YES(O(1),O(1))
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 We are left with following problem, upon which TcT provides the
12.58/4.93 certificate YES(O(1),O(1)).
12.58/4.93
12.58/4.93 Strict Trs:
12.58/4.93 { g(X) -> u(h(X), h(X), X)
12.58/4.93 , u(d(), c(Y), X) -> k(Y)
12.58/4.93 , h(d()) -> c(a())
12.58/4.93 , h(d()) -> c(b())
12.58/4.93 , f(k(a()), k(b()), X) -> f(X, X, X) }
12.58/4.93 Obligation:
12.58/4.93 runtime complexity
12.58/4.93 Answer:
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 We add the following weak dependency pairs:
12.58/4.93
12.58/4.93 Strict DPs:
12.58/4.93 { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.58/4.93 , u^#(d(), c(Y), X) -> c_2(Y)
12.58/4.93 , h^#(d()) -> c_3()
12.58/4.93 , h^#(d()) -> c_4()
12.58/4.93 , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.58/4.93
12.58/4.93 and mark the set of starting terms.
12.58/4.93
12.58/4.93 We are left with following problem, upon which TcT provides the
12.58/4.93 certificate YES(O(1),O(1)).
12.58/4.93
12.58/4.93 Strict DPs:
12.58/4.93 { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.58/4.93 , u^#(d(), c(Y), X) -> c_2(Y)
12.58/4.93 , h^#(d()) -> c_3()
12.58/4.93 , h^#(d()) -> c_4()
12.58/4.93 , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.58/4.93 Strict Trs:
12.58/4.93 { g(X) -> u(h(X), h(X), X)
12.58/4.93 , u(d(), c(Y), X) -> k(Y)
12.58/4.93 , h(d()) -> c(a())
12.58/4.93 , h(d()) -> c(b())
12.58/4.93 , f(k(a()), k(b()), X) -> f(X, X, X) }
12.58/4.93 Obligation:
12.58/4.93 runtime complexity
12.58/4.93 Answer:
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 We replace rewrite rules by usable rules:
12.58/4.93
12.58/4.93 Strict Usable Rules:
12.58/4.93 { h(d()) -> c(a())
12.58/4.93 , h(d()) -> c(b()) }
12.58/4.93
12.58/4.93 We are left with following problem, upon which TcT provides the
12.58/4.93 certificate YES(O(1),O(1)).
12.58/4.93
12.58/4.93 Strict DPs:
12.58/4.93 { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.58/4.93 , u^#(d(), c(Y), X) -> c_2(Y)
12.58/4.93 , h^#(d()) -> c_3()
12.58/4.93 , h^#(d()) -> c_4()
12.58/4.93 , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.58/4.93 Strict Trs:
12.58/4.93 { h(d()) -> c(a())
12.58/4.93 , h(d()) -> c(b()) }
12.58/4.93 Obligation:
12.58/4.93 runtime complexity
12.58/4.93 Answer:
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 The weightgap principle applies (using the following constant
12.58/4.93 growth matrix-interpretation)
12.58/4.93
12.58/4.93 The following argument positions are usable:
12.58/4.93 Uargs(c_1) = {1}, Uargs(u^#) = {1, 2}
12.58/4.93
12.58/4.93 TcT has computed the following constructor-restricted matrix
12.58/4.93 interpretation.
