WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) U31(tt()) -> 0() U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) plus(N,0()) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0()) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1 ,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,activate,and,isNat,plus,s ,x} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. plus(N,0()) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) x(N,0()) -> U31(isNat(N)) x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),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(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) U31(tt()) -> 0() U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1 ,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,activate,and,isNat,plus,s ,x} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs 0#() -> c_1() U11#(tt(),N) -> c_2(activate#(N)) U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U31#(tt()) -> c_4(0#()) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__0()) -> c_14() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() 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(),N) -> c_2(activate#(N)) U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U31#(tt()) -> c_4(0#()) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__0()) -> c_14() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Strict TRS: 0() -> n__0() U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) U31(tt()) -> 0() U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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(),N) -> c_2(activate#(N)) U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U31#(tt()) -> c_4(0#()) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__0()) -> c_14() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),N) -> c_2(activate#(N)) U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U31#(tt()) -> c_4(0#()) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__0()) -> c_14() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(and) = {1,2}, uargs(isNat) = {1}, uargs(n__isNat) = {1}, uargs(plus) = {1,2}, uargs(x) = {1,2}, uargs(and#) = {1,2}, uargs(isNat#) = {1}, uargs(plus#) = {1,2}, uargs(s#) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(U11) = [0] p(U21) = [0] p(U31) = [0] p(U41) = [0] p(activate) = [1] x1 + [2] p(and) = [1] x1 + [1] x2 + [0] p(isNat) = [1] x1 + [1] p(n__0) = [4] p(n__isNat) = [1] x1 + [0] p(n__plus) = [1] x1 + [1] x2 + [6] p(n__s) = [1] x1 + [4] p(n__x) = [1] x1 + [1] x2 + [5] p(plus) = [1] x1 + [1] x2 + [7] p(s) = [1] x1 + [5] p(tt) = [4] p(x) = [1] x1 + [1] x2 + [6] p(0#) = [0] p(U11#) = [1] x2 + [0] p(U21#) = [2] x2 + [2] x3 + [5] p(U31#) = [0] p(U41#) = [1] x2 + [3] x3 + [0] p(activate#) = [1] x1 + [0] p(and#) = [1] x1 + [1] x2 + [0] p(isNat#) = [1] x1 + [0] p(plus#) = [1] x1 + [1] x2 + [0] p(s#) = [1] x1 + [0] p(x#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [3] p(c_13) = [0] p(c_14) = [0] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] Following rules are strictly oriented: activate#(n__0()) = [4] > [0] = c_7(0#()) activate#(n__plus(X1,X2)) = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [0] = c_9(plus#(X1,X2)) activate#(n__s(X)) = [1] X + [4] > [1] X + [0] = c_10(s#(X)) activate#(n__x(X1,X2)) = [1] X1 + [1] X2 + [5] > [1] X1 + [0] = c_11(x#(X1,X2)) and#(tt(),X) = [1] X + [4] > [1] X + [3] = c_12(activate#(X)) isNat#(n__0()) = [4] > [0] = c_14() isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [6] > [1] V1 + [1] V2 + [5] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [1] V1 + [4] > [1] V1 + [2] = c_16(isNat#(activate(V1))) 0() = [5] > [4] = n__0() activate(X) = [1] X + [2] > [1] X + [0] = X activate(n__0()) = [6] > [5] = 0() activate(n__isNat(X)) = [1] X + [2] > [1] X + [1] = isNat(X) activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [8] > [1] X1 + [1] X2 + [7] = plus(X1,X2) activate(n__s(X)) = [1] X + [6] > [1] X + [5] = s(X) activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [7] > [1] X1 + [1] X2 + [6] = x(X1,X2) and(tt(),X) = [1] X + [4] > [1] X + [2] = activate(X) isNat(X) = [1] X + [1] > [1] X + [0] = n__isNat(X) isNat(n__0()) = [5] > [4] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [7] > [1] V1 + [1] V2 + [5] = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = [1] V1 + [5] > [1] V1 + [3] = isNat(activate(V1)) isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [6] > [1] V1 + [1] V2 + [5] = and(isNat(activate(V1)),n__isNat(activate(V2))) plus(X1,X2) = [1] X1 + [1] X2 + [7] > [1] X1 + [1] X2 + [6] = n__plus(X1,X2) s(X) = [1] X + [5] > [1] X + [4] = n__s(X) x(X1,X2) = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [5] = n__x(X1,X2) Following rules are (at-least) weakly oriented: 0#() = [0] >= [0] = c_1() U11#(tt(),N) = [1] N + [0] >= [1] N + [0] = c_2(activate#(N)) U21#(tt(),M,N) = [2] M + [2] N + [5] >= [1] M + [1] N + [11] = c_3(s#(plus(activate(N),activate(M)))) U31#(tt()) = [0] >= [0] = c_4(0#()) U41#(tt(),M,N) = [1] M + [3] N + [0] >= [1] M + [2] N + [12] = c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) = [1] X + [0] >= [0] = c_6() activate#(n__isNat(X)) = [1] X + [0] >= [1] X + [0] = c_8(isNat#(X)) isNat#(X) = [1] X + [0] >= [0] = c_13() isNat#(n__x(V1,V2)) = [1] V1 + [1] V2 + [5] >= [1] V1 + [1] V2 + [5] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = c_18() s#(X) = [1] X + [0] >= [0] = c_19() x#(X1,X2) = [1] X1 + [0] >= [0] = c_20() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),N) -> c_2(activate#(N)) U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U31#(tt()) -> c_4(0#()) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Weak DPs: activate#(n__0()) -> c_7(0#()) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__0()) -> c_14() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:0#() -> c_1() 2:S:U11#(tt(),N) -> c_2(activate#(N)) -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):16 -->_1 activate#(n__s(X)) -> c_10(s#(X)):15 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):14 -->_1 activate#(n__0()) -> c_7(0#()):13 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):7 -->_1 activate#(X) -> c_6():6 3:S:U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_19():11 4:S:U31#(tt()) -> c_4(0#()) -->_1 0#() -> c_1():1 5:S:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) -->_1 plus#(X1,X2) -> c_18():10 6:S:activate#(X) -> c_6() 7:S:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):20 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):19 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):9 -->_1 isNat#(n__0()) -> c_14():18 -->_1 isNat#(X) -> c_13():8 8:S:isNat#(X) -> c_13() 9:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):17 10:S:plus#(X1,X2) -> c_18() 11:S:s#(X) -> c_19() 12:S:x#(X1,X2) -> c_20() 13:W:activate#(n__0()) -> c_7(0#()) -->_1 0#() -> c_1():1 14:W:activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_18():10 15:W:activate#(n__s(X)) -> c_10(s#(X)) -->_1 s#(X) -> c_19():11 16:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():12 17:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):16 -->_1 activate#(n__s(X)) -> c_10(s#(X)):15 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):14 -->_1 activate#(n__0()) -> c_7(0#()):13 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):7 -->_1 activate#(X) -> c_6():6 18:W:isNat#(n__0()) -> c_14() 19:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):17 20:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):20 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):19 -->_1 isNat#(n__0()) -> c_14():18 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):9 -->_1 isNat#(X) -> c_13():8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 18: isNat#(n__0()) -> c_14() * Step 6: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),N) -> c_2(activate#(N)) U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U31#(tt()) -> c_4(0#()) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Weak DPs: activate#(n__0()) -> c_7(0#()) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:0#() -> c_1() 2:S:U11#(tt(),N) -> c_2(activate#(N)) -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):16 -->_1 activate#(n__s(X)) -> c_10(s#(X)):15 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):14 -->_1 activate#(n__0()) -> c_7(0#()):13 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):7 -->_1 activate#(X) -> c_6():6 3:S:U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_19():11 4:S:U31#(tt()) -> c_4(0#()) -->_1 0#() -> c_1():1 5:S:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) -->_1 plus#(X1,X2) -> c_18():10 6:S:activate#(X) -> c_6() 7:S:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):20 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):19 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):9 -->_1 isNat#(X) -> c_13():8 8:S:isNat#(X) -> c_13() 9:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):17 10:S:plus#(X1,X2) -> c_18() 11:S:s#(X) -> c_19() 12:S:x#(X1,X2) -> c_20() 13:W:activate#(n__0()) -> c_7(0#()) -->_1 0#() -> c_1():1 14:W:activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_18():10 15:W:activate#(n__s(X)) -> c_10(s#(X)) -->_1 s#(X) -> c_19():11 16:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():12 17:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):16 -->_1 activate#(n__s(X)) -> c_10(s#(X)):15 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):14 -->_1 activate#(n__0()) -> c_7(0#()):13 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):7 -->_1 activate#(X) -> c_6():6 19:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):17 20:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):20 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):19 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):9 -->_1 isNat#(X) -> c_13():8 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(2,U11#(tt(),N) -> c_2(activate#(N))),(4,U31#(tt()) -> c_4(0#()))] * Step 7: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Weak DPs: activate#(n__0()) -> c_7(0#()) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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: 0#() -> c_1() - Weak DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} Problem (S) - Strict DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Weak DPs: 0#() -> c_1() activate#(n__0()) -> c_7(0#()) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} ** Step 7.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 0#() -> c_1() - Weak DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:0#() -> c_1() 3:W:U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_19():11 5:W:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) -->_1 plus#(X1,X2) -> c_18():10 6:W:activate#(X) -> c_6() 7:W:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):20 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):19 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):9 -->_1 isNat#(X) -> c_13():8 8:W:isNat#(X) -> c_13() 9:W:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):17 10:W:plus#(X1,X2) -> c_18() 11:W:s#(X) -> c_19() 12:W:x#(X1,X2) -> c_20() 13:W:activate#(n__0()) -> c_7(0#()) -->_1 0#() -> c_1():1 14:W:activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_18():10 15:W:activate#(n__s(X)) -> c_10(s#(X)) -->_1 s#(X) -> c_19():11 16:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():12 17:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(X) -> c_6():6 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):7 -->_1 activate#(n__0()) -> c_7(0#()):13 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):14 -->_1 activate#(n__s(X)) -> c_10(s#(X)):15 -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):16 19:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):17 20:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(X) -> c_13():8 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):9 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):19 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):20 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: isNat#(X) -> c_13() 14: activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) 15: activate#(n__s(X)) -> c_10(s#(X)) 16: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) 12: x#(X1,X2) -> c_20() 6: activate#(X) -> c_6() 5: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) 10: plus#(X1,X2) -> c_18() 3: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) 11: s#(X) -> c_19() ** Step 7.a:2: PredecessorEstimationCP WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 0#() -> c_1() - Weak DPs: activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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: 1: 0#() -> c_1() The strictly oriented rules are moved into the weak component. *** Step 7.a:2.