YES(O(1),O(n^2))
31.51/18.77 YES(O(1),O(n^2))
31.51/18.77
31.51/18.77 We are left with following problem, upon which TcT provides the
31.51/18.77 certificate YES(O(1),O(n^2)).
31.51/18.77
31.51/18.77 Strict Trs:
31.51/18.77 { a__first(X1, X2) -> first(X1, X2)
31.51/18.77 , a__first(0(), X) -> nil()
31.51/18.77 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
31.51/18.77 , mark(0()) -> 0()
31.51/18.77 , mark(nil()) -> nil()
31.51/18.77 , mark(s(X)) -> s(mark(X))
31.51/18.77 , mark(cons(X1, X2)) -> cons(mark(X1), X2)
31.51/18.77 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
31.51/18.77 , mark(from(X)) -> a__from(mark(X))
31.51/18.77 , a__from(X) -> cons(mark(X), from(s(X)))
31.51/18.77 , a__from(X) -> from(X) }
31.51/18.77 Obligation:
31.51/18.77 innermost runtime complexity
31.51/18.77 Answer:
31.51/18.77 YES(O(1),O(n^2))
31.51/18.77
31.51/18.77 The weightgap principle applies (using the following nonconstant
31.51/18.77 growth matrix-interpretation)
31.51/18.77
31.51/18.77 The following argument positions are usable:
31.51/18.77 Uargs(a__first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {1},
31.51/18.77 Uargs(a__from) = {1}
31.51/18.77
31.51/18.77 TcT has computed the following matrix interpretation satisfying
31.51/18.77 not(EDA) and not(IDA(1)).
31.51/18.77
31.51/18.77 [a__first](x1, x2) = [1] x1 + [1] x2 + [1]
31.51/18.77
31.51/18.77 [0] = [0]
31.51/18.77
31.51/18.77 [nil] = [6]
31.51/18.77
31.51/18.77 [s](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [cons](x1, x2) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [mark](x1) = [0]
31.51/18.77
31.51/18.77 [first](x1, x2) = [1] x1 + [1] x2 + [0]
31.51/18.77
31.51/18.77 [a__from](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [from](x1) = [1] x1 + [7]
31.51/18.77
31.51/18.77 The order satisfies the following ordering constraints:
31.51/18.77
31.51/18.77 [a__first(X1, X2)] = [1] X1 + [1] X2 + [1]
31.51/18.77 > [1] X1 + [1] X2 + [0]
31.51/18.77 = [first(X1, X2)]
31.51/18.77
31.51/18.77 [a__first(0(), X)] = [1] X + [1]
31.51/18.77 ? [6]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [a__first(s(X), cons(Y, Z))] = [1] X + [1] Y + [1]
31.51/18.77 > [0]
31.51/18.77 = [cons(mark(Y), first(X, Z))]
31.51/18.77
31.51/18.77 [mark(0())] = [0]
31.51/18.77 >= [0]
31.51/18.77 = [0()]
31.51/18.77
31.51/18.77 [mark(nil())] = [0]
31.51/18.77 ? [6]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [mark(s(X))] = [0]
31.51/18.77 >= [0]
31.51/18.77 = [s(mark(X))]
31.51/18.77
31.51/18.77 [mark(cons(X1, X2))] = [0]
31.51/18.77 >= [0]
31.51/18.77 = [cons(mark(X1), X2)]
31.51/18.77
31.51/18.77 [mark(first(X1, X2))] = [0]
31.51/18.77 ? [1]
31.51/18.77 = [a__first(mark(X1), mark(X2))]
31.51/18.77
31.51/18.77 [mark(from(X))] = [0]
31.51/18.77 >= [0]
31.51/18.77 = [a__from(mark(X))]
31.51/18.77
31.51/18.77 [a__from(X)] = [1] X + [0]
31.51/18.77 >= [0]
31.51/18.77 = [cons(mark(X), from(s(X)))]
31.51/18.77
31.51/18.77 [a__from(X)] = [1] X + [0]
31.51/18.77 ? [1] X + [7]
31.51/18.77 = [from(X)]
31.51/18.77
31.51/18.77
31.51/18.77 Further, it can be verified that all rules not oriented are covered by the weightgap condition.
