Theorems · Theorem · general topology
LipschitzWith.dist_iterate_succ_le_geometric
∀ {α : Type u} [inst : PseudoMetricSpace α] {K : NNReal} {f : α → α},
LipschitzWith K f → ∀ (x : α) (n : ℕ), dist (f^[n] x) (f^[n + 1] x) ≤ dist x (f x) * ↑K ^ n- Defined in
- Mathlib.Topology.MetricSpace.Lipschitz
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 152 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NNRealstatement and proof · cited by 4,310
- mul_commproof · cited by 2,262
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement and proof · cited by 1,539
- NNReal.toRealstatement and proof · cited by 1,260
- Nat.iteratestatement and proof · cited by 740
- LipschitzWithstatement and proof · cited by 316
- LipschitzWith.dist_le_mulproof · cited by 20
- Function.iterate_succproof · cited by 19
- LipschitzWith.iterateproof · cited by 4
Cited by2
Results whose statement or proof uses this declaration.
- ODE.FunSpace.dist_iterate_iterate_next_le_of_lipschitzWithproof · cited by 1
- ContractingWith.apriori_dist_iterate_fixedPoint_leproof · cited by 0