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 0#() -> c_1() - Weak DPs: activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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_7) = {1}, uargs(c_8) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U21) = [8] p(U31) = [1] p(U41) = [2] x1 + [2] x2 + [8] x3 + [2] p(activate) = [1] p(and) = [8] x1 + [12] p(isNat) = [0] p(n__0) = [4] p(n__isNat) = [0] p(n__plus) = [0] p(n__s) = [0] p(n__x) = [2] p(plus) = [8] x1 + [2] x2 + [0] p(s) = [1] x1 + [1] p(tt) = [2] p(x) = [1] x1 + [1] p(0#) = [5] p(U11#) = [1] x1 + [0] p(U21#) = [1] x1 + [4] x3 + [0] p(U31#) = [4] x1 + [0] p(U41#) = [1] x1 + [1] x2 + [1] x3 + [8] p(activate#) = [4] x1 + [0] p(and#) = [8] x2 + [0] p(isNat#) = [0] p(plus#) = [2] x1 + [1] p(s#) = [2] x1 + [1] p(x#) = [1] x1 + [8] x2 + [0] p(c_1) = [4] p(c_2) = [2] p(c_3) = [8] p(c_4) = [1] p(c_5) = [2] x1 + [4] p(c_6) = [1] p(c_7) = [1] x1 + [11] p(c_8) = [8] x1 + [0] p(c_9) = [1] x1 + [4] p(c_10) = [2] p(c_11) = [1] x1 + [1] p(c_12) = [2] x1 + [0] p(c_13) = [2] p(c_14) = [1] p(c_15) = [2] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [8] x1 + [0] p(c_18) = [2] p(c_19) = [2] p(c_20) = [1] Following rules are strictly oriented: 0#() = [5] > [4] = c_1() Following rules are (at-least) weakly oriented: activate#(n__0()) = [16] >= [16] = c_7(0#()) activate#(n__isNat(X)) = [0] >= [0] = c_8(isNat#(X)) and#(tt(),X) = [8] X + [0] >= [8] X + [0] = c_12(activate#(X)) isNat#(n__plus(V1,V2)) = [0] >= [0] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [0] >= [0] = c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [0] >= [0] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) *** Step 7.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 0#() -> c_1() activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 7.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 0#() -> c_1() activate#(n__0()) -> c_7(0#()) activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:0#() -> c_1() 2:W:activate#(n__0()) -> c_7(0#()) -->_1 0#() -> c_1():1 3:W:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):7 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 4:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):3 -->_1 activate#(n__0()) -> c_7(0#()):2 5:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):4 6:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):7 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 7:W:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: activate#(n__isNat(X)) -> c_8(isNat#(X)) 4: and#(tt(),X) -> c_12(activate#(X)) 7: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 6: isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) 5: isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 2: activate#(n__0()) -> c_7(0#()) 1: 0#() -> c_1() *** Step 7.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Weak DPs: 0#() -> c_1() activate#(n__0()) -> c_7(0#()) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_19():8 2:S:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) -->_1 plus#(X1,X2) -> c_18():7 3:S:activate#(X) -> c_6() 4:S:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):17 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):16 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(X) -> c_13():5 5:S:isNat#(X) -> c_13() 6:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):15 7:S:plus#(X1,X2) -> c_18() 8:S:s#(X) -> c_19() 9:S:x#(X1,X2) -> c_20() 10:W:0#() -> c_1() 11:W:activate#(n__0()) -> c_7(0#()) -->_1 0#() -> c_1():10 12:W:activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_18():7 13:W:activate#(n__s(X)) -> c_10(s#(X)) -->_1 s#(X) -> c_19():8 14:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():9 15:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):14 -->_1 activate#(n__s(X)) -> c_10(s#(X)):13 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):12 -->_1 activate#(n__0()) -> c_7(0#()):11 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4 -->_1 activate#(X) -> c_6():3 16:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):15 17:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):17 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):16 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(X) -> c_13():5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: activate#(n__0()) -> c_7(0#()) 10: 0#() -> c_1() ** Step 7.b:2: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() s#(X) -> c_19() x#(X1,X2) -> c_20() - Weak DPs: activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) s#(X) -> c_19() - Weak DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} Problem (S) - Strict DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() x#(X1,X2) -> c_20() - Weak DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) s#(X) -> c_19() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} *** Step 7.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) s#(X) -> c_19() - Weak DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_19():8 2:W:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) -->_1 plus#(X1,X2) -> c_18():7 3:W:activate#(X) -> c_6() 4:W:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):17 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):16 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(X) -> c_13():5 5:W:isNat#(X) -> c_13() 6:W:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):15 7:W:plus#(X1,X2) -> c_18() 8:S:s#(X) -> c_19() 9:W:x#(X1,X2) -> c_20() 12:W:activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_18():7 13:W:activate#(n__s(X)) -> c_10(s#(X)) -->_1 s#(X) -> c_19():8 14:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():9 15:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(X) -> c_6():3 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):12 -->_1 activate#(n__s(X)) -> c_10(s#(X)):13 -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):14 16:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):15 17:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(X) -> c_13():5 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):16 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):17 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: isNat#(X) -> c_13() 12: activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) 14: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) 9: x#(X1,X2) -> c_20() 3: activate#(X) -> c_6() 2: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) 7: plus#(X1,X2) -> c_18() *** Step 7.b:2.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) s#(X) -> c_19() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__s(X)) -> c_10(s#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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: 1: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) The strictly oriented rules are moved into the weak component. **** Step 7.b:2.