YES(O(1),O(n^1)) 0.00/0.87 YES(O(1),O(n^1)) 0.00/0.87 0.00/0.87 We are left with following problem, upon which TcT provides the 0.00/0.87 certificate YES(O(1),O(n^1)). 0.00/0.87 0.00/0.87 Strict Trs: 0.00/0.87 { t(N) -> cs(r(q(N)), nt(ns(N))) 0.00/0.87 , t(X) -> nt(X) 0.00/0.87 , q(0()) -> 0() 0.00/0.87 , q(s(X)) -> s(p(q(X), d(X))) 0.00/0.87 , s(X) -> ns(X) 0.00/0.87 , p(X, 0()) -> X 0.00/0.87 , p(0(), X) -> X 0.00/0.87 , p(s(X), s(Y)) -> s(s(p(X, Y))) 0.00/0.87 , d(0()) -> 0() 0.00/0.87 , d(s(X)) -> s(s(d(X))) 0.00/0.87 , f(X1, X2) -> nf(X1, X2) 0.00/0.87 , f(0(), X) -> nil() 0.00/0.87 , f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) 0.00/0.87 , a(X) -> X 0.00/0.87 , a(nt(X)) -> t(a(X)) 0.00/0.87 , a(ns(X)) -> s(a(X)) 0.00/0.87 , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 0.00/0.87 Obligation: 0.00/0.87 innermost runtime complexity 0.00/0.87 Answer: 0.00/0.87 YES(O(1),O(n^1)) 0.00/0.87 0.00/0.87 Arguments of following rules are not normal-forms: 0.00/0.87 0.00/0.87 { q(s(X)) -> s(p(q(X), d(X))) 0.00/0.87 , p(s(X), s(Y)) -> s(s(p(X, Y))) 0.00/0.87 , d(s(X)) -> s(s(d(X))) 0.00/0.87 , f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) } 0.00/0.87 0.00/0.87 All above mentioned rules can be savely removed. 0.00/0.87 0.00/0.87 We are left with following problem, upon which TcT provides the 0.00/0.87 certificate YES(O(1),O(n^1)). 0.00/0.87 0.00/0.87 Strict Trs: 0.00/0.87 { t(N) -> cs(r(q(N)), nt(ns(N))) 0.00/0.87 , t(X) -> nt(X) 0.00/0.87 , q(0()) -> 0() 0.00/0.87 , s(X) -> ns(X) 0.00/0.87 , p(X, 0()) -> X 0.00/0.87 , p(0(), X) -> X 0.00/0.87 , d(0()) -> 0() 0.00/0.87 , f(X1, X2) -> nf(X1, X2) 0.00/0.87 , f(0(), X) -> nil() 0.00/0.87 , a(X) -> X 0.00/0.87 , a(nt(X)) -> t(a(X)) 0.00/0.87 , a(ns(X)) -> s(a(X)) 0.00/0.87 , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 0.00/0.87 Obligation: 0.00/0.87 innermost runtime complexity 0.00/0.87 Answer: 0.00/0.87 YES(O(1),O(n^1)) 0.00/0.87 0.00/0.87 We add the following weak dependency pairs: 0.00/0.87 0.00/0.87 Strict DPs: 0.00/0.87 { t^#(N) -> c_1(q^#(N)) 0.00/0.87 , t^#(X) -> c_2() 0.00/0.87 , q^#(0()) -> c_3() 0.00/0.87 , s^#(X) -> c_4() 0.00/0.87 , p^#(X, 0()) -> c_5() 0.00/0.87 , p^#(0(), X) -> c_6() 0.00/0.87 , d^#(0()) -> c_7() 0.00/0.87 , f^#(X1, X2) -> c_8() 0.00/0.87 , f^#(0(), X) -> c_9() 0.00/0.87 , a^#(X) -> c_10() 0.00/0.87 , a^#(nt(X)) -> c_11(t^#(a(X))) 0.00/0.87 , a^#(ns(X)) -> c_12(s^#(a(X))) 0.00/0.87 , a^#(nf(X1, X2)) -> c_13(f^#(a(X1), a(X2))) } 0.00/0.87 0.00/0.87 and mark the set of starting terms. 