31.51/18.77
31.51/18.77 We are left with following problem, upon which TcT provides the
31.51/18.77 certificate YES(O(1),O(n^2)).
31.51/18.77
31.51/18.77 Strict Trs:
31.51/18.77 { a__first(0(), X) -> nil()
31.51/18.77 , mark(0()) -> 0()
31.51/18.77 , mark(nil()) -> nil()
31.51/18.77 , mark(s(X)) -> s(mark(X))
31.51/18.77 , mark(cons(X1, X2)) -> cons(mark(X1), X2)
31.51/18.77 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
31.51/18.77 , mark(from(X)) -> a__from(mark(X))
31.51/18.77 , a__from(X) -> cons(mark(X), from(s(X)))
31.51/18.77 , a__from(X) -> from(X) }
31.51/18.77 Weak Trs:
31.51/18.77 { a__first(X1, X2) -> first(X1, X2)
31.51/18.77 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) }
31.51/18.77 Obligation:
31.51/18.77 innermost runtime complexity
31.51/18.77 Answer:
31.51/18.77 YES(O(1),O(n^2))
31.51/18.77
31.51/18.77 The weightgap principle applies (using the following nonconstant
31.51/18.77 growth matrix-interpretation)
31.51/18.77
31.51/18.77 The following argument positions are usable:
31.51/18.77 Uargs(a__first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {1},
31.51/18.77 Uargs(a__from) = {1}
31.51/18.77
31.51/18.77 TcT has computed the following matrix interpretation satisfying
31.51/18.77 not(EDA) and not(IDA(1)).
31.51/18.77
31.51/18.77 [a__first](x1, x2) = [1] x1 + [1] x2 + [0]
31.51/18.77
31.51/18.77 [0] = [4]
31.51/18.77
31.51/18.77 [nil] = [3]
31.51/18.77
31.51/18.77 [s](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [cons](x1, x2) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [mark](x1) = [0]
31.51/18.77
31.51/18.77 [first](x1, x2) = [1] x1 + [1] x2 + [0]
31.51/18.77
31.51/18.77 [a__from](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [from](x1) = [1] x1 + [7]
31.51/18.77
31.51/18.77 The order satisfies the following ordering constraints:
31.51/18.77
31.51/18.77 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0]
31.51/18.77 >= [1] X1 + [1] X2 + [0]
31.51/18.77 = [first(X1, X2)]
31.51/18.77
31.51/18.77 [a__first(0(), X)] = [1] X + [4]
31.51/18.77 > [3]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [a__first(s(X), cons(Y, Z))] = [1] X + [1] Y + [0]
31.51/18.77 >= [0]
31.51/18.77 = [cons(mark(Y), first(X, Z))]
31.51/18.77
31.51/18.77 [mark(0())] = [0]
31.51/18.77 ? [4]
31.51/18.77 = [0()]
31.51/18.77
31.51/18.77 [mark(nil())] = [0]
31.51/18.77 ? [3]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [mark(s(X))] = [0]
31.51/18.77 >= [0]
31.51/18.77 = [s(mark(X))]
31.51/18.77
31.51/18.77 [mark(cons(X1, X2))] = [0]
31.51/18.77 >= [0]
31.51/18.77 = [cons(mark(X1), X2)]
31.51/18.77
31.51/18.77 [mark(first(X1, X2))] = [0]
31.51/18.77 >= [0]
31.51/18.77 = [a__first(mark(X1), mark(X2))]
31.51/18.77
31.51/18.77 [mark(from(X))] = [0]
31.51/18.77 >= [0]
31.51/18.77 = [a__from(mark(X))]
31.51/18.77
31.51/18.77 [a__from(X)] = [1] X + [0]
31.51/18.77 >= [0]
31.51/18.77 = [cons(mark(X), from(s(X)))]
31.51/18.77
31.51/18.77 [a__from(X)] = [1] X + [0]
31.51/18.77 ? [1] X + [7]
31.51/18.77 = [from(X)]
31.51/18.77
31.51/18.77
31.51/18.77 Further, it can be verified that all rules not oriented are covered by the weightgap condition.
