import Mathlib.Data.Finset.Basic import Mathlib.GroupTheory.GroupAction.Basic import Mathlib.GroupTheory.Exponent import Rubin.Tactic namespace Rubin.Period variable {G a : Type _} variable [Group G] variable [MulAction G α] noncomputable def period (p : α) (g : G) : ℕ := sInf {n : ℕ | n > 0 ∧ g ^ n • p = p} #align period Rubin.Period.period theorem period_le_fix {p : α} {g : G} {m : ℕ} (m_pos : m > 0) (gmp_eq_p : g ^ m • p = p) : 0 < Rubin.Period.period p g ∧ Rubin.Period.period p g ≤ m := by constructor · by_contra h'; have period_zero : Rubin.Period.period p g = 0; linarith; rcases Nat.sInf_eq_zero.1 period_zero with ⟨cont, h_1⟩ | h; linarith; exact Set.eq_empty_iff_forall_not_mem.mp h ↑m ⟨m_pos, gmp_eq_p⟩ exact Nat.sInf_le ⟨m_pos, gmp_eq_p⟩ #align period_le_fix Rubin.Period.period_le_fix theorem notfix_le_period {p : α} {g : G} {n : ℕ} (n_pos : n > 0) (period_pos : Rubin.Period.period p g > 0) (pmoves : ∀ i : ℕ, 0 < i → i < n → g ^ i • p ≠ p) : n ≤ Rubin.Period.period p g := by by_contra period_le exact (pmoves (Rubin.Period.period p g) period_pos (not_le.mp period_le)) (Nat.sInf_mem (Nat.nonempty_of_pos_sInf period_pos)).2 #align notfix_le_period Rubin.Period.notfix_le_period theorem notfix_le_period' {p : α} {g : G} {n : ℕ} (n_pos : n > 0) (period_pos : Rubin.Period.period p g > 0) (pmoves : ∀ i : Fin n, 0 < (i : ℕ) → g ^ (i : ℕ) • p ≠ p) : n ≤ Rubin.Period.period p g := Rubin.Period.notfix_le_period n_pos period_pos fun (i : ℕ) (i_pos : 0 < i) (i_lt_n : i < n) => pmoves (⟨i, i_lt_n⟩ : Fin n) i_pos #align notfix_le_period' Rubin.Period.notfix_le_period' theorem period_neutral_eq_one (p : α) : Rubin.Period.period p (1 : G) = 1 := by have : 0 < Rubin.Period.period p (1 : G) ∧ Rubin.Period.period p (1 : G) ≤ 1 := Rubin.Period.period_le_fix (by norm_num : 1 > 0) (by group_action : (1 : G) ^ 1 • p = p) linarith #align period_neutral_eq_one Rubin.Period.period_neutral_eq_one def periods (U : Set α) (H : Subgroup G) : Set ℕ := {n : ℕ | ∃ (p : α) (g : H), p ∈ U ∧ Rubin.Period.period (p : α) (g : G) = n} #align periods Rubin.Period.periods -- TODO: split into multiple lemmas theorem periods_lemmas {U : Set α} (U_nonempty : Set.Nonempty U) {H : Subgroup G} (exp_ne_zero : Monoid.exponent H ≠ 0) : (Rubin.Period.periods U H).Nonempty ∧ BddAbove (Rubin.Period.periods U H) ∧ ∃ (m : ℕ) (m_pos : m > 0), ∀ (p : α) (g : H), g ^ m • p = p := by rcases Monoid.exponentExists_iff_ne_zero.2 exp_ne_zero with ⟨m, m_pos, gm_eq_one⟩ have gmp_eq_p : ∀ (p : α) (g : H), g ^ m • p = p := by intro p g; rw [gm_eq_one g]; group_action have periods_nonempty : (Rubin.Period.periods U H).Nonempty := by use 1 let p := Set.Nonempty.some U_nonempty; use p use 1 constructor · exact Set.Nonempty.some_mem U_nonempty · exact Rubin.Period.period_neutral_eq_one p have periods_bounded : BddAbove (Rubin.Period.periods U H) := by use m; intro some_period hperiod; rcases hperiod with ⟨p, g, hperiod⟩ rw [← hperiod.2] exact (Rubin.Period.period_le_fix m_pos (gmp_eq_p p g)).2 exact ⟨periods_nonempty, periods_bounded, m, m_pos, gmp_eq_p⟩ #align period_lemma Rubin.Period.periods_lemmas theorem period_from_exponent (U : Set α) (U_nonempty : U.Nonempty) {H : Subgroup G} (exp_ne_zero : Monoid.exponent H ≠ 0) : ∃ (p : α) (g : H) (n : ℕ), p ∈ U ∧ n > 0 ∧ Rubin.Period.period (p : α) (g : G) = n ∧ n = sSup (Rubin.Period.periods U H) := by rcases Rubin.Period.periods_lemmas U_nonempty exp_ne_zero with ⟨periods_nonempty, periods_bounded, m, m_pos, gmp_eq_p⟩ rcases Nat.sSup_mem periods_nonempty periods_bounded with ⟨p, g, hperiod⟩ use p use g use sSup (Rubin.Period.periods U H) -- TODO: cleanup? exact ⟨ hperiod.1, hperiod.2 ▸ (Rubin.Period.period_le_fix m_pos (gmp_eq_p p g)).1, hperiod.2, rfl ⟩ #align period_from_exponent Rubin.Period.period_from_exponent theorem zero_lt_period_le_Sup_periods {U : Set α} (U_nonempty : U.Nonempty) {H : Subgroup G} (exp_ne_zero : Monoid.exponent H ≠ 0) : ∀ (p : U) (g : H), 0 < Rubin.Period.period (p : α) (g : G) ∧ Rubin.Period.period (p : α) (g : G) ≤ sSup (Rubin.Period.periods U H) := by rcases Rubin.Period.periods_lemmas U_nonempty exp_ne_zero with ⟨periods_nonempty, periods_bounded, m, m_pos, gmp_eq_p⟩ intro p g have period_in_periods : Rubin.Period.period (p : α) (g : G) ∈ Rubin.Period.periods U H := by use p; use g simp exact ⟨(Rubin.Period.period_le_fix m_pos (gmp_eq_p p g)).1, le_csSup periods_bounded period_in_periods⟩ #align zero_lt_period_le_Sup_periods Rubin.Period.zero_lt_period_le_Sup_periods theorem pow_period_fix (p : α) (g : G) : g ^ Rubin.Period.period p g • p = p := by cases eq_zero_or_neZero (Rubin.Period.period p g) with | inl h => rw [h]; simp | inr h => exact (Nat.sInf_mem (Nat.nonempty_of_pos_sInf (Nat.pos_of_ne_zero (@NeZero.ne _ _ (Rubin.Period.period p g) h)))).2 #align pow_period_fix Rubin.Period.pow_period_fix end Rubin.Period