WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() U41(tt(),N) -> activate(N) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0()) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0()) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1 ,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate ,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. plus(N,0()) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) x(N,0()) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() U41(tt(),N) -> activate(N) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1 ,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate ,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U32#(tt()) -> c_6() U41#(tt(),N) -> c_7(activate#(N)) U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U61#(tt()) -> c_10(0#()) U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(X) -> c_13() activate#(n__0()) -> c_14(0#()) activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) isNat#(n__0()) -> c_18() isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) plus#(X1,X2) -> c_22() s#(X) -> c_23() x#(X1,X2) -> c_24() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U32#(tt()) -> c_6() U41#(tt(),N) -> c_7(activate#(N)) U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U61#(tt()) -> c_10(0#()) U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(X) -> c_13() activate#(n__0()) -> c_14(0#()) activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) isNat#(n__0()) -> c_18() isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) plus#(X1,X2) -> c_22() s#(X) -> c_23() x#(X1,X2) -> c_24() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() U41(tt(),N) -> activate(N) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5 ,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U32#(tt()) -> c_6() U41#(tt(),N) -> c_7(activate#(N)) U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U61#(tt()) -> c_10(0#()) U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(X) -> c_13() activate#(n__0()) -> c_14(0#()) activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) isNat#(n__0()) -> c_18() isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) plus#(X1,X2) -> c_22() s#(X) -> c_23() x#(X1,X2) -> c_24() * Step 4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U32#(tt()) -> c_6() U41#(tt(),N) -> c_7(activate#(N)) U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U61#(tt()) -> c_10(0#()) U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(X) -> c_13() activate#(n__0()) -> c_14(0#()) activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) isNat#(n__0()) -> c_18() isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) plus#(X1,X2) -> c_22() s#(X) -> c_23() x#(X1,X2) -> c_24() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5 ,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,6,13,18,22,23,24} by application of Pre({1,3,4,6,13,18,22,23,24}) = {2,5,7,8,9,10,11,12,14,15,16,17,19,20,21}. Here rules are labelled as follows: 1: 0#() -> c_1() 2: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 3: U12#(tt()) -> c_3() 4: U21#(tt()) -> c_4() 5: U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 6: U32#(tt()) -> c_6() 7: U41#(tt(),N) -> c_7(activate#(N)) 8: U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 9: U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 10: U61#(tt()) -> c_10(0#()) 11: U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 12: U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 13: activate#(X) -> c_13() 14: activate#(n__0()) -> c_14(0#()) 15: activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) 16: activate#(n__s(X)) -> c_16(s#(X)) 17: activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) 18: isNat#(n__0()) -> c_18() 19: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 20: isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 21: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 22: plus#(X1,X2) -> c_22() 23: s#(X) -> c_23() 24: x#(X1,X2) -> c_24() * Step 5: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U41#(tt(),N) -> c_7(activate#(N)) U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U61#(tt()) -> c_10(0#()) U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(n__0()) -> c_14(0#()) activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak DPs: 0#() -> c_1() U12#(tt()) -> c_3() U21#(tt()) -> c_4() U32#(tt()) -> c_6() activate#(X) -> c_13() isNat#(n__0()) -> c_18() plus#(X1,X2) -> c_22() s#(X) -> c_23() x#(X1,X2) -> c_24() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5 ,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {6,9,10,11,12} by application of Pre({6,9,10,11,12}) = {1,2,3,4,5,7,8,13,14,15}. Here rules are labelled as follows: 1: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 2: U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 3: U41#(tt(),N) -> c_7(activate#(N)) 4: U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 5: U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 6: U61#(tt()) -> c_10(0#()) 7: