Local confluence

Conjectura.Defs.ComputerScience.Rewriting.LocallyConfluent

/-
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.Joinable

/-! # Local confluence

## 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

/-- A relation is **locally confluent** when any two *single* steps out of a common element are
`Joinable`. Weaker than `Confluent`; Newman's lemma says the two coincide for terminating
relations. -/
def LocallyConfluent {α : Type*} (r : α → α → Prop) : Prop :=
  ∀ ⦃a b c : α⦄, r a b → r a c → Joinable r b c

end Conjectura.Rewriting