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) s#(X) -> c_19() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__s(X)) -> c_10(s#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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_3) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U21) = [0] p(U31) = [0] p(U41) = [0] p(activate) = [1] p(and) = [1] x1 + [0] p(isNat) = [0] p(n__0) = [0] p(n__isNat) = [1] x1 + [5] p(n__plus) = [0] p(n__s) = [0] p(n__x) = [0] p(plus) = [0] p(s) = [0] p(tt) = [3] p(x) = [0] p(0#) = [0] p(U11#) = [0] p(U21#) = [5] p(U31#) = [0] p(U41#) = [0] p(activate#) = [0] p(and#) = [0] p(isNat#) = [0] p(plus#) = [0] p(s#) = [0] p(x#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [4] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [4] x1 + [0] p(c_9) = [0] p(c_10) = [2] x1 + [0] p(c_11) = [4] x1 + [0] p(c_12) = [4] x1 + [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [4] x1 + [0] p(c_16) = [2] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] Following rules are strictly oriented: U21#(tt(),M,N) = [5] > [4] = c_3(s#(plus(activate(N),activate(M)))) Following rules are (at-least) weakly oriented: activate#(n__isNat(X)) = [0] >= [0] = c_8(isNat#(X)) activate#(n__s(X)) = [0] >= [0] = c_10(s#(X)) and#(tt(),X) = [0] >= [0] = c_12(activate#(X)) isNat#(n__plus(V1,V2)) = [0] >= [0] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [0] >= [0] = c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [0] >= [0] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) s#(X) = [0] >= [0] = c_19() **** Step 7.b:2.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: s#(X) -> c_19() - Weak DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__s(X)) -> c_10(s#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.b:2.a:2.b:1: PredecessorEstimationCP WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: s#(X) -> c_19() - Weak DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__s(X)) -> c_10(s#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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 = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: s#(X) -> c_19() Consider the set of all dependency pairs 1: s#(X) -> c_19() 2: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) 3: activate#(n__isNat(X)) -> c_8(isNat#(X)) 4: activate#(n__s(X)) -> c_10(s#(X)) 5: and#(tt(),X) -> c_12(activate#(X)) 6: isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 7: isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) 8: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) Processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 7.b:2.a:2.b:1.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: s#(X) -> c_19() - Weak DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__s(X)) -> c_10(s#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U21) = [0] p(U31) = [0] p(U41) = [0] p(activate) = [1] x1 + [4] p(and) = [0] p(isNat) = [0] p(n__0) = [0] p(n__isNat) = [0] p(n__plus) = [0] p(n__s) = [4] p(n__x) = [0] p(plus) = [0] p(s) = [0] p(tt) = [0] p(x) = [0] p(0#) = [0] p(U11#) = [0] p(U21#) = [4] p(U31#) = [0] p(U41#) = [0] p(activate#) = [2] x1 + [0] p(and#) = [2] x2 + [0] p(isNat#) = [0] p(plus#) = [0] p(s#) = [4] p(x#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [4] x1 + [0] p(c_9) = [0] p(c_10) = [2] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [4] x1 + [0] p(c_18) = [2] p(c_19) = [0] p(c_20) = [0] Following rules are strictly oriented: s#(X) = [4] > [0] = c_19() Following rules are (at-least) weakly oriented: U21#(tt(),M,N) = [4] >= [4] = c_3(s#(plus(activate(N),activate(M)))) activate#(n__isNat(X)) = [0] >= [0] = c_8(isNat#(X)) activate#(n__s(X)) = [8] >= [8] = c_10(s#(X)) and#(tt(),X) = [2] X + [0] >= [2] X + [0] = c_12(activate#(X)) isNat#(n__plus(V1,V2)) = [0] >= [0] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [0] >= [0] = c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [0] >= [0] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) ***** Step 7.b:2.a:2.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__s(X)) -> c_10(s#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) s#(X) -> c_19() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 7.b:2.a:2.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__s(X)) -> c_10(s#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) s#(X) -> c_19() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_19():8 2:W:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):7 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 3:W:activate#(n__s(X)) -> c_10(s#(X)) -->_1 s#(X) -> c_19():8 4:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__s(X)) -> c_10(s#(X)):3 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):2 5:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):4 6:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):7 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 7:W:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):4 8:W:s#(X) -> c_19() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: activate#(n__isNat(X)) -> c_8(isNat#(X)) 4: and#(tt(),X) -> c_12(activate#(X)) 7: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 6: isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) 5: isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 3: activate#(n__s(X)) -> c_10(s#(X)) 1: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) 8: s#(X) -> c_19() ***** Step 7.b:2.a:2.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 7.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() x#(X1,X2) -> c_20() - Weak DPs: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__s(X)) -> c_10(s#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) s#(X) -> c_19() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) -->_1 plus#(X1,X2) -> c_18():6 2:S:activate#(X) -> c_6() 3:S:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):14 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):13 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 -->_1 isNat#(X) -> c_13():4 4:S:isNat#(X) -> c_13() 5:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):12 6:S:plus#(X1,X2) -> c_18() 7:S:x#(X1,X2) -> c_20() 8:W:U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_19():15 9:W:activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_18():6 10:W:activate#(n__s(X)) -> c_10(s#(X)) -->_1 s#(X) -> c_19():15 11:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():7 12:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):11 -->_1 activate#(n__s(X)) -> c_10(s#(X)):10 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):9 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):3 -->_1 activate#(X) -> c_6():2 13:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):12 14:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):14 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):13 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 -->_1 isNat#(X) -> c_13():4 15:W:s#(X) -> c_19() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))) 10: activate#(n__s(X)) -> c_10(s#(X)) 15: s#(X) -> c_19() *** Step 7.b:2.b:2: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() x#(X1,X2) -> c_20() - Weak DPs: activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) plus#(X1,X2) -> c_18() - Weak DPs: activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} Problem (S) - Strict DPs: activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) plus#(X1,X2) -> c_18() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} **** Step 7.