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 { member(x', Cons(x, xs)) ->
0.00/0.87 member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
0.00/0.87 , member(x, Nil()) -> False()
0.00/0.87 , notEmpty(Cons(x, xs)) -> True()
0.00/0.87 , notEmpty(Nil()) -> False()
0.00/0.87 , goal(x, xs) -> member(x, xs) }
0.00/0.87 Weak Trs:
0.00/0.87 { !EQ(S(x), S(y)) -> !EQ(x, y)
0.00/0.87 , !EQ(S(x), 0()) -> False()
0.00/0.87 , !EQ(0(), S(y)) -> False()
0.00/0.87 , !EQ(0(), 0()) -> True()
0.00/0.87 , member[Ite][True][Ite](True(), x, xs) -> True()
0.00/0.87 , member[Ite][True][Ite](False(), x', Cons(x, xs)) ->
0.00/0.87 member(x', xs) }
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 { member^#(x', Cons(x, xs)) ->
0.00/0.87 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))
0.00/0.87 , member^#(x, Nil()) -> c_2()
0.00/0.87 , notEmpty^#(Cons(x, xs)) -> c_3()
0.00/0.87 , notEmpty^#(Nil()) -> c_4()
0.00/0.87 , goal^#(x, xs) -> c_5(member^#(x, xs)) }
0.00/0.87 Weak DPs:
0.00/0.87 { member[Ite][True][Ite]^#(True(), x, xs) -> c_10()
0.00/0.87 , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.87 c_11(member^#(x', xs))
0.00/0.87 , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y))
0.00/0.87 , !EQ^#(S(x), 0()) -> c_7()
0.00/0.87 , !EQ^#(0(), S(y)) -> c_8()
0.00/0.87 , !EQ^#(0(), 0()) -> c_9() }
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 { member^#(x', Cons(x, xs)) ->
0.00/0.87 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))
0.00/0.87 , member^#(x, Nil()) -> c_2()
0.00/0.87 , notEmpty^#(Cons(x, xs)) -> c_3()
0.00/0.87 , notEmpty^#(Nil()) -> c_4()
0.00/0.87 , goal^#(x, xs) -> c_5(member^#(x, xs)) }
0.00/0.87 Strict Trs:
0.00/0.87 { member(x', Cons(x, xs)) ->
0.00/0.87 member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
0.00/0.87 , member(x, Nil()) -> False()
0.00/0.87 , notEmpty(Cons(x, xs)) -> True()
0.00/0.87 , notEmpty(Nil()) -> False()
0.00/0.87 , goal(x, xs) -> member(x, xs) }
0.00/0.87 Weak DPs:
0.00/0.87 { member[Ite][True][Ite]^#(True(), x, xs) -> c_10()
0.00/0.87 , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.87 c_11(member^#(x', xs))
0.00/0.87 , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y))
0.00/0.87 , !EQ^#(S(x), 0()) -> c_7()
0.00/0.87 , !EQ^#(0(), S(y)) -> c_8()
0.00/0.87 , !EQ^#(0(), 0()) -> c_9() }
0.00/0.87 Weak Trs:
0.00/0.87 { !EQ(S(x), S(y)) -> !EQ(x, y)
0.00/0.87 , !EQ(S(x), 0()) -> False()
0.00/0.87 , !EQ(0(), S(y)) -> False()
0.00/0.87 , !EQ(0(), 0()) -> True()
0.00/0.87 , member[Ite][True][Ite](True(), x, xs) -> True()
0.00/0.87 , member[Ite][True][Ite](False(), x', Cons(x, xs)) ->
0.00/0.87 member(x', xs) }
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 Weak Usable Rules:
0.00/0.87 { !EQ(S(x), S(y)) -> !EQ(x, y)
0.00/0.87 , !EQ(S(x), 0()) -> False()
0.00/0.87 , !EQ(0(), S(y)) -> False()
0.00/0.87 , !EQ(0(), 0()) -> True() }
