*** 1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,from/1} / {cons/2,cons1/2,n__from/1,s/1} Obligation: Innermost basic terms: {2nd,activate,from}/{cons,cons1,n__from,s} Applied Processor: DependencyPairs {dpKind_ = DT} Proof: We add the following dependency tuples: Strict DPs 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) 2nd#(cons1(X,cons(Y,Z))) -> c_2() activate#(X) -> c_3() activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5() from#(X) -> c_6() Weak DPs and mark the set of starting terms. *** 1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) 2nd#(cons1(X,cons(Y,Z))) -> c_2() activate#(X) -> c_3() activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5() from#(X) -> c_6() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) 2nd(cons1(X,cons(Y,Z))) -> Y activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} Obligation: Innermost basic terms: {2nd#,activate#,from#}/{cons,cons1,n__from,s} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) 2nd#(cons1(X,cons(Y,Z))) -> c_2() activate#(X) -> c_3() activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5() from#(X) -> c_6() *** 1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: 2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) 2nd#(cons1(X,cons(Y,Z))) -> c_2() activate#(X) -> c_3() activate#(n__from(X)) -> c_4(from#(X)) from#(X) -> c_5() from#(X) -> c_6() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} Obligation: Innermost basic terms: {2nd#,activate#,from#}/{cons,cons1,n__from,s} Applied Processor: Trivial Proof: Consider the dependency graph 1:S:2nd#(cons(X,X1)) -> c_1(2nd#(cons1(X,activate(X1))),activate#(X1)) -->_2 activate#(n__from(X)) -> c_4(from#(X)):4 -->_2 activate#(X) -> c_3():3 -->_1 2nd#(cons1(X,cons(Y,Z))) -> c_2():2 2:S:2nd#(cons1(X,cons(Y,Z))) -> c_2() 3:S:activate#(X) -> c_3() 4:S:activate#(n__from(X)) -> c_4(from#(X)) -->_1 from#(X) -> c_6():6 -->_1 from#(X) -> c_5():5 5:S:from#(X) -> c_5() 6:S:from#(X) -> c_6() The dependency graph contains no loops, we remove all dependency pairs. *** 1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {2nd/1,activate/1,from/1,2nd#/1,activate#/1,from#/1} / {cons/2,cons1/2,n__from/1,s/1,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} Obligation: Innermost basic terms: {2nd#,activate#,from#}/{cons,cons1,n__from,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).