MAYBE TRS: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} DP: Strict: { f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3), active#(f(X1, X2, X3)) -> f#(X1, X2, active(X3)), active#(f(X1, X2, X3)) -> f#(active(X1), X2, X3), active#(f(X1, X2, X3)) -> active#(X1), active#(f(X1, X2, X3)) -> active#(X3), active#(f(a(), b(), X)) -> f#(X, X, X), proper#(f(X1, X2, X3)) -> f#(proper(X1), proper(X2), proper(X3)), proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X2), proper#(f(X1, X2, X3)) -> proper#(X3), top#(mark(X)) -> proper#(X), top#(mark(X)) -> top#(proper(X)), top#(ok(X)) -> active#(X), top#(ok(X)) -> top#(active(X))} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} EDG: {(proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X3)) (proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X2)) (proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X1)) (proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> f#(proper(X1), proper(X2), proper(X3))) (top#(ok(X)) -> top#(active(X)), top#(ok(X)) -> top#(active(X))) (top#(ok(X)) -> top#(active(X)), top#(ok(X)) -> active#(X)) (top#(ok(X)) -> top#(active(X)), top#(mark(X)) -> top#(proper(X))) (top#(ok(X)) -> top#(active(X)), top#(mark(X)) -> proper#(X)) (active#(f(X1, X2, X3)) -> f#(X1, X2, active(X3)), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3)) (active#(f(X1, X2, X3)) -> f#(X1, X2, active(X3)), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)) (active#(f(X1, X2, X3)) -> f#(X1, X2, active(X3)), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)) (active#(f(X1, X2, X3)) -> active#(X3), active#(f(a(), b(), X)) -> f#(X, X, X)) (active#(f(X1, X2, X3)) -> active#(X3), active#(f(X1, X2, X3)) -> active#(X3)) (active#(f(X1, X2, X3)) -> active#(X3), active#(f(X1, X2, X3)) -> active#(X1)) (active#(f(X1, X2, X3)) -> active#(X3), active#(f(X1, X2, X3)) -> f#(active(X1), X2, X3)) (active#(f(X1, X2, X3)) -> active#(X3), active#(f(X1, X2, X3)) -> f#(X1, X2, active(X3))) (active#(f(a(), b(), X)) -> f#(X, X, X), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3)) (active#(f(a(), b(), X)) -> f#(X, X, X), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)) (active#(f(a(), b(), X)) -> f#(X, X, X), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)) (top#(ok(X)) -> active#(X), active#(f(a(), b(), X)) -> f#(X, X, X)) (top#(ok(X)) -> active#(X), active#(f(X1, X2, X3)) -> active#(X3)) (top#(ok(X)) -> active#(X), active#(f(X1, X2, X3)) -> active#(X1)) (top#(ok(X)) -> active#(X), active#(f(X1, X2, X3)) -> f#(active(X1), X2, X3)) (top#(ok(X)) -> active#(X), active#(f(X1, X2, X3)) -> f#(X1, X2, active(X3))) (f#(mark(X1), X2, X3) -> f#(X1, X2, X3), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3)) (f#(mark(X1), X2, X3) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)) (f#(mark(X1), X2, X3) -> f#(X1, X2, X3), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)) (active#(f(X1, X2, X3)) -> f#(active(X1), X2, X3), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3)) (active#(f(X1, X2, X3)) -> f#(active(X1), X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)) (active#(f(X1, X2, X3)) -> f#(active(X1), X2, X3), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)) (f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)) (f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)) (f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3)) (f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)) (f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)) (f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3)) (top#(mark(X)) -> proper#(X), proper#(f(X1, X2, X3)) -> f#(proper(X1), proper(X2), proper(X3))) (top#(mark(X)) -> proper#(X), proper#(f(X1, X2, X3)) -> proper#(X1)) (top#(mark(X)) -> proper#(X), proper#(f(X1, X2, X3)) -> proper#(X2)) (top#(mark(X)) -> proper#(X), proper#(f(X1, X2, X3)) -> proper#(X3)) (proper#(f(X1, X2, X3)) -> proper#(X3), proper#(f(X1, X2, X3)) -> f#(proper(X1), proper(X2), proper(X3))) (proper#(f(X1, X2, X3)) -> proper#(X3), proper#(f(X1, X2, X3)) -> proper#(X1)) (proper#(f(X1, X2, X3)) -> proper#(X3), proper#(f(X1, X2, X3)) -> proper#(X2)) (proper#(f(X1, X2, X3)) -> proper#(X3), proper#(f(X1, X2, X3)) -> proper#(X3)) (proper#(f(X1, X2, X3)) -> f#(proper(X1), proper(X2), proper(X3)), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)) (proper#(f(X1, X2, X3)) -> f#(proper(X1), proper(X2), proper(X3)), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)) (proper#(f(X1, X2, X3)) -> f#(proper(X1), proper(X2), proper(X3)), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3)) (proper#(f(X1, X2, X3)) -> proper#(X2), proper#(f(X1, X2, X3)) -> f#(proper(X1), proper(X2), proper(X3))) (proper#(f(X1, X2, X3)) -> proper#(X2), proper#(f(X1, X2, X3)) -> proper#(X1)) (proper#(f(X1, X2, X3)) -> proper#(X2), proper#(f(X1, X2, X3)) -> proper#(X2)) (proper#(f(X1, X2, X3)) -> proper#(X2), proper#(f(X1, X2, X3)) -> proper#(X3)) (top#(mark(X)) -> top#(proper(X)), top#(mark(X)) -> proper#(X)) (top#(mark(X)) -> top#(proper(X)), top#(mark(X)) -> top#(proper(X))) (top#(mark(X)) -> top#(proper(X)), top#(ok(X)) -> active#(X)) (top#(mark(X)) -> top#(proper(X)), top#(ok(X)) -> top#(active(X))) (active#(f(X1, X2, X3)) -> active#(X1), active#(f(X1, X2, X3)) -> f#(X1, X2, active(X3))) (active#(f(X1, X2, X3)) -> active#(X1), active#(f(X1, X2, X3)) -> f#(active(X1), X2, X3)) (active#(f(X1, X2, X3)) -> active#(X1), active#(f(X1, X2, X3)) -> active#(X1)) (active#(f(X1, X2, X3)) -> active#(X1), active#(f(X1, X2, X3)) -> active#(X3)) (active#(f(X1, X2, X3)) -> active#(X1), active#(f(a(), b(), X)) -> f#(X, X, X))} SCCS: Scc: {top#(mark(X)) -> top#(proper(X)), top#(ok(X)) -> top#(active(X))} Scc: {proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X2), proper#(f(X1, X2, X3)) -> proper#(X3)} Scc: {active#(f(X1, X2, X3)) -> active#(X1), active#(f(X1, X2, X3)) -> active#(X3)} Scc: { f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3)} SCC: Strict: {top#(mark(X)) -> top#(proper(X)), top#(ok(X)) -> top#(active(X))} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} Fail SCC: Strict: {proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X2), proper#(f(X1, X2, X3)) -> proper#(X3)} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} SPSC: Simple Projection: pi(proper#) = 0 Strict: {proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X3)} EDG: {(proper#(f(X1, X2, X3)) -> proper#(X3), proper#(f(X1, X2, X3)) -> proper#(X3)) (proper#(f(X1, X2, X3)) -> proper#(X3), proper#(f(X1, X2, X3)) -> proper#(X1)) (proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X1)) (proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X3))} SCCS: Scc: {proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X3)} SCC: Strict: {proper#(f(X1, X2, X3)) -> proper#(X1), proper#(f(X1, X2, X3)) -> proper#(X3)} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} SPSC: Simple Projection: pi(proper#) = 0 Strict: {proper#(f(X1, X2, X3)) -> proper#(X3)} EDG: {(proper#(f(X1, X2, X3)) -> proper#(X3), proper#(f(X1, X2, X3)) -> proper#(X3))} SCCS: Scc: {proper#(f(X1, X2, X3)) -> proper#(X3)} SCC: Strict: {proper#(f(X1, X2, X3)) -> proper#(X3)} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} SPSC: Simple Projection: pi(proper#) = 0 Strict: {} Qed SCC: Strict: {active#(f(X1, X2, X3)) -> active#(X1), active#(f(X1, X2, X3)) -> active#(X3)} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} SPSC: Simple Projection: pi(active#) = 0 Strict: {active#(f(X1, X2, X3)) -> active#(X3)} EDG: {(active#(f(X1, X2, X3)) -> active#(X3), active#(f(X1, X2, X3)) -> active#(X3))} SCCS: Scc: {active#(f(X1, X2, X3)) -> active#(X3)} SCC: Strict: {active#(f(X1, X2, X3)) -> active#(X3)} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} SPSC: Simple Projection: pi(active#) = 0 Strict: {} Qed SCC: Strict: { f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3), f#(ok(X1), ok(X2), ok(X3)) -> f#(X1, X2, X3)} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} SPSC: Simple Projection: pi(f#) = 0 Strict: {f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)} EDG: {(f#(mark(X1), X2, X3) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)) (f#(mark(X1), X2, X3) -> f#(X1, X2, X3), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)) (f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)) (f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3))} SCCS: Scc: {f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)} SCC: Strict: {f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(mark(X1), X2, X3) -> f#(X1, X2, X3)} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} SPSC: Simple Projection: pi(f#) = 0 Strict: {f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)} EDG: {(f#(X1, X2, mark(X3)) -> f#(X1, X2, X3), f#(X1, X2, mark(X3)) -> f#(X1, X2, X3))} SCCS: Scc: {f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)} SCC: Strict: {f#(X1, X2, mark(X3)) -> f#(X1, X2, X3)} Weak: { f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)), f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)), f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)), active(f(X1, X2, X3)) -> f(X1, X2, active(X3)), active(f(X1, X2, X3)) -> f(active(X1), X2, X3), active(f(a(), b(), X)) -> mark(f(X, X, X)), active(c()) -> mark(a()), active(c()) -> mark(b()), proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)), proper(a()) -> ok(a()), proper(b()) -> ok(b()), proper(c()) -> ok(c()), top(mark(X)) -> top(proper(X)), top(ok(X)) -> top(active(X))} SPSC: Simple Projection: pi(f#) = 2 Strict: {} Qed