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 { member^#(x', Cons(x, xs)) ->
0.00/0.87 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))
0.00/0.87 , member^#(x, Nil()) -> c_2()
0.00/0.87 , notEmpty^#(Cons(x, xs)) -> c_3()
0.00/0.87 , notEmpty^#(Nil()) -> c_4()
0.00/0.87 , goal^#(x, xs) -> c_5(member^#(x, xs)) }
0.00/0.87 Weak DPs:
0.00/0.87 { member[Ite][True][Ite]^#(True(), x, xs) -> c_10()
0.00/0.87 , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.87 c_11(member^#(x', xs))
0.00/0.87 , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y))
0.00/0.87 , !EQ^#(S(x), 0()) -> c_7()
0.00/0.87 , !EQ^#(0(), S(y)) -> c_8()
0.00/0.87 , !EQ^#(0(), 0()) -> c_9() }
0.00/0.87 Weak Trs:
0.00/0.87 { !EQ(S(x), S(y)) -> !EQ(x, y)
0.00/0.87 , !EQ(S(x), 0()) -> False()
0.00/0.87 , !EQ(0(), S(y)) -> False()
0.00/0.87 , !EQ(0(), 0()) -> True() }
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(c_1) = {1}, Uargs(member[Ite][True][Ite]^#) = {1},
0.00/0.87 Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_11) = {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 [!EQ](x1, x2) = [0]
0.00/0.87 [0]
0.00/0.87
0.00/0.87 [True] = [0]
0.00/0.87 [0]
0.00/0.87
0.00/0.87 [S](x1) = [1 0] x1 + [0]
0.00/0.87 [0 0] [0]
0.00/0.87
0.00/0.87 [Cons](x1, x2) = [1 0] x2 + [0]
0.00/0.87 [0 0] [0]
0.00/0.87
0.00/0.87 [Nil] = [2]
0.00/0.87 [2]
0.00/0.87
0.00/0.87 [0] = [0]
0.00/0.87 [0]
0.00/0.87
0.00/0.87 [False] = [0]
0.00/0.87 [0]
0.00/0.87
0.00/0.87 [member^#](x1, x2) = [0 0] x1 + [1]
0.00/0.87 [1 1] [1]
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 [member[Ite][True][Ite]^#](x1, x2, x3) = [2 0] x1 + [0 0] x2 + [1]
0.00/0.87 [0 0] [1 1] [1]
0.00/0.87
0.00/0.87 [c_2] = [0]
0.00/0.87 [1]
0.00/0.87
0.00/0.87 [notEmpty^#](x1) = [1 1] x1 + [2]
0.00/0.87 [1 2] [1]
0.00/0.87
0.00/0.87 [c_3] = [1]
0.00/0.87 [1]
0.00/0.87
0.00/0.87 [c_4] = [1]
0.00/0.87 [1]
0.00/0.87
0.00/0.87 [goal^#](x1, x2) = [2 2] x1 + [1 1] x2 + [2]
0.00/0.87 [1 2] [2 2] [2]
0.00/0.87
0.00/0.87 [c_5](x1) = [1 0] x1 + [0]
0.00/0.87 [0 1] [1]
0.00/0.87
0.00/0.87 [!EQ^#](x1, x2) = [0]
0.00/0.87 [1]
0.00/0.87
0.00/0.87 [c_6](x1) = [1 0] x1 + [0]
0.00/0.87 [0 1] [0]
0.00/0.87
0.00/0.87 [c_7] = [0]
0.00/0.87 [1]
0.00/0.87
0.00/0.87 [c_8] = [0]
0.00/0.87 [1]
0.00/0.87
0.00/0.87 [c_9] = [0]
0.00/0.87 [1]
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 + [0]
0.00/0.87 [0 1] [0]
0.00/0.87
0.00/0.87 The order satisfies the following ordering constraints:
0.00/0.87
0.00/0.87 [!EQ(S(x), S(y))] = [0]
0.00/0.87 [0]
0.00/0.87 >= [0]
0.00/0.87 [0]
0.00/0.87 = [!EQ(x, y)]
0.00/0.87
0.00/0.87 [!EQ(S(x), 0())] = [0]
0.00/0.87 [0]
0.00/0.87 >= [0]
0.00/0.87 [0]
0.00/0.87 = [False()]
0.00/0.87
0.00/0.87 [!EQ(0(), S(y))] = [0]
0.00/0.87 [0]
0.00/0.87 >= [0]
0.00/0.87 [0]
0.00/0.87 = [False()]
0.00/0.87
0.00/0.87 [!EQ(0(), 0())] = [0]
0.00/0.87 [0]
0.00/0.87 >= [0]
0.00/0.87 [0]
0.00/0.87 = [True()]
0.00/0.87
0.00/0.87 [member^#(x', Cons(x, xs))] = [0 0] x' + [1]