12.58/4.93
12.58/4.93 [h](x1) = [2]
12.58/4.93 [0]
12.58/4.93
12.58/4.93 [d] = [0]
12.58/4.93 [0]
12.58/4.93
12.58/4.93 [c](x1) = [0]
12.58/4.93 [0]
12.58/4.93
12.58/4.93 [k](x1) = [0]
12.58/4.93 [0]
12.58/4.93
12.58/4.93 [a] = [0]
12.58/4.93 [0]
12.58/4.93
12.58/4.93 [b] = [0]
12.58/4.93 [0]
12.58/4.93
12.58/4.93 [g^#](x1) = [1 1] x1 + [1]
12.58/4.93 [1 1] [2]
12.58/4.93
12.58/4.93 [c_1](x1) = [1 0] x1 + [0]
12.58/4.93 [0 1] [2]
12.58/4.93
12.58/4.93 [u^#](x1, x2, x3) = [1 0] x1 + [2 0] x2 + [0 0] x3 + [1]
12.58/4.93 [1 1] [0 0] [1 1] [2]
12.58/4.93
12.58/4.93 [c_2](x1) = [0 0] x1 + [1]
12.58/4.93 [1 1] [1]
12.58/4.93
12.58/4.93 [h^#](x1) = [1 2] x1 + [2]
12.58/4.93 [2 1] [2]
12.58/4.93
12.58/4.93 [c_3] = [1]
12.58/4.93 [1]
12.58/4.93
12.58/4.93 [c_4] = [1]
12.58/4.93 [1]
12.58/4.93
12.58/4.93 [f^#](x1, x2, x3) = [1]
12.58/4.93 [2]
12.58/4.93
12.58/4.93 [c_5](x1) = [2]
12.58/4.93 [2]
12.58/4.93
12.58/4.93 The order satisfies the following ordering constraints:
12.58/4.93
12.58/4.93 [h(d())] = [2]
12.58/4.93 [0]
12.58/4.93 > [0]
12.58/4.93 [0]
12.58/4.93 = [c(a())]
12.58/4.93
12.58/4.93 [h(d())] = [2]
12.58/4.93 [0]
12.58/4.93 > [0]
12.58/4.93 [0]
12.58/4.93 = [c(b())]
12.58/4.93
12.58/4.93 [g^#(X)] = [1 1] X + [1]
12.58/4.93 [1 1] [2]
12.58/4.93 ? [0 0] X + [7]
12.58/4.93 [1 1] [6]
12.58/4.93 = [c_1(u^#(h(X), h(X), X))]
12.58/4.93
12.58/4.93 [u^#(d(), c(Y), X)] = [0 0] X + [1]
12.58/4.93 [1 1] [2]
12.58/4.93 ? [0 0] Y + [1]
12.58/4.93 [1 1] [1]
12.58/4.93 = [c_2(Y)]
12.58/4.93
12.58/4.93 [h^#(d())] = [2]
12.58/4.93 [2]
12.58/4.93 > [1]
12.58/4.93 [1]
12.58/4.93 = [c_3()]
12.58/4.93
12.58/4.93 [h^#(d())] = [2]
12.58/4.93 [2]
12.58/4.93 > [1]
12.58/4.93 [1]
12.58/4.93 = [c_4()]
12.58/4.93
12.58/4.93 [f^#(k(a()), k(b()), X)] = [1]
12.58/4.93 [2]
12.58/4.93 ? [2]
12.58/4.93 [2]
12.58/4.93 = [c_5(f^#(X, X, X))]
12.58/4.93
12.58/4.93
12.58/4.93 Further, it can be verified that all rules not oriented are covered by the weightgap condition.
12.58/4.93
12.58/4.93 We are left with following problem, upon which TcT provides the
12.58/4.93 certificate YES(O(1),O(1)).
12.58/4.93
12.58/4.93 Strict DPs:
12.58/4.93 { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.58/4.93 , u^#(d(), c(Y), X) -> c_2(Y)
12.58/4.93 , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.58/4.93 Weak DPs:
12.58/4.93 { h^#(d()) -> c_3()
12.58/4.93 , h^#(d()) -> c_4() }
12.58/4.93 Weak Trs:
12.58/4.93 { h(d()) -> c(a())
12.58/4.93 , h(d()) -> c(b()) }
12.58/4.93 Obligation:
12.58/4.93 runtime complexity
12.58/4.93 Answer:
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 We estimate the number of application of {1,3} by applications of
12.58/4.93 Pre({1,3}) = {2}. Here rules are labeled as follows:
12.58/4.93
12.58/4.93 DPs:
12.58/4.93 { 1: g^#(X) -> c_1(u^#(h(X), h(X), X))
12.58/4.93 , 2: u^#(d(), c(Y), X) -> c_2(Y)
12.58/4.93 , 3: f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X))
12.58/4.93 , 4: h^#(d()) -> c_3()
12.58/4.93 , 5: h^#(d()) -> c_4() }
12.58/4.93
12.58/4.93 We are left with following problem, upon which TcT provides the
12.58/4.93 certificate YES(O(1),O(1)).
