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  "body": "\n## Statement\n\nLet $(P, \\le)$ be an [[omega-complete partial order|T4BCQHR68Y]]: every increasing sequence $x_0 \\le x_1 \\le x_2 \\le \\cdots$ in $P$ has a [[supremum|0R3H5PANRQ]] in $P$.\nAssume that $P$ has a [[least element|0AANKKNHS4]] $\\bot$.\nLet $f \\colon P \\to P$ be [[continuous|C6GAHS1Z2P]] in this sense: $f$ is monotone, and $f\\left(\\sup_n x_n\\right) = \\sup_n f(x_n)$ for every increasing sequence $(x_n)$.\nThen $f$ has a [[least fixed point|DBD64TH52D]], and\n$$\\operatorname{lfp}(f) = \\sup_{n \\ge 0} f^n(\\bot),$$\nwhere $f^0(\\bot) = \\bot$ and $f^{n+1}(\\bot) = f(f^n(\\bot))$.\n\n## Remarks\n\nProof idea: $\\bot \\le f(\\bot)$, and monotonicity gives $f^n(\\bot) \\le f^{n+1}(\\bot)$ by induction, so the iterates form an increasing sequence with supremum $x$.\nContinuity gives $f(x) = \\sup_n f^{n+1}(\\bot) = x$.\nIf $y$ is any fixed point, then $\\bot \\le y$ and induction gives $f^n(\\bot) \\le f^n(y) = y$, so $x \\le y$.\nThe same statement holds for a directed-complete partial order with least element and a Scott-continuous $f$.\nExample: on the power set of $\\N$, the map $f(S) = \\set{0} \\cup \\set{n + 2 : n \\in S}$ preserves unions of increasing sequences.\nIts iterates from $\\emptyset$ are $\\set{0}$, $\\set{0, 2}$, $\\set{0, 2, 4}$, and so on, and their union, the even numbers, is the least fixed point.\nContinuity cannot be dropped from the formula: on the chain $\\N \\cup \\set{\\omega, \\omega + 1}$, the monotone map with $f(n) = n + 1$, $f(\\omega) = \\omega + 1$ and $f(\\omega + 1) = \\omega + 1$ has least fixed point $\\omega + 1$, while $\\sup_n f^n(0) = \\omega$.\n\n## Explanations\n\n### L1 human\n\nStart at the bottom and apply $f$ again and again.\nThe values climb up step by step, and their limit is the least fixed point of $f$.\nThis gives a way to compute least fixed points, not only to know that they exist.\nIt is how the meaning of a recursive program is defined: run the recursion $0, 1, 2, \\ldots$ levels deep and take the limit.\n\n### L2 human\n\nThe theorem is the main tool of denotational semantics, where recursive definitions are interpreted as least fixed points of continuous maps on domains.\nCompared with the [[Knaster–Tarski theorem|RHXP3HA64M]], it needs less of the order, only suprema of increasing sequences, but more of the map, namely continuity, and it gives an explicit construction.\nFor a monotone map that is not continuous, the iteration has to continue transfinitely, through $f^\\omega(\\bot) = \\sup_n f^n(\\bot)$, $f^{\\omega + 1}(\\bot)$, and so on, until it stabilises.\nIn Datalog and in dataflow analysis on finite lattices, the iteration stops after finitely many steps.\n",
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  "name": "Kleene fixed-point theorem",
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