# Toward Structured Proofs for Dynamic Logics

## TL;DR

Kaisar is a structured proof language for differential dynamic logic that introduces nominals to simplify reasoning about historical states in safety-critical cyber-physical systems, supported by formal metatheory.

## Contribution

The paper introduces Kaisar, a novel structured proof language with first-class nominals for differential dynamic logic, enhancing CPS proofs and providing formal semantics and metatheory.

## Key findings

- Successfully implemented Kaisar in KeYmaera X
- Reproduced safety proofs for parachute and robot control
- Proved soundness, completeness, and formal properties of Kaisar

## Abstract

We present Kaisar, a structured interactive proof language for differential dynamic logic (dL), for safety-critical cyber-physical systems (CPS). The defining feature of Kaisar is *nominal terms*, which simplify CPS proofs by making the frequently needed historical references to past program states first-class. To support nominals, we extend the notion of structured proof with a first-class notion of *structured symbolic execution* of CPS models. We implement Kaisar in the theorem prover KeYmaera X and reproduce an example on the safe operation of a parachute and a case study on ground robot control. We show how nominals simplify common CPS reasoning tasks when combined with other features of structured proof. We develop an extensive metatheory for Kaisar. In addition to soundness and completeness, we show a formal specification for Kaisar's nominals and relate Kaisar to a nominal variant of dL.