0.00/0.87 0.00/0.87 We are left with following problem, upon which TcT provides the 0.00/0.87 certificate YES(O(1),O(n^1)). 0.00/0.87 0.00/0.87 Strict DPs: 0.00/0.87 { t^#(N) -> c_1(q^#(N)) 0.00/0.87 , t^#(X) -> c_2() 0.00/0.87 , q^#(0()) -> c_3() 0.00/0.87 , s^#(X) -> c_4() 0.00/0.87 , p^#(X, 0()) -> c_5() 0.00/0.87 , p^#(0(), X) -> c_6() 0.00/0.87 , d^#(0()) -> c_7() 0.00/0.87 , f^#(X1, X2) -> c_8() 0.00/0.87 , f^#(0(), X) -> c_9() 0.00/0.87 , a^#(X) -> c_10() 0.00/0.87 , a^#(nt(X)) -> c_11(t^#(a(X))) 0.00/0.87 , a^#(ns(X)) -> c_12(s^#(a(X))) 0.00/0.87 , a^#(nf(X1, X2)) -> c_13(f^#(a(X1), a(X2))) } 0.00/0.87 Strict Trs: 0.00/0.87 { t(N) -> cs(r(q(N)), nt(ns(N))) 0.00/0.87 , t(X) -> nt(X) 0.00/0.87 , q(0()) -> 0() 0.00/0.87 , s(X) -> ns(X) 0.00/0.87 , p(X, 0()) -> X 0.00/0.87 , p(0(), X) -> X 0.00/0.87 , d(0()) -> 0() 0.00/0.87 , f(X1, X2) -> nf(X1, X2) 0.00/0.87 , f(0(), X) -> nil() 0.00/0.87 , a(X) -> X 0.00/0.87 , a(nt(X)) -> t(a(X)) 0.00/0.87 , a(ns(X)) -> s(a(X)) 0.00/0.87 , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 0.00/0.87 Obligation: 0.00/0.87 innermost runtime complexity 0.00/0.87 Answer: 0.00/0.87 YES(O(1),O(n^1)) 0.00/0.87 0.00/0.87 We replace rewrite rules by usable rules: 0.00/0.87 0.00/0.87 Strict Usable Rules: 0.00/0.87 { t(N) -> cs(r(q(N)), nt(ns(N))) 0.00/0.87 , t(X) -> nt(X) 0.00/0.87 , q(0()) -> 0() 0.00/0.87 , s(X) -> ns(X) 0.00/0.87 , f(X1, X2) -> nf(X1, X2) 0.00/0.87 , f(0(), X) -> nil() 0.00/0.87 , a(X) -> X 0.00/0.87 , a(nt(X)) -> t(a(X)) 0.00/0.87 , a(ns(X)) -> s(a(X)) 0.00/0.87 , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 0.00/0.87 0.00/0.87 We are left with following problem, upon which TcT provides the 0.00/0.87 certificate YES(O(1),O(n^1)). 0.00/0.87 0.00/0.87 Strict DPs: 0.00/0.87 { t^#(N) -> c_1(q^#(N)) 0.00/0.87 , t^#(X) -> c_2() 0.00/0.87 , q^#(0()) -> c_3() 0.00/0.87 , s^#(X) -> c_4() 0.00/0.87 , p^#(X, 0()) -> c_5() 0.00/0.87 , p^#(0(), X) -> c_6() 0.00/0.87 , d^#(0()) -> c_7() 0.00/0.87 , f^#(X1, X2) -> c_8() 0.00/0.87 , f^#(0(), X) -> c_9() 0.00/0.87 , a^#(X) -> c_10() 0.00/0.87 , a^#(nt(X)) -> c_11(t^#(a(X))) 0.00/0.87 , a^#(ns(X)) -> c_12(s^#(a(X))) 0.00/0.87 , a^#(nf(X1, X2)) -> c_13(f^#(a(X1), a(X2))) } 0.00/0.87 Strict Trs: 0.00/0.87 { t(N) -> cs(r(q(N)), nt(ns(N))) 0.00/0.87 , t(X) -> nt(X) 0.00/0.87 , q(0()) -> 0() 0.00/0.87 , s(X) -> ns(X) 0.00/0.87 , f(X1, X2) -> nf(X1, X2) 