YES(O(1),O(n^1)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs)) , member(x, Nil()) -> False() , notEmpty(Cons(x, xs)) -> True() , notEmpty(Nil()) -> False() , goal(x, xs) -> member(x, xs) } Weak Trs: { !EQ(S(x), S(y)) -> !EQ(x, y) , !EQ(S(x), 0()) -> False() , !EQ(0(), S(y)) -> False() , !EQ(0(), 0()) -> True() , member[Ite][True][Ite](True(), x, xs) -> True() , member[Ite][True][Ite](False(), x', Cons(x, xs)) -> member(x', xs) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add the following innermost weak dependency pairs: Strict DPs: { member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) , member^#(x, Nil()) -> c_2() , notEmpty^#(Cons(x, xs)) -> c_3() , notEmpty^#(Nil()) -> c_4() , goal^#(x, xs) -> c_5(member^#(x, xs)) } Weak DPs: { member[Ite][True][Ite]^#(True(), x, xs) -> c_10() , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y)) , !EQ^#(S(x), 0()) -> c_7() , !EQ^#(0(), S(y)) -> c_8() , !EQ^#(0(), 0()) -> c_9() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) , member^#(x, Nil()) -> c_2() , notEmpty^#(Cons(x, xs)) -> c_3() , notEmpty^#(Nil()) -> c_4() , goal^#(x, xs) -> c_5(member^#(x, xs)) } Strict Trs: { member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs)) , member(x, Nil()) -> False() , notEmpty(Cons(x, xs)) -> True() , notEmpty(Nil()) -> False() , goal(x, xs) -> member(x, xs) } Weak DPs: { member[Ite][True][Ite]^#(True(), x, xs) -> c_10() , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y)) , !EQ^#(S(x), 0()) -> c_7() , !EQ^#(0(), S(y)) -> c_8() , !EQ^#(0(), 0()) -> c_9() } Weak Trs: { !EQ(S(x), S(y)) -> !EQ(x, y) , !EQ(S(x), 0()) -> False() , !EQ(0(), S(y)) -> False() , !EQ(0(), 0()) -> True() , member[Ite][True][Ite](True(), x, xs) -> True() , member[Ite][True][Ite](False(), x', Cons(x, xs)) -> member(x', xs) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Weak Usable Rules: { !EQ(S(x), S(y)) -> !EQ(x, y) , !EQ(S(x), 0()) -> False() , !EQ(0(), S(y)) -> False() , !EQ(0(), 0()) -> True() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) , member^#(x, Nil()) -> c_2() , notEmpty^#(Cons(x, xs)) -> c_3() , notEmpty^#(Nil()) -> c_4() , goal^#(x, xs) -> c_5(member^#(x, xs)) } Weak DPs: { member[Ite][True][Ite]^#(True(), x, xs) -> c_10() , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y)) , !EQ^#(S(x), 0()) -> c_7() , !EQ^#(0(), S(y)) -> c_8() , !EQ^#(0(), 0()) -> c_9() } Weak Trs: { !EQ(S(x), S(y)) -> !EQ(x, y) , !EQ(S(x), 0()) -> False() , !EQ(0(), S(y)) -> False() , !EQ(0(), 0()) -> True() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_1) = {1}, Uargs(member[Ite][True][Ite]^#) = {1}, Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_11) = {1} TcT has computed the following constructor-restricted matrix interpretation. [!EQ](x1, x2) = [0] [0] [True] = [0] [0] [S](x1) = [1 0] x1 + [0] [0 0] [0] [Cons](x1, x2) = [1 0] x2 + [0] [0 0] [0] [Nil] = [2] [2] [0] = [0] [0] [False] = [0] [0] [member^#](x1, x2) = [0 0] x1 + [1] [1 1] [1] [c_1](x1) = [1 0] x1 + [1] [0 1] [1] [member[Ite][True][Ite]^#](x1, x2, x3) = [2 0] x1 + [0 0] x2 + [1] [0 0] [1 1] [1] [c_2] = [0] [1] [notEmpty^#](x1) = [1 1] x1 + [2] [1 2] [1] [c_3] = [1] [1] [c_4] = [1] [1] [goal^#](x1, x2) = [2 2] x1 + [1 1] x2 + [2] [1 2] [2 2] [2] [c_5](x1) = [1 0] x1 + [0] [0 1] [1] [!EQ^#](x1, x2) = [0] [1] [c_6](x1) = [1 0] x1 + [0] [0 1] [0] [c_7] = [0] [1] [c_8] = [0] [1] [c_9] = [0] [1] [c_10] = [1] [1] [c_11](x1) = [1 0] x1 + [0] [0 1] [0] The following symbols are considered usable {!EQ, member^#, member[Ite][True][Ite]^#, notEmpty^#, goal^#, !EQ^#} The order satisfies the following ordering constraints: [!EQ(S(x), S(y))] = [0] [0] >= [0] [0] = [!EQ(x, y)] [!EQ(S(x), 0())] = [0] [0] >= [0] [0] = [False()] [!EQ(0(), S(y))] = [0] [0] >= [0] [0] = [False()] [!EQ(0(), 0())] = [0] [0] >= [0] [0] = [True()] [member^#(x', Cons(x, xs))] = [0 0] x' + [1] [1 1] [1] ? [0 0] x' + [2] [1 1] [2] = [c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))] [member^#(x, Nil())] = [0 0] x + [1] [1 1] [1] > [0] [1] = [c_2()] [member[Ite][True][Ite]^#(True(), x, xs)] = [0 0] x + [1] [1 1] [1] >= [1] [1] = [c_10()] [member[Ite][True][Ite]^#(False(), x', Cons(x, xs))] = [0 0] x' + [1] [1 1] [1] >= [0 0] x' + [1] [1 1] [1] = [c_11(member^#(x', xs))] [notEmpty^#(Cons(x, xs))] = [1 0] xs + [2] [1 0] [1] > [1] [1] = [c_3()] [notEmpty^#(Nil())] = [6] [7] > [1] [1] = [c_4()] [goal^#(x, xs)] = [2 