Theorems · Theorem · order theory

reflTransGen_of_succ

∀ {α : Type u_1} [inst : LinearOrder α] [inst_1 : SuccOrder α] [IsSuccArchimedean α] (r : α → α → Prop) {n m : α},
  (∀ i ∈ Set.Ico n m, r i (Order.succ i)) → (∀ i ∈ Set.Ico m n, r (Order.succ i) i) → Relation.ReflTransGen r n m

(n, m) is in the reflexive-transitive closure of ~ if i ~ succ i and succ i ~ i for all i between n and m.

Defined in

Mathlib.Order.SuccPred.Relation

Cited by

6 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderSuccOrderIsSuccArchimedean