rfunctions | a set of R functions | Functional Programming library

You can simply link to them in the corresponding fields. To go one directory up, you can use ../.

In ZFC, the real numbers satisfy two useful properties:

1. there is a function e : R * R -> bool that returns true if and only if its two arguments are equal, and

2. the order relation is antisymmetric: if x <= y and y <= x, then x = y.

Both of these properties would fail in Coq if the real numbers were defined without additional axioms in terms of Cauchy sequences or Dedekind cuts. For example, a Dedekind cut can be defined as a pair of predicates P Q : rational -> Prop that satisfy certain properties. It is impossible to write a Coq function that decides whether two cuts are equal, because equality of predicates on rationals is undecidable. And any reasonable notion of ordering on cuts would fail to satisfy antisymmetry because equality on predicates is not extensional: it is not the case that forall x, P x <-> Q x implies P = Q.

As for your second question, it is true that there can be only countably many Coq terms of type R -> Prop. However, the same is true of ZFC: there are only countably many formulas for defining subsets of the real numbers. This is connected to the Löwenheim-Skolem paradox, which implies that if ZFC is consistent it has a countable model -- which, in particular, would have only countably many real numbers. Both in ZFC and in Coq, however, it is impossible to define a function that enumerates all real numbers: they are countable from our own external perspective on the theory, but uncountable from the theory's point of view.

I used dplyr however result for plot3 is different and I dont know why. Could you provide your results for each calculations or check mine and let me know where the mistake is?

Also. If your are interested in calculating diversity indices you can get familiar with vegan package and especially diversity() function

You can define CXXFLAGS in your make file: