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Fungrim entry: d0cb24

Symbol: Minimum minxSf(x)\mathop{\min}\limits_{x \in S} f(x) Minimum value of a set or function
Minimum(S), rendered min(S)\min\left(S\right), represents the minimum 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 minimum exists.
Minimum(f(x), ForElement(x, S)), rendered minxSf(x)\mathop{\min}\limits_{x \in S} f(x), represents min{f(x):xS}\min \left\{ f(x) : x \in S \right\}.
Minimum(f(x), ForElement(x, S), P(x)), rendered minxS,P(x)f(x)\mathop{\min}\limits_{x \in S,\,P(x)} f(x), represents min{f(x):xSandP(x)}\min \left\{ f(x) : x \in S \,\mathbin{\operatorname{and}}\, P(x) \right\}.
Minimum(f(x), For(x), P(x)), rendered minP(x)f(x)\mathop{\min}\limits_{P(x)} f(x), represents min{f(x):P(x)}\min \left\{ f(x) : P(x) \right\}.
Minimum(f(x, y), For(Tuple(x, y)), P(x, y)), rendered minP(x,y)f ⁣(x,y)\mathop{\min}\limits_{P\left(x, y\right)} f\!\left(x, y\right), represents min{f ⁣(x,y):P ⁣(x,y)}\min \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
MinimumminxSf(x)\mathop{\min}\limits_{x \in S} f(x) Minimum value of a set or function
RRR\mathbb{R} Real numbers
Infinity\infty Positive infinity
Source code for this entry:
Entry(ID("d0cb24"),
    SymbolDefinition(Minimum, Minimum(f(x), ForElement(x, S)), "Minimum value of a set or function"),
    Description(SourceForm(Minimum(S)), ", rendered", Minimum(S), ", represents the minimum 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 minimum exists."),
    Description(SourceForm(Minimum(f(x), ForElement(x, S))), ", rendered", Minimum(f(x), ForElement(x, S)), ", represents", Minimum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Minimum(f(x), ForElement(x, S), P(x))), ", rendered", Minimum(f(x), ForElement(x, S), P(x)), ", represents", Minimum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Minimum(f(x), For(x), P(x))), ", rendered", Minimum(f(x), For(x), P(x)), ", represents", Minimum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Minimum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Minimum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Minimum(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