CIRC: A Circular Coinductive Prover

Abstract

Abstract. CIRC is an automated circular coinductive prover implemented as an extension of Maude. The circular coinductive technique that forms the core of CIRC is discussed, together with a high-level implementation using metalevel capabilities of rewriting logic. To reflect the strength of CIRC in automatically proving behavioral properties, an example defining and proving properties about infinite streams of infinite binary trees is shown. CIRC also provides limited support for automated inductive proving, which can be used in combination with coinduction.

Well-Definedness of Streams by Termination

“…We presented a technique by which well-definedness of stream specifications like f (0 : σ) = 1 : f (σ) f (1 : σ) = 0 : f (f (σ) c = 1 : c can be proved fully automatically, where a tool like Circ [8] fails, and the productivity tool [3] fails to prove productivity of f (c). The main idea is to prove well-definedness by proving termination of a transformed system Obs(R s ), in this way exploiting the power of present termination provers.…”

“…Both [3] and [11] fail to prove that the identity is the only stream function satisfying the equation for f . By essentially choosing Obs(R s ) as the input and adding information about special contexts, the tool Circ [8] is able to prove that f is the identity.…”

“…Finding such an arbitrary bisimulation is a hard problem in the general setting, but restricting to circular co-induction [6] finding this automatically is tractable. A strong tool doing so is Circ [8]. The tool Circ focuses on proving equality, but proving well-definedness of a function f can also be proved by equality as long as the equations for f are orthogonal: take a copy f of f with the same equations, and prove f = f .…”

A Tool Proving Well-Definedness of Streams Using Termination Tools

“…An approach to prove productivity is given in [1], together with a corresponding tool. The second s by proving equality of two copies of the same specification by the tool Circ [4]. In the latter it is essential that the specification is deterministic.…”

Circular Coinduction: A Proof Theoretical Foundation

“…At our knowledge, variants of circular coinduction have been implemented in three systems so far: in a behavioral extension of OBJ called BOBJ [13] (not maintained anymore), in Isabelle/HOL for CoCasl [7], and in CIRC [9]. The latter implements a more general circular reasoning principle that will be discussed elsewhere in its full generality; the coinductive projection of this general principle is the subject of this paper and will be discussed in detail in the sequel.…”