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Fungrim entry: 6ec976

Symbol: Supremum supP(x)f ⁣(x)\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right) Supremum of a set or function
This operator can be called with 1 or 3 arguments.
Called with 1 argument, Supremum(S), rendered sup(S)\operatorname{sup}\left(S\right), represents the supremum of the set SS. This operator is only defined if SS is a subset of R{,+}\mathbb{R} \cup \left\{-\infty, +\infty\right\}. The supremum 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 sup(S)=b\operatorname{sup}\left(S\right) = b.
Called with 3 arguments, Supremum(f(x), Var(x), P(x)), rendered supP(x)f ⁣(x)\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right), represents sup{f ⁣(x):P ⁣(x)}\operatorname{sup} \left\{ f\!\left(x\right) : P\!\left(x\right) \right\} where P ⁣(x)P\!\left(x\right) is a predicate defining the range of xx.
Supremum(f(x, y), Var(x, y), P(x, y)), rendered supP(x,y)f ⁣(x,y)\mathop{\operatorname{sup}}\limits_{P\left(x, y\right)} f\!\left(x, y\right), represents sup{f ⁣(x,y):P ⁣(x,y)}\operatorname{sup} \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 Var(x) declares x as a locally bound variable within the scope of the arguments to this operator. The corresponding predicate P ⁣(x)P\!\left(x\right) 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, Var(x, y), Var(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
SupremumsupP(x)f ⁣(x)\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right) Supremum of a set or function
RRR\mathbb{R} Real numbers
Infinity\infty Positive infinity
OpenInterval(a,b)\left(a, b\right) Open interval
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
Source code for this entry:
Entry(ID("6ec976"),
    SymbolDefinition(Supremum, Supremum(f(x), Var(x), P(x)), "Supremum of a set or function"),
    Description("This operator can be called with 1 or 3 arguments."),
    Description("Called with 1 argument, ", SourceForm(Supremum(S)), ", rendered", Supremum(S), ", represents the supremum of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), ".", "The supremum does not need to be an element of", S, "itself; in particular, for an open interval", Equal(S, OpenInterval(a, b)), ", we have", Equal(Supremum(S), b), "."),
    Description("Called with 3 arguments, ", SourceForm(Supremum(f(x), Var(x), P(x))), ", rendered", Supremum(f(x), Var(x), P(x)), ", represents", Supremum(SetBuilder(f(x), x, P(x))), "where", P(x), "is a predicate defining the range of", x, "."),
    Description(SourceForm(Supremum(f(x, y), Var(x, y), P(x, y))), ", rendered", Supremum(f(x, y), Var(x, y), P(x, y)), ", represents", Supremum(SetBuilder(f(x, y), x, 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(Var(x)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "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(Var(x, y)), ", ", SourceForm(Var(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.

2019-09-19 20:12:49.583742 UTC