YES(?,POLY) 'Pop* (timeout of 60.0 seconds)' -------------------------------- Answer: YES(?,POLY) Input Problem: innermost runtime-complexity with respect to Rules: { a__f(f(X)) -> a__c(f(g(f(X)))) , a__c(X) -> d(X) , a__h(X) -> a__c(d(X)) , mark(f(X)) -> a__f(mark(X)) , mark(c(X)) -> a__c(X) , mark(h(X)) -> a__h(mark(X)) , mark(g(X)) -> g(X) , mark(d(X)) -> d(X) , a__f(X) -> f(X) , a__c(X) -> c(X) , a__h(X) -> h(X)} Proof Output: The input was oriented with the instance of POP* as induced by the precedence a__f > a__c, a__f > a__h, a__h > a__c, mark > a__f, mark > a__c, mark > a__h and safe mapping safe(a__f) = {1}, safe(f) = {1}, safe(a__c) = {1}, safe(g) = {1}, safe(d) = {1}, safe(a__h) = {1}, safe(mark) = {}, safe(c) = {1}, safe(h) = {1} . For your convenience, here is the input in predicative notation: Rules: { a__f(; f(; X)) -> a__c(; f(; g(; f(; X)))) , a__c(; X) -> d(; X) , a__h(; X) -> a__c(; d(; X)) , mark(f(; X);) -> a__f(; mark(X;)) , mark(c(; X);) -> a__c(; X) , mark(h(; X);) -> a__h(; mark(X;)) , mark(g(; X);) -> g(; X) , mark(d(; X);) -> d(; X) , a__f(; X) -> f(; X) , a__c(; X) -> c(; X) , a__h(; X) -> h(; X)}