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  "body": "\n## Definition\n\nLet $\\sigma$ be a [[signature|WDVVEYD6AT]] with only function symbols and let $X$ be a set of variables.\nLet $A$ be a [[structure|983GZBQYT4]] of signature $\\sigma$, that is, an algebra, with operations $f^A$.\nLet $\\alpha \\colon X \\to A$ be a map, called an *assignment*.\nThe *value* $t^A[\\alpha]$ of a [[term|6D3WA7N2W4]] $t$ over $\\sigma$ and $X$ is defined by recursion on $t$:\n\n1. if $t$ is a variable $x \\in X$, then $t^A[\\alpha] = \\alpha(x)$;\n2. if $t$ is a constant symbol $c$, then $t^A[\\alpha] = c^A$;\n3. if $t = f(t_1, \\ldots, t_n)$ for a function symbol $f$ of arity $n \\ge 1$, then $t^A[\\alpha] = f^A(t_1^A[\\alpha], \\ldots, t_n^A[\\alpha])$.\n\n## Remarks\n\nThe recursion is well defined because every term has a unique reading.\nThe value depends only on the values of $\\alpha$ at the variables that occur in $t$.\nIf these variables are among $x_1, \\ldots, x_n$ and $\\alpha(x_i) = a_i$, one writes $t^A(a_1, \\ldots, a_n)$.\n\nExample: in the ring $(\\Z, +, \\cdot, 0, 1)$, let $t = x \\cdot (y + 1)$ and $\\alpha(x) = 3$, $\\alpha(y) = 4$.\nClause 1 gives $x \\mapsto 3$ and $y \\mapsto 4$, clause 2 gives $1 \\mapsto 1$, and clause 3 twice gives $y + 1 \\mapsto 5$ and $t \\mapsto 15$.\n\nNon-example: computing $x \\cdot (y + 1)$ at the same assignment as $3 \\cdot 4 + 1 = 13$ is not term evaluation.\nThe outermost symbol of $t$ is $\\cdot$ applied to $x$ and $y + 1$, so clause 3 requires the product of the values $3$ and $5$.\n\nHomomorphisms commute with evaluation: if $h \\colon A \\to B$ is a homomorphism, then $h(t^A[\\alpha]) = t^B[h \\circ \\alpha]$.\nIn Mathlib term evaluation is `FirstOrder.Language.Term.realize`.\n",
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