Theorems · Theorem · order theory
Function.id_le_iterate_of_id_le
∀ {α : Type u_1} [inst : Preorder α] {f : α → α}, id ≤ f → ∀ (n : ℕ), id ≤ f^[n]If $x ≤ f x$ for all $x$ (we write this as id ≤ f), then the same is true for any iterate
f^[n] of f.
- Defined in
- Mathlib.Order.Iterate
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 24 from the axioms · uses propext, Quot.sound
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- Nat.iteratestatement and proof · cited by 740
- monotone_idproof · cited by 31
- Function.iterate_idproof · cited by 8
- Monotone.iterate_le_of_leproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- Succ.recproof · cited by 10
- Order.le_succ_iterateproof · cited by 5
- SzemerediRegularity.initialBound_le_boundproof · cited by 3
- exists_succ_iterate_iff_leproof · cited by 0