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In formal reasoning, in particular in mathematical logic, computer algebra, and automated theorem proving, a fresh variable is a variable that did not occur in the context considered so far.^{[1][citation needed]} The concept is often used without explanation.^{[2][citation needed]}

Fresh variables may be used to replace other variables, to eliminate variable shadowing or capture. For instance, in alpha-conversion, the processing of terms in the lambda calculus into equivalent terms with renamed variables, replacing variables with fresh variables can be helpful as a way to avoid accidentally capturing variables that should be free.^[3] Another use for fresh variables involves the development of loop invariants in formal program verification, where it is sometimes useful to replace constants by newly introduced fresh variables.^[4]

For example, in term rewriting, before applying a rule ${\displaystyle l\to r}$ to a given term ${\displaystyle t}$, each variable in ${\displaystyle l\to r}$ should be replaced by a fresh one to avoid clashes with variables occurring in ${\displaystyle t}$.^{[citation needed]} Given the rule $${\displaystyle \operatorname {append} (\operatorname {cons} (x,y),z)\to \operatorname {cons} (x,\operatorname {append} (y,z))}$$ and the term $${\displaystyle \operatorname {append} (\operatorname {cons} (x,\operatorname {cons} (y,\mathrm {nil} )),\operatorname {cons} (3,\mathrm {nil} )),}$$ attempting to find a matching substitution of the rule's left-hand side, ${\displaystyle \operatorname {append} (\operatorname {cons} (x,y),z)}$, within ${\displaystyle \operatorname {append} (\operatorname {cons} (x,\operatorname {cons} (y,\mathrm {nil} )),\operatorname {cons} (3,\mathrm {nil} ))}$ will fail, since ${\displaystyle y}$ cannot match ${\displaystyle \operatorname {cons} (y,\mathrm {nil} )}$. However, if the rule is replaced by a fresh copy^[a] $${\displaystyle \operatorname {append} (\operatorname {cons} (v_{1},v_{2}),v_{3})\to \operatorname {cons} (v_{1},\operatorname {append} (v_{2},v_{3}))}$$ before, matching will succeed with the answer substitution $${\displaystyle \{v_{1}\mapsto x,\;v_{2}\mapsto \operatorname {cons} (y,\mathrm {nil} ),\;v_{3}\mapsto \operatorname {cons} (3,\mathrm {nil} )\}.}$$

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