Charles Conley (1978) has introduced a very weak form of recurrence for flows, which he called “chain recurrence”. Roughly speaking, a point is chain recurrent if it returns to itself by following the flow for an arbitrarily long time, making arbitrarily small jumps, or errors, along the way. He then proved the existence of what he termed as complete Lyapunov function, a real-valued function strictly decreasing everywhere except on components of the chain-recurrent set, where it is constant. This result nowadays is called “The Fundamental Theorem of Dynamical Systems” (D. Norton, 1995). The Conley theory has been generalized for dispersive semiflows by I. U. Bronstein and A. Y. Kopanski (1984-85), and for relations by E. Akin (1993). We are concerned with inclusions xn+1 2 F(xn); n 2 Z; with upper semicontinuous F in a complete metric space (X; d). A sequence (xn); n 2 Z; which satisfies this inclusion is called a chain. If instead of inclusion, the inequality %(xn+1;F(xn)) · ± is satisfied by x = x0; x1; : : : ; xn = y , then one speaks about a ±-chain connecting x and y (here %(a;B) denotes the Hausdorff semidistance). The chain-recurrent, or Conley’s relation CF; is defined as follows: y 2 CF(x) () 8" > 0 there is an "-chain beginning at x and ending at y. This relation is closed and transitive (E. Akin, 1993). The chain-recurrent set jCFj defined as the set of points x, such that x 2 CF(x); is partitioned by the equivalence relation CF \ (CF)¡1 into equivalence classes, coined as basic sets. Our aim is twofold: to generalize the Conley theory for discrete inclusions, and to adapt it to the subdynamics on weakly invariant (viable) subsets. A nonempty subset ¤ ½ X is said to be viable on Z, if for every x 2 ¤ there exists at least a chain xn; n 2 Z, such that x0 = x and xn 2 ¤ for all n 2 Z: In the particular case when F consists of a finite family of affine maps, whose linear parts satisfy the “generalized cone condition”, the maximal compact viable set coincides with the closure of the subset of periodic points, and the dynamics on it is complex enough, say, it is mixing (GG, 2017). A continuous real valued function L on X is called a Lyapunov function for the inclusion xn+1 2 F(xn) (n 2 Z); if y 2 F(x) ) L(y) · L(x): If, in addition, L(x) = L(y) if and only if x and y lie in the same basic set of the chain recurrent set jCFj, then L is said to be complete.Given a compact viable subset ¤ ½ X, we prove that there exists a complete Lyapunov function with respect to Conley’s relation C(Fj¤) of the restriction of the initial relation F on the viable subset ¤. In other words, the complete Lyapunov function decreases strictly beyond the chain recurrent subset of ¤ and separates basic sets in C(Fj¤). This means that the discrete inclusion behaves as a laminar flow behind the chain recurrent set, while the entire complexity is concentrated on the last set.