12.58/4.93
12.58/4.93 Strict DPs: { u^#(d(), c(Y), X) -> c_2(Y) }
12.58/4.93 Weak DPs:
12.58/4.93 { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.58/4.93 , h^#(d()) -> c_3()
12.58/4.93 , h^#(d()) -> c_4()
12.58/4.93 , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.58/4.93 Weak Trs:
12.58/4.93 { h(d()) -> c(a())
12.58/4.93 , h(d()) -> c(b()) }
12.58/4.93 Obligation:
12.58/4.93 runtime complexity
12.58/4.93 Answer:
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 The following weak DPs constitute a sub-graph of the DG that is
12.58/4.93 closed under successors. The DPs are removed.
12.58/4.93
12.58/4.93 { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.58/4.93 , h^#(d()) -> c_3()
12.58/4.93 , h^#(d()) -> c_4()
12.58/4.93 , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.58/4.93
12.58/4.93 We are left with following problem, upon which TcT provides the
12.58/4.93 certificate YES(O(1),O(1)).
12.58/4.93
12.58/4.93 Strict DPs: { u^#(d(), c(Y), X) -> c_2(Y) }
12.58/4.93 Weak Trs:
12.58/4.93 { h(d()) -> c(a())
12.58/4.93 , h(d()) -> c(b()) }
12.58/4.93 Obligation:
12.58/4.93 runtime complexity
12.58/4.93 Answer:
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 No rule is usable, rules are removed from the input problem.
12.58/4.93
12.58/4.93 We are left with following problem, upon which TcT provides the
12.58/4.93 certificate YES(O(1),O(1)).
12.58/4.93
12.58/4.93 Strict DPs: { u^#(d(), c(Y), X) -> c_2(Y) }
12.58/4.93 Obligation:
12.58/4.93 runtime complexity
12.58/4.93 Answer:
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 We use the processor 'matrix interpretation of dimension 1' to
12.58/4.93 orient following rules strictly.
12.58/4.93
12.58/4.93 DPs:
12.58/4.93 { 1: u^#(d(), c(Y), X) -> c_2(Y) }
12.58/4.93
12.58/4.93 Sub-proof:
12.58/4.93 ----------
12.58/4.93 The following argument positions are usable:
12.58/4.93 none
12.58/4.93
12.58/4.93 TcT has computed the following constructor-restricted matrix
12.58/4.93 interpretation. Note that the diagonal of the component-wise maxima
12.58/4.93 of interpretation-entries (of constructors) contains no more than 0
12.58/4.93 non-zero entries.
12.58/4.93
12.58/4.93 [d] = [7]
12.58/4.93
12.58/4.93 [c](x1) = [0]
12.58/4.93
12.58/4.93 [u^#](x1, x2, x3) = [7] x2 + [7] x3 + [7]
12.58/4.93
12.58/4.93 [c_2](x1) = [5]
12.58/4.93
12.58/4.93 The order satisfies the following ordering constraints:
12.58/4.93
12.58/4.93 [u^#(d(), c(Y), X)] = [7] X + [7]
12.58/4.93 > [5]
12.58/4.93 = [c_2(Y)]
12.58/4.93
12.58/4.93
12.58/4.93 The strictly oriented rules are moved into the weak component.
12.58/4.93
12.58/4.93 We are left with following problem, upon which TcT provides the
12.58/4.93 certificate YES(O(1),O(1)).
12.58/4.93
12.58/4.93 Weak DPs: { u^#(d(), c(Y), X) -> c_2(Y) }
12.58/4.93 Obligation:
12.58/4.93 runtime complexity
12.58/4.93 Answer:
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 The following weak DPs constitute a sub-graph of the DG that is
12.58/4.93 closed under successors. The DPs are removed.
12.58/4.93
12.58/4.93 { u^#(d(), c(Y), X) -> c_2(Y) }
12.58/4.93
12.58/4.93 We are left with following problem, upon which TcT provides the
12.58/4.93 certificate YES(O(1),O(1)).
12.58/4.93
12.58/4.93 Rules: Empty
12.58/4.93 Obligation:
12.58/4.93 runtime complexity
12.58/4.93 Answer:
12.58/4.93 YES(O(1),O(1))
12.58/4.93
12.58/4.93 Empty rules are trivially bounded
12.58/4.93
12.58/4.93 Hurray, we answered YES(O(1),O(1))
12.58/4.94 EOF