WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() U41(tt(),N) -> activate(N) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(N,0()) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(N,0()) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1 ,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate ,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. plus(N,0()) -> U41(isNat(N),N) plus(N,s(M)) -> U51(isNat(M),M,N) x(N,0()) -> U61(isNat(N)) x(N,s(M)) -> U71(isNat(M),M,N) All above mentioned rules can be savely removed. * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() U41(tt(),N) -> activate(N) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1 ,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate ,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(U31) = {1,2}, uargs(U32) = {1}, uargs(U52) = {1,2,3}, uargs(U72) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1}, uargs(x) = {1,2} Following symbols are considered usable: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x} TcT has computed the following interpretation: p(0) = [2] p(U11) = [1] x1 + [4] x2 + [0] p(U12) = [1] x1 + [0] p(U21) = [1] x1 + [15] p(U31) = [1] x1 + [4] x2 + [0] p(U32) = [1] x1 + [0] p(U41) = [1] x2 + [0] p(U51) = [1] x1 + [1] x2 + [12] x3 + [9] p(U52) = [1] x1 + [1] x2 + [8] x3 + [7] p(U61) = [12] p(U71) = [2] x1 + [8] x2 + [12] x3 + [0] p(U72) = [1] x1 + [1] x2 + [8] x3 + [15] p(activate) = [1] x1 + [0] p(isNat) = [4] x1 + [1] p(n__0) = [2] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [6] p(n__x) = [1] x1 + [1] x2 + [0] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [6] p(tt) = [8] p(x) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: U11(tt(),V2) = [4] V2 + [8] > [4] V2 + [1] = U12(isNat(activate(V2))) U21(tt()) = [23] > [8] = tt() U31(tt(),V2) = [4] V2 + [8] > [4] V2 + [1] = U32(isNat(activate(V2))) U51(tt(),M,N) = [1] M + [12] N + [17] > [1] M + [12] N + [8] = U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) = [1] M + [8] N + [15] > [1] M + [1] N + [6] = s(plus(activate(N),activate(M))) U61(tt()) = [12] > [2] = 0() U72(tt(),M,N) = [1] M + [8] N + [23] > [1] M + [2] N + [0] = plus(x(activate(N),activate(M)),activate(N)) isNat(n__0()) = [9] > [8] = tt() isNat(n__s(V1)) = [4] V1 + [25] > [4] V1 + [16] = U21(isNat(activate(V1))) Following rules are (at-least) weakly oriented: 0() = [2] >= [2] = n__0() U12(tt()) = [8] >= [8] = tt() U32(tt()) = [8] >= [8] = tt() U41(tt(),N) = [1] N + [0] >= [1] N + [0] = activate(N) U71(tt(),M,N) = [8] M + [12] N + [16] >= [1] M + [12] N + [16] = U72(isNat(activate(N)),activate(M),activate(N)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [2] >= [2] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(X1,X2) activate(n__s(X)) = [1] X + [6] >= [1] X + [6] = s(X) activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = x(X1,X2) isNat(n__plus(V1,V2)) = [4] V1 + [4] V2 + [1] >= [4] V1 + [4] V2 + [1] = U11(isNat(activate(V1)),activate(V2)) isNat(n__x(V1,V2)) = [4] V1 + [4] V2 + [1] >= [4] V1 + [4] V2 + [1] = U31(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 + [6] >= [1] X + [6] = n__s(X) x(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__x(X1,X2) * Step 3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() U12(tt()) -> tt() U32(tt()) -> tt() U41(tt(),N) -> activate(N) U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Weak TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) isNat(n__0()) -> tt() isNat(n__s(V1)) -> U21(isNat(activate(V1))) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1 ,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate ,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(U31) = {1,2}, uargs(U32) = {1}, uargs(U52) = {1,2,3}, uargs(U72) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1}, uargs(x) = {1,2} Following