Theorems · Theorem · order theory

reflTransGen_of_pred

∀ {α : Type u_1} [inst : LinearOrder α] [inst_1 : PredOrder α] [IsPredArchimedean α] (r : α → α → Prop) {n m : α},
  (∀ i ∈ Set.Ioc m n, r i (Order.pred i)) → (∀ i ∈ Set.Ioc n m, r (Order.pred i) i) → Relation.ReflTransGen r n m

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

Defined in

Mathlib.Order.SuccPred.Relation

Cited by

1 results in Mathlib

Foundations

Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderPredOrderIsPredArchimedean