Diamond property of a reduction relation

A reduction relation $\to$ has the diamond property if $a\to b$ and $a\to c$ imply a common one-step successor $b\to d\leftarrow c$. For relations allowing an identity step, the same definition includes zero-length joins. A finite grid of these diamonds proves confluence of ${\to }^{\ast }$. Ordinary one-step beta reduction fails this property because contracting before duplication can save two separate contractions.

A reduction relation has the Church-Rosser property when its equivalence relation generated by forward and backward steps satisfies $b{\leftrightarrow }^{\ast }c\Rightarrow \exists d(b{\to }^{\ast }d{\leftarrow }^{\ast }c)$. It is equivalent to confluence of ${\to }^{\ast }$. The Church-Rosser theorem establishes it for beta reduction.