31.51/18.77
31.51/18.77 We are left with following problem, upon which TcT provides the
31.51/18.77 certificate YES(O(1),O(n^2)).
31.51/18.77
31.51/18.77 Strict Trs:
31.51/18.77 { mark(0()) -> 0()
31.51/18.77 , mark(nil()) -> nil()
31.51/18.77 , mark(s(X)) -> s(mark(X))
31.51/18.77 , mark(cons(X1, X2)) -> cons(mark(X1), X2)
31.51/18.77 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
31.51/18.77 , mark(from(X)) -> a__from(mark(X))
31.51/18.77 , a__from(X) -> cons(mark(X), from(s(X)))
31.51/18.77 , a__from(X) -> from(X) }
31.51/18.77 Weak Trs:
31.51/18.77 { a__first(X1, X2) -> first(X1, X2)
31.51/18.77 , a__first(0(), X) -> nil()
31.51/18.77 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) }
31.51/18.77 Obligation:
31.51/18.77 innermost runtime complexity
31.51/18.77 Answer:
31.51/18.77 YES(O(1),O(n^2))
31.51/18.77
31.51/18.77 The weightgap principle applies (using the following nonconstant
31.51/18.77 growth matrix-interpretation)
31.51/18.77
31.51/18.77 The following argument positions are usable:
31.51/18.77 Uargs(a__first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {1},
31.51/18.77 Uargs(a__from) = {1}
31.51/18.77
31.51/18.77 TcT has computed the following matrix interpretation satisfying
31.51/18.77 not(EDA) and not(IDA(1)).
31.51/18.77
31.51/18.77 [a__first](x1, x2) = [1] x1 + [1] x2 + [0]
31.51/18.77
31.51/18.77 [0] = [4]
31.51/18.77
31.51/18.77 [nil] = [3]
31.51/18.77
31.51/18.77 [s](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [cons](x1, x2) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [mark](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [first](x1, x2) = [1] x1 + [1] x2 + [0]
31.51/18.77
31.51/18.77 [a__from](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [from](x1) = [1] x1 + [4]
31.51/18.77
31.51/18.77 The order satisfies the following ordering constraints:
31.51/18.77
31.51/18.77 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0]
31.51/18.77 >= [1] X1 + [1] X2 + [0]
31.51/18.77 = [first(X1, X2)]
31.51/18.77
31.51/18.77 [a__first(0(), X)] = [1] X + [4]
31.51/18.77 > [3]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [a__first(s(X), cons(Y, Z))] = [1] X + [1] Y + [0]
31.51/18.77 >= [1] Y + [0]
31.51/18.77 = [cons(mark(Y), first(X, Z))]
31.51/18.77
31.51/18.77 [mark(0())] = [4]
31.51/18.77 >= [4]
31.51/18.77 = [0()]
31.51/18.77
31.51/18.77 [mark(nil())] = [3]
31.51/18.77 >= [3]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [mark(s(X))] = [1] X + [0]
31.51/18.77 >= [1] X + [0]
31.51/18.77 = [s(mark(X))]
31.51/18.77
31.51/18.77 [mark(cons(X1, X2))] = [1] X1 + [0]
31.51/18.77 >= [1] X1 + [0]
31.51/18.77 = [cons(mark(X1), X2)]
31.51/18.77
31.51/18.77 [mark(first(X1, X2))] = [1] X1 + [1] X2 + [0]
31.51/18.77 >= [1] X1 + [1] X2 + [0]
31.51/18.77 = [a__first(mark(X1), mark(X2))]
31.51/18.77
31.51/18.77 [mark(from(X))] = [1] X + [4]
31.51/18.77 > [1] X + [0]
31.51/18.77 = [a__from(mark(X))]
31.51/18.77
31.51/18.77 [a__from(X)] = [1] X + [0]
31.51/18.77 >= [1] X + [0]
31.51/18.77 = [cons(mark(X), from(s(X)))]
31.51/18.77
31.51/18.77 [a__from(X)] = [1] X + [0]
31.51/18.77 ? [1] X + [4]
31.51/18.77 = [from(X)]
31.51/18.77
31.51/18.77
31.51/18.77 Further, it can be verified that all rules not oriented are covered by the weightgap condition.
31.51/18.77
31.51/18.77 We are left with following problem, upon which TcT provides the
31.51/18.77 certificate YES(O(1),O(n^2)).
31.51/18.77
31.51/18.77 Strict Trs:
31.51/18.77 { mark(0()) -> 0()
31.51/18.77 , mark(nil()) -> nil()
31.51/18.77 , mark(s(X)) -> s(mark(X))
31.51/18.77 , mark(cons(X1, X2)) -> cons(mark(X1), X2)
31.51/18.77 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
31.51/18.77 , a__from(X) -> cons(mark(X), from(s(X)))
31.51/18.77 , a__from(X) -> from(X) }
31.51/18.77 Weak Trs:
31.51/18.77 { a__first(X1, X2) -> first(X1, X2)
31.51/18.77 , a__first(0(), X) -> nil()
31.51/18.77 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
31.51/18.77 , mark(from(X)) -> a__from(mark(X)) }
31.51/18.77 Obligation:
31.51/18.77 innermost runtime complexity
31.51/18.77 Answer:
31.51/18.77 YES(O(1),O(n^2))
31.51/18.77
31.51/18.77 The weightgap principle applies (using the following nonconstant
31.51/18.77 growth matrix-interpretation)
31.51/18.77
31.51/18.77 The following argument positions are usable:
31.51/18.77 Uargs(a__first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {1},
31.51/18.77 Uargs(a__from) = {1}
31.51/18.77
31.51/18.77 TcT has computed the following matrix interpretation satisfying
31.51/18.77 not(EDA) and not(IDA(1)).
31.51/18.77
31.51/18.77 [a__first](x1, x2) = [1] x1 + [1] x2 + [0]
31.51/18.77
31.51/18.77 [0] = [4]
31.51/18.77
31.51/18.77 [nil] = [3]
31.51/18.77
31.51/18.77 [s](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [cons](x1, x2) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [mark](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [first](x1, x2) = [1] x1 + [1] x2 + [0]
31.51/18.77
31.51/18.77 [a__from](x1) = [1] x1 + [1]
31.51/18.77
31.51/18.77 [from](x1) = [1] x1 + [4]
31.51/18.77
31.51/18.77 The order satisfies the following ordering constraints:
31.51/18.77
31.51/18.77 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0]
31.51/18.77 >= [1] X1 + [1] X2 + [0]
31.51/18.77 = [first(X1, X2)]
31.51/18.77
31.51/18.77 [a__first(0(), X)] = [1] X + [4]
31.51/18.77 > [3]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [a__first(s(X), cons(Y, Z))] = [1] X + [1] Y + [0]
31.51/18.77 >= [1] Y + [0]
31.51/18.77 = [cons(mark(Y), first(X, Z))]
31.51/18.77
31.51/18.77 [mark(0())] = [4]
31.51/18.77 >= [4]
31.51/18.77 = [0()]
31.51/18.77
31.51/18.77 [mark(nil())] = [3]
31.51/18.77 >= [3]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [mark(s(X))] = [1] X + [0]
31.51/18.77 >= [1] X + [0]
31.51/18.77 = [s(mark(X))]
31.51/18.77
31.51/18.77 [mark(cons(X1, X2))] = [1] X1 + [0]
31.51/18.77 >= [1] X1 + [0]
31.51/18.77 = [cons(mark(X1), X2)]
31.51/18.77
31.51/18.77 [mark(first(X1, X2))] = [1] X1 + [1] X2 + [0]
31.51/18.77 >= [1] X1 + [1] X2 + [0]
31.51/18.77 = [a__first(mark(X1), mark(X2))]
31.51/18.77
31.51/18.77 [mark(from(X))] = [1] X + [4]
31.51/18.77 > [1] X + [1]
31.51/18.77 = [a__from(mark(X))]
31.51/18.77
31.51/18.77 [a__from(X)] = [1] X + [1]
31.51/18.77 > [1] X + [0]
31.51/18.77 = [cons(mark(X), from(s(X)))]
31.51/18.77
31.51/18.77 [a__from(X)] = [1] X + [1]
31.51/18.77 ? [1] X + [4]
31.51/18.77 = [from(X)]
31.51/18.77
31.51/18.77
31.51/18.77 Further, it can be verified that all rules not oriented are covered by the weightgap condition.
