http://brightprogrammer.in/tags/equivalence-relation/Recent content in Equivalence-Relation on Siddharth MishraHugoenSun, 14 Jun 2026 21:53:10 -0700http://brightprogrammer.in/posts/2-lambda-calculus-equivalences-and-reductions/Thu, 06 Feb 2025 00:00:00 +0000http://brightprogrammer.in/posts/2-lambda-calculus-equivalences-and-reductions/<p>This section is important, because we&rsquo;ll learn how are applications performed on given lambda expressions.</p> <div class="notice notice-info"> <p class="notice-title">Info</p> <div class="notice-body">TL;DR, to apply a lambda term in a lambda expression, we use a series of reductions. These reductions come from corresponding equivalence relations. A reduction is just another way of replacing the lambda expression with a corresponding equivalent expression.</div> </div> <h1 id="equivalence">Equivalence</h1> <p>When considering equivalence of two expressions, they can be equivalent in four ways</p> <ul> <li>$\alpha$-equivalence - Renaming of bound variables</li> <li>$\beta$-equivalence - Function application and reduction</li> <li>$\eta$-equivalence - Function extensionality</li> <li>syntactic equivalence</li> </ul> <p>These equivalences are basically the equivalence relations we talked about above. Using a series of these equivalence relations between two lambda expressions, we can apply reductions to source lambda expression, and then reach a final, target expression from a source expression.</p>