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ho : 0 < ↑(orderOf x) ⊢ i % ↑(orderOf x) ≡ i [ZMOD ↑(orderOf x)]	Please generate a tactic in lean4 to solve the state. STATE: case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ho : 0 < ↑(orderOf x) ⊢ x ^ (i % ↑(orderOf x)) = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	exact Int.mod_modEq _ _	case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ho : 0 < ↑(orderOf x) ⊢ i % ↑(orderOf x) ≡ i [ZMOD ↑(orderOf x)]	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mp.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ho : 0 < ↑(orderOf x) ⊢ i % ↑(orderOf x) ≡ i [ZMOD ↑(orderOf x)] TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	obtain ⟨n, hnonneg, hlt, heq⟩ := this	G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y this : ∃ n, 0 ≤ n ∧ n < ↑(orderOf x) ∧ x ^ n = y ⊢ ∃ n < orderOf x, x ^ n = y	case intro.intro.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y n : ℤ hnonneg : 0 ≤ n hlt : n < ↑(orderOf x) heq : x ^ n = y ⊢ ∃ n < orderOf x, x ^ n = y	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y this : ∃ n, 0 ≤ n ∧ n < ↑(orderOf x) ∧ x ^ n = y ⊢ ∃ n < orderOf x, x ^ n = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	refine' ⟨n.natAbs, _, _⟩	case intro.intro.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y n : ℤ hnonneg : 0 ≤ n hlt : n < ↑(orderOf x) heq : x ^ n = y ⊢ ∃ n < orderOf x, x ^ n = y	case intro.intro.intro.refine'_1 G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y n : ℤ hnonneg : 0 ≤ n hlt : n < ↑(orderOf x) heq : x ^ n = y ⊢ Int.natAbs n < orderOf x case intro.intro.intro.refine'_2 G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y n : ℤ hnonneg : 0 ≤ n hlt : n < ↑(orderOf x) heq : x ^ n = y ⊢ x ^ Int.natAbs n = y	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y n : ℤ hnonneg : 0 ≤ n hlt : n < ↑(orderOf x) heq : x ^ n = y ⊢ ∃ n < orderOf x, x ^ n = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	rwa [← Int.ofNat_lt, Int.natAbs_of_nonneg hnonneg]	case intro.intro.intro.refine'_1 G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y n : ℤ hnonneg : 0 ≤ n hlt : n < ↑(orderOf x) heq : x ^ n = y ⊢ Int.natAbs n < orderOf x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.refine'_1 G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y n : ℤ hnonneg : 0 ≤ n hlt : n < ↑(orderOf x) heq : x ^ n = y ⊢ Int.natAbs n < orderOf x TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	rwa [← zpow_ofNat, Int.natAbs_of_nonneg hnonneg]	case intro.intro.intro.refine'_2 G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y n : ℤ hnonneg : 0 ≤ n hlt : n < ↑(orderOf x) heq : x ^ n = y ⊢ x ^ Int.natAbs n = y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.refine'_2 G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y n : ℤ hnonneg : 0 ≤ n hlt : n < ↑(orderOf x) heq : x ^ n = y ⊢ x ^ Int.natAbs n = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	rwa [Int.coe_nat_pos, pos_iff_ne_zero, Ne.def, orderOf_eq_zero_iff, Classical.not_not]	G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ⊢ 0 < ↑(orderOf x)	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℤ hi : (fun x_1 => x ^ x_1) i = y ⊢ 0 < ↑(orderOf x) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	use i	case mpr.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℕ hi : i < orderOf x ∧ x ^ i = y ⊢ y ∈ zpowers x	case h G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℕ hi : i < orderOf x ∧ x ^ i = y ⊢ (fun x_1 => x ^ x_1) ↑i = y	Please generate a tactic in lean4 to solve the state. STATE: case mpr.intro G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℕ hi : i < orderOf x ∧ x ^ i = y ⊢ y ∈ zpowers x TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	Subgroup.mem_closure_singleton'	[273, 1]	[289, 33]	simp only [zpow_ofNat, hi.2]	case h G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℕ hi : i < orderOf x ∧ x ^ i = y ⊢ (fun x_1 => x ^ x_1) ↑i = y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h G : Type u_1 inst✝ : Group G x y : G h : IsOfFinOrder x i : ℕ hi : i < orderOf x ∧ x ^ i = y ⊢ (fun x_1 => x ^ x_1) ↑i = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_i'	[293, 1]	[294, 139]	ext	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g ⊢ ↑(Subgroup.closure {g}) = {x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x}	case h G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g x✝ : G ⊢ x✝ ∈ ↑(Subgroup.closure {g}) ↔ x✝ ∈ {x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g ⊢ ↑(Subgroup.closure {g}) = {x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_i'	[293, 1]	[294, 139]	simp [Subgroup.mem_closure_singleton' h]	case h G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g