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  "body": "\n## Definition\n\n*Intuitionistic type theory*, or Martin-Löf type theory, is the dependent [[type theory|42A2M9S2C1]] with the following parts.\n\n1. Four forms of judgment in a context $\\Gamma$. They say that $A$ is a type, or that $A$ and $B$ are equal types. They also say that $a$ has type $A$, written $a : A$, or that $a$ and $b$ are equal terms of type $A$.\n2. Dependent function types $\\prod_{x : A} B(x)$, whose terms are functions $\\lambda x.\\, b(x)$ with $b(x) : B(x)$ for each $x : A$.\n3. Dependent pair types $\\sum_{x : A} B(x)$, whose terms are pairs $(a, b)$ with $a : A$ and $b : B(a)$.\n4. Identity types $\\mathrm{Id}_A(a, b)$ for $a, b : A$, with the reflexivity term $\\mathrm{refl}_a : \\mathrm{Id}_A(a, a)$ and an induction rule.\n5. Base types: the empty type $\\mathbf{0}$, the unit type $\\mathbf{1}$, the type $\\mathbf{2}$ of booleans and the natural numbers $\\N$. Each has introduction rules and an induction rule.\n6. Universes $U_0 : U_1 : \\cdots$, types whose terms are codes of types.\n\nEach type former comes with formation, introduction, elimination and computation rules.\n\n## Remarks\n\nUnder the [[Curry–Howard correspondence|HTF9E5W7EZ]], $\\prod_{x : A} B(x)$ reads as \"for all $x$ in $A$, $B(x)$\", $\\sum_{x : A} B(x)$ as \"there is $x$ in $A$ with $B(x)$\", and $\\mathbf{0}$ as falsity.\nExample: the term $\\lambda n.\\, (n + 1, \\mathrm{refl}_{n+1})$ has type $\\prod_{n : \\N} \\sum_{m : \\N} \\mathrm{Id}_{\\N}(m, n + 1)$, so it proves that every natural number has a successor.\nNon-example: there is no closed term of type $\\prod_{A : U_0} (A + (A \\to \\mathbf{0}))$, so the [[intuitionistic logic|Q11N8X99FK]] of the theory does not include excluded middle.\nMartin-Löf gave several versions, with intensional or extensional identity types.\n",
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