Integral of Non Positive Functions

Abstract

Summary.In this article, we formalize in the Mizar system [1,7] PreliminariesLet X be a non empty set and f be a non-negative partial function from X to R. Observe that −f is non-positive.Let f be a non-positive partial function from X to R. One can check that −f is non-negative. Now we state the propositions: (1) Let us consider a non empty set X, a non-positive partial function f from X to R, and a set E. Then f E is non-positive. (2) Let us consider a non empty set X, a set A, a real number r, and a partial function f from X to R. (ii) if c 0, then c · f is non-negative.

A Coq Formalization of the Bochner integral

“…Measure theory and nonnegative Lebesgue integration has been formalized in formal proof assistants such as Mizar 1 , PVS [14], Isabelle/HOL [13], HOL4 2 , Lean [8,16], and Coq 3 . We may cite [2,9] in Mizar, dedicated libraries in PVS 4 , Isabelle/HOL 5 , and Lean 6 , [11] in HOL4, and dedicated libraries 7 and [6] in Coq.…”