0.00/0.87 , f(0(), X) -> nil() 0.00/0.87 , a(X) -> X 0.00/0.87 , a(nt(X)) -> t(a(X)) 0.00/0.87 , a(ns(X)) -> s(a(X)) 0.00/0.87 , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 0.00/0.87 Obligation: 0.00/0.87 innermost runtime complexity 0.00/0.87 Answer: 0.00/0.87 YES(O(1),O(n^1)) 0.00/0.87 0.00/0.87 The weightgap principle applies (using the following constant 0.00/0.87 growth matrix-interpretation) 0.00/0.87 0.00/0.87 The following argument positions are usable: 0.00/0.87 Uargs(t) = {1}, Uargs(cs) = {1}, Uargs(r) = {1}, Uargs(s) = {1}, 0.00/0.87 Uargs(f) = {1, 2}, Uargs(t^#) = {1}, Uargs(c_1) = {1}, 0.00/0.87 Uargs(s^#) = {1}, Uargs(f^#) = {1, 2}, Uargs(c_11) = {1}, 0.00/0.87 Uargs(c_12) = {1}, Uargs(c_13) = {1} 0.00/0.87 0.00/0.87 TcT has computed the following constructor-restricted matrix 0.00/0.87 interpretation. 0.00/0.87 0.00/0.87 [t](x1) = [1 1] x1 + [1] 0.00/0.87 [0 0] [2] 0.00/0.87 0.00/0.87 [cs](x1, x2) = [1 0] x1 + [0] 0.00/0.87 [0 0] [1] 0.00/0.87 0.00/0.87 [r](x1) = [1 0] x1 + [0] 0.00/0.87 [0 0] [2] 0.00/0.87 0.00/0.87 [q](x1) = [0 1] x1 + [0] 0.00/0.87 [0 0] [1] 0.00/0.87 0.00/0.87 [nt](x1) = [1 1] x1 + [0] 0.00/0.87 [0 0] [2] 0.00/0.87 0.00/0.87 [ns](x1) = [1 0] x1 + [1] 0.00/0.87 [0 1] [2] 0.00/0.87 0.00/0.87 [0] = [0] 0.00/0.87 [1] 0.00/0.87 0.00/0.87 [s](x1) = [1 0] x1 + [2] 0.00/0.87 [0 1] [2] 0.00/0.87 0.00/0.87 [f](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 0.00/0.87 [0 1] [0 1] [2] 0.00/0.87 0.00/0.87 [nil] = [1] 0.00/0.87 [1] 0.00/0.87 0.00/0.87 [nf](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 0.00/0.87 [0 1] [0 1] [2] 0.00/0.87 0.00/0.87 [a](x1) = [2 1] x1 + [1] 0.00/0.87 [0 1] [0] 0.00/0.87 0.00/0.87 [t^#](x1) = [1 0] x1 + [1] 0.00/0.87 [0 0] [2] 0.00/0.87 0.00/0.87 [c_1](x1) = [1 0] x1 + [1] 0.00/0.87 [0 1] [1] 0.00/0.87 0.00/0.87 [q^#](x1) = [1 0] x1 + [2] 0.00/0.87 [1 2] [2] 0.00/0.87 0.00/0.87 [c_2] = [0] 0.00/0.87 [1] 0.00/0.87 0.00/0.87 [c_3] = [1] 0.00/0.87 [0] 0.00/0.87 0.00/0.87 [s^#](x1) = [1 0] x1 + [1] 0.00/0.87 [0 0] [2] 0.00/0.87 0.00/0.87 [c_4] = [0] 0.00/0.87 [1] 0.00/0.87 0.00/0.87 [p^#](x1, x2) = [2 2] x1 + [1 2] x2 + [2] 0.00/0.87 [1 2] [1 2] [2] 0.00/0.87 0.00/0.87 [c_5] = [1] 0.00/0.87 [0] 0.00/0.87 0.00/0.87 [c_6] = [1] 0.00/0.87 [0] 0.00/0.87 0.00/0.87 [d^#](x1) = [1 2] x1 + [2] 0.00/0.87 [2 2] [2] 0.00/0.87 0.00/0.87 [c_7] = [1] 0.00/0.87 [0] 0.00/0.87 0.00/0.87 [f^#](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 0.00/0.87 [0 