Documentation

Mathlib.Order.Interval.Finset.SuccPred

Finset intervals in a successor-predecessor order #

This file proves relations between the various finset intervals in a successor/predecessor order.

Notes #

Please keep in sync with:

TODO #

Copy over insert lemmas from Mathlib/Order/Interval/Finset/Nat.lean.

Two-sided intervals #

Orders possibly with maximal elements #

Equalities of intervals #

theorem Finset.Ico_succ_left_eq_Ioo {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] (a b : α) :

Ico (Order.succ a) b = Ioo a b

theorem Finset.Icc_succ_left_eq_Ioc_of_not_isMax {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {a : α} (ha : ¬IsMax a) (b : α) :

Icc (Order.succ a) b = Ioc a b

theorem Finset.Ico_succ_right_eq_Icc_of_not_isMax {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {b : α} (hb : ¬IsMax b) (a : α) :

Ico a (Order.succ b) = Icc a b

theorem Finset.Ioo_succ_right_eq_Ioc_of_not_isMax {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {b : α} (hb : ¬IsMax b) (a : α) :

Ioo a (Order.succ b) = Ioc a b

theorem Finset.Ico_succ_succ_eq_Ioc_of_not_isMax {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {b : α} (hb : ¬IsMax b) (a : α) :

Ico (Order.succ a) (Order.succ b) = Ioc a b

Inserting into intervals #

theorem Finset.insert_Icc_succ_left_eq_Icc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {a b : α} (h : a ≤ b) :

insert a (Icc (Order.succ a) b) = Icc a b

theorem Finset.insert_Icc_right_eq_Icc_succ {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {a b : α} (h : a ≤ Order.succ b) :

insert (Order.succ b) (Icc a b) = Icc a (Order.succ b)

theorem Finset.insert_Ico_right_eq_Ico_succ_of_not_isMax {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {a b : α} (h : a ≤ b) (hb : ¬IsMax b) :

insert b (Ico a b) = Ico a (Order.succ b)

theorem Finset.insert_Ico_succ_left_eq_Ico {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {a b : α} (h : a < b) :

insert a (Ico (Order.succ a) b) = Ico a b

theorem Finset.insert_Ioc_right_eq_Ioc_succ_of_not_isMax {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {a b : α} (h : a ≤ b) (hb : ¬IsMax b) :

insert (Order.succ b) (Ioc a b) = Ioc a (Order.succ b)

theorem Finset.insert_Ioc_succ_left_eq_Ioc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {a b : α} (h : a < b) :

insert (Order.succ a) (Ioc (Order.succ a) b) = Ioc a b

Orders with no maximal elements #

Equalities of intervals #

theorem Finset.Icc_succ_left_eq_Ioc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] [NoMaxOrder α] (a b : α) :

Icc (Order.succ a) b = Ioc a b

theorem Finset.Ico_succ_right_eq_Icc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] [NoMaxOrder α] (a b : α) :

Ico a (Order.succ b) = Icc a b

theorem Finset.Ioo_succ_right_eq_Ioc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] [NoMaxOrder α] (a b : α) :

Ioo a (Order.succ b) = Ioc a b

theorem Finset.Ico_succ_succ_eq_Ioc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] [NoMaxOrder α] (a b : α) :

Ico (Order.succ a) (Order.succ b) = Ioc a b

Inserting into intervals #

theorem Finset.insert_Ico_right_eq_Ico_succ {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {a b : α} [NoMaxOrder α] (h : a ≤ b) :

insert b (Ico a b) = Ico a (Order.succ b)

theorem Finset.insert_Ioc_right_eq_Ioc_succ {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] {a b : α} [NoMaxOrder α] (h : a ≤ b) :

insert (Order.succ b) (Ioc a b) = Ioc a (Order.succ b)

Orders possibly with minimal elements #

Equalities of intervals #

theorem Finset.Ioc_pred_right_eq_Ioo {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] (a b : α) :

Ioc a (Order.pred b) = Ioo a b

theorem Finset.Icc_pred_right_eq_Ico_of_not_isMin {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {b : α} (hb : ¬IsMin b) (a : α) :

Icc a (Order.pred b) = Ico a b

theorem Finset.Ioc_pred_left_eq_Icc_of_not_isMin {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a : α} (ha : ¬IsMin a) (b : α) :

Ioc (Order.pred a) b = Icc a b

theorem Finset.Ioo_pred_left_eq_Ioc_of_not_isMin {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a : α} (ha : ¬IsMin a) (b : α) :

Ioo (Order.pred a) b = Ico a b

theorem Finset.Ioc_pred_pred_eq_Ico_of_not_isMin {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a : α} (ha : ¬IsMin a) (b : α) :

