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rubin-lean4/Rubin/Period.lean

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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]
-- TODO: move to Rubin.Period
noncomputable def period (p : α) (g : G) : :=
sInf {n : | n > 0 ∧ g ^ n • p = p}
#align period Rubin.Period.period
theorem period_le_fix {p : α} {g : G} {m : } (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_pos : n > 0)
(period_pos : Rubin.Period.period p g > 0) (pmoves : ∀ i : , 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_pos : n > 0)
(period_pos : 0 < Rubin.Period.period p g)
(pmoves : ∀ i : Fin n, 0 < (i : ) → g ^ (i : ) • p ≠ p) : n ≤ Rubin.Period.period p g :=
Rubin.Period.notfix_le_period n_pos period_pos fun (i : ) (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
theorem moves_within_period {n : } (g : G) (x : α) :
0 < n → n < period x g → g^n • x ≠ x :=
by
intro n_pos n_lt_period
unfold period at n_lt_period
apply Nat.not_mem_of_lt_sInf at n_lt_period
simp at n_lt_period
apply n_lt_period
exact n_pos
-- Variant of moves_within_period, which works with integers
theorem moves_within_period' {z : } (g : G) (x : α) :
0 < z → z < period x g → g^z • x ≠ x :=
by
intro n_pos n_lt_period
rw [<-Int.ofNat_natAbs_eq_of_nonneg _ (Int.le_of_lt n_pos)]
rw [zpow_ofNat]
apply moves_within_period
· rw [<-Int.natAbs_zero]
apply Int.natAbs_lt_natAbs_of_nonneg_of_lt
rfl
assumption
· rw [<-Int.natAbs_cast (period x g)]
apply Int.natAbs_lt_natAbs_of_nonneg_of_lt
exact Int.le_of_lt n_pos
assumption
def periods (U : Set α) (H : Subgroup G) : Set :=
{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 : ) (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 : ),
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 period_pos {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) :=
fun p g =>
(zero_lt_period_le_Sup_periods U_nonempty exp_ne_zero p g).1
theorem period_pos' {U : Set α} (U_nonempty : U.Nonempty) {H : Subgroup G}
(exp_ne_zero : Monoid.exponent H ≠ 0) :
∀ (p : α) (g : G), p ∈ U → g ∈ H → 0 < Rubin.Period.period (p : α) (g : G) :=
fun p g p_in_U g_in_H => period_pos U_nonempty exp_ne_zero ⟨p, p_in_U⟩ ⟨g, g_in_H⟩
theorem 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), Rubin.Period.period (p : α) (g : G) ≤ sSup (Rubin.Period.periods U H) :=
fun p g =>
(zero_lt_period_le_Sup_periods U_nonempty exp_ne_zero p g).2
theorem period_le_Sup_periods' {U : Set α} (U_nonempty : U.Nonempty)
{H : Subgroup G} (exp_ne_zero : Monoid.exponent H ≠ 0) :
∀ (p : α) (g : G), p ∈ U → g ∈ H → Rubin.Period.period p g ≤ sSup (Rubin.Period.periods U H) :=
fun p g p_in_U g_in_H => period_le_Sup_periods U_nonempty exp_ne_zero ⟨p, p_in_U⟩ ⟨g, g_in_H⟩
-- TODO: rename to pow_period_fixes
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