0] [0 0] [2] 0.00/0.87 0.00/0.87 [c_8] = [1] 0.00/0.87 [1] 0.00/0.87 0.00/0.87 [c_9] = [1] 0.00/0.87 [1] 0.00/0.87 0.00/0.87 [a^#](x1) = [2 1] x1 + [0] 0.00/0.87 [0 0] [0] 0.00/0.87 0.00/0.87 [c_10] = [1] 0.00/0.87 [1] 0.00/0.87 0.00/0.87 [c_11](x1) = [1 0] x1 + [2] 0.00/0.87 [0 1] [2] 0.00/0.87 0.00/0.87 [c_12](x1) = [1 0] x1 + [2] 0.00/0.87 [0 1] [2] 0.00/0.87 0.00/0.87 [c_13](x1) = [1 0] x1 + [2] 0.00/0.87 [0 1] [2] 0.00/0.87 0.00/0.87 The order satisfies the following ordering constraints: 0.00/0.87 0.00/0.87 [t(N)] = [1 1] N + [1] 0.00/0.87 [0 0] [2] 0.00/0.87 > [0 1] N + [0] 0.00/0.87 [0 0] [1] 0.00/0.87 = [cs(r(q(N)), nt(ns(N)))] 0.00/0.87 0.00/0.87 [t(X)] = [1 1] X + [1] 0.00/0.87 [0 0] [2] 0.00/0.87 > [1 1] X + [0] 0.00/0.87 [0 0] [2] 0.00/0.87 = [nt(X)] 0.00/0.87 0.00/0.87 [q(0())] = [1] 0.00/0.87 [1] 0.00/0.87 > [0] 0.00/0.87 [1] 0.00/0.87 = [0()] 0.00/0.87 0.00/0.87 [s(X)] = [1 0] X + [2] 0.00/0.87 [0 1] [2] 0.00/0.87 > [1 0] X + [1] 0.00/0.87 [0 1] [2] 0.00/0.87 = [ns(X)] 0.00/0.87 0.00/0.87 [f(X1, X2)] = [1 0] X1 + [1 0] X2 + [2] 0.00/0.87 [0 1] [0 1] [2] 0.00/0.87 > [1 0] X1 + [1 0] X2 + [1] 0.00/0.87 [0 1] [0 1] [2] 0.00/0.87 = [nf(X1, X2)] 0.00/0.87 0.00/0.87 [f(0(), X)] = [1 0] X + [2] 0.00/0.87 [0 1] [3] 0.00/0.87 > [1] 0.00/0.87 [1] 0.00/0.87 = [nil()] 0.00/0.87 0.00/0.87 [a(X)] = [2 1] X + [1] 0.00/0.87 [0 1] [0] 0.00/0.87 > [1 0] X + [0] 0.00/0.87 [0 1] [0] 0.00/0.87 = [X] 0.00/0.87 0.00/0.87 [a(nt(X))] = [2 2] X + [3] 0.00/0.87 [0 0] [2] 0.00/0.87 > [2 2] X + [2] 0.00/0.87 [0 0] [2] 0.00/0.87 = [t(a(X))] 0.00/0.87 0.00/0.87 [a(ns(X))] = [2 1] X + [5] 0.00/0.87 [0 1] [2] 0.00/0.87 > [2 1] X + [3] 0.00/0.87 [0 1] [2] 0.00/0.87 = [s(a(X))] 0.00/0.87 0.00/0.87 [a(nf(X1, X2))] = [2 1] X1 + [2 1] X2 + [5] 0.00/0.87 [0 1] [0 1] [2] 0.00/0.87 > [2 1] X1 + [2 1] X2 + [4] 0.00/0.87 [0 1] [0 1] [2] 0.00/0.87 = [f(a(X1), a(X2))] 0.00/0.87 0.00/0.87 [t^#(N)] = [1 0] N + [1] 0.00/0.87 [0 0] [2] 0.00/0.87 ? [1 0] N + [3] 0.00/0.87 [1 2] [3] 0.00/0.87 = [c_1(q^#(N))] 0.00/0.87 0.00/0.87 [t^#(X)] = [1 0] X + [1] 0.00/0.87 [0 0] [2] 0.00/0.87 > [0] 0.00/0.87 [1] 0.00/0.87 = [c_2()] 0.00/0.87 0.00/0.87 [q^#(0())] = [2] 0.00/0.87 [4] 0.00/0.87 > [1] 0.00/0.87 [0] 0.00/0.87 = [c_3()] 0.00/0.87 0.00/0.87 [s^#(X)] = [1 0] X + [1] 0.00/0.87 [0 0] [2] 0.00/0.87 > [0] 0.00/0.87 [1] 0.00/0.87 = [c_4()] 0.00/0.87 0.00/0.87 [p^#(X, 0())] = [2 2] X + [4] 0.00/0.87 [1 2] [4] 0.00/0.87 > [1] 0.00/0.87 [0] 0.00/0.87 = [c_5()] 