*** 1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1} / {n__cons/2,n__from/1,s/1} Obligation: Innermost basic terms: {2nd,activate,cons,from}/{n__cons,n__from,s} Applied Processor: InnermostRuleRemoval Proof: Arguments of following rules are not normal-forms. 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) All above mentioned rules can be savely removed. *** 1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1} / {n__cons/2,n__from/1,s/1} Obligation: Innermost basic terms: {2nd,activate,cons,from}/{n__cons,n__from,s} Applied Processor: DependencyPairs {dpKind_ = DT} Proof: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) cons#(X1,X2) -> c_4() from#(X) -> c_5(cons#(X,n__from(s(X)))) from#(X) -> c_6() Weak DPs and mark the set of starting terms. *** 1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) cons#(X1,X2) -> c_4() from#(X) -> c_5(cons#(X,n__from(s(X)))) from#(X) -> c_6() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0} Obligation: Innermost basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) cons#(X1,X2) -> c_4() from#(X) -> c_5(cons#(X,n__from(s(X)))) from#(X) -> c_6() *** 1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2)) activate#(n__from(X)) -> c_3(from#(X)) cons#(X1,X2) -> c_4() from#(X) -> c_5(cons#(X,n__from(s(X)))) from#(X) -> c_6() Strict TRS Rules: Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0} Obligation: Innermost basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s} Applied Processor: Trivial Proof: Consider the dependency graph 1:S:activate#(X) -> c_1() 2:S:activate#(n__cons(X1,X2)) -> c_2(cons#(X1,X2)) -->_1 cons#(X1,X2) -> c_4():4 3:S:activate#(n__from(X)) -> c_3(from#(X)) -->_1 from#(X) -> c_5(cons#(X,n__from(s(X)))):5 -->_1 from#(X) -> c_6():6 4:S:cons#(X1,X2) -> c_4() 5:S:from#(X) -> c_5(cons#(X,n__from(s(X)))) -->_1 cons#(X1,X2) -> c_4():4 6:S:from#(X) -> c_6() The dependency graph contains no loops, we remove all dependency pairs. *** 1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: Weak TRS Rules: Signature: {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0} Obligation: Innermost basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).