  This paper introduces a new proof calculus for differential dynamic logic
(dL) that is entirely based on uniform substitution, a proof rule that
substitutes a formula for a predicate symbol everywhere. Uniform substitutions
make it possible to rely on axioms rather than axiom schemata, substantially
simplifying implementations. Instead of nontrivial schema variables and
soundness-critical side conditions on the occurrence patterns of variables, the
resulting calculus adopts only a finite number of ordinary dL formulas as
axioms. The static semantics of differential dynamic logic is captured
exclusively in uniform substitutions and bound variable renamings as opposed to
being spread in delicate ways across the prover implementation. In addition to
sound uniform substitutions, this paper introduces differential forms for
differential dynamic logic that make it possible to internalize differential
invariants, differential substitutions, and derivations as first-class axioms
in dL.