0.00/0.87 0.00/0.87 [p^#(0(), X)] = [1 2] X + [4] 0.00/0.87 [1 2] [4] 0.00/0.87 > [1] 0.00/0.87 [0] 0.00/0.87 = [c_6()] 0.00/0.87 0.00/0.87 [d^#(0())] = [4] 0.00/0.87 [4] 0.00/0.87 > [1] 0.00/0.87 [0] 0.00/0.87 = [c_7()] 0.00/0.87 0.00/0.87 [f^#(X1, X2)] = [1 0] X1 + [1 0] X2 + [2] 0.00/0.87 [0 0] [0 0] [2] 0.00/0.87 > [1] 0.00/0.87 [1] 0.00/0.87 = [c_8()] 0.00/0.87 0.00/0.87 [f^#(0(), X)] = [1 0] X + [2] 0.00/0.87 [0 0] [2] 0.00/0.87 > [1] 0.00/0.87 [1] 0.00/0.87 = [c_9()] 0.00/0.87 0.00/0.87 [a^#(X)] = [2 1] X + [0] 0.00/0.87 [0 0] [0] 0.00/0.87 ? [1] 0.00/0.87 [1] 0.00/0.87 = [c_10()] 0.00/0.87 0.00/0.87 [a^#(nt(X))] = [2 2] X + [2] 0.00/0.87 [0 0] [0] 0.00/0.87 ? [2 1] X + [4] 0.00/0.87 [0 0] [4] 0.00/0.87 = [c_11(t^#(a(X)))] 0.00/0.87 0.00/0.87 [a^#(ns(X))] = [2 1] X + [4] 0.00/0.87 [0 0] [0] 0.00/0.87 ? [2 1] X + [4] 0.00/0.87 [0 0] [4] 0.00/0.87 = [c_12(s^#(a(X)))] 0.00/0.87 0.00/0.87 [a^#(nf(X1, X2))] = [2 1] X1 + [2 1] X2 + [4] 0.00/0.87 [0 0] [0 0] [0] 0.00/0.87 ? [2 1] X1 + [2 1] X2 + [6] 0.00/0.87 [0 0] [0 0] [4] 0.00/0.87 = [c_13(f^#(a(X1), a(X2)))] 0.00/0.87 0.00/0.87 0.00/0.87 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.87 0.00/0.87 We are left with following problem, upon which TcT provides the 0.00/0.87 certificate YES(O(1),O(1)). 0.00/0.87 0.00/0.87 Strict DPs: 0.00/0.87 { t^#(N) -> c_1(q^#(N)) 0.00/0.87 , a^#(X) -> c_10() 0.00/0.87 , a^#(nt(X)) -> c_11(t^#(a(X))) 0.00/0.87 , a^#(ns(X)) -> c_12(s^#(a(X))) 0.00/0.87 , a^#(nf(X1, X2)) -> c_13(f^#(a(X1), a(X2))) } 0.00/0.87 Weak DPs: 0.00/0.87 { t^#(X) -> c_2() 0.00/0.87 , q^#(0()) -> c_3() 0.00/0.87 , s^#(X) -> c_4() 0.00/0.87 , p^#(X, 0()) -> c_5() 0.00/0.87 , p^#(0(), X) -> c_6() 0.00/0.87 , d^#(0()) -> c_7() 0.00/0.87 , f^#(X1, X2) -> c_8() 0.00/0.87 , f^#(0(), X) -> c_9() } 0.00/0.87 Weak Trs: 0.00/0.87 { t(N) -> cs(r(q(N)), nt(ns(N))) 0.00/0.87 , t(X) -> nt(X) 0.00/0.87 , q(0()) -> 0() 0.00/0.87 , s(X) -> ns(X) 0.00/0.87 , f(X1, X2) -> nf(X1, X2) 0.00/0.87 , f(0(), X) -> nil() 0.00/0.87 , a(X) -> X 0.00/0.87 , a(nt(X)) -> t(a(X)) 0.00/0.87 , a(ns(X)) -> s(a(X)) 0.00/0.87 , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 0.00/0.87 Obligation: 0.00/0.87 innermost runtime complexity 0.00/0.87 Answer: 0.00/0.87 YES(O(1),O(1)) 0.00/0.87 0.00/0.87 We estimate the number of application of {1,2,4,5} by applications 0.00/0.87 of Pre({1,2,4,5}) = {3}. Here rules are labeled as follows: 0.00/0.87 0.00/0.87 DPs: 0.00/0.87 { 1: t^#(N) -> c_1(q^#(N)) 