31.51/18.77
31.51/18.77 We are left with following problem, upon which TcT provides the
31.51/18.77 certificate YES(O(1),O(n^2)).
31.51/18.77
31.51/18.77 Strict Trs:
31.51/18.77 { mark(0()) -> 0()
31.51/18.77 , mark(nil()) -> nil()
31.51/18.77 , mark(s(X)) -> s(mark(X))
31.51/18.77 , mark(cons(X1, X2)) -> cons(mark(X1), X2)
31.51/18.77 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
31.51/18.77 , a__from(X) -> from(X) }
31.51/18.77 Weak Trs:
31.51/18.77 { a__first(X1, X2) -> first(X1, X2)
31.51/18.77 , a__first(0(), X) -> nil()
31.51/18.77 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
31.51/18.77 , mark(from(X)) -> a__from(mark(X))
31.51/18.77 , a__from(X) -> cons(mark(X), from(s(X))) }
31.51/18.77 Obligation:
31.51/18.77 innermost runtime complexity
31.51/18.77 Answer:
31.51/18.77 YES(O(1),O(n^2))
31.51/18.77
31.51/18.77 The weightgap principle applies (using the following nonconstant
31.51/18.77 growth matrix-interpretation)
31.51/18.77
31.51/18.77 The following argument positions are usable:
31.51/18.77 Uargs(a__first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {1},
31.51/18.77 Uargs(a__from) = {1}
31.51/18.77
31.51/18.77 TcT has computed the following matrix interpretation satisfying
31.51/18.77 not(EDA) and not(IDA(1)).
31.51/18.77
31.51/18.77 [a__first](x1, x2) = [1] x1 + [1] x2 + [4]
31.51/18.77
31.51/18.77 [0] = [0]
31.51/18.77
31.51/18.77 [nil] = [2]
31.51/18.77
31.51/18.77 [s](x1) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [cons](x1, x2) = [1] x1 + [0]
31.51/18.77
31.51/18.77 [mark](x1) = [1] x1 + [1]
31.51/18.77
31.51/18.77 [first](x1, x2) = [1] x1 + [1] x2 + [0]
31.51/18.77
31.51/18.77 [a__from](x1) = [1] x1 + [2]
31.51/18.77
31.51/18.77 [from](x1) = [1] x1 + [7]
31.51/18.77
31.51/18.77 The order satisfies the following ordering constraints:
31.51/18.77
31.51/18.77 [a__first(X1, X2)] = [1] X1 + [1] X2 + [4]
31.51/18.77 > [1] X1 + [1] X2 + [0]
31.51/18.77 = [first(X1, X2)]
31.51/18.77
31.51/18.77 [a__first(0(), X)] = [1] X + [4]
31.51/18.77 > [2]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [a__first(s(X), cons(Y, Z))] = [1] X + [1] Y + [4]
31.51/18.77 > [1] Y + [1]
31.51/18.77 = [cons(mark(Y), first(X, Z))]
31.51/18.77
31.51/18.77 [mark(0())] = [1]
31.51/18.77 > [0]
31.51/18.77 = [0()]
31.51/18.77
31.51/18.77 [mark(nil())] = [3]
31.51/18.77 > [2]
31.51/18.77 = [nil()]
31.51/18.77
31.51/18.77 [mark(s(X))] = [1] X + [1]
31.51/18.77 >= [1] X + [1]
31.51/18.77 = [s(mark(X))]
31.51/18.77
31.51/18.77 [mark(cons(X1, X2))] = [1] X1 + [1]
31.51/18.77 >= [1] X1 + [1]
31.51/18.77 = [cons(mark(X1), X2)]
31.51/18.77
31.51/18.78 [mark(first(X1, X2))] = [1] X1 + [1] X2 + [1]
31.51/18.78 ? [1] X1 + [1] X2 + [6]
31.51/18.78 = [a__first(mark(X1), mark(X2))]
31.51/18.78
31.51/18.78 [mark(from(X))] = [1] X + [8]
31.51/18.78 > [1] X + [3]
31.51/18.78 = [a__from(mark(X))]
31.51/18.78
31.51/18.78 [a__from(X)] = [1] X + [2]
31.51/18.78 > [1] X + [1]
31.51/18.78 = [cons(mark(X), from(s(X)))]
31.51/18.78
31.51/18.78 [a__from(X)] = [1] X + [2]
31.51/18.78 ? [1] X + [7]
31.51/18.78 = [from(X)]
31.51/18.78
31.51/18.78
31.51/18.78 Further, it can be verified that all rules not oriented are covered by the weightgap condition.
