Theorems · Theorem
Function.IsFixedPt.eq
∀ {α : Type u₁} {f : α → α} {x : α}, Function.IsFixedPt f x → f x = xIf x is a fixed point of f, then f x = x. This is useful, e.g., for rw or simp.
- Defined in
- Mathlib.Logic.Function.Defs
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 2 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites1
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Function.IsFixedPtstatement and proof · cited by 84
Cited by14
Results whose statement or proof uses this declaration.
- ContractingWith.eq_or_edist_eq_top_of_fixedPointsproof · cited by 3
- OrdinalApprox.lfpApprox_le_of_mem_fixedPointsproof · cited by 3
- ContractingWith.dist_le_of_fixedPointproof · cited by 2
- Function.IsPeriodicPt.eq_of_apply_eq_sameproof · cited by 2
- Function.isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterateproof · cited by 2
- Function.IsPeriodicPt.iterate_mod_applyproof · cited by 2
- Function.periodic_iterate_iffproof · cited by 2
- RootPairing.span_root_image_eq_top_of_forall_orthogonalproof · cited by 1
- Function.IsFixedPt.birkhoffSum_eqproof · cited by 1
- ContractingWith.edist_le_of_fixedPointproof · cited by 1
- Function.bijOn_ptsOfPeriodproof · cited by 1
- Equiv.Perm.SameCycle.eq_of_leftproof · cited by 1