Documentation

Mathlib.Order.SuccPred.Archimedean

Archimedean successor and predecessor #

class IsSuccArchimedean (α : Type u_3) [Preorder α] [SuccOrder α] :

A SuccOrder is succ-archimedean if one can go from any two comparable elements by iterating succ

  • exists_succ_iterate_of_le : ∀ {a b : α}, a b∃ (n : ), Order.succ^[n] a = b

    If a ≤ b then one can get to a from b by iterating succ

Instances
    theorem IsSuccArchimedean.exists_succ_iterate_of_le {α : Type u_3} :
    ∀ {inst : Preorder α} {inst_1 : SuccOrder α} [self : IsSuccArchimedean α] {a b : α}, a b∃ (n : ), Order.succ^[n] a = b

    If a ≤ b then one can get to a from b by iterating succ

    class IsPredArchimedean (α : Type u_3) [Preorder α] [PredOrder α] :

    A PredOrder is pred-archimedean if one can go from any two comparable elements by iterating pred

    • exists_pred_iterate_of_le : ∀ {a b : α}, a b∃ (n : ), Order.pred^[n] b = a

      If a ≤ b then one can get to b from a by iterating pred

    Instances
      theorem IsPredArchimedean.exists_pred_iterate_of_le {α : Type u_3} :
      ∀ {inst : Preorder α} {inst_1 : PredOrder α} [self : IsPredArchimedean α] {a b : α}, a b∃ (n : ), Order.pred^[n] b = a

      If a ≤ b then one can get to b from a by iterating pred

      theorem LE.le.exists_succ_iterate {α : Type u_1} [Preorder α] [SuccOrder α] [IsSuccArchimedean α] {a : α} {b : α} (h : a b) :
      ∃ (n : ), Order.succ^[n] a = b
      theorem exists_succ_iterate_iff_le {α : Type u_1} [Preorder α] [SuccOrder α] [IsSuccArchimedean α] {a : α} {b : α} :
      (∃ (n : ), Order.succ^[n] a = b) a b
      theorem Succ.rec {α : Type u_1} [Preorder α] [SuccOrder α] [IsSuccArchimedean α] {P : αProp} {m : α} (h0 : P m) (h1 : ∀ (n : α), m nP nP (Order.succ n)) ⦃n : α (hmn : m n) :
      P n

      Induction principle on a type with a SuccOrder for all elements above a given element m.

      theorem Succ.rec_iff {α : Type u_1} [Preorder α] [SuccOrder α] [IsSuccArchimedean α] {p : αProp} (hsucc : ∀ (a : α), p a p (Order.succ a)) {a : α} {b : α} (h : a b) :
      p a p b
      theorem LE.le.exists_pred_iterate {α : Type u_1} [Preorder α] [PredOrder α] [IsPredArchimedean α] {a : α} {b : α} (h : a b) :
      ∃ (n : ), Order.pred^[n] b = a
      theorem exists_pred_iterate_iff_le {α : Type u_1} [Preorder α] [PredOrder α] [IsPredArchimedean α] {a : α} {b : α} :
      (∃ (n : ), Order.pred^[n] b = a) a b
      theorem Pred.rec {α : Type u_1} [Preorder α] [PredOrder α] [IsPredArchimedean α] {P : αProp} {m : α} (h0 : P m) (h1 : nm, P nP (Order.pred n)) ⦃n : α (hmn : n m) :
      P n

      Induction principle on a type with a PredOrder for all elements below a given element m.