31.51/18.78
31.51/18.78 We are left with following problem, upon which TcT provides the
31.51/18.78 certificate YES(O(1),O(n^2)).
31.51/18.78
31.51/18.78 Strict Trs:
31.51/18.78 { mark(s(X)) -> s(mark(X))
31.51/18.78 , mark(cons(X1, X2)) -> cons(mark(X1), X2)
31.51/18.78 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
31.51/18.78 , a__from(X) -> from(X) }
31.51/18.78 Weak Trs:
31.51/18.78 { a__first(X1, X2) -> first(X1, X2)
31.51/18.78 , a__first(0(), X) -> nil()
31.51/18.78 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
31.51/18.78 , mark(0()) -> 0()
31.51/18.78 , mark(nil()) -> nil()
31.51/18.78 , mark(from(X)) -> a__from(mark(X))
31.51/18.78 , a__from(X) -> cons(mark(X), from(s(X))) }
31.51/18.78 Obligation:
31.51/18.78 innermost runtime complexity
31.51/18.78 Answer:
31.51/18.78 YES(O(1),O(n^2))
31.51/18.78
31.51/18.78 We use the processor 'matrix interpretation of dimension 2' to
31.51/18.78 orient following rules strictly.
31.51/18.78
31.51/18.78 Trs:
31.51/18.78 { mark(s(X)) -> s(mark(X))
31.51/18.78 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) }
31.51/18.78
31.51/18.78 The induced complexity on above rules (modulo remaining rules) is
31.51/18.78 YES(?,O(n^2)) . These rules are moved into the corresponding weak
31.51/18.78 component(s).
31.51/18.78
31.51/18.78 Sub-proof:
31.51/18.78 ----------
31.51/18.78 The following argument positions are usable:
31.51/18.78 Uargs(a__first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {1},
31.51/18.78 Uargs(a__from) = {1}
31.51/18.78
31.51/18.78 TcT has computed the following constructor-based matrix
31.51/18.78 interpretation satisfying not(EDA).
31.51/18.78
31.51/18.78 [a__first](x1, x2) = [1 0] x1 + [1 4] x2 + [4]
31.51/18.78 [0 1] [0 1] [4]
31.51/18.78
31.51/18.78 [0] = [0]
31.51/18.78 [0]
31.51/18.78
31.51/18.78 [nil] = [0]
31.51/18.78 [0]
31.51/18.78
31.51/18.78 [s](x1) = [1 0] x1 + [0]
31.51/18.78 [0 1] [4]
31.51/18.78
31.51/18.78 [cons](x1, x2) = [1 5] x1 + [0]
31.51/18.78 [0 1] [0]
31.51/18.78
31.51/18.78 [mark](x1) = [1 1] x1 + [0]
31.51/18.78 [0 1] [0]
31.51/18.78
31.51/18.78 [first](x1, x2) = [1 0] x1 + [1 4] x2 + [4]
31.51/18.78 [0 1] [0 1] [4]
31.51/18.78
31.51/18.78 [a__from](x1) = [1 6] x1 + [0]
31.51/18.78 [0 1] [0]
31.51/18.78
31.51/18.78 [from](x1) = [1 6] x1 + [0]
31.51/18.78 [0 1] [0]
31.51/18.78
31.51/18.78 The order satisfies the following ordering constraints:
31.51/18.78
31.51/18.78 [a__first(X1, X2)] = [1 0] X1 + [1 4] X2 + [4]
31.51/18.78 [0 1] [0 1] [4]
31.51/18.78 >= [1 0] X1 + [1 4] X2 + [4]
31.51/18.78 [0 1] [0 1] [4]
31.51/18.78 = [first(X1, X2)]
31.51/18.78
31.51/18.78 [a__first(0(), X)] = [1 4] X + [4]
31.51/18.78 [0 1] [4]
31.51/18.78 > [0]
31.51/18.78 [0]
31.51/18.78 = [nil()]
31.51/18.78