2] x + [1 1] xs + [2] [1 2] [2 2] [2] > [0 0] x + [1] [1 1] [2] = [c_5(member^#(x, xs))] [!EQ^#(S(x), S(y))] = [0] [1] >= [0] [1] = [c_6(!EQ^#(x, y))] [!EQ^#(S(x), 0())] = [0] [1] >= [0] [1] = [c_7()] [!EQ^#(0(), S(y))] = [0] [1] >= [0] [1] = [c_8()] [!EQ^#(0(), 0())] = [0] [1] >= [0] [1] = [c_9()] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) } Weak DPs: { member^#(x, Nil()) -> c_2() , member[Ite][True][Ite]^#(True(), x, xs) -> c_10() , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) , notEmpty^#(Cons(x, xs)) -> c_3() , notEmpty^#(Nil()) -> c_4() , goal^#(x, xs) -> c_5(member^#(x, xs)) , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y)) , !EQ^#(S(x), 0()) -> c_7() , !EQ^#(0(), S(y)) -> c_8() , !EQ^#(0(), 0()) -> c_9() } Weak Trs: { !EQ(S(x), S(y)) -> !EQ(x, y) , !EQ(S(x), 0()) -> False() , !EQ(0(), S(y)) -> False() , !EQ(0(), 0()) -> True() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { member^#(x, Nil()) -> c_2() , member[Ite][True][Ite]^#(True(), x, xs) -> c_10() , notEmpty^#(Cons(x, xs)) -> c_3() , notEmpty^#(Nil()) -> c_4() , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y)) , !EQ^#(S(x), 0()) -> c_7() , !EQ^#(0(), S(y)) -> c_8() , !EQ^#(0(), 0()) -> c_9() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) } Weak DPs: { member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) , goal^#(x, xs) -> c_5(member^#(x, xs)) } Weak Trs: { !EQ(S(x), S(y)) -> !EQ(x, y) , !EQ(S(x), 0()) -> False() , !EQ(0(), S(y)) -> False() , !EQ(0(), 0()) -> True() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) Consider the dependency graph 1: member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) -->_1 member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) :2 2: member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) -->_1 member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) :1 3: goal^#(x, xs) -> c_5(member^#(x, xs)) -->_1 member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) :1 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). { goal^#(x, xs) -> c_5(member^#(x, xs)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) } Weak DPs: { member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) } Weak Trs: { !EQ(S(x), S(y)) -> !EQ(x, y) , !EQ(S(x), 0()) -> False() , !EQ(0(), S(y)) -> False() , !EQ(0(), 0()) -> True() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) , 2: member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_11) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [member](x1, x2) = [7] x1 + [7] x2 + [0] [!EQ](x1, x2) = [0] [True] = [0] [S](x1) = [1] x1 + [0] [Cons](x1, x2) = [1] x2 + [4] [member[Ite][True][Ite]](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [Nil] = [0] [0] = [0] [notEmpty](x1) = [7] x1 + [0] [goal](x1, x2) = [7] x1 + [7] x2 + [0] [False] = [0] [member^#](x1, x2) = [2] x2 + [4] [c_1](x1) = [1] x1 + [3] [member[Ite][True][Ite]^#](x1, x2, x3) = [2] x3 + [0] [c_2] = [0] [notEmpty^#](x1) = [7] x1 + [0] [c_3] = [0] [c_4] = [0] [goal^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_5](x1) = [7] x1 + [0] [!EQ^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_6](x1) = [7] x1 + [0] [c_7] = [0] [c_8] = [0] [c_9] = [0] [c_10] = [0] [c_11](x1) = [1] x1 + [1] The following symbols are considered usable {!EQ, member^#, member[Ite][True][Ite]^#} The order satisfies the following ordering constraints: [!EQ(S(x), S(y))] = [0] >= [0] = [!EQ(x, y)] [!EQ(S(x), 0())] = [0] >= [0] = [False()] [!EQ(0(), S(y))] = [0] >= [0] = [False()] [!EQ(0(), 0())] = [0] >= [0] = [True()] [member^#(x', Cons(x, xs))] = [2] xs + [12] > [2] xs + [11] = [c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))] [member[Ite][True][Ite]^#(False(), x', Cons(x, xs))] = [2] xs + [8] > [2] xs + [5] = [c_11(member^#(x', xs))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) } Weak Trs: { !EQ(S(x), S(y)) -> !EQ(x, y) , !EQ(S(x), 0()) -> False() , !EQ(0(), S(y)) -> False() , !EQ(0(), 0()) -> True() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { member^#(x', Cons(x, xs)) -> c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) -> c_11(member^#(x', xs)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { !EQ(S(x), S(y)) -> !EQ(x, y) , !EQ(S(x), 0()) -> False() , !EQ(0(), S(y)) -> False() , !EQ(0(), 0()) -> True() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))