      theorem Pred.rec_iff {α : Type u_1} [Preorder α] [PredOrder α] [IsPredArchimedean α] {p : αProp} (hsucc : ∀ (a : α), p a p (Order.pred a)) {a : α} {b : α} (h : a b) :
      p a p b
      theorem succ_max {α : Type u_1} [LinearOrder α] [SuccOrder α] (a : α) (b : α) :
      theorem succ_min {α : Type u_1} [LinearOrder α] [SuccOrder α] (a : α) (b : α) :
      theorem exists_succ_iterate_or {α : Type u_1} [LinearOrder α] [SuccOrder α] [IsSuccArchimedean α] {a : α} {b : α} :
      (∃ (n : ), Order.succ^[n] a = b) ∃ (n : ), Order.succ^[n] b = a
      theorem Succ.rec_linear {α : Type u_1} [LinearOrder α] [SuccOrder α] [IsSuccArchimedean α] {p : αProp} (hsucc : ∀ (a : α), p a p (Order.succ a)) (a : α) (b : α) :
      p a p b
      theorem pred_max {α : Type u_1} [LinearOrder α] [PredOrder α] (a : α) (b : α) :
      theorem pred_min {α : Type u_1} [LinearOrder α] [PredOrder α] (a : α) (b : α) :
      theorem exists_pred_iterate_or {α : Type u_1} [LinearOrder α] [PredOrder α] [IsPredArchimedean α] {a : α} {b : α} :
      (∃ (n : ), Order.pred^[n] b = a) ∃ (n : ), Order.pred^[n] a = b
      theorem Pred.rec_linear {α : Type u_1} [LinearOrder α] [PredOrder α] [IsPredArchimedean α] {p : αProp} (hsucc : ∀ (a : α), p a p (Order.pred a)) (a : α) (b : α) :
      p a p b
      @[deprecated StrictMono.not_bddAbove_range_of_isSuccArchimedean]
      theorem StrictMono.not_bddAbove_range {α : Type u_1} {β : Type u_2} [Preorder α] [Nonempty α] [Preorder β] {f : αβ} [NoMaxOrder α] [SuccOrder β] [IsSuccArchimedean β] (hf : StrictMono f) :

      Alias of StrictMono.not_bddAbove_range_of_isSuccArchimedean.

      @[deprecated StrictMono.not_bddBelow_range_of_isPredArchimedean]
      theorem StrictMono.not_bddBelow_range {α : Type u_1} {β : Type u_2} [Preorder α] [Nonempty α] [Preorder β] {f : αβ} [NoMinOrder α] [PredOrder β] [IsPredArchimedean β] (hf : StrictMono f) :

      Alias of StrictMono.not_bddBelow_range_of_isPredArchimedean.

      @[deprecated StrictAnti.not_bddBelow_range_of_isSuccArchimedean]
      theorem StrictAnti.not_bddAbove_range {α : Type u_1} {β : Type u_2} [Preorder α] [Nonempty α] [Preorder β] {f : αβ} [NoMinOrder α] [SuccOrder β] [IsSuccArchimedean β] (hf : StrictAnti f) :

      Alias of StrictAnti.not_bddBelow_range_of_isSuccArchimedean.

      @[deprecated StrictAnti.not_bddBelow_range_of_isPredArchimedean]
      theorem StrictAnti.not_bddBelow_range {α : Type u_1} {β : Type u_2} [Preorder α] [Nonempty α] [Preorder β] {f : αβ} [NoMaxOrder α] [PredOrder β] [IsPredArchimedean β] (hf : StrictAnti f) :

      Alias of StrictAnti.not_bddBelow_range_of_isPredArchimedean.

      @[instance 100]
      Equations
      • =
      @[instance 100]
      Equations
      • =
      theorem Succ.rec_bot {α : Type u_1} [Preorder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α] (p : αProp) (hbot : p ) (hsucc : ∀ (a : α), p ap (Order.succ a)) (a : α) :
      p a
      theorem Pred.rec_top {α : Type u_1} [Preorder α] [OrderTop α] [PredOrder α] [IsPredArchimedean α] (p : αProp) (htop : p ) (hpred : ∀ (a : α), p ap (Order.pred a)) (a : α) :
      p a
      theorem SuccOrder.forall_ne_bot_iff {α : Type u_1} [Nontrivial α] [PartialOrder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α] (P : αProp) :
      (∀ (i : α), i P i) ∀ (i : α), P (SuccOrder.succ i)

      IsSuccArchimedean transfers across equivalences between SuccOrders.

      IsPredArchimedean transfers across equivalences between PredOrders.