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  "body": "\n## Summary\n\nThe Curry–Howard correspondence is the match between proofs in [[intuitionistic logic|Q11N8X99FK]] and programs in typed [[lambda calculus|BD756KR8FM]].\nA proposition corresponds to a type, and a proof of the proposition corresponds to a program of that type.\nSimplifying a proof corresponds to running the program.\nIt is the basis of proof assistants built on [[type theory|Q48KGV33AH]], where checking a proof means checking the type of a program.\n\n## Explanations\n\n### L1 human\n\nThe slogan is \"propositions are types, and proofs are programs\".\nA proof of \"if $A$ then $B$\" is a function that turns any proof of $A$ into a proof of $B$.\nA proof of \"$A$ and $B$\" is a pair made of a proof of $A$ and a proof of $B$.\nFor example, the program that takes $x$ and $y$ and returns $x$ has type $A \\to (B \\to A)$, and it is also a proof of that formula.\nCurry noticed such a match for combinators, and Howard extended it to natural deduction.\n",
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