Intuitionistic ancestral logic

In this article we define pure intuitionistic Ancestral Logic(
$iAL$
), extending pure intuitionistic First-Order Logic (
$iFOL$
). This logic is a dependently typed abstract programming languagewith computational functionality beyond
$iFOL$
given by its realizer for the transitive closure, TC. We derive this operator from the natural type theoretic definition of TC using intersection. We show that provable formulas in
$iAL$
are uniformly realizable, thus
$iAL$
is sound with respect to constructive type theory. We further show that
$iAL$
subsumes Kleene Algebras with tests and thus serves as a natural programming logic for proving properties of program schemes. We also extract schemes from proofs that
$iAL$
specifications are solvable.