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) plus#(X1,X2) -> c_18() - Weak DPs: activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) -->_1 plus#(X1,X2) -> c_18():6 2:W:activate#(X) -> c_6() 3:W:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):14 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):13 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 -->_1 isNat#(X) -> c_13():4 4:W:isNat#(X) -> c_13() 5:W:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):12 6:S:plus#(X1,X2) -> c_18() 7:W:x#(X1,X2) -> c_20() 9:W:activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_18():6 11:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():7 12:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(X) -> c_6():2 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):3 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):9 -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):11 13:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):12 14:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(X) -> c_13():4 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):13 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):14 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: isNat#(X) -> c_13() 11: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) 7: x#(X1,X2) -> c_20() 2: activate#(X) -> c_6() **** Step 7.b:2.b:2.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) plus#(X1,X2) -> c_18() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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: 1: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) The strictly oriented rules are moved into the weak component. ***** Step 7.b:2.b:2.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) plus#(X1,X2) -> c_18() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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_5) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U21) = [0] p(U31) = [0] p(U41) = [0] p(activate) = [3] p(and) = [0] p(isNat) = [4] x1 + [0] p(n__0) = [0] p(n__isNat) = [0] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [0] p(n__x) = [1] x1 + [1] x2 + [0] p(plus) = [0] p(s) = [0] p(tt) = [0] p(x) = [0] p(0#) = [0] p(U11#) = [0] p(U21#) = [4] x1 + [0] p(U31#) = [4] p(U41#) = [4] p(activate#) = [0] p(and#) = [4] x2 + [0] p(isNat#) = [0] p(plus#) = [0] p(s#) = [0] p(x#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [4] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [4] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [4] x1 + [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [4] x1 + [0] p(c_16) = [2] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] Following rules are strictly oriented: U41#(tt(),M,N) = [4] > [0] = c_5(plus#(x(activate(N),activate(M)),activate(N))) Following rules are (at-least) weakly oriented: activate#(n__isNat(X)) = [0] >= [0] = c_8(isNat#(X)) activate#(n__plus(X1,X2)) = [0] >= [0] = c_9(plus#(X1,X2)) and#(tt(),X) = [4] X + [0] >= [0] = c_12(activate#(X)) isNat#(n__plus(V1,V2)) = [0] >= [0] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [0] >= [0] = c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [0] >= [0] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) = [0] >= [0] = c_18() ***** Step 7.b:2.b:2.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: plus#(X1,X2) -> c_18() - Weak DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 7.b:2.b:2.a:2.b:1: PredecessorEstimationCP WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: plus#(X1,X2) -> c_18() - Weak DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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 = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: plus#(X1,X2) -> c_18() Consider the set of all dependency pairs 1: plus#(X1,X2) -> c_18() 2: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) 3: activate#(n__isNat(X)) -> c_8(isNat#(X)) 4: activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) 5: and#(tt(),X) -> c_12(activate#(X)) 6: isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 7: isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) 8: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) Processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 7.b:2.b:2.a:2.b:1.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: plus#(X1,X2) -> c_18() - Weak DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U21) = [0] p(U31) = [0] p(U41) = [0] p(activate) = [2] p(and) = [0] p(isNat) = [2] x1 + [4] p(n__0) = [0] p(n__isNat) = [0] p(n__plus) = [4] p(n__s) = [0] p(n__x) = [4] p(plus) = [2] x1 + [4] p(s) = [1] x1 + [0] p(tt) = [0] p(x) = [3] x1 + [2] p(0#) = [0] p(U11#) = [0] p(U21#) = [2] x2 + [2] p(U31#) = [1] p(U41#) = [4] x2 + [4] p(activate#) = [1] x1 + [0] p(and#) = [4] x2 + [0] p(isNat#) = [0] p(plus#) = [2] p(s#) = [0] p(x#) = [4] x2 + [4] p(c_1) = [1] p(c_2) = [2] x1 + [1] p(c_3) = [0] p(c_4) = [0] p(c_5) = [2] x1 + [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [4] x1 + [0] p(c_9) = [2] x1 + [0] p(c_10) = [2] x1 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [4] x1 + [0] p(c_13) = [1] p(c_14) = [0] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [2] x1 + [0] p(c_18) = [1] p(c_19) = [4] p(c_20) = [1] Following rules are strictly oriented: plus#(X1,X2) = [2] > [1] = c_18() Following rules are (at-least) weakly oriented: U41#(tt(),M,N) = [4] M + [4] >= [4] = c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(n__isNat(X)) = [0] >= [0] = c_8(isNat#(X)) activate#(n__plus(X1,X2)) = [4] >= [4] = c_9(plus#(X1,X2)) and#(tt(),X) = [4] X + [0] >= [4] X + [0] = c_12(activate#(X)) isNat#(n__plus(V1,V2)) = [0] >= [0] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [0] >= [0] = c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [0] >= [0] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) ****** Step 7.b:2.b:2.a:2.