WORST_CASE(?,O(n^2)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) All above mentioned rules can be savely removed. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(U42) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [1] x2 + [0] p(U12) = [1] x1 + [0] p(U21) = [1] x1 + [0] p(U31) = [1] x2 + [0] p(U41) = [1] x2 + [2] x3 + [0] p(U42) = [1] x1 + [1] x2 + [1] x3 + [9] p(activate) = [1] x1 + [0] p(isNat) = [1] x1 + [8] p(n__0) = [1] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [13] p(tt) = [0] Following rules are strictly oriented: activate(n__0()) = [1] > [0] = 0() isNat(n__0()) = [9] > [0] = tt() s(X) = [1] X + [13] > [1] X + [0] = n__s(X) Following rules are (at-least) weakly oriented: 0() = [0] >= [1] = n__0() U11(tt(),V2) = [1] V2 + [0] >= [1] V2 + [8] = U12(isNat(activate(V2))) U12(tt()) = [0] >= [0] = tt() U21(tt()) = [0] >= [0] = tt() U31(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U41(tt(),M,N) = [1] M + [2] N + [0] >= [1] M + [2] N + [17] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1] M + [1] N + [9] >= [1] M + [1] N + [13] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [0] >= [1] X + [13] = s(activate(X)) isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [8] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [1] V1 + [8] >= [1] V1 + [8] = U21(isNat(activate(V1))) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) - Weak TRS: activate(n__0()) -> 0() isNat(n__0()) -> tt() s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(U42) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [1] x2 + [0] p(U12) = [1] x1 + [0] p(U21) = [1] x1 + [0] p(U31) = [1] x2 + [0] p(U41) = [1] x2 + [2] x3 + [0] p(U42) = [1] x1 + [1] x2 + [1] x3 + [0] p(activate) = [1] x1 + [0] p(isNat) = [1] x1 + [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [9] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [9] p(tt) = [0] Following rules are strictly oriented: isNat(n__s(V1)) = [1] V1 + [9] > [1] V1 + [0] = U21(isNat(activate(V1))) Following rules are (at-least) weakly oriented: 0() = [0] >= [0] = n__0() U11(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = U12(isNat(activate(V2))) U12(tt()) = [0] >= [0] = tt() U21(tt()) = [0] >= [0] = tt() U31(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U41(tt(),M,N) = [1] M + [2] N + [0] >= [1] M + [2] N + [0] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1] M + [1] N + [0] >= [1] M + [1] N + [9] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [9] >= [1] X + [9] = s(activate(X)) isNat(n__0()) = [0] >= [0] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0] >= [1] V1 + [1] V2 + [0] = U11(isNat(activate(V1)),activate(V2)) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) s(X) = [1] X + [9] >= [1] X + [9] = n__s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) - Weak TRS: activate(n__0()) -> 0() isNat(n__0()) -> tt() isNat(n__s(V1)) -> U21(isNat(activate(V1))) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(U42) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [1] x2 + [0] p(U12) = [1] x1 + [0] p(U21) = [1] x1 + [0] p(U31) = [1] x2 + [3] p(U41) = [5] x1 + [1] x2 + [2] x3 + [0] p(U42) = [1] x1 + [1] x2 + [1] x3 + [0] p(activate) = [1] x1 + [9] p(isNat) = [1] x1 + [2] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [9] p(plus) = [1] x1 + [1] x2 + [3] p(s) = [1] x1 + [9] p(tt) = [2] Following rules are strictly oriented: activate(X) = [1] X + [9] > [1] X + [0] = X plus(X1,X2) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [0] = n__plus(X1,X2) Following rules are (at-least) weakly oriented: 0() = [0] >= [0] = n__0() U11(tt(),V2) = [1] V2 + [2] >= [1] V2 + [11] = U12(isNat(activate(V2))) U12(tt()) = [2] >= [2] = tt() U21(tt()) = [2] >= [2] = tt() U31(tt(),N) = [1] N + [3] >= [1] N + [9] = activate(N) U41(tt(),M,N) = [1] M + [2] N + [10] >= [1] M + [2] N + [29] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [30] = s(plus(activate(N),activate(M))) activate(n__0()) = [9] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [9] >= [1] X1 + [1] X2 + [21] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [18] >= [1] X + [18] = s(activate(X)) isNat(n__0()) = [2] >= [2] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [2] >= [1] V1 + [1] V2 + [20] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [1] V1 + [11] >= [1] V1 + [11] = U21(isNat(activate(V1))) s(X) = [1] X + [9] >= [1] X + [9] = n__s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) - Weak TRS: activate(X) -> X activate(n__0()) -> 0() isNat(n__0()) -> tt() isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(U42) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [1] x2 + [0] p(U12) = [1] x1 + [0] p(U21) = [1] x1 + [1] p(U31) = [1] x2 + [1] p(U41) = [1] x2 + [2] x3 + [0] p(U42) = [1] x1 + [1] x2 + [1] x3 + [7] p(activate) = [1] x1 + [0] p(isNat) = [1] x1 + [0] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [3] p(n__s) = [1] x1 + [1] p(plus) = [1] x1 + [1] x2 + [3] p(s) = [1] x1 + [1] p(tt) = [0] Following rules are strictly oriented: U21(tt()) = [1] > [0] = tt() U31(tt(),N) = [1] N + [1] > [1] N + [0] = activate(N) U42(tt(),M,N) = [1] M + [1] N + [7] > [1] M + [1] N + [4] = s(plus(activate(N),activate(M))) isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3] > [1] V1 + [1] V2 + [0] = U11(isNat(activate(V1)),activate(V2)) Following rules are (at-least) weakly oriented: 0() = [0] >= [0] = n__0() U11(tt(),V2) = [1] V2 + [0] >= [1] V2 + [0] = U12(isNat(activate(V2))) U12(tt()) = [0] >= [0] = tt() U41(tt(),M,N) = [1] M + [2] N + [0] >= [1] M + [2] N + [7] = U42(isNat(activate(N)),activate(M),activate(N)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [1] >= [1] X + [1] = s(activate(X)) isNat(n__0()) = [0] >= [0] = tt() isNat(n__s(V1)) = [1] V1 + [1] >= [1] V1 + [1] = U21(isNat(activate(V1))) plus(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = n__plus(X1,X2) s(X) = [1] X + [1] >= [1] X + [1] = n__s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) - Weak TRS: U21(tt()) -> tt() U31(tt(),N) -> activate(N) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(U42) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [1] x2 + [4] p(U12) = [1] x1 + [1] p(U21) = [1] x1 + [0] p(U31) = [1] x2 + [0] p(U41) = [1] x2 + [2] x3 + [0] p(U42) = [1] x1 + [1] x2 + [1] x3 + [0] p(activate) = [1] x1 + [0] p(isNat) = [1] x1 + [0] p(n__0) = [5] p(n__plus) = [1] x1 + [1] x2 + [4] p(n__s) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [4] p(s) = [1] x1 + [1] p(tt) = [5] Following rules are strictly oriented: U11(tt(),V2) = [1] V2 + [9] > [1] V2 + [1] = U12(isNat(activate(V2))) U12(tt()) = [6] > [5] = tt() Following rules are (at-least) weakly oriented: 0() = [0] >= [5] = n__0() U21(tt()) = [5] >= [5] = tt() U31(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U41(tt(),M,N) = [1] M + [2] N + [0] >= [1] M + [2] N + [0] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1] M + [1] N + [5] >= [1] M + [1] N + [5] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [5] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [0] >= [1] X + [1] = s(activate(X)) isNat(n__0()) = [5] >= [5] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [4] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [1] V1 + [0] >= [1] V1 + [0] = U21(isNat(activate(V1))) plus(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = n__plus(X1,X2) s(X) = [1] X + [1] >= [1] X + [0] = n__s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) - Weak TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(U42) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [1] x2 + [0] p(U12) = [1] x1 + [0] p(U21) = [1] x1 + [0] p(U31) = [1] x2 + [4] p(U41) = [3] x2 + [4] x3 + [4] p(U42) = [1] x1 + [1] x2 + [1] x3 + [1] p(activate) = [1] x1 + [0] p(isNat) = [1] x1 + [2] p(n__0) = [1] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [2] p(plus) = [1] x1 + [1] x2 + [1] p(s) = [1] x1 + [3] p(tt) = [3] Following rules are strictly oriented: U41(tt(),M,N) = [3] M + [4] N + [4] > [1] M + [2] N + [3] = U42(isNat(activate(N)),activate(M),activate(N)) Following rules are (at-least) weakly oriented: 0() = [0] >= [1] = n__0() U11(tt(),V2) = [1] V2 + [3] >= [1] V2 + [2] = U12(isNat(activate(V2))) U12(tt()) = [3] >= [3] = tt() U21(tt()) = [3] >= [3] = tt() U31(tt(),N) = [1] N + [4] >= [1] N + [0] = activate(N) U42(tt(),M,N) = [1] M + [1] N + [4] >= [1] M + [1] N + [4] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [1] >= [0] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [1] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [2] >= [1] X + [3] = s(activate(X)) isNat(n__0()) = [3] >= [3] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [2] >= [1] V1 + [1] V2 + [2] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [1] V1 + [4] >= [1] V1 + [2] = U21(isNat(activate(V1))) plus(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) s(X) = [1] X + [3] >= [1] X + [2] = n__s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) - Weak TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(U42) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [3] p(U11) = [1] x1 + [1] x2 + [5] p(U12) = [1] x1 + [0] p(U21) = [1] x1 + [0] p(U31) = [1] x1 + [1] x2 + [5] p(U41) = [3] x1 + [1] x2 + [2] x3 + [2] p(U42) = [1] x1 + [1] x2 + [1] x3 + [7] p(activate) = [1] x1 + [1] p(isNat) = [1] x1 + [1] p(n__0) = [2] p(n__plus) = [1] x1 + [1] x2 + [7] p(n__s) = [1] x1 + [1] p(plus) = [1] x1 + [1] x2 + [7] p(s) = [1] x1 + [1] p(tt) = [3] Following rules are strictly oriented: 0() = [3] > [2] = n__0() Following rules are (at-least) weakly oriented: U11(tt(),V2) = [1] V2 + [8] >= [1] V2 + [2] = U12(isNat(activate(V2))) U12(tt()) = [3] >= [3] = tt() U21(tt()) = [3] >= [3] = tt() U31(tt(),N) = [1] N + [8] >= [1] N + [1] = activate(N) U41(tt(),M,N) = [1] M + [2] N + [11] >= [1] M + [2] N + [11] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1] M + [1] N + [10] >= [1] M + [1] N + [10] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n__0()) = [3] >= [3] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [9] = plus(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [2] >= [1] X + [2] = s(activate(X)) isNat(n__0()) = [3] >= [3] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [8] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [1] V1 + [2] >= [1] V1 + [2] = U21(isNat(activate(V1))) plus(X1,X2) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [7] = n__plus(X1,X2) s(X) = [1] X + [1] >= [1] X + [1] = n__s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(U42) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s} TcT has computed the following interpretation: p(0) = [3] [1] p(U11) = [1 0] x1 + [2 2] x2 + [0] [0 1] [4 4] [0] p(U12) = [1 0] x1 + [0] [2 0] [0] p(U21) = [1 0] x1 + [0] [0 0] [0] p(U31) = [1 0] x1 + [2 4] x2 + [7] [4 0] [1 4] [3] p(U41) = [0 0] x1 + [1 6] x2 + [3 7] x3 + [4] [6 0] [4 4] [4 1] [1] p(U42) = [1 0] x1 + [1 5] x2 + [1 4] x3 + [4] [0 0] [0 2] [0 1] [1] p(activate) = [1 1] x1 + [0] [0 1] [0] p(isNat) = [2 0] x1 + [0] [4 0] [1] p(n__0) = [2] [1] p(n__plus) = [1 1] x1 + [1 2] x2 + [0] [0 1] [0 1] [0] p(n__s) = [1 2] x1 + [0] [0 1] [1] p(plus) = [1 1] x1 + [1 2] x2 + [0] [0 1] [0 1] [0] p(s) = [1 2] x1 + [0] [0 1] [1] p(tt) = [2] [0] Following rules are strictly