Ioc (Order.pred a) (Order.pred b) = Ico a b

Inserting into intervals #

theorem Finset.insert_Icc_pred_right_eq_Icc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a b : α} (h : a ≤ b) :

insert b (Icc a (Order.pred b)) = Icc a b

theorem Finset.insert_Icc_left_eq_Icc_pred {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a b : α} (h : Order.pred a ≤ b) :

insert (Order.pred a) (Icc a b) = Icc (Order.pred a) b

theorem Finset.insert_Ioc_left_eq_Ioc_pred_of_not_isMin {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a b : α} (h : a ≤ b) (ha : ¬IsMin a) :

insert a (Ioc a b) = Ioc (Order.pred a) b

theorem Finset.insert_Ioc_pred_right_eq_Ioc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a b : α} (h : a < b) :

insert b (Ioc a (Order.pred b)) = Ioc a b

theorem Finset.insert_Ico_left_eq_Ico_pred_of_not_isMin {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a b : α} (h : a ≤ b) (ha : ¬IsMin a) :

insert (Order.pred a) (Ico a b) = Ico (Order.pred a) b

theorem Finset.insert_Ico_pred_right_eq_Ico {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a b : α} (h : a < b) :

insert (Order.pred b) (Ico a (Order.pred b)) = Ico a b

Orders with no minimal elements #

Equalities of intervals #

theorem Finset.Icc_pred_right_eq_Ico {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] [NoMinOrder α] (a b : α) :

Icc a (Order.pred b) = Ico a b

theorem Finset.Ioc_pred_left_eq_Icc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] [NoMinOrder α] (a b : α) :

Ioc (Order.pred a) b = Icc a b

theorem Finset.Ioo_pred_left_eq_Ioc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] [NoMinOrder α] (a b : α) :

Ioo (Order.pred a) b = Ico a b

theorem Finset.Ioc_pred_pred_eq_Ico {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] [NoMinOrder α] (a b : α) :

Ioc (Order.pred a) (Order.pred b) = Ico a b

Inserting into intervals #

theorem Finset.insert_Ioc_left_eq_Ioc_pred {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a b : α} [NoMinOrder α] (h : a ≤ b) :

insert a (Ioc a b) = Ioc (Order.pred a) b

theorem Finset.insert_Ico_left_eq_Ico_pred {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [PredOrder α] {a b : α} [NoMinOrder α] (h : a ≤ b) :

insert (Order.pred a) (Ico a b) = Ico (Order.pred a) b

theorem Finset.Icc_succ_pred_eq_Ioo {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] [SuccOrder α] [PredOrder α] [Nontrivial α] (a b : α) :

Icc (Order.succ a) (Order.pred b) = Ioo a b

One-sided interval towards `⊥` #

theorem Finset.Iio_succ_eq_Iic_of_not_isMax {α : Type u_1} [LinearOrder α] [LocallyFiniteOrderBot α] [SuccOrder α] {b : α} (hb : ¬IsMax b) :

Iio (Order.succ b) = Iic b

theorem Finset.Iio_succ_eq_Iic {α : Type u_1} [LinearOrder α] [LocallyFiniteOrderBot α] [SuccOrder α] [NoMaxOrder α] (b : α) :

Iio (Order.succ b) = Iic b

theorem Finset.Iic_pred_eq_Iio_of_not_isMin {α : Type u_1} [LinearOrder α] [LocallyFiniteOrderBot α] [PredOrder α] {b : α} (hb : ¬IsMin b) :

Iic (Order.pred b) = Iio b

theorem Finset.Iic_pred_eq_Iio {α : Type u_1} [LinearOrder α] [LocallyFiniteOrderBot α] [PredOrder α] [NoMinOrder α] (b : α) :

Iic (Order.pred b) = Iio b

One-sided interval towards `⊤` #

theorem Finset.Ici_succ_eq_Ioi_of_not_isMax {α : Type u_1} [LinearOrder α] [LocallyFiniteOrderTop α] [SuccOrder α] {a : α} (ha : ¬IsMax a) :

Ici (Order.succ a) = Ioi a

theorem Finset.Ici_succ_eq_Ioi {α : Type u_1} [LinearOrder α] [LocallyFiniteOrderTop α] [SuccOrder α] [NoMaxOrder α] (a : α) :

Ici (Order.succ a) = Ioi a

theorem Finset.Ioi_pred_eq_Ici_of_not_isMin {α : Type u_1} [LinearOrder α] [LocallyFiniteOrderTop α] [PredOrder α] {a : α} (ha : ¬IsMin a) :

Ioi (Order.pred a) = Ici a

theorem Finset.Ioi_pred_eq_Ici {α : Type u_1} [LinearOrder α] [LocallyFiniteOrderTop α] [PredOrder α] [NoMinOrder α] (a : α) :

Ioi (Order.pred a) = Ici a