0.00/0.87 [1 1] [1]
0.00/0.87 ? [0 0] x' + [2]
0.00/0.87 [1 1] [2]
0.00/0.87 = [c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))]
0.00/0.87
0.00/0.87 [member^#(x, Nil())] = [0 0] x + [1]
0.00/0.87 [1 1] [1]
0.00/0.87 > [0]
0.00/0.87 [1]
0.00/0.87 = [c_2()]
0.00/0.87
0.00/0.87 [member[Ite][True][Ite]^#(True(), x, xs)] = [0 0] x + [1]
0.00/0.87 [1 1] [1]
0.00/0.87 >= [1]
0.00/0.87 [1]
0.00/0.87 = [c_10()]
0.00/0.87
0.00/0.87 [member[Ite][True][Ite]^#(False(), x', Cons(x, xs))] = [0 0] x' + [1]
0.00/0.87 [1 1] [1]
0.00/0.87 >= [0 0] x' + [1]
0.00/0.87 [1 1] [1]
0.00/0.87 = [c_11(member^#(x', xs))]
0.00/0.87
0.00/0.87 [notEmpty^#(Cons(x, xs))] = [1 0] xs + [2]
0.00/0.87 [1 0] [1]
0.00/0.87 > [1]
0.00/0.87 [1]
0.00/0.87 = [c_3()]
0.00/0.87
0.00/0.87 [notEmpty^#(Nil())] = [6]
0.00/0.87 [7]
0.00/0.87 > [1]
0.00/0.87 [1]
0.00/0.87 = [c_4()]
0.00/0.87
0.00/0.87 [goal^#(x, xs)] = [2 2] x + [1 1] xs + [2]
0.00/0.87 [1 2] [2 2] [2]
0.00/0.87 > [0 0] x + [1]
0.00/0.87 [1 1] [2]
0.00/0.87 = [c_5(member^#(x, xs))]
0.00/0.87
0.00/0.87 [!EQ^#(S(x), S(y))] = [0]
0.00/0.87 [1]
0.00/0.87 >= [0]
0.00/0.87 [1]
0.00/0.87 = [c_6(!EQ^#(x, y))]
0.00/0.87
0.00/0.87 [!EQ^#(S(x), 0())] = [0]
0.00/0.87 [1]
0.00/0.87 >= [0]
0.00/0.87 [1]
0.00/0.87 = [c_7()]
0.00/0.87
0.00/0.87 [!EQ^#(0(), S(y))] = [0]
0.00/0.87 [1]
0.00/0.87 >= [0]
0.00/0.87 [1]
0.00/0.87 = [c_8()]
0.00/0.87
0.00/0.87 [!EQ^#(0(), 0())] = [0]
0.00/0.87 [1]
0.00/0.87 >= [0]
0.00/0.87 [1]
0.00/0.87 = [c_9()]
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(n^1)).
0.00/0.87
0.00/0.87 Strict DPs:
0.00/0.87 { member^#(x', Cons(x, xs)) ->
0.00/0.87 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) }
0.00/0.87 Weak DPs:
0.00/0.87 { member^#(x, Nil()) -> c_2()
0.00/0.87 , member[Ite][True][Ite]^#(True(), x, xs) -> c_10()
0.00/0.87 , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.87 c_11(member^#(x', xs))
0.00/0.87 , notEmpty^#(Cons(x, xs)) -> c_3()
0.00/0.87 , notEmpty^#(Nil()) -> c_4()
0.00/0.87 , goal^#(x, xs) -> c_5(member^#(x, xs))
0.00/0.87 , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y))
0.00/0.87 , !EQ^#(S(x), 0()) -> c_7()
0.00/0.87 , !EQ^#(0(), S(y)) -> c_8()
0.00/0.87 , !EQ^#(0(), 0()) -> c_9() }
0.00/0.87 Weak Trs:
0.00/0.87 { !EQ(S(x), S(y)) -> !EQ(x, y)
0.00/0.87 , !EQ(S(x), 0()) -> False()
0.00/0.87 , !EQ(0(), S(y)) -> False()
0.00/0.87 , !EQ(0(), 0()) -> True() }
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 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 { member^#(x, Nil()) -> c_2()
0.00/0.87 , member[Ite][True][Ite]^#(True(), x, xs) -> c_10()
0.00/0.87 , notEmpty^#(Cons(x, xs)) -> c_3()
0.00/0.87 , notEmpty^#(Nil()) -> c_4()
0.00/0.87 , !EQ^#(S(x), S(y)) -> c_6(!EQ^#(x, y))
0.00/0.87 , !EQ^#(S(x), 0()) -> c_7()
0.00/0.87 , !EQ^#(0(), S(y)) -> c_8()
0.00/0.87 , !EQ^#(0(), 0()) -> 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(n^1)).