oriented: activate(n__s(X)) = [1 3] X + [1] [0 1] [1] > [1 3] X + [0] [0 1] [1] = s(activate(X)) Following rules are (at-least) weakly oriented: 0() = [3] [1] >= [2] [1] = n__0() U11(tt(),V2) = [2 2] V2 + [2] [4 4] [0] >= [2 2] V2 + [0] [4 4] [0] = U12(isNat(activate(V2))) U12(tt()) = [2] [4] >= [2] [0] = tt() U21(tt()) = [2] [0] >= [2] [0] = tt() U31(tt(),N) = [2 4] N + [9] [1 4] [11] >= [1 1] N + [0] [0 1] [0] = activate(N) U41(tt(),M,N) = [1 6] M + [3 7] N + [4] [4 4] [4 1] [13] >= [1 6] M + [3 7] N + [4] [0 2] [0 1] [1] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1 5] M + [1 4] N + [6] [0 2] [0 1] [1] >= [1 5] M + [1 4] N + [0] [0 1] [0 1] [1] = s(plus(activate(N),activate(M))) activate(X) = [1 1] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [3] [1] >= [3] [1] = 0() activate(n__plus(X1,X2)) = [1 2] X1 + [1 3] X2 + [0] [0 1] [0 1] [0] >= [1 2] X1 + [1 3] X2 + [0] [0 1] [0 1] [0] = plus(activate(X1),activate(X2)) isNat(n__0()) = [4] [9] >= [2] [0] = tt() isNat(n__plus(V1,V2)) = [2 2] V1 + [2 4] V2 + [0] [4 4] [4 8] [1] >= [2 2] V1 + [2 4] V2 + [0] [4 4] [4 8] [1] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [2 4] V1 + [0] [4 8] [1] >= [2 2] V1 + [0] [0 0] [0] = U21(isNat(activate(V1))) plus(X1,X2) = [1 1] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] >= [1 1] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] = n__plus(X1,X2) s(X) = [1 2] X + [0] [0 1] [1] >= [1 2] X + [0] [0 1] [1] = n__s(X) * Step 10: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(U42) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s} TcT has computed the following interpretation: p(0) = [0] [0] p(U11) = [1 0] x1 + [1 1] x2 + [0] [0 0] [0 0] [0] p(U12) = [1 0] x1 + [0] [0 0] [0] p(U21) = [1 2] x1 + [0] [0 0] [0] p(U31) = [0 0] x1 + [1 1] x2 + [1] [2 1] [4 1] [0] p(U41) = [2 0] x1 + [1 5] x2 + [3 6] x3 + [5] [2 1] [4 5] [4 3] [2] p(U42) = [2 0] x1 + [1 4] x2 + [1 3] x3 + [1] [2 4] [4 1] [0 1] [2] p(activate) = [1 1] x1 + [0] [0 1] [0] p(isNat) = [1 0] x1 + [4] [0 0] [0] p(n__0) = [0] [0] p(n__plus) = [1 1] x1 + [1 2] x2 + [0] [0 1] [0 1] [7] p(n__s) = [1 1] x1 + [2] [0 1] [2] p(plus) = [1 1] x1 + [1 2] x2 + [0] [0 1] [0 1] [7] p(s) = [1 1] x1 + [2] [0 1] [2] p(tt) = [4] [0] Following rules are strictly oriented: activate(n__plus(X1,X2)) = [1 2] X1 + [1 3] X2 + [7] [0 1] [0 1] [7] > [1 2] X1 + [1 3] X2 + [0] [0 1] [0 1] [7] = plus(activate(X1),activate(X2)) Following rules are (at-least) weakly oriented: 0() = [0] [0] >= [0] [0] = n__0() U11(tt(),V2) = [1 1] V2 + [4] [0 0] [0] >= [1 1] V2 + [4] [0 0] [0] = U12(isNat(activate(V2))) U12(tt()) = [4] [0] >= [4] [0] = tt() U21(tt()) = [4] [0] >= [4] [0] = tt() U31(tt(),N) = [1 1] N + [1] [4 1] [8] >= [1 1] N + [0] [0 1] [0] = activate(N) U41(tt(),M,N) = [1 5] M + [3 6] N + [13] [4 5] [4 3] [10] >= [1 5] M + [3 6] N + [9] [4 5] [2 3] [10] = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = [1 4] M + [1 3] N + [9] [4 1] [0 1] [10] >= [1 4] M + [1 3] N + [9] [0 1] [0 1] [9] = s(plus(activate(N),activate(M))) activate(X) = [1 1] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [0] [0] >= [0] [0] = 0() activate(n__s(X)) = [1 2] X + [4] [0 1] [2] >= [1 2] X + [2] [0 1] [2] = s(activate(X)) isNat(n__0()) = [4] [0] >= [4] [0] = tt() isNat(n__plus(V1,V2)) = [1 1] V1 + [1 2] V2 + [4] [0 0] [0 0] [0] >= [1 1] V1 + [1 2] V2 + [4] [0 0] [0 0] [0] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [1 1] V1 + [6] [0 0] [0] >= [1 1] V1 + [4] [0 0] [0] = U21(isNat(activate(V1))) plus(X1,X2) = [1 1] X1 + [1 2] X2 + [0] [0 1] [0 1] [7] >= [1 1] X1 + [1 2] X2 + [0] [0 1] [0 1] [7] = n__plus(X1,X2) s(X) = [1 1] X + [2] [0 1] [2] >= [1 1] X + [2] [0 1] [2] = n__s(X) * Step 11: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))