x✝ : G ⊢ x✝ ∈ ↑(Subgroup.closure {g}) ↔ x✝ ∈ {x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x}	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g x✝ : G ⊢ x✝ ∈ ↑(Subgroup.closure {g}) ↔ x✝ ∈ {x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_ii	[296, 1]	[299, 69]	rw [← orderOf_eq_zero_iff] at h	G : Type u_1 inst✝ : Group G g : G i j : ℤ h : ¬IsOfFinOrder g ⊢ g ^ i = g ^ j ↔ i = j	G : Type u_1 inst✝ : Group G g : G i j : ℤ h : orderOf g = 0 ⊢ g ^ i = g ^ j ↔ i = j	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G i j : ℤ h : ¬IsOfFinOrder g ⊢ g ^ i = g ^ j ↔ i = j TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_ii	[296, 1]	[299, 69]	rw [zpow_eq_zpow_iff_modEq, h, Int.ofNat_zero, Int.modEq_zero_iff]	G : Type u_1 inst✝ : Group G g : G i j : ℤ h : orderOf g = 0 ⊢ g ^ i = g ^ j ↔ i = j	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G i j : ℤ h : orderOf g = 0 ⊢ g ^ i = g ^ j ↔ i = j TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_ii'	[301, 1]	[304, 39]	rw [← Subgroup.zpowers_eq_closure]	G : Type u_1 inst✝ : Group G g : G _h : ¬IsOfFinOrder g ⊢ ↑(Subgroup.closure {g}) = {x \| ∃ i, g ^ i = x}	G : Type u_1 inst✝ : Group G g : G _h : ¬IsOfFinOrder g ⊢ ↑(Subgroup.zpowers g) = {x \| ∃ i, g ^ i = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G _h : ¬IsOfFinOrder g ⊢ ↑(Subgroup.closure {g}) = {x \| ∃ i, g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_ii'	[301, 1]	[304, 39]	ext	G : Type u_1 inst✝ : Group G g : G _h : ¬IsOfFinOrder g ⊢ ↑(Subgroup.zpowers g) = {x \| ∃ i, g ^ i = x}	case h G : Type u_1 inst✝ : Group G g : G _h : ¬IsOfFinOrder g x✝ : G ⊢ x✝ ∈ ↑(Subgroup.zpowers g) ↔ x✝ ∈ {x \| ∃ i, g ^ i = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G _h : ¬IsOfFinOrder g ⊢ ↑(Subgroup.zpowers g) = {x \| ∃ i, g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_ii'	[301, 1]	[304, 39]	simp [Subgroup.mem_zpowers_iff]	case h G : Type u_1 inst✝ : Group G g : G _h : ¬IsOfFinOrder g x✝ : G ⊢ x✝ ∈ ↑(Subgroup.zpowers g) ↔ x✝ ∈ {x \| ∃ i, g ^ i = x}	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h G : Type u_1 inst✝ : Group G g : G _h : ¬IsOfFinOrder g x✝ : G ⊢ x✝ ∈ ↑(Subgroup.zpowers g) ↔ x✝ ∈ {x \| ∃ i, g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	have := (IsOfFinOrder.orderOf_pos h).ne'	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g ⊢ #↑↑(Subgroup.closure {g}) = ↑(orderOf g)	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑↑(Subgroup.closure {g}) = ↑(orderOf g)	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g ⊢ #↑↑(Subgroup.closure {g}) = ↑(orderOf g) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	rw [l1_2_20_i' _ _ h]	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑↑(Subgroup.closure {g}) = ↑(orderOf g)	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑{x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x} = ↑(orderOf g)	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑↑(Subgroup.closure {g}) = ↑(orderOf g) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	trans #((Set.Ico 0 (orderOf g)).image fun i => g ^ i)	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑{x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x} = ↑(orderOf g)	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑{x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x} = #↑((fun i => g ^ i) '' Set.Ico 0 (orderOf g)) G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑((fun i => g ^ i) '' Set.Ico 0 (orderOf g)) = ↑(orderOf g)	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑{x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x} = ↑(orderOf g) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	simp only [Cardinal.mk_fintype, Nat.cast_inj, Fintype.card_ofFinset]	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑((fun i => g ^ i) '' Set.Ico 0 (orderOf g)) = ↑(orderOf g)	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ (Finset.image (fun i => g ^ i) (Set.toFinset (Set.Ico 0 (orderOf g)))).card = orderOf g	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑((fun i => g ^ i) '' Set.Ico 0 (orderOf g)) = ↑(orderOf g) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	rw [Finset.card_image_of_injOn]	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ (Finset.image (fun i => g ^ i) (Set.toFinset (Set.Ico 0 (orderOf g)))).card = orderOf g	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ (Set.toFinset (Set.Ico 0 (orderOf g))).card = orderOf g G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ Set.InjOn (fun i => g ^ i) ↑(Set.toFinset (Set.Ico 0 (orderOf g)))	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ (Finset.image (fun i => g ^ i) (Set.toFinset (Set.Ico 0 (orderOf g)))).card = orderOf g TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	simp only [tsub_zero, eq_self_iff_true, Set.toFinset_card, Nat.card_fintypeIco]	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ (Set.toFinset (Set.Ico 0 (orderOf g))).card = orderOf g G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ Set.InjOn (fun i => g ^ i) ↑(Set.toFinset (Set.Ico 0 (orderOf g)))	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ Set.InjOn (fun i => g ^ i) ↑(Set.toFinset (Set.Ico 0 (orderOf g)))	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ (Set.toFinset (Set.Ico 0 (orderOf g))).card = orderOf g G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ Set.InjOn (fun i => g ^ i) ↑(Set.toFinset (Set.Ico 0 (orderOf g))) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	intro x hx y hy hxy	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ Set.InjOn (fun i => g ^ i) ↑(Set.toFinset (Set.Ico 0 (orderOf g)))	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : (fun i => g ^ i) x = (fun i => g ^ i) y ⊢ x = y	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ Set.InjOn (fun i => g ^ i) ↑(Set.toFinset (Set.Ico 0 (orderOf g))) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	rwa [pow_eq_pow_iff_modEq, Nat.ModEq, Nat.mod_eq_of_lt, Nat.mod_eq_of_lt] at hxy	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : (fun i => g ^ i) x = (fun i => g ^ i) y ⊢ x = y	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : x = y % orderOf g ⊢ y < orderOf g G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : x % orderOf g = y % orderOf g ⊢ x < orderOf g	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : (fun i => g ^ i) x = (fun i => g ^ i) y ⊢ x = y TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	simpa using hy	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : x = y % orderOf g ⊢ y < orderOf g G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : x % orderOf g = y % orderOf g ⊢ x < orderOf g	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : x % orderOf g = y % orderOf g ⊢ x < orderOf g	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : x = y % orderOf g ⊢ y < orderOf g G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : x % orderOf g = y % orderOf g ⊢ x < orderOf g TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	simpa using hx	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : x % orderOf g = y % orderOf g ⊢ x < orderOf g	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 x : ℕ hx : x ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) y : ℕ hy : y ∈ ↑(Set.toFinset (Set.Ico 0 (orderOf g))) hxy : x % orderOf g = y % orderOf g ⊢ x < orderOf g TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii	[306, 1]	[319, 17]	congr	G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑{x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x} = #↑((fun i => g ^ i) '' Set.Ico 0 (orderOf g))	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : IsOfFinOrder g this : orderOf g ≠ 0 ⊢ #↑{x \| ∃ i ∈ Set.Ico 0 (orderOf g), g ^ i = x} = #↑((fun i => g ^ i) '' Set.Ico 0 (orderOf g)) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	zpow_inj_iff_of_orderOf_eq_zero	[331, 1]	[333, 29]	rw [orderOf_eq_zero_iff] at h	G : Type u A : Type v x y : G a b : A n✝ m✝ : ℕ inst✝¹ : Group G inst✝ : AddGroup A i : ℤ h : orderOf x = 0 n m : ℤ ⊢ x ^ n = x ^ m ↔ n = m	G : Type u A : Type v x y : G a b : A n✝ m✝ : ℕ inst✝¹ : Group G inst✝ : AddGroup A i : ℤ h : ¬IsOfFinOrder x n m : ℤ ⊢ x ^ n = x ^ m ↔ n = m	Please generate a tactic in lean4 to solve the state. STATE: G : Type u A : Type v x y : G a b : A n✝ m✝ : ℕ inst✝¹ : Group G inst✝ : AddGroup A i : ℤ h : orderOf x = 0 n m : ℤ ⊢ x ^ n = x ^ m ↔ n = m TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	zpow_inj_iff_of_orderOf_eq_zero	[331, 1]	[333, 29]	exact l1_2_20_ii _ _ n m h	G : Type u A : Type v x y : G a b : A n✝ m✝ : ℕ inst✝¹ : Group G inst✝ : AddGroup A i : ℤ h : ¬IsOfFinOrder x n m : ℤ ⊢ x ^ n = x ^ m ↔ n = m	no goals	Please generate a tactic in lean4 to solve the state. STATE: G : Type u A : Type v x y : G a b : A n✝ m✝ : ℕ inst✝¹ : Group G inst✝ : AddGroup A i : ℤ h : ¬IsOfFinOrder x n m : ℤ ⊢ x ^ n = x ^ m ↔ n = m TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	rw [l1_2_20_ii' _ _ h]	G : Type u_1 inst✝ : Group G g : G h : ¬IsOfFinOrder g ⊢ Set.Infinite ↑(Subgroup.closure {g})	G : Type u_1 inst✝ : Group G g : G h : ¬IsOfFinOrder g ⊢ Set.Infinite {x \| ∃ i, g ^ i = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : ¬IsOfFinOrder g ⊢ Set.Infinite ↑(Subgroup.closure {g}) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	rw [← orderOf_eq_zero_iff] at h	G : Type u_1 inst✝ : Group G g : G h : ¬IsOfFinOrder g ⊢ Set.Infinite {x \| ∃ i, g ^ i = x}	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Set.Infinite {x \| ∃ i, g ^ i = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : ¬IsOfFinOrder g ⊢ Set.Infinite {x \| ∃ i, g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	have : ((Set.univ : Set ℕ).image fun i => g ^ i).Infinite := by simp apply Set.infinite_range_of_injective intro x y hxy rwa [pow_inj_iff_of_orderOf_eq_zero h] at hxy	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Set.Infinite {x \| ∃ i, g ^ i = x}	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) ⊢ Set.Infinite {x \| ∃ i, g ^ i = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Set.Infinite {x \| ∃ i, g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	apply Set.Infinite.mono _ this	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) ⊢ Set.Infinite {x \| ∃ i, g ^ i = x}	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) ⊢ (fun i => g ^ i) '' Set.univ ⊆ {x \| ∃ i, g ^ i = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) ⊢ Set.Infinite {x \| ∃ i, g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	intro x hx	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) ⊢ (fun i => g ^ i) '' Set.univ ⊆ {x \| ∃ i, g ^ i = x}	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G hx : x ∈ (fun i => g ^ i) '' Set.univ ⊢ x ∈ {x \| ∃ i, g ^ i = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) ⊢ (fun i => g ^ i) '' Set.univ ⊆ {x \| ∃ i, g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	simp only [Set.image_univ, Set.mem_range] at hx	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G hx : x ∈ (fun i => g ^ i) '' Set.univ ⊢ x ∈ {x \| ∃ i, g ^ i = x}	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G hx : ∃ y, g ^ y = x ⊢ x ∈ {x \| ∃ i, g ^ i = x}	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G hx : x ∈ (fun i => g ^ i) '' Set.univ ⊢ x ∈ {x \| ∃ i, g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	simp only [Set.mem_setOf]	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G hx : ∃ y, g ^ y = x ⊢ x ∈ {x \| ∃ i, g ^ i = x}	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G hx : ∃ y, g ^ y = x ⊢ ∃ i, g ^ i = x	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G hx : ∃ y, g ^ y = x ⊢ x ∈ {x \| ∃ i, g ^ i = x} TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	obtain ⟨n, hn⟩ := hx	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G hx : ∃ y, g ^ y = x ⊢ ∃ i, g ^ i = x	case intro G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G n : ℕ hn : g ^ n = x ⊢ ∃ i, g ^ i = x	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G hx : ∃ y, g ^ y = x ⊢ ∃ i, g ^ i = x TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	use n	case intro G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G n : ℕ hn : g ^ n = x ⊢ ∃ i, g ^ i = x	case h G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G n : ℕ hn : g ^ n = x ⊢ g ^ ↑n = x	Please generate a tactic in lean4 to solve the state. STATE: case intro G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G n : ℕ hn : g ^ n = x ⊢ ∃ i, g ^ i = x TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	simp only [zpow_ofNat, hn]	case h G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G n : ℕ hn : g ^ n = x ⊢ g ^ ↑n = x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 this : Set.Infinite ((fun i => g ^ i) '' Set.univ) x : G n : ℕ hn : g ^ n = x ⊢ g ^ ↑n = x TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	simp	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Set.Infinite ((fun i => g ^ i) '' Set.univ)	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Set.Infinite (Set.range fun i => g ^ i)	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Set.Infinite ((fun i => g ^ i) '' Set.univ) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	apply Set.infinite_range_of_injective	G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Set.Infinite (Set.range fun i => g ^ i)	case hi G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Function.Injective fun i => g ^ i	Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Set.Infinite (Set.range fun i => g ^ i) TACTIC:
https://github.com/Ruben-VandeVelde/algebra-i.git	b4852b5538060d7d33a834a4c349d255e7e643dc	AlgebraI/Subgroups.lean	l1_2_20_iii'	[337, 1]	[352, 29]	intro x y hxy	case hi G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Function.Injective fun i => g ^ i	case hi G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 x y : ℕ hxy : (fun i => g ^ i) x = (fun i => g ^ i) y ⊢ x = y	Please generate a tactic in lean4 to solve the state. STATE: case hi G : Type u_1 inst✝ : Group G g : G h : orderOf g = 0 ⊢ Function.Injective fun i => g ^ i TACTIC:

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