U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 8: U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 9: activate#(n__0()) -> c_14(0#()) 10: activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) 11: activate#(n__s(X)) -> c_16(s#(X)) 12: activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) 13: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 14: isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 15: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 16: 0#() -> c_1() 17: U12#(tt()) -> c_3() 18: U21#(tt()) -> c_4() 19: U32#(tt()) -> c_6() 20: activate#(X) -> c_13() 21: isNat#(n__0()) -> c_18() 22: plus#(X1,X2) -> c_22() 23: s#(X) -> c_23() 24: x#(X1,X2) -> c_24() * Step 6: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U41#(tt(),N) -> c_7(activate#(N)) U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak DPs: 0#() -> c_1() U12#(tt()) -> c_3() U21#(tt()) -> c_4() U32#(tt()) -> c_6() U61#(tt()) -> c_10(0#()) activate#(X) -> c_13() activate#(n__0()) -> c_14(0#()) activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) isNat#(n__0()) -> c_18() plus#(X1,X2) -> c_22() s#(X) -> c_23() x#(X1,X2) -> c_24() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5 ,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3,5,7} by application of Pre({3,5,7}) = {4,6}. Here rules are labelled as follows: 1: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 2: U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) 3: U41#(tt(),N) -> c_7(activate#(N)) 4: U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 5: U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 6: U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) 7: U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 8: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 9: isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) 10: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) 11: 0#() -> c_1() 12: U12#(tt()) -> c_3() 13: U21#(tt()) -> c_4() 14: U32#(tt()) -> c_6() 15: U61#(tt()) -> c_10(0#()) 16: activate#(X) -> c_13() 17: activate#(n__0()) -> c_14(0#()) 18: activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) 19: activate#(n__s(X)) -> c_16(s#(X)) 20: activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) 21: isNat#(n__0()) -> c_18() 22: plus#(X1,X2) -> c_22() 23: s#(X) -> c_23() 24: x#(X1,X2) -> c_24() * Step 7: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak DPs: 0#() -> c_1() U12#(tt()) -> c_3() U21#(tt()) -> c_4() U32#(tt()) -> c_6() U41#(tt(),N) -> c_7(activate#(N)) U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) U61#(tt()) -> c_10(0#()) U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(X) -> c_13() activate#(n__0()) -> c_14(0#()) activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) activate#(n__s(X)) -> c_16(s#(X)) activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) isNat#(n__0()) -> c_18() plus#(X1,X2) -> c_22() s#(X) -> c_23() x#(X1,X2) -> c_24() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5 ,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_3 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_3 activate#(n__s(X)) -> c_16(s#(X)):19 -->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_3 activate#(n__0()) -> c_14(0#()):17 -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 -->_2 isNat#(n__0()) -> c_18():21 -->_3 activate#(X) -> c_13():16 -->_1 U12#(tt()) -> c_3():9 2:S:U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_3 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_3 activate#(n__s(X)) -> c_16(s#(X)):19 -->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_3 activate#(n__0()) -> c_14(0#()):17 -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 -->_2 isNat#(n__0()) -> c_18():21 -->_3 activate#(X) -> c_13():16 -->_1 U32#(tt()) -> c_6():11 3:S:U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_4 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_3 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_5 activate#(n__s(X)) -> c_16(s#(X)):19 -->_4 activate#(n__s(X)) -> c_16(s#(X)):19 -->_3 activate#(n__s(X)) -> c_16(s#(X)):19 -->_5 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_5 activate#(n__0()) -> c_14(0#()):17 -->_4 activate#(n__0()) -> c_14(0#()):17 -->_3 activate#(n__0()) -> c_14(0#()):17 -->_1 U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)):13 -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 -->_2 isNat#(n__0()) -> c_18():21 -->_5 activate#(X) -> c_13():16 -->_4 activate#(X) -> c_13():16 -->_3 activate#(X) -> c_13():16 4:S:U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_4 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_3 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_5 activate#(n__s(X)) -> c_16(s#(X)):19 -->_4 activate#(n__s(X)) -> c_16(s#(X)):19 -->_3 activate#(n__s(X)) -> c_16(s#(X)):19 -->_5 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_5 activate#(n__0()) -> c_14(0#()):17 -->_4 activate#(n__0()) -> c_14(0#()):17 -->_3 activate#(n__0()) -> c_14(0#()):17 -->_1 