0.00/0.87 , 2: a^#(X) -> c_10() 0.00/0.87 , 3: a^#(nt(X)) -> c_11(t^#(a(X))) 0.00/0.87 , 4: a^#(ns(X)) -> c_12(s^#(a(X))) 0.00/0.87 , 5: a^#(nf(X1, X2)) -> c_13(f^#(a(X1), a(X2))) 0.00/0.87 , 6: t^#(X) -> c_2() 0.00/0.87 , 7: q^#(0()) -> c_3() 0.00/0.87 , 8: s^#(X) -> c_4() 0.00/0.87 , 9: p^#(X, 0()) -> c_5() 0.00/0.87 , 10: p^#(0(), X) -> c_6() 0.00/0.87 , 11: d^#(0()) -> c_7() 0.00/0.87 , 12: f^#(X1, X2) -> c_8() 0.00/0.87 , 13: f^#(0(), X) -> c_9() } 0.00/0.87 0.00/0.87 We are left with following problem, upon which TcT provides the 0.00/0.87 certificate YES(O(1),O(1)). 0.00/0.87 0.00/0.87 Strict DPs: { a^#(nt(X)) -> c_11(t^#(a(X))) } 0.00/0.87 Weak DPs: 0.00/0.87 { t^#(N) -> c_1(q^#(N)) 0.00/0.87 , t^#(X) -> c_2() 0.00/0.87 , q^#(0()) -> c_3() 0.00/0.87 , s^#(X) -> c_4() 0.00/0.87 , p^#(X, 0()) -> c_5() 0.00/0.87 , p^#(0(), X) -> c_6() 0.00/0.87 , d^#(0()) -> c_7() 0.00/0.87 , f^#(X1, X2) -> c_8() 0.00/0.87 , f^#(0(), X) -> c_9() 0.00/0.87 , a^#(X) -> c_10() 0.00/0.87 , a^#(ns(X)) -> c_12(s^#(a(X))) 0.00/0.87 , a^#(nf(X1, X2)) -> c_13(f^#(a(X1), a(X2))) } 0.00/0.87 Weak Trs: 0.00/0.87 { t(N) -> cs(r(q(N)), nt(ns(N))) 0.00/0.87 , t(X) -> nt(X) 0.00/0.87 , q(0()) -> 0() 0.00/0.87 , s(X) -> ns(X) 0.00/0.87 , f(X1, X2) -> nf(X1, X2) 0.00/0.87 , f(0(), X) -> nil() 0.00/0.87 , a(X) -> X 0.00/0.87 , a(nt(X)) -> t(a(X)) 0.00/0.87 , a(ns(X)) -> s(a(X)) 0.00/0.87 , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 0.00/0.87 Obligation: 0.00/0.87 innermost runtime complexity 0.00/0.87 Answer: 0.00/0.87 YES(O(1),O(1)) 0.00/0.87 0.00/0.87 We estimate the number of application of {1} by applications of 0.00/0.87 Pre({1}) = {}. Here rules are labeled as follows: 0.00/0.87 0.00/0.87 DPs: 0.00/0.87 { 1: a^#(nt(X)) -> c_11(t^#(a(X))) 0.00/0.87 , 2: t^#(N) -> c_1(q^#(N)) 0.00/0.87 , 3: t^#(X) -> c_2() 0.00/0.87 , 4: q^#(0()) -> c_3() 0.00/0.87 , 5: s^#(X) -> c_4() 0.00/0.87 , 6: p^#(X, 0()) -> c_5() 0.00/0.87 , 7: p^#(0(), X) -> c_6() 0.00/0.87 , 8: d^#(0()) -> c_7() 0.00/0.87 , 9: f^#(X1, X2) -> c_8() 0.00/0.87 , 10: f^#(0(), X) -> c_9() 0.00/0.87 , 11: a^#(X) -> c_10() 0.00/0.87 , 12: a^#(ns(X)) -> c_12(s^#(a(X))) 0.00/0.87 , 13: a^#(nf(X1, X2)) -> c_13(f^#(a(X1), a(X2))) } 0.00/0.87 0.00/0.87 We are left with following problem, upon which TcT provides the 0.00/0.87 certificate YES(O(1),O(1)). 