Theorems · Theorem · order theory
Monotone.monotone_iterate_of_le_map
∀ {α : Type u_1} [inst : Preorder α] {f : α → α} {x : α}, Monotone f → x ≤ f x → Monotone fun n => f^[n] xIf f is a monotone map and x ≤ f x at some point x, then the iterates f^[n] x form
a monotone sequence.
- Defined in
- Mathlib.Order.Iterate
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 25 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Preorderstatement and proof · cited by 7,952
- Monotonestatement and proof · cited by 1,397
- Nat.iteratestatement and proof · cited by 740
- monotone_nat_of_le_succproof · cited by 45
- Function.iterate_succ_applyproof · cited by 18
- Monotone.iterateproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- toZ_strictMonoproof · cited by 3
- Order.isMax_iterate_succ_of_eq_of_ltproof · cited by 2
- Monotone.antitone_iterate_of_map_leproof · cited by 1