In a dependent type theory satisfying the propositions as
types correspondence together with the proofs-as-programs paradigm,
the validity of the unique choice rule or even more of the choice rule says
that the extraction of a computable witness from an existential statement
under hypothesis can be performed within the same theory.
Here we show that the unique choice rule, and hence the choice rule,
are not valid both in Coquand\u2019s Calculus of Constructions with indexed
sum types, list types and binary disjoint sums and in its predicative
version implemented in the intensional level of the Minimalist Founda-
tion. This means that in these theories the extraction of computational
witnesses from existential statements must be performed in a more ex-
pressive proofs-as-programs theory
