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Fungrim entry: bbeb35

Symbol: Infimum infxSf(x)\mathop{\operatorname{inf}}\limits_{x \in S} f(x) Infimum of a set or function
Infimum(S), rendered inf(S)\operatorname{inf}\left(S\right), represents the infimum of the set SS. This operator is only defined if SS is a subset of R{,+}\mathbb{R} \cup \left\{-\infty, +\infty\right\}. The infimum does not need to be an element of SS itself; in particular, for an open interval S=(a,b)S = \left(a, b\right), we have inf(S)=b\operatorname{inf}\left(S\right) = b.
Infimum(f(x), ForElement(x, S)), rendered infxSf(x)\mathop{\operatorname{inf}}\limits_{x \in S} f(x), represents inf{f(x):xS}\operatorname{inf} \left\{ f(x) : x \in S \right\}.
Infimum(f(x), ForElement(x, S), P(x)), rendered infxS,P(x)f(x)\mathop{\operatorname{inf}}\limits_{x \in S,\,P(x)} f(x), represents inf{f(x):xSandP(x)}\operatorname{inf} \left\{ f(x) : x \in S \,\mathbin{\operatorname{and}}\, P(x) \right\}.
Infimum(f(x), For(x), P(x)), rendered infP(x)f(x)\mathop{\operatorname{inf}}\limits_{P(x)} f(x), represents inf{f(x):P(x)}\operatorname{inf} \left\{ f(x) : P(x) \right\}.
Infimum(f(x, y), For(Tuple(x, y)), P(x, y)), rendered infP(x,y)f ⁣(x,y)\mathop{\operatorname{inf}}\limits_{P\left(x, y\right)} f\!\left(x, y\right), represents inf{f ⁣(x,y):P ⁣(x,y)}\operatorname{inf} \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
InfimuminfxSf(x)\mathop{\operatorname{inf}}\limits_{x \in S} f(x) Infimum of a set or function
RRR\mathbb{R} Real numbers
Infinity\infty Positive infinity
OpenInterval(a,b)\left(a, b\right) Open interval
Source code for this entry:
Entry(ID("bbeb35"),
    SymbolDefinition(Infimum, Infimum(f(x), ForElement(x, S)), "Infimum of a set or function"),
    Description(SourceForm(Infimum(S)), ", rendered", Infimum(S), ", represents the infimum of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), ".", "The infimum does not need to be an element of", S, "itself; in particular, for an open interval", Equal(S, OpenInterval(a, b)), ", we have", Equal(Infimum(S), b), "."),
    Description(SourceForm(Infimum(f(x), ForElement(x, S))), ", rendered", Infimum(f(x), ForElement(x, S)), ", represents", Infimum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Infimum(f(x), ForElement(x, S), P(x))), ", rendered", Infimum(f(x), ForElement(x, S), P(x)), ", represents", Infimum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Infimum(f(x), For(x), P(x))), ", rendered", Infimum(f(x), For(x), P(x)), ", represents", Infimum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Infimum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Infimum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Infimum(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.

2020-01-31 18:09:28.494564 UTC