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:2.b:2.a:2.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) plus#(X1,X2) -> c_18() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) -->_1 plus#(X1,X2) -> c_18():8 2:W:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):7 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 3:W:activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_18():8 4:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):3 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):2 5:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):4 6:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):7 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):6 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):5 7:W:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):4 8:W:plus#(X1,X2) -> c_18() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: activate#(n__isNat(X)) -> c_8(isNat#(X)) 4: and#(tt(),X) -> c_12(activate#(X)) 7: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 6: isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) 5: isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 3: activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) 1: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) 8: plus#(X1,X2) -> c_18() ****** Step 7.b:2.b:2.a:2.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 7.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak DPs: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) plus#(X1,X2) -> c_18() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(X) -> c_6() 2:S:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):11 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):10 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):4 -->_1 isNat#(X) -> c_13():3 3:S:isNat#(X) -> c_13() 4:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):9 5:S:x#(X1,X2) -> c_20() 6:W:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) -->_1 plus#(X1,X2) -> c_18():12 7:W:activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_18():12 8:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():5 9:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):8 -->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)):7 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):2 -->_1 activate#(X) -> c_6():1 10:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):9 11:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):11 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):10 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):4 -->_1 isNat#(X) -> c_13():3 12:W:plus#(X1,X2) -> c_18() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))) 7: activate#(n__plus(X1,X2)) -> c_9(plus#(X1,X2)) 12: plus#(X1,X2) -> c_18() **** Step 7.b:2.b:2.b:2: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak DPs: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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: activate#(X) -> c_6() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} Problem (S) - Strict DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak DPs: activate#(X) -> c_6() activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} ***** Step 7.b:2.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_6() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(X) -> c_6() 2:W:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):11 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):10 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):4 -->_1 isNat#(X) -> c_13():3 3:W:isNat#(X) -> c_13() 4:W:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):9 5:W:x#(X1,X2) -> c_20() 8:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():5 9:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(X) -> c_6():1 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):2 -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):8 10:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):9 11:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(X) -> c_13():3 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):4 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):10 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):11 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: isNat#(X) -> c_13() 8: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) 5: x#(X1,X2) -> c_20() ***** Step 7.b:2.b:2.b:2.a:2: PredecessorEstimationCP WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_6() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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 = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: activate#(X) -> c_6() The strictly oriented rules are moved into the weak component. ****** Step 7.b:2.b:2.b:2.a:2.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_6() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_8) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [1] p(U11) = [2] x1 + [2] x2 + [8] p(U21) = [4] x2 + [4] x3 + [8] p(U31) = [1] p(U41) = [1] x1 + [2] x3 + [1] p(activate) = [0] p(and) = [1] p(isNat) = [0] p(n__0) = [1] p(n__isNat) = [4] p(n__plus) = [0] p(n__s) = [4] p(n__x) = [0] p(plus) = [2] x1 + [1] x2 + [1] p(s) = [1] x1 + [0] p(tt) = [0] p(x) = [2] x1 + [2] p(0#) = [1] p(U11#) = [1] x2 + [1] p(U21#) = [4] x2 + [0] p(U31#) = [2] p(U41#) = [4] x1 + [1] x2 + [1] p(activate#) = [2] x1 + [1] p(and#) = [2] x2 + [1] p(isNat#) = [9] p(plus#) = [0] p(s#) = [2] p(x#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [2] x1 + [4] p(c_3) = [4] p(c_4) = [1] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] x1 + [1] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [1] p(c_10) = [1] x1 + [4] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [8] p(c_14) = [1] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [4] p(c_19) = [0] p(c_20) = [1] Following rules are strictly oriented: activate#(X) = [2] X + [1] > [0] = c_6() Following rules are (at-least) weakly oriented: activate#(n__isNat(X)) = [9] >= [9] = c_8(isNat#(X)) and#(tt(),X) = [2] X + [1] >= [2] X + [1] = c_12(activate#(X)) isNat#(n__plus(V1,V2)) = [9] >= [9] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [9] >= [9] = c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [9] >= [9] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) ****** Step 7.b:2.b:2.b:2.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:2.b:2.b:2.