0.00/0.87 0.00/0.87 Weak DPs: 0.00/0.87 { t^#(N) -> c_1(q^#(N)) 0.00/0.87 , t^#(X) -> c_2() 0.00/0.87 , q^#(0()) -> c_3() 0.00/0.87 , s^#(X) -> c_4() 0.00/0.87 , p^#(X, 0()) -> c_5() 0.00/0.87 , p^#(0(), X) -> c_6() 0.00/0.87 , d^#(0()) -> c_7() 0.00/0.87 , f^#(X1, X2) -> c_8() 0.00/0.87 , f^#(0(), X) -> c_9() 0.00/0.87 , a^#(X) -> c_10() 0.00/0.87 , a^#(nt(X)) -> c_11(t^#(a(X))) 0.00/0.87 , a^#(ns(X)) -> c_12(s^#(a(X))) 0.00/0.87 , a^#(nf(X1, X2)) -> c_13(f^#(a(X1), a(X2))) } 0.00/0.87 Weak Trs: 0.00/0.87 { t(N) -> cs(r(q(N)), nt(ns(N))) 0.00/0.87 , t(X) -> nt(X) 0.00/0.87 , q(0()) -> 0() 0.00/0.87 , s(X) -> ns(X) 0.00/0.87 , f(X1, X2) -> nf(X1, X2) 0.00/0.87 , f(0(), X) -> nil() 0.00/0.87 , a(X) -> X 0.00/0.87 , a(nt(X)) -> t(a(X)) 0.00/0.87 , a(ns(X)) -> s(a(X)) 0.00/0.87 , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 0.00/0.87 Obligation: 0.00/0.87 innermost runtime complexity 0.00/0.87 Answer: 0.00/0.87 YES(O(1),O(1)) 0.00/0.87 0.00/0.87 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.87 closed under successors. The DPs are removed. 0.00/0.87 0.00/0.87 { t^#(N) -> c_1(q^#(N)) 0.00/0.87 , t^#(X) -> c_2() 0.00/0.87 , q^#(0()) -> c_3() 0.00/0.87 , s^#(X) -> c_4() 0.00/0.87 , p^#(X, 0()) -> c_5() 0.00/0.87 , p^#(0(), X) -> c_6() 0.00/0.87 , d^#(0()) -> c_7() 0.00/0.87 , f^#(X1, X2) -> c_8() 0.00/0.87 , f^#(0(), X) -> c_9() 0.00/0.87 , a^#(X) -> c_10() 0.00/0.87 , a^#(nt(X)) -> c_11(t^#(a(X))) 0.00/0.87 , a^#(ns(X)) -> c_12(s^#(a(X))) 0.00/0.87 , a^#(nf(X1, X2)) -> c_13(f^#(a(X1), a(X2))) } 0.00/0.87 0.00/0.87 We are left with following problem, upon which TcT provides the 0.00/0.87 certificate YES(O(1),O(1)). 0.00/0.87 0.00/0.87 Weak Trs: 0.00/0.87 { t(N) -> cs(r(q(N)), nt(ns(N))) 0.00/0.87 , t(X) -> nt(X) 0.00/0.87 , q(0()) -> 0() 0.00/0.87 , s(X) -> ns(X) 0.00/0.87 , f(X1, X2) -> nf(X1, X2) 0.00/0.87 , f(0(), X) -> nil() 0.00/0.87 , a(X) -> X 0.00/0.87 , a(nt(X)) -> t(a(X)) 0.00/0.87 , a(ns(X)) -> s(a(X)) 0.00/0.87 , a(nf(X1, X2)) -> f(a(X1), a(X2)) } 0.00/0.87 Obligation: 0.00/0.87 innermost runtime complexity 0.00/0.87 Answer: 0.00/0.87 YES(O(1),O(1)) 0.00/0.87 0.00/0.87 No rule is usable, rules are removed from the input problem. 0.00/0.87 0.00/0.87 We are left with following problem, upon which TcT provides the 0.00/0.87 certificate YES(O(1),O(1)). 0.00/0.87 0.00/0.87 Rules: Empty 0.00/0.87 Obligation: 0.00/0.87 innermost runtime complexity 0.00/0.87 Answer: 0.00/0.87 YES(O(1),O(1)) 0.00/0.87 0.00/0.87 Empty rules are trivially bounded 0.00/0.87 0.00/0.87 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.88 EOF