U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)):15 -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 -->_2 isNat#(n__0()) -> c_18():21 -->_5 activate#(X) -> c_13():16 -->_4 activate#(X) -> c_13():16 -->_3 activate#(X) -> c_13():16 5:S:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_3 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_4 activate#(n__s(X)) -> c_16(s#(X)):19 -->_3 activate#(n__s(X)) -> c_16(s#(X)):19 -->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_4 activate#(n__0()) -> c_14(0#()):17 -->_3 activate#(n__0()) -> c_14(0#()):17 -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__0()) -> c_18():21 -->_4 activate#(X) -> c_13():16 -->_3 activate#(X) -> c_13():16 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 -->_1 U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):1 6:S:isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_3 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_3 activate#(n__s(X)) -> c_16(s#(X)):19 -->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_3 activate#(n__0()) -> c_14(0#()):17 -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__0()) -> c_18():21 -->_3 activate#(X) -> c_13():16 -->_1 U21#(tt()) -> c_4():10 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 7:S:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_4 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_3 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_4 activate#(n__s(X)) -> c_16(s#(X)):19 -->_3 activate#(n__s(X)) -> c_16(s#(X)):19 -->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_4 activate#(n__0()) -> c_14(0#()):17 -->_3 activate#(n__0()) -> c_14(0#()):17 -->_2 isNat#(n__0()) -> c_18():21 -->_4 activate#(X) -> c_13():16 -->_3 activate#(X) -> c_13():16 -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 -->_1 U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):2 8:W:0#() -> c_1() 9:W:U12#(tt()) -> c_3() 10:W:U21#(tt()) -> c_4() 11:W:U32#(tt()) -> c_6() 12:W:U41#(tt(),N) -> c_7(activate#(N)) -->_1 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_1 activate#(n__s(X)) -> c_16(s#(X)):19 -->_1 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_1 activate#(n__0()) -> c_14(0#()):17 -->_1 activate#(X) -> c_13():16 13:W:U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) -->_4 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_3 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_4 activate#(n__s(X)) -> c_16(s#(X)):19 -->_3 activate#(n__s(X)) -> c_16(s#(X)):19 -->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_4 activate#(n__0()) -> c_14(0#()):17 -->_3 activate#(n__0()) -> c_14(0#()):17 -->_1 s#(X) -> c_23():23 -->_2 plus#(X1,X2) -> c_22():22 -->_4 activate#(X) -> c_13():16 -->_3 activate#(X) -> c_13():16 14:W:U61#(tt()) -> c_10(0#()) -->_1 0#() -> c_1():8 15:W:U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_5 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_4 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_3 activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)):20 -->_5 activate#(n__s(X)) -> c_16(s#(X)):19 -->_4 activate#(n__s(X)) -> c_16(s#(X)):19 -->_3 activate#(n__s(X)) -> c_16(s#(X)):19 -->_5 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)):18 -->_5 activate#(n__0()) -> c_14(0#()):17 -->_4 activate#(n__0()) -> c_14(0#()):17 -->_3 activate#(n__0()) -> c_14(0#()):17 -->_2 x#(X1,X2) -> c_24():24 -->_1 plus#(X1,X2) -> c_22():22 -->_5 activate#(X) -> c_13():16 -->_4 activate#(X) -> c_13():16 -->_3 activate#(X) -> c_13():16 16:W:activate#(X) -> c_13() 17:W:activate#(n__0()) -> c_14(0#()) -->_1 0#() -> c_1():8 18:W:activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_22():22 19:W:activate#(n__s(X)) -> c_16(s#(X)) -->_1 s#(X) -> c_23():23 20:W:activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) -->_1 x#(X1,X2) -> c_24():24 21:W:isNat#(n__0()) -> c_18() 22:W:plus#(X1,X2) -> c_22() 23:W:s#(X) -> c_23() 24:W:x#(X1,X2) -> c_24() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: U61#(tt()) -> c_10(0#()) 12: U41#(tt(),N) -> c_7(activate#(N)) 15: U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 13: U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))) ,plus#(activate(N),activate(M)) ,activate#(N) ,activate#(M)) 9: U12#(tt()) -> c_3() 11: U32#(tt()) -> c_6() 10: U21#(tt()) -> c_4() 16: activate#(X) -> c_13() 21: isNat#(n__0()) -> c_18() 17: activate#(n__0()) -> c_14(0#()) 8: 0#() -> c_1() 18: activate#(n__plus(X1,X2)) -> c_15(plus#(X1,X2)) 22: plus#(X1,X2) -> c_22() 19: activate#(n__s(X)) -> c_16(s#(X)) 23: s#(X) -> c_23() 20: activate#(n__x(X1,X2)) -> c_17(x#(X1,X2)) 24: x#(X1,X2) -> c_24() * Step 8: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5 ,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 2:S:U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 3:S:U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 4:S:U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)) ,isNat#(activate(N)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 5:S:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 -->_1 U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):1 6:S:isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 7:S:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):7 -->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)) ,isNat#(activate(V1)) ,activate#(V1) ,activate#(V2)):5 -->_1 U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) * Step 9: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U31#(tt(),V2) -> c_5(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71# ,U72#,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} Problem (S) - Strict DPs: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) - Weak DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U31#(tt(),V2) -> c_5(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71# ,U72#,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} ** Step 9.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U31#(tt(),V2) -> c_5(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) 6: isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) Consider the set of all dependency pairs 1: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) 2: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) 3: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) 4: U71#(tt(),M,N) -> c_11(isNat#(activate(N))) 5: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) 7: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2,6} These cover all (indirect) predecessors of dependency pairs {2,3,4,6} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 9.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U31#(tt(),V2) -> c_5(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_11) = {1}, uargs(c_19) = {1,2}, uargs(c_20) = {1}, uargs(c_21) = {1,2} Following symbols are considered usable: {0,U11,U12,U21,U31,U32,activate,isNat,plus,s,x,0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] p(U12) = [1] p(U21) = [1] x1 + [0] p(U31) = [0] p(U32) = [0] p(U41) = [0] p(U51) = [0] p(U52) = [1] x1 + [1] x2 + [4] x3 + [1] p(U61) = [1] p(U71) = [1] x2 + [1] x3 + [0] p(U72) = [1] x1 + [1] x2 + [0] p(activate) = [1] x1 + [0] p(isNat) = [1] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [3] p(n__s) = [1] x1 + [2] p(n__x) = [1] x1 + [1] x2 + [5] p(plus) = [1] x1 + [1] x2 + [3] p(s) = [1] x1 + [2] p(tt) = [0] p(x) = [1] x1 + [1] x2 + [5] p(0#) = [0] p(U11#) = [2] x1 + [1] x2 + [1] p(U12#) = [2] x1 + [0] p(U21#) = [0] p(U31#) = [1] x2 + [4] p(U32#) = [0] p(U41#) = [0] p(U51#) = [4] x3 + [4] p(U52#) = [0] p(U61#) = [0] p(U71#) = [1] x1 + [1] x3 + [1] p(U72#) = [1] x2 + [4] x3 + [0] p(activate#) = [0] p(isNat#) = [1] x1 + [1] p(plus#) = [1] x1 + [1] x2 + [1] p(s#) = [0] p(x#) = [1] x1 + [4] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [4] p(c_8) = [4] x1 + [0] p(c_9) = [2] x1 + [1] x2 + [4] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [4] x3 + [1] p(c_13) = [0] p(c_14) = [4] x1 + [1] p(c_15) = [1] x1 + [1] p(c_16) = [0] p(c_17) = [1] p(c_18) = [4] p(c_19) = [1] x1 + [1] x2 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [1] x1 + [1] x2 + [1] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] Following rules are strictly oriented: U31#(tt(),V2) = [1] V2 + [4] > [1] V2 + [1] = c_5(isNat#(activate(V2))) isNat#(n__s(V1)) = [1] V1 + [3] > [1] V1 + [1] = c_20(isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [1] V2 + [1] >= [1] V2 + [1] = c_2(isNat#(activate(V2))) U51#(tt(),M,N) = [4] N + [4] >= [4] N + [4] = c_8(isNat#(activate(N))) U71#(tt(),M,N) = [1] N + [1] >= [1] N + [1] = c_11(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [4] = c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [1] V1 + [1] V2 + [6] >= [1] V1 + [1] V2 + [6] = c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 0() = [0] >= [0] = n__0() U11(tt(),V2) = [1] >= [1] = U12(isNat(activate(V2))) U12(tt()) = [1] >= [0] = tt() U21(tt()) = [0] >= [0] = tt() U31(tt(),V2) = [0] >= [0] = U32(isNat(activate(V2))) U32(tt()) = [0] >= [0] = tt() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = plus(X1,X2) activate(n__s(X)) = [1] X + [2] >= [1] X + [2] = s(X) activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = x(X1,X2) isNat(n__0()) = [1] >= [0] = tt() isNat(n__plus(V1,V2)) = [1] >= [1] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [1] >= [1] = U21(isNat(activate(V1))) isNat(n__x(V1,V2)) = [1] >= [0] = U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = n__plus(X1,X2) s(X) = [1] X + [2] >= [1] X + [2] = n__s(X) x(X1,X2) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = n__x(X1,X2) *** Step 9.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 9.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Consider the set of all dependency pairs 1: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) 2: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 3: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 4: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) 5: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) 6: U71#(tt(),M,N) -> c_11(isNat#(activate(N))) 7: isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {3} These cover all (indirect) predecessors of dependency pairs {3,4,5,6} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 9.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_11) = {1}, uargs(c_19) = {1,2}, uargs(c_20) = {1}, uargs(c_21) = {1,2} Following symbols are considered usable: {0,activate,plus,s,x,0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72#,activate#,isNat#,plus#,s# ,x#} TcT has computed the following interpretation: p(0) = [4] p(U11) = [5] x2 + [2] p(U12) = [6] p(U21) = [4] p(U31) = [1] x2 + [0] p(U32) = [1] x1 + [1] p(U41) = [4] x1 + [1] x2 + [0] p(U51) = [4] x2 + [1] x3 + [4] p(U52) = [4] x2 + [1] x3 + [2] p(U61) = [1] x1 + [2] p(U71) = [1] p(U72) = [1] x1 + [2] x2 + [4] x3 + [0] p(activate) = [1] x1 + [0] p(isNat) = [0] p(n__0) = [4] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [0] p(n__x) = [1] x1 + [1] x2 + [2] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [0] p(tt) = [0] p(x) = [1] x1 + [1] x2 + [2] p(0#) = [0] p(U11#) = [4] x2 + [0] p(U12#) = [0] p(U21#) = [1] x1 + [1] p(U31#) = [4] x2 + [0] p(U32#) = [1] x1 + [0] p(U41#) = [2] x1 + [1] x2 + [1] p(U51#) = [4] x3 + [3] p(U52#) = [1] x1 + [1] x3 + [1] p(U61#) = [1] x1 + [1] p(U71#) = [1] x1 + [4] x3 + [2] p(U72#) = [1] x2 + [2] x3 + [1] p(activate#) = [2] p(isNat#) = [4] x1 + [0] p(plus#) = [4] x2 + [0] p(s#) = [1] p(x#) = [1] x2 + [0] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [2] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [4] p(c_8) = [1] x1 + [3] p(c_9) = [1] x1 + [1] x2 + [2] x4 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x3 + [1] x4 + [1] x5 + [0] p(c_13) = [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] p(c_18) = [0] p(c_19) = [1] x1 + [1] x2 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [1] x1 + [1] x2 + [1] p(c_22) = [1] p(c_23) = [1] p(c_24) = [0] Following rules are strictly oriented: isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [8] > [4] V1 + [4] V2 + [1] = c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = c_2(isNat#(activate(V2))) U31#(tt(),V2) = [4] V2 + [0] >= [4] V2 + [0] = c_5(isNat#(activate(V2))) U51#(tt(),M,N) = [4] N + [3] >= [4] N + [3] = c_8(isNat#(activate(N))) U71#(tt(),M,N) = [4] N + [2] >= [4] N + [0] = c_11(isNat#(activate(N))) isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0] >= [4] V1 + [4] V2 + [0] = c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) = [4] V1 + [0] >= [4] V1 + [0] = c_20(isNat#(activate(V1))) 0() = [4] >= [4] = n__0() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [4] >= [4] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(X1,X2) activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(X) activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = x(X1,X2) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) x(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__x(X1,X2) **** Step 9.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 9.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Consider the set of all dependency pairs 1: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) 2: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 3: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) 4: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) 5: U71#(tt(),M,N) -> c_11(isNat#(activate(N))) 6: isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) 7: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {1,2,4,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 9.a:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak DPs: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_11) = {1}, uargs(c_19) = {1,2}, uargs(c_20) = {1}, uargs(c_21) = {1,2} Following symbols are considered usable: {0,activate,plus,s,x,0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72#,activate#,isNat#,plus#,s# ,x#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U12) = [0] p(U21) = [0] p(U31) = [1] x2 + [0] p(U32) = [0] p(U41) = [0] p(U51) = [0] p(U52) = [0] p(U61) = [0] p(U71) = [0] p(U72) = [0] p(activate) = [1] x1 + [0] p(isNat) = [6] x1 + [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [1] p(n__s) = [1] x1 + [0] p(n__x) = [1] x1 + [1] x2 + [2] p(plus) = [1] x1 + [1] x2 + [1] p(s) = [1] x1 + [0] p(tt) = [0] p(x) = [1] x1 + [1] x2 + [2] p(0#) = [0] p(U11#) = [1] x2 + [0] p(U12#) = [0] p(U21#) = [0] p(U31#) = [1] x2 + [0] p(U32#) = [0] p(U41#) = [0] p(U51#) = [6] x3 + [2] p(U52#) = [0] p(U61#) = [0] p(U71#) = [1] x3 + [1] p(U72#) = [0] p(activate#) = [0] p(isNat#) = [1] x1 + [0] p(plus#) = [0] p(s#) = [0] p(x#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [1] p(c_8) = [4] x1 + [1] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [4] x4 + [1] x5 + [2] p(c_13) = [0] p(c_14) = [1] x1 + [1] p(c_15) = [1] p(c_16) = [0] p(c_17) = [0] p(c_18) = [1] p(c_19) = [1] x1 + [1] x2 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [1] x1 + [1] x2 + [2] p(c_22) = [0] p(c_23) = [1] p(c_24) = [1] Following rules are strictly oriented: isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [1] > [1] V1 + [1] V2 + [0] = c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) Following rules are (at-least) weakly oriented: U11#(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = c_2(isNat#(activate(V2))) U31#(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = c_5(isNat#(activate(V2))) U51#(tt(),M,N) = [6] N + [2] >= [4] N + [1] = c_8(isNat#(activate(N))) U71#(tt(),M,N) = [1] N + [1] >= [1] N + [1] = c_11(isNat#(activate(N))) isNat#(n__s(V1)) = [1] V1 + [0] >= [1] V1 + [0] = c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [1] V1 + [1] V2 + [2] >= [1] V1 + [1] V2 + [2] = c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 0() = [0] >= [0] = n__0() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = plus(X1,X2) activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(X) activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = x(X1,X2) plus(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = n__plus(X1,X2) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) x(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__x(X1,X2) ***** Step 9.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) - Weak DPs: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 9.a:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U11#(tt(),V2) -> c_2(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 2:W:U31#(tt(),V2) -> c_5(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 3:W:U51#(tt(),M,N) -> c_8(isNat#(activate(N))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 4:W:U71#(tt(),M,N) -> c_11(isNat#(activate(N))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 5:W:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 -->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2))):1 6:W:isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 7:W:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 -->_1 U31#(tt(),V2) -> c_5(isNat#(activate(V2))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: U71#(tt(),M,N) -> c_11(isNat#(activate(N))) 3: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) 1: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) 5: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 7: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) 2: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) ***** Step 9.a:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 9.b:1: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) - Weak DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U31#(tt(),V2) -> c_5(isNat#(activate(V2))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2} by application of Pre({1,2}) = {}. Here rules are labelled as follows: 1: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) 2: U71#(tt(),M,N) -> c_11(isNat#(activate(N))) 3: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) 4: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) 5: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) 7: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) ** Step 9.b:2: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) U31#(tt(),V2) -> c_5(isNat#(activate(V2))) U51#(tt(),M,N) -> c_8(isNat#(activate(N))) U71#(tt(),M,N) -> c_11(isNat#(activate(N))) isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U11#(tt(),V2) -> c_2(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 2:W:U31#(tt(),V2) -> c_5(isNat#(activate(V2))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 3:W:U51#(tt(),M,N) -> c_8(isNat#(activate(N))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 4:W:U71#(tt(),M,N) -> c_11(isNat#(activate(N))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 5:W:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 -->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2))):1 6:W:isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 7:W:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) -->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7 -->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6 -->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5 -->_1 U31#(tt(),V2) -> c_5(isNat#(activate(V2))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: U71#(tt(),M,N) -> c_11(isNat#(activate(N))) 3: U51#(tt(),M,N) -> c_8(isNat#(activate(N))) 1: U11#(tt(),V2) -> c_2(isNat#(activate(V2))) 5: isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 7: isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))) 6: isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))) 2: U31#(tt(),V2) -> c_5(isNat#(activate(V2))) ** Step 9.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2 ,0#/0,U11#/2,U12#/1,U21#/1,U31#/2,U32#/1,U41#/2,U51#/3,U52#/3,U61#/1,U71#/3,U72#/3,activate#/1,isNat#/1 ,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1 ,c_9/4,c_10/1,c_11/1,c_12/5,c_13/0,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0 ,c_24/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U32#,U41#,U51#,U52#,U61#,U71#,U72# ,activate#,isNat#,plus#,s#,x#} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))