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Fungrim entry: 65ccf2

Symbol: Maximum maxxSf(x)\mathop{\max}\limits_{x \in S} f(x) Maximum value of a set or function
Called with 1 argument, Maximum(S), rendered max(S)\max\left(S\right), represents the maximum element of the set SS. This operator is only defined if SS is a subset of R{,+}\mathbb{R} \cup \left\{-\infty, +\infty\right\} and the maximum exists.
Maximum(f(x), ForElement(x, S)), rendered maxxSf(x)\mathop{\max}\limits_{x \in S} f(x), represents max{f(x):xS}\max \left\{ f(x) : x \in S \right\}.
Maximum(f(x), ForElement(x, S), P(x)), rendered maxxS,P(x)f(x)\mathop{\max}\limits_{x \in S,\,P(x)} f(x), represents max{f(x):xSandP(x)}\max \left\{ f(x) : x \in S \,\mathbin{\operatorname{and}}\, P(x) \right\}.
Maximum(f(x), For(x), P(x)), rendered maxP(x)f(x)\mathop{\max}\limits_{P(x)} f(x), represents max{f(x):P(x)}\max \left\{ f(x) : P(x) \right\}.
Maximum(f(x, y), For(Tuple(x, y)), P(x, y)), rendered maxP(x,y)f ⁣(x,y)\mathop{\max}\limits_{P\left(x, y\right)} f\!\left(x, y\right), represents max{f ⁣(x,y):P ⁣(x,y)}\max \left\{ f\!\left(x, y\right) : P\!\left(x, y\right) \right\} where P ⁣(x,y)P\!\left(x, y\right) is a predicate defining the range of xx and yy, and similarly for any number n2n \ge 2 of variables.
The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate P(x)P(x) must define the domain of xx unambiguously; that is, it must include a statement such as xSx \in S where SS is a known set. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate P ⁣(x,y)P\!\left(x, y\right), P ⁣(x,y,z)P\!\left(x, y, z\right), etc.
Definitions:
Fungrim symbol Notation Short description
MaximummaxxSf(x)\mathop{\max}\limits_{x \in S} f(x) Maximum value of a set or function
RRR\mathbb{R} Real numbers
Infinity\infty Positive infinity
Source code for this entry:
Entry(ID("65ccf2"),
    SymbolDefinition(Maximum, Maximum(f(x), ForElement(x, S)), "Maximum value of a set or function"),
    Description("Called with 1 argument, ", SourceForm(Maximum(S)), ", rendered", Maximum(S), ", represents the maximum element of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), " and the maximum exists."),
    Description(SourceForm(Maximum(f(x), ForElement(x, S))), ", rendered", Maximum(f(x), ForElement(x, S)), ", represents", Maximum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Maximum(f(x), ForElement(x, S), P(x))), ", rendered", Maximum(f(x), ForElement(x, S), P(x)), ", represents", Maximum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Maximum(f(x), For(x), P(x))), ", rendered", Maximum(f(x), For(x), P(x)), ", represents", Maximum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Maximum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Maximum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Maximum(Set(f(x, y), For(Tuple(x, y)), P(x, y))), "where", P(x, y), "is a predicate defining the range of", x, "and", y, ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", the corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC