Normalizing element

Conjectura.Defs.ComputerScience.Rewriting.Normalizing

/-
Copyright (c) 2025 Fabrizio Montesi and Thomas Waring. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fabrizio Montesi, Thomas Waring, Chris Henson
-/

import Conjectura.Defs.ComputerScience.Rewriting.Normal

/-! # Normalizing element

## Source

Adapted for Conjectura from [cslib](https://github.com/leanprover/cslib), `Cslib/Foundations/Relation/Defs.lean`, split into one concept per module. Released by its authors under Apache 2.0.
-/

namespace Conjectura.Rewriting

/-- An element is **normalizing** when at least one finite rewrite sequence starting from it
reaches a `Normal` form. It may still have other, non-terminating sequences. -/
def Normalizing {α : Type*} (r : α → α → Prop) (x : α) : Prop :=
  ∃ y, Relation.ReflTransGen r x y ∧ Normal r y

end Conjectura.Rewriting