symbols are considered usable: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x} TcT has computed the following interpretation: p(0) = [2] p(U11) = [1] x1 + [8] x2 + [0] p(U12) = [1] x1 + [0] p(U21) = [1] x1 + [10] p(U31) = [1] x1 + [8] x2 + [0] p(U32) = [1] x1 + [1] p(U41) = [2] x1 + [1] x2 + [12] p(U51) = [2] x1 + [3] x2 + [10] x3 + [8] p(U52) = [1] x1 + [2] x2 + [2] x3 + [1] p(U61) = [2] x1 + [4] p(U71) = [8] x1 + [9] x2 + [10] x3 + [4] p(U72) = [1] x1 + [8] x2 + [2] x3 + [9] p(activate) = [1] x1 + [0] p(isNat) = [8] x1 + [0] p(n__0) = [2] p(n__plus) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [2] p(n__x) = [1] x1 + [1] x2 + [2] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [2] p(tt) = [1] p(x) = [1] x1 + [1] x2 + [2] Following rules are strictly oriented: U32(tt()) = [2] > [1] = tt() U41(tt(),N) = [1] N + [14] > [1] N + [0] = activate(N) U71(tt(),M,N) = [9] M + [10] N + [12] > [8] M + [10] N + [9] = U72(isNat(activate(N)),activate(M),activate(N)) isNat(n__x(V1,V2)) = [8] V1 + [8] V2 + [16] > [8] V1 + [8] V2 + [0] = U31(isNat(activate(V1)),activate(V2)) Following rules are (at-least) weakly oriented: 0() = [2] >= [2] = n__0() U11(tt(),V2) = [8] V2 + [1] >= [8] V2 + [0] = U12(isNat(activate(V2))) U12(tt()) = [1] >= [1] = tt() U21(tt()) = [11] >= [1] = tt() U31(tt(),V2) = [8] V2 + [1] >= [8] V2 + [1] = U32(isNat(activate(V2))) U51(tt(),M,N) = [3] M + [10] N + [10] >= [2] M + [10] N + [1] = U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) = [2] M + [2] N + [2] >= [1] M + [1] N + [2] = s(plus(activate(N),activate(M))) U61(tt()) = [6] >= [2] = 0() U72(tt(),M,N) = [8] M + [2] N + [10] >= [1] M + [2] N + [2] = plus(x(activate(N),activate(M)),activate(N)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [2] >= [2] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(X1,X2) activate(n__s(X)) = [1] X + [2] >= [1] X + [2] = s(X) activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = x(X1,X2) isNat(n__0()) = [16] >= [1] = tt() isNat(n__plus(V1,V2)) = [8] V1 + [8] V2 + [0] >= [8] V1 + [8] V2 + [0] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [8] V1 + [16] >= [8] V1 + [10] = U21(isNat(activate(V1))) plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__plus(X1,X2) s(X) = [1] X + [2] >= [1] X + [2] = n__s(X) x(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__x(X1,X2) * Step 4: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() U12(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Weak TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() U41(tt(),N) -> activate(N) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) isNat(n__0()) -> tt() isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1 ,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate ,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(U31) = {1,2}, uargs(U32) = {1}, uargs(U52) = {1,2,3}, uargs(U72) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1}, uargs(x) = {1,2} Following symbols are considered usable: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x} TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [2] x2 + [11] p(U12) = [1] x1 + [3] p(U21) = [1] x1 + [0] p(U31) = [1] x1 + [2] x2 + [0] p(U32) = [1] x1 + [0] p(U41) = [4] x1 + [1] x2 + [14] p(U51) = [5] x1 + [1] x2 + [6] x3 + [9] p(U52) = [2] x1 + [1] x2 + [2] x3 + [12] p(U61) = [6] x1 + [0] p(U71) = [2] x1 + [8] x2 + [8] x3 + [12] p(U72) = [1] x1 + [8] x2 + [4] x3 + [8] p(activate) = [1] x1 + [0] p(isNat) = [2] x1 + [1] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [8] p(n__s) = [1] x1 + [0] p(n__x) = [1] x1 + [1] x2 + [0] p(plus) = [1] x1 + [1] x2 + [8] p(s) = [1] x1 + [0] p(tt) = [1] p(x) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: U12(tt()) = [4] > [1] = tt() isNat(n__plus(V1,V2)) = [2] V1 + [2] V2 + [17] > [2] V1 + [2] V2 + [12] = U11(isNat(activate(V1)),activate(V2)) Following rules are (at-least) weakly oriented: 0() = [0] >= [0] = n__0() U11(tt(),V2) = [2] V2 + [12] >= [2] V2 + [4] = U12(isNat(activate(V2))) U21(tt()) = [1] >= [1] = tt() U31(tt(),V2) = [2] V2 + [1] >= [2] V2 + [1] = U32(isNat(activate(V2))) U32(tt()) = [1] >= [1] = tt() U41(tt(),N) = [1] N + [18] >= [1] N + [0] = activate(N) U51(tt(),M,N) = [1] M + [6] N + [14] >= [1] M + [6] N + [14] = U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) = [1] M + [2] N + [14] >= [1] M + [1] N + [8] = s(plus(activate(N),activate(M))) U61(tt()) = [6] >= [0] = 0() U71(tt(),M,N) = [8] M + [8] N + [14] >= [8] M + [6] N + [9] = U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) = [8] M + [4] N + [9] >= [1] M + [2] N + [8] = plus(x(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 + [8] >= [1] X1 + [1] X2 + [8] = plus(X1,X2) activate(n__s(X)) = [1] X + [0] >= [1] X + [0] = s(X) activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = x(X1,X2) isNat(n__0()) = [1] >= [1] = tt() isNat(n__s(V1)) = [2] V1 + [1] >= [2] V1 + [1] = U21(isNat(activate(V1))) isNat(n__x(V1,V2)) = [2] V1 + [2] V2 + [1] >= [2] V1 + [2] V2 + [1] = U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = n__plus(X1,X2) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) x(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__x(X1,X2) * Step 5: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Weak TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() U41(tt(),N) -> activate(N) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1 ,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate ,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(U31) = {1,2}, uargs(U32) = {1}, uargs(U52) = {1,2,3}, uargs(U72) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1}, uargs(x) = {1,2} Following symbols are considered usable: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x} TcT has computed the following interpretation: p(0) = [1] p(U11) = [1] x1 + [4] x2 + [8] p(U12) = [1] x1 + [3] p(U21) = [1] x1 + [0] p(U31) = [1] x1 + [4] x2 + [12] p(U32) = [1] x1 + [0] p(U41) = [1] x1 + [1] x2 + [10] p(U51) = [2] x1 + [1] x2 + [10] x3 + [11] p(U52) = [2] x1 + [1] x2 + [2] x3 + [0] p(U61) = [1] p(U71) = [2] x1 + [2] x2 + [10] x3 + [11] p(U72) = [2] x1 + [1] x2 + [2] x3 + [0] p(activate) = [1] x1 + [1] p(isNat) = [4] x1 + [8] p(n__0) = [0] p(n__plus) = [1] x1 + [1] x2 + [4] p(n__s) = [1] x1 + [4] p(n__x) = [1] x1 + [1] x2 + [5] p(plus) = [1] x1 + [1] x2 + [5] p(s) = [1] x1 + [5] p(tt) = [8] p(x) = [1] x1 + [1] x2 + [5] Following rules are strictly oriented: 0() = [1] > [0] = n__0() activate(X) = [1] X + [1] > [1] X + [0] = X activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [5] = x(X1,X2) plus(X1,X2) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [4] = n__plus(X1,X2) s(X) = [1] X + [5] > [1] X + [4] = n__s(X) Following rules are (at-least) weakly oriented: U11(tt(),V2) = [4] V2 + [16] >= [4] V2 + [15] = U12(isNat(activate(V2))) U12(tt()) = [11] >= [8] = tt() U21(tt()) = [8] >= [8] = tt() U31(tt(),V2) = [4] V2 + [20] >= [4] V2 + [12] = U32(isNat(activate(V2))) U32(tt()) = [8] >= [8] = tt() U41(tt(),N) = [1] N + [18] >= [1] N + [1] = activate(N) U51(tt(),M,N) = [1] M + [10] N + [27] >= [1] M + [10] N + [27] = U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) = [1] M + [2] N + [16] >= [1] M + [1] N + [12] = s(plus(activate(N),activate(M))) U61(tt()) = [1] >= [1] = 0() U71(tt(),M,N) = [2] M + [10] N + [27] >= [1] M + [10] N + [27] = U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) = [1] M + [2] N + [16] >= [1] M + [2] N + [13] = plus(x(activate(N),activate(M)),activate(N)) activate(n__0()) = [1] >= [1] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = plus(X1,X2) activate(n__s(X)) = [1] X + [5] >= [1] X + [5] = s(X) isNat(n__0()) = [8] >= [8] = tt() isNat(n__plus(V1,V2)) = [4] V1 + [4] V2 + [24] >= [4] V1 + [4] V2 + [24] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [4] V1 + [24] >= [4] V1 + [12] = U21(isNat(activate(V1))) isNat(n__x(V1,V2)) = [4] V1 + [4] V2 + [28] >= [4] V1 + [4] V2 + [28] = U31(isNat(activate(V1)),activate(V2)) x(X1,X2) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = n__x(X1,X2) * Step 6: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) x(X1,X2) -> n__x(X1,X2) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() U41(tt(),N) -> activate(N) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1 ,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate ,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(U31) = {1,2}, uargs(U32) = {1}, uargs(U52) = {1,2,3}, uargs(U72) = {1,2,3}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(s) = {1}, uargs(x) = {1,2} Following symbols are considered usable: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x} TcT has computed the following interpretation: p(0) = [8] p(U11) = [1] x1 + [1] x2 + [0] p(U12) = [1] x1 + [2] p(U21) = [1] x1 + [0] p(U31) = [1] x1 + [1] x2 + [1] p(U32) = [1] x1 + [5] p(U41) = [5] x1 + [1] x2 + [6] p(U51) = [5] x1 + [14] x2 + [12] x3 + [4] p(U52) = [2] x1 + [12] x2 + [8] x3 + [2] p(U61) = [2] x1 + [15] p(U71) = [3] x2 + [6] x3 + [12] p(U72) = [3] x1 + [2] x2 + [3] x3 + [4] p(activate) = [1] x1 + [1] p(isNat) = [1] x1 + [0] p(n__0) = [8] p(n__plus) = [1] x1 + [1] x2 + [8] p(n__s) = [1] x1 + [2] p(n__x) = [1] x1 + [1] x2 + [4] p(plus) = [1] x1 + [1] x2 + [8] p(s) = [1] x1 + [2] p(tt) = [5] p(x) = [1] x1 + [1] x2 + [5] Following rules are strictly oriented: activate(n__0()) = [9] > [8] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [9] > [1] X1 + [1] X2 + [8] = plus(X1,X2) activate(n__s(X)) = [1] X + [3] > [1] X + [2] = s(X) x(X1,X2) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [4] = n__x(X1,X2) Following rules are (at-least) weakly oriented: 0() = [8] >= [8] = n__0() U11(tt(),V2) = [1] V2 + [5] >= [1] V2 + [3] = U12(isNat(activate(V2))) U12(tt()) = [7] >= [5] = tt() U21(tt()) = [5] >= [5] = tt() U31(tt(),V2) = [1] V2 + [6] >= [1] V2 + [6] = U32(isNat(activate(V2))) U32(tt()) = [10] >= [5] = tt() U41(tt(),N) = [1] N + [31] >= [1] N + [1] = activate(N) U51(tt(),M,N) = [14] M + [12] N + [29] >= [12] M + [10] N + [24] = U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) = [12] M + [8] N + [12] >= [1] M + [1] N + [12] = s(plus(activate(N),activate(M))) U61(tt()) = [25] >= [8] = 0() U71(tt(),M,N) = [3] M + [6] N + [12] >= [2] M + [6] N + [12] = U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) = [2] M + [3] N + [19] >= [1] M + [2] N + [16] = plus(x(activate(N),activate(M)),activate(N)) activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [5] = x(X1,X2) isNat(n__0()) = [8] >= [5] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8] >= [1] V1 + [1] V2 + [2] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [1] V1 + [2] >= [1] V1 + [1] = U21(isNat(activate(V1))) isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [4] >= [1] V1 + [1] V2 + [3] = U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = n__plus(X1,X2) s(X) = [1] X + [2] >= [1] X + [2] = n__s(X) * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),V2) -> U32(isNat(activate(V2))) U32(tt()) -> tt() U41(tt(),N) -> activate(N) U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) U52(tt(),M,N) -> s(plus(activate(N),activate(M))) U61(tt()) -> 0() U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) activate(n__x(X1,X2)) -> x(X1,X2) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) x(X1,X2) -> n__x(X1,X2) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1 ,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate ,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))