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(X) -> c_6() activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(X) -> c_6() 2:W:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):5 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):4 3:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):2 -->_1 activate#(X) -> c_6():1 4:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):3 5:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):5 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):4 6:W:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: activate#(n__isNat(X)) -> c_8(isNat#(X)) 3: and#(tt(),X) -> c_12(activate#(X)) 6: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 5: isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) 4: isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 1: activate#(X) -> c_6() ****** Step 7.b:2.b:2.b:2.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ***** Step 7.b:2.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak DPs: activate#(X) -> c_6() activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):9 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):8 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):3 -->_1 isNat#(X) -> c_13():2 2:S:isNat#(X) -> c_13() 3:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):7 4:S:x#(X1,X2) -> c_20() 5:W:activate#(X) -> c_6() 6:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():4 7:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):6 -->_1 activate#(X) -> c_6():5 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):1 8:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):7 9:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):9 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):8 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):3 -->_1 isNat#(X) -> c_13():2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: activate#(X) -> c_6() ***** Step 7.b:2.b:2.b:2.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak DPs: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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: 4: x#(X1,X2) -> c_20() The strictly oriented rules are moved into the weak component. ****** Step 7.b:2.b:2.b:2.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) x#(X1,X2) -> c_20() - Weak DPs: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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_8) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [1] p(U11) = [4] x1 + [2] x2 + [1] p(U21) = [1] x1 + [8] x2 + [0] p(U31) = [1] p(U41) = [8] x1 + [4] x2 + [4] x3 + [1] p(activate) = [2] x1 + [2] p(and) = [10] p(isNat) = [8] x1 + [11] p(n__0) = [0] p(n__isNat) = [0] p(n__plus) = [0] p(n__s) = [1] x1 + [0] p(n__x) = [1] x1 + [1] p(plus) = [1] p(s) = [1] p(tt) = [0] p(x) = [1] x1 + [8] p(0#) = [2] p(U11#) = [1] x1 + [8] x2 + [0] p(U21#) = [1] x2 + [8] x3 + [0] p(U31#) = [1] p(U41#) = [1] x1 + [1] x2 + [0] p(activate#) = [1] x1 + [0] p(and#) = [1] x2 + [0] p(isNat#) = [0] p(plus#) = [1] x2 + [8] p(s#) = [2] x1 + [8] p(x#) = [1] x1 + [1] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [2] x1 + [8] p(c_8) = [8] x1 + [0] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [1] p(c_15) = [8] x1 + [0] p(c_16) = [2] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [0] p(c_19) = [2] p(c_20) = [0] Following rules are strictly oriented: x#(X1,X2) = [1] X1 + [1] > [0] = c_20() Following rules are (at-least) weakly oriented: activate#(n__isNat(X)) = [0] >= [0] = c_8(isNat#(X)) activate#(n__x(X1,X2)) = [1] X1 + [1] >= [1] X1 + [1] = c_11(x#(X1,X2)) and#(tt(),X) = [1] X + [0] >= [1] X + [0] = c_12(activate#(X)) isNat#(X) = [0] >= [0] = c_13() isNat#(n__plus(V1,V2)) = [0] >= [0] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [0] >= [0] = c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [0] >= [0] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) ****** Step 7.b:2.b:2.b:2.b:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak DPs: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:2.b:2.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak DPs: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) x#(X1,X2) -> c_20() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):7 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):3 -->_1 isNat#(X) -> c_13():2 2:S:isNat#(X) -> c_13() 3:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):5 4:W:activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) -->_1 x#(X1,X2) -> c_20():8 5:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)):4 -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):1 6:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):5 7:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):7 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):3 -->_1 isNat#(X) -> c_13():2 8:W:x#(X1,X2) -> c_20() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: activate#(n__x(X1,X2)) -> c_11(x#(X1,X2)) 8: x#(X1,X2) -> c_20() ****** Step 7.b:2.b:2.b:2.b:2.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak DPs: and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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: 1: activate#(n__isNat(X)) -> c_8(isNat#(X)) 3: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) The strictly oriented rules are moved into the weak component. ******* Step 7.b:2.b:2.b:2.b:2.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) isNat#(X) -> c_13() isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak DPs: and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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_8) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: {0,activate,and,isNat,plus,s,x,0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [2] p(U11) = [2] x2 + [1] p(U21) = [4] x1 + [1] x2 + [0] p(U31) = [0] p(U41) = [2] x1 + [2] x2 + [1] p(activate) = [1] x1 + [2] p(and) = [1] x2 + [2] p(isNat) = [1] x1 + [3] p(n__0) = [0] p(n__isNat) = [1] x1 + [1] p(n__plus) = [1] x1 + [1] x2 + [3] p(n__s) = [1] x1 + [3] p(n__x) = [1] x2 + [3] p(plus) = [1] x1 + [1] x2 + [3] p(s) = [1] x1 + [3] p(tt) = [3] p(x) = [1] x2 + [5] p(0#) = [0] p(U11#) = [1] x2 + [2] p(U21#) = [0] p(U31#) = [2] x1 + [8] p(U41#) = [8] x1 + [1] x2 + [2] x3 + [8] p(activate#) = [8] x1 + [0] p(and#) = [8] x2 + [0] p(isNat#) = [8] x1 + [1] p(plus#) = [1] x1 + [1] x2 + [0] p(s#) = [4] x1 + [0] p(x#) = [0] p(c_1) = [0] p(c_2) = [2] x1 + [0] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [2] p(c_6) = [0] p(c_7) = [1] p(c_8) = [1] x1 + [6] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [2] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [1] p(c_15) = [1] x1 + [1] p(c_16) = [1] x1 + [3] p(c_17) = [1] x1 + [0] p(c_18) = [2] p(c_19) = [4] p(c_20) = [2] Following rules are strictly oriented: activate#(n__isNat(X)) = [8] X + [8] > [8] X + [7] = c_8(isNat#(X)) isNat#(n__x(V1,V2)) = [8] V2 + [25] > [8] V2 + [24] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) Following rules are (at-least) weakly oriented: and#(tt(),X) = [8] X + [0] >= [8] X + [0] = c_12(activate#(X)) isNat#(X) = [8] X + [1] >= [1] = c_13() isNat#(n__plus(V1,V2)) = [8] V1 + [8] V2 + [25] >= [8] V2 + [25] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [8] V1 + [25] >= [8] V1 + [20] = c_16(isNat#(activate(V1))) 0() = [2] >= [0] = n__0() activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__0()) = [2] >= [2] = 0() activate(n__isNat(X)) = [1] X + [3] >= [1] X + [3] = isNat(X) activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [3] = plus(X1,X2) activate(n__s(X)) = [1] X + [5] >= [1] X + [3] = s(X) activate(n__x(X1,X2)) = [1] X2 + [5] >= [1] X2 + [5] = x(X1,X2) and(tt(),X) = [1] X + [2] >= [1] X + [2] = activate(X) isNat(X) = [1] X + [3] >= [1] X + [1] = n__isNat(X) isNat(n__0()) = [3] >= [3] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [6] >= [1] V2 + [5] = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = [1] V1 + [6] >= [1] V1 + [5] = isNat(activate(V1)) isNat(n__x(V1,V2)) = [1] V2 + [6] >= [1] V2 + [5] = and(isNat(activate(V1)),n__isNat(activate(V2))) plus(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = n__plus(X1,X2) s(X) = [1] X + [3] >= [1] X + [3] = n__s(X) x(X1,X2) = [1] X2 + [5] >= [1] X2 + [3] = n__x(X1,X2) ******* Step 7.b:2.b:2.b:2.b:2.b:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: isNat#(X) -> c_13() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 7.b:2.b:2.b:2.b:2.b:2.b:1: PredecessorEstimationCP WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: isNat#(X) -> c_13() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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 = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: isNat#(X) -> c_13() The strictly oriented rules are moved into the weak component. ******** Step 7.b:2.b:2.b:2.b:2.b:2.b:1.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: isNat#(X) -> c_13() - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_8) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1} Following symbols are considered usable: {0,activate,and,isNat,plus,s,x,0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#,x#} TcT has computed the following interpretation: p(0) = [5] p(U11) = [1] x1 + [1] x2 + [0] p(U21) = [1] x1 + [2] x3 + [8] p(U31) = [1] p(U41) = [2] x1 + [1] x2 + [0] p(activate) = [1] x1 + [0] p(and) = [1] x2 + [0] p(isNat) = [1] p(n__0) = [5] p(n__isNat) = [1] p(n__plus) = [8] p(n__s) = [0] p(n__x) = [0] p(plus) = [8] p(s) = [0] p(tt) = [1] p(x) = [0] p(0#) = [0] p(U11#) = [1] p(U21#) = [1] x1 + [2] x3 + [0] p(U31#) = [8] x1 + [4] p(U41#) = [1] x1 + [1] x2 + [1] x3 + [0] p(activate#) = [4] x1 + [4] p(and#) = [4] x1 + [4] x2 + [0] p(isNat#) = [8] p(plus#) = [1] x1 + [4] p(s#) = [0] p(x#) = [1] x1 + [8] x2 + [0] p(c_1) = [1] p(c_2) = [2] x1 + [8] p(c_3) = [2] x1 + [1] p(c_4) = [1] x1 + [2] p(c_5) = [1] x1 + [4] p(c_6) = [1] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [0] p(c_13) = [5] p(c_14) = [8] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [1] Following rules are strictly oriented: isNat#(X) = [8] > [5] = c_13() Following rules are (at-least) weakly oriented: activate#(n__isNat(X)) = [8] >= [8] = c_8(isNat#(X)) and#(tt(),X) = [4] X + [4] >= [4] X + [4] = c_12(activate#(X)) isNat#(n__plus(V1,V2)) = [8] >= [8] = c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) = [8] >= [8] = c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) = [8] >= [8] = c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 0() = [5] >= [5] = n__0() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [5] >= [5] = 0() activate(n__isNat(X)) = [1] >= [1] = isNat(X) activate(n__plus(X1,X2)) = [8] >= [8] = plus(X1,X2) activate(n__s(X)) = [0] >= [0] = s(X) activate(n__x(X1,X2)) = [0] >= [0] = x(X1,X2) and(tt(),X) = [1] X + [0] >= [1] X + [0] = activate(X) isNat(X) = [1] >= [1] = n__isNat(X) isNat(n__0()) = [1] >= [1] = tt() isNat(n__plus(V1,V2)) = [1] >= [1] = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = [1] >= [1] = isNat(activate(V1)) isNat(n__x(V1,V2)) = [1] >= [1] = and(isNat(activate(V1)),n__isNat(activate(V2))) plus(X1,X2) = [8] >= [8] = n__plus(X1,X2) s(X) = [0] >= [0] = n__s(X) x(X1,X2) = [0] >= [0] = n__x(X1,X2) ******** Step 7.b:2.b:2.b:2.b:2.b:2.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 7.b:2.b:2.b:2.b:2.b:2.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__isNat(X)) -> c_8(isNat#(X)) and#(tt(),X) -> c_12(activate#(X)) isNat#(X) -> c_13() isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(n__isNat(X)) -> c_8(isNat#(X)) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):5 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):4 -->_1 isNat#(X) -> c_13():3 2:W:and#(tt(),X) -> c_12(activate#(X)) -->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):1 3:W:isNat#(X) -> c_13() 4:W:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):2 5:W:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) -->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))):6 -->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))):5 -->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))):4 -->_1 isNat#(X) -> c_13():3 6:W:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) -->_1 and#(tt(),X) -> c_12(activate#(X)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: activate#(n__isNat(X)) -> c_8(isNat#(X)) 2: and#(tt(),X) -> c_12(activate#(X)) 6: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 5: isNat#(n__s(V1)) -> c_16(isNat#(activate(V1))) 4: isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))) 3: isNat#(X) -> c_13() ******** Step 7.b:2.b:2.b:2.b:2.b:2.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) and(tt(),X) -> activate(X) isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(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,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3 ,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0 ,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1 ,c_18/0,c_19/0,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s# ,x#} and constructors {n__0,n__isNat,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))