0.00/0.87
0.00/0.87 Strict DPs:
0.00/0.87 { member^#(x', Cons(x, xs)) ->
0.00/0.87 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) }
0.00/0.87 Weak DPs:
0.00/0.87 { member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.87 c_11(member^#(x', xs))
0.00/0.87 , goal^#(x, xs) -> c_5(member^#(x, xs)) }
0.00/0.87 Weak Trs:
0.00/0.87 { !EQ(S(x), S(y)) -> !EQ(x, y)
0.00/0.87 , !EQ(S(x), 0()) -> False()
0.00/0.87 , !EQ(0(), S(y)) -> False()
0.00/0.87 , !EQ(0(), 0()) -> True() }
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 Consider the dependency graph
0.00/0.87
0.00/0.87 1: member^#(x', Cons(x, xs)) ->
0.00/0.87 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))
0.00/0.87 -->_1 member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.87 c_11(member^#(x', xs)) :2
0.00/0.87
0.00/0.87 2: member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.87 c_11(member^#(x', xs))
0.00/0.87 -->_1 member^#(x', Cons(x, xs)) ->
0.00/0.87 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) :1
0.00/0.87
0.00/0.87 3: goal^#(x, xs) -> c_5(member^#(x, xs))
0.00/0.87 -->_1 member^#(x', Cons(x, xs)) ->
0.00/0.87 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) :1
0.00/0.87
0.00/0.87
0.00/0.87 Following roots of the dependency graph are removed, as the
0.00/0.87 considered set of starting terms is closed under reduction with
0.00/0.87 respect to these rules (modulo compound contexts).
0.00/0.87
0.00/0.87 { goal^#(x, xs) -> c_5(member^#(x, xs)) }
0.00/0.87
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 { member^#(x', Cons(x, xs)) ->
0.00/0.87 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs))) }
0.00/0.87 Weak DPs:
0.00/0.87 { member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.87 c_11(member^#(x', xs)) }
0.00/0.87 Weak Trs:
0.00/0.87 { !EQ(S(x), S(y)) -> !EQ(x, y)
0.00/0.87 , !EQ(S(x), 0()) -> False()
0.00/0.87 , !EQ(0(), S(y)) -> False()
0.00/0.87 , !EQ(0(), 0()) -> True() }
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 use the processor 'matrix interpretation of dimension 1' to
0.00/0.87 orient following rules strictly.
0.00/0.87
0.00/0.87 DPs:
0.00/0.87 { 2: member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.87 c_11(member^#(x', xs)) }
0.00/0.87
0.00/0.87 Sub-proof:
0.00/0.87 ----------
0.00/0.87 The following argument positions are usable:
0.00/0.87 Uargs(c_1) = {1}, Uargs(c_11) = {1}
0.00/0.87
0.00/0.87 TcT has computed the following constructor-based matrix
0.00/0.87 interpretation satisfying not(EDA).
