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  "body": "\n## Definition\n\nConsider intuitionistic propositional logic in [[natural deduction|6KT3BJZ7D1]], with [[implication|0ZNX6DQ3EJ]], [[conjunction|0M28ASWG8T]] and [[disjunction|0H5GKDTB01]].\nConsider the simply typed [[lambda calculus|BD756KR8FM]] with function types $A \\to B$, product types $A \\times B$ and sum types $A + B$.\nThe *Curry–Howard correspondence* is the map between the two systems given by these clauses.\n\n1. A formula is read as a type: $A \\to B$ as the function type, $A \\wedge B$ as $A \\times B$, and $A \\vee B$ as $A + B$.\n2. An assumption $A$ with label $x$ is read as a variable $x : A$.\n3. Each introduction rule is read as a term constructor: $\\to$-introduction as $\\lambda$-abstraction, $\\wedge$-introduction as pairing, and $\\vee$-introduction as an injection.\n4. Each elimination rule is read as a destructor: $\\to$-elimination as application, $\\wedge$-elimination as a projection, and $\\vee$-elimination as case analysis.\n\nUnder this map, a derivation of $B$ from assumptions $A_1, \\dots, A_n$ corresponds to a term $M$ with $x_1 : A_1, \\dots, x_n : A_n \\vdash M : B$, and conversely.\nA detour in a derivation is an introduction followed at once by an elimination.\nIt corresponds to a redex, and removing the detour corresponds to one reduction step, such as $(\\lambda x.\\, M) N \\to M[N/x]$.\n\n## Remarks\n\nExample: the term $\\lambda f.\\, \\lambda x.\\, f\\, (f\\, x)$ has type $(A \\to A) \\to (A \\to A)$.\nRead as a derivation, it uses $\\to$-elimination twice and $\\to$-introduction twice, and it proves the formula $(A \\to A) \\to (A \\to A)$.\nNon-example: Peirce's law $((A \\to B) \\to A) \\to A$ is a classical tautology but has no closed term of its type in the simply typed lambda calculus.\nThis matches the fact that it is not provable in [[intuitionistic logic|Q11N8X99FK]].\nThe correspondence extends to quantifiers with dependent types, and to classical logic with control operators.\n",
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