On the Strong Co-induction in Coq

Using theorem prover Coq [4] for the verification of concurrent systems usually includes the formalization of the modal μ–calculus [8] in that logical framework [13] [11] [15]. These implementations build upon the following interpretation: given the state space S of a system, the meaning of a formula is the subset of states where it is satisfied. Then, a formula α(X) with a free variable X can be seen as a function αX : 2 S → 2 which maps each T ∈ 2 to the meaning of α(X) when X is given the value T. If X occurs only positively in α(X) then αX is a monotone function and the theory of monotone functions on the complete lattice 2 applies. In particular, the Knaster–Tarski theorem [14] on the existence of least (μS.αX(S)) and greatest (νS.αX(S)) fixed points of αX . In this paper, we provide a library in Coq containing intuitionistic proofs of some facts that are on the basis of formal verification tools such as Model Checking or Theorem Proving: the Reduction Lemma [8] [17] and the Well–founded induction on minimum fixed points [1]. In order to improve usability, most of the proofs are given in a general frame of partial order relations and not only in the specific complete lattice of a power-set. Find the complete Coq code in http://lfcia.org/publications.shtml