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))) 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) 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)) 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) - Signature: {0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus ,s} and constructors {n__0,n__isNat,n__plus,n__s,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) All above mentioned rules can be savely removed. * Step 2: WeightGap 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))) 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) 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)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus ,s} and constructors {n__0,n__isNat,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(and) = {1,2}, uargs(isNat) = {1}, uargs(n__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) = [4] x1 + [2] x2 + [0] p(U21) = [5] x1 + [1] x2 + [1] x3 + [6] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [5] p(isNat) = [1] x1 + [1] p(n__0) = [0] p(n__isNat) = [1] x1 + [9] p(n__plus) = [1] x1 + [1] x2 + [1] p(n__s) = [1] x1 + [5] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [0] p(tt) = [4] Following rules are strictly oriented: U11(tt(),N) = [2] N + [16] > [1] N + [0] = activate(N) U21(tt(),M,N) = [1] M + [1] N + [26] > [1] M + [1] N + [0] = s(plus(activate(N),activate(M))) activate(n__isNat(X)) = [1] X + [9] > [1] X + [1] = isNat(X) activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = plus(X1,X2) activate(n__s(X)) = [1] X + [5] > [1] X + [0] = s(X) and(tt(),X) = [1] X + [9] > [1] X + [0] = activate(X) isNat(n__s(V1)) = [1] V1 + [6] > [1] V1 + [1] = isNat(activate(V1)) Following rules are (at-least) weakly oriented: 0() = [0] >= [0] = n__0() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() isNat(X) = [1] X + [1] >= [1] X + [9] = n__isNat(X) isNat(n__0()) = [1] >= [4] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [2] >= [1] V1 + [1] V2 + [15] = and(isNat(activate(V1)),n__isNat(activate(V2))) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [1] = n__plus(X1,X2) s(X) = [1] X + [0] >= [1] X + [5] = n__s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() isNat(X) -> n__isNat(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Weak TRS: U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) and(tt(),X) -> activate(X) isNat(n__s(V1)) -> isNat(activate(V1)) - Signature: {0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus ,s} and constructors {n__0,n__isNat,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(and) = {1,2}, uargs(isNat) = {1}, uargs(n__isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(U11) = [1] x2 + [0] p(U21) = [1] x2 + [1] x3 + [4] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(isNat) = [1] x1 + [4] p(n__0) = [0] p(n__isNat) = [1] x1 + [4] 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: 0() = [1] > [0] = n__0() isNat(n__0()) = [4] > [0] = tt() Following rules are (at-least) weakly oriented: U11(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U21(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()) = [0] >= [1] = 0() activate(n__isNat(X)) = [1] X + [4] >= [1] X + [4] = isNat(X) activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = plus(X1,X2) activate(n__s(X)) = [1] X + [1] >= [1] X + [1] = s(X) and(tt(),X) = [1] X + [0] >= [1] X + [0] = activate(X) isNat(X) = [1] X + [4] >= [1] X + [4] = n__isNat(X) isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [7] >= [1] V1 + [1] V2 + [8] = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = [1] V1 + [5] >= [1] V1 + [4] = 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 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__0()) -> 0() isNat(X) -> n__isNat(X) isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Weak TRS: 0() -> n__0() U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__s(V1)) -> isNat(activate(V1)) - Signature: {0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus ,s} and constructors {n__0,n__isNat,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(and) = {1,2}, uargs(isNat) = {1}, uargs(n__isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(U11) = [1] x2 + [0] p(U21) = [1] x2 + [1] x3 + [7] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(isNat) = [1] x1 + [1] p(n__0) = [0] p(n__isNat) = [1] x1 + [1] 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 + [0] p(tt) = [1] Following rules are strictly oriented: isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8] > [1] V1 + [1] V2 + [2] = and(isNat(activate(V1)),n__isNat(activate(V2))) Following rules are (at-least) weakly oriented: 0() = [1] >= [0] = n__0() U11(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U21(tt(),M,N) = [1] M + [1] N + [7] >= [1] M + [1] N + [7] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [1] = 0() activate(n__isNat(X)) = [1] X + [1] >= [1] X + [1] = isNat(X) activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [7] = plus(X1,X2) activate(n__s(X)) = [1] X + [1] >= [1] X + [0] = s(X) and(tt(),X) = [1] X + [1] >= [1] X + [0] = activate(X) isNat(X) = [1] X + [1] >= [1] X + [1] = n__isNat(X) isNat(n__0()) = [1] >= [1] = tt() isNat(n__s(V1)) = [1] V1 + [2] >= [1] V1 + [1] = isNat(activate(V1)) plus(X1,X2) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [7] = n__plus(X1,X2) s(X) = [1] X + [0] >= [1] X + [1] = 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^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__0()) -> 0() isNat(X) -> n__isNat(X) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Weak TRS: 0() -> n__0() U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(n__isNat(X)) -> isNat(X) activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) and(tt(),X) -> activate(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)) - Signature: {0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus ,s} and constructors {n__0,n__isNat,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(and) = {1,2}, uargs(isNat) = {1}, uargs(n__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) = [4] x2 + [4] p(U21) = [2] x1 + [4] x2 + [1] x3 + [6] p(activate) = [1] x1 + [1] p(and) = [1] x1 + [1] x2 + [0] p(isNat) = [1] x1 + [1] p(n__0) = [0] p(n__isNat) = [1] x1 + [0] p(n__plus) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [2] p(plus) = [1] x1 + [1] x2 + [3] p(s) = [1] x1 + [3] p(tt) = [1] Following rules are strictly oriented: activate(X) = [1] X + [1] > [1] X + [0] = X activate(n__0()) = [1] > [0] = 0() isNat(X) = [1] X + [1] > [1] X + [0] = n__isNat(X) plus(X1,X2) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [2] = n__plus(X1,X2) s(X) = [1] X + [3] > [1] X + [2] = n__s(X) Following rules are (at-least) weakly oriented: 0() = [0] >= [0] = n__0() U11(tt(),N) = [4] N + [4] >= [1] N + [1] = activate(N) U21(tt(),M,N) = [4] M + [1] N + [8] >= [1] M + [1] N + [8] = s(plus(activate(N),activate(M))) activate(n__isNat(X)) = [1] X + [1] >= [1] X + [1] = isNat(X) activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = plus(X1,X2) activate(n__s(X)) = [1] X + [3] >= [1] X + [3] = s(X) and(tt(),X) = [1] X + [1] >= [1] X + [1] = activate(X) isNat(n__0()) = [1] >= [1] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3] >= [1] V1 + [1] V2 + [3] = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = [1] V1 + [3] >= [1] V1 + [2] = isNat(activate(V1)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) 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) 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)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus ,s} and constructors {n__0,n__isNat,n__plus,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))