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  "body": "\n## Definition\n\nFix a countably infinite set $V$ of variables.\nThe *lambda terms* are the strings defined by these rules:\n\n1. every variable $x \\in V$ is a term;\n2. if $M$ is a term and $x \\in V$, then $(\\lambda x.\\, M)$ is a term, called an *abstraction*;\n3. if $M$ and $N$ are terms, then $(M\\, N)$ is a term, called an *application*.\n\nIn $\\lambda x.\\, M$ the variable $x$ is bound in $M$.\nTerms that differ only in the names of bound variables are identified ($\\alpha$-equivalence).\nLet $M[x := N]$ be the term obtained from $M$ by replacing the free occurrences of $x$ by $N$.\nBefore the replacement, bound variables of $M$ are renamed so that no free variable of $N$ gets bound.\nThe *untyped lambda calculus* is the set of terms together with *$\\beta$-reduction*.\nThis is the relation generated by the rule\n$$(\\lambda x.\\, M)\\, N \\to_\\beta M[x := N],$$\napplied to any subterm of a term.\n\n## Remarks\n\nExample: $(\\lambda x.\\, x\\, x)(\\lambda y.\\, y)$ is a term by rules 1 to 3, and\n$$(\\lambda x.\\, x\\, x)(\\lambda y.\\, y) \\to_\\beta (\\lambda y.\\, y)(\\lambda y.\\, y) \\to_\\beta \\lambda y.\\, y.$$\nThe last term has no further reduction, so it is a *normal form*.\n\nNon-example: the string $\\lambda x$ alone is not a term, since rule 2 requires a body $M$ after the dot.\n\nThe term $\\Omega = (\\lambda x.\\, x\\, x)(\\lambda x.\\, x\\, x)$ reduces only to itself, so it has no normal form.\nBy the Church–Rosser theorem, a term has at most one normal form up to $\\alpha$-equivalence.\n",
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