Formalization of real analysis: a survey of proof assistants and libraries
For simplicity, we adopt the following rules: a, a1, a2, b, b1, b2 are real numbers, p is a finite sequence, F , G, H are finite sequences of elements of R, i, j, k are natural numbers, f is a function from R into R, and x1 is a set. Let I1 be a subset of R. We say that I1 is closed-interval if and only if: (Def. 1) There exist real numbers a, b such that a ¬ b and I1 = [a, b]. Let us mention that there exists a subset of R which is closed-interval. In the sequel A, A1, A2 are closed-interval subsets of R. The following propositions are true: (1) Every closed-interval subset of R is compact.