0.00/0.87
0.00/0.87 [!EQ](x1, x2) = [0]
0.00/0.87
0.00/0.87 [True] = [7]
0.00/0.87
0.00/0.87 [S](x1) = [1] x1 + [0]
0.00/0.87
0.00/0.87 [Cons](x1, x2) = [1] x2 + [4]
0.00/0.87
0.00/0.87 [0] = [0]
0.00/0.87
0.00/0.87 [False] = [0]
0.00/0.87
0.00/0.87 [member^#](x1, x2) = [2] x2 + [0]
0.00/0.87
0.00/0.87 [c_1](x1) = [1] x1 + [0]
0.00/0.87
0.00/0.87 [member[Ite][True][Ite]^#](x1, x2, x3) = [2] x3 + [0]
0.00/0.87
0.00/0.87 [c_11](x1) = [1] x1 + [1]
0.00/0.87
0.00/0.87 The order satisfies the following ordering constraints:
0.00/0.87
0.00/0.87 [!EQ(S(x), S(y))] = [0]
0.00/0.87 >= [0]
0.00/0.87 = [!EQ(x, y)]
0.00/0.87
0.00/0.87 [!EQ(S(x), 0())] = [0]
0.00/0.87 >= [0]
0.00/0.87 = [False()]
0.00/0.87
0.00/0.87 [!EQ(0(), S(y))] = [0]
0.00/0.87 >= [0]
0.00/0.87 = [False()]
0.00/0.87
0.00/0.87 [!EQ(0(), 0())] = [0]
0.00/0.87 ? [7]
0.00/0.87 = [True()]
0.00/0.87
0.00/0.87 [member^#(x', Cons(x, xs))] = [2] xs + [8]
0.00/0.87 >= [2] xs + [8]
0.00/0.87 = [c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))]
0.00/0.88
0.00/0.88 [member[Ite][True][Ite]^#(False(), x', Cons(x, xs))] = [2] xs + [8]
0.00/0.88 > [2] xs + [1]
0.00/0.88 = [c_11(member^#(x', xs))]
0.00/0.88
0.00/0.88
0.00/0.88 We return to the main proof. Consider the set of all dependency
0.00/0.88 pairs
0.00/0.88
0.00/0.88 :
0.00/0.88 { 1: member^#(x', Cons(x, xs)) ->
0.00/0.88 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))
0.00/0.88 , 2: member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.88 c_11(member^#(x', xs)) }
0.00/0.88
0.00/0.88 Processor 'matrix interpretation of dimension 1' induces the
0.00/0.88 complexity certificate YES(?,O(n^1)) on application of dependency
0.00/0.88 pairs {2}. These cover all (indirect) predecessors of dependency
0.00/0.88 pairs {1,2}, their number of application is equally bounded. The
0.00/0.88 dependency pairs are shifted into the weak component.
0.00/0.88
0.00/0.88 We are left with following problem, upon which TcT provides the
0.00/0.88 certificate YES(O(1),O(1)).
0.00/0.88
0.00/0.88 Weak DPs:
0.00/0.88 { member^#(x', Cons(x, xs)) ->
0.00/0.88 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))
0.00/0.88 , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.88 c_11(member^#(x', xs)) }
0.00/0.88 Weak Trs:
0.00/0.88 { !EQ(S(x), S(y)) -> !EQ(x, y)
0.00/0.88 , !EQ(S(x), 0()) -> False()
0.00/0.88 , !EQ(0(), S(y)) -> False()
0.00/0.88 , !EQ(0(), 0()) -> True() }
0.00/0.88 Obligation:
0.00/0.88 innermost runtime complexity
0.00/0.88 Answer:
0.00/0.88 YES(O(1),O(1))
0.00/0.88
0.00/0.88 The following weak DPs constitute a sub-graph of the DG that is
0.00/0.88 closed under successors. The DPs are removed.
0.00/0.88
0.00/0.88 { member^#(x', Cons(x, xs)) ->
0.00/0.88 c_1(member[Ite][True][Ite]^#(!EQ(x', x), x', Cons(x, xs)))
0.00/0.88 , member[Ite][True][Ite]^#(False(), x', Cons(x, xs)) ->
0.00/0.88 c_11(member^#(x', xs)) }
0.00/0.88
0.00/0.88 We are left with following problem, upon which TcT provides the
0.00/0.88 certificate YES(O(1),O(1)).
0.00/0.88
0.00/0.88 Weak Trs:
0.00/0.88 { !EQ(S(x), S(y)) -> !EQ(x, y)
0.00/0.88 , !EQ(S(x), 0()) -> False()
0.00/0.88 , !EQ(0(), S(y)) -> False()
0.00/0.88 , !EQ(0(), 0()) -> True() }
0.00/0.88 Obligation:
0.00/0.88 innermost runtime complexity
0.00/0.88 Answer:
0.00/0.88 YES(O(1),O(1))
0.00/0.88
0.00/0.88 No rule is usable, rules are removed from the input problem.
0.00/0.88
0.00/0.88 We are left with following problem, upon which TcT provides the
0.00/0.88 certificate YES(O(1),O(1)).
0.00/0.88
0.00/0.88 Rules: Empty
0.00/0.88 Obligation:
0.00/0.88 innermost runtime complexity
0.00/0.88 Answer:
0.00/0.88 YES(O(1),O(1))
0.00/0.88
0.00/0.88 Empty rules are trivially bounded
0.00/0.88
0.00/0.88 Hurray, we answered YES(O(1),O(n^1))
0.00/0.88 EOF