31.51/18.78 [a__first(s(X), cons(Y, Z))] = [1 0] X + [1 9] Y + [4]
31.51/18.78 [0 1] [0 1] [8]
31.51/18.78 > [1 6] Y + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 = [cons(mark(Y), first(X, Z))]
31.51/18.78
31.51/18.78 [mark(0())] = [0]
31.51/18.78 [0]
31.51/18.78 >= [0]
31.51/18.78 [0]
31.51/18.78 = [0()]
31.51/18.78
31.51/18.78 [mark(nil())] = [0]
31.51/18.78 [0]
31.51/18.78 >= [0]
31.51/18.78 [0]
31.51/18.78 = [nil()]
31.51/18.78
31.51/18.78 [mark(s(X))] = [1 1] X + [4]
31.51/18.78 [0 1] [4]
31.51/18.78 > [1 1] X + [0]
31.51/18.78 [0 1] [4]
31.51/18.78 = [s(mark(X))]
31.51/18.78
31.51/18.78 [mark(cons(X1, X2))] = [1 6] X1 + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 >= [1 6] X1 + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 = [cons(mark(X1), X2)]
31.51/18.78
31.51/18.78 [mark(first(X1, X2))] = [1 1] X1 + [1 5] X2 + [8]
31.51/18.78 [0 1] [0 1] [4]
31.51/18.78 > [1 1] X1 + [1 5] X2 + [4]
31.51/18.78 [0 1] [0 1] [4]
31.51/18.78 = [a__first(mark(X1), mark(X2))]
31.51/18.78
31.51/18.78 [mark(from(X))] = [1 7] X + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 >= [1 7] X + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 = [a__from(mark(X))]
31.51/18.78
31.51/18.78 [a__from(X)] = [1 6] X + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 >= [1 6] X + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 = [cons(mark(X), from(s(X)))]
31.51/18.78
31.51/18.78 [a__from(X)] = [1 6] X + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 >= [1 6] X + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 = [from(X)]
31.51/18.78
31.51/18.78
31.51/18.78 We return to the main proof.
31.51/18.78
31.51/18.78 We are left with following problem, upon which TcT provides the
31.51/18.78 certificate YES(O(1),O(n^2)).
31.51/18.78
31.51/18.78 Strict Trs:
31.51/18.78 { mark(cons(X1, X2)) -> cons(mark(X1), X2)
31.51/18.78 , a__from(X) -> from(X) }
31.51/18.78 Weak Trs:
31.51/18.78 { a__first(X1, X2) -> first(X1, X2)
31.51/18.78 , a__first(0(), X) -> nil()
31.51/18.78 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
31.51/18.78 , mark(0()) -> 0()
31.51/18.78 , mark(nil()) -> nil()
31.51/18.78 , mark(s(X)) -> s(mark(X))
31.51/18.78 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
31.51/18.78 , mark(from(X)) -> a__from(mark(X))
31.51/18.78 , a__from(X) -> cons(mark(X), from(s(X))) }
31.51/18.78 Obligation:
31.51/18.78 innermost runtime complexity
31.51/18.78 Answer:
31.51/18.78 YES(O(1),O(n^2))
31.51/18.78
31.51/18.78 We use the processor 'matrix interpretation of dimension 2' to
31.51/18.78 orient following rules strictly.
31.51/18.78
31.51/18.78 Trs:
31.51/18.78 { mark(cons(X1, X2)) -> cons(mark(X1), X2)
31.51/18.78 , a__from(X) -> from(X) }
31.51/18.78
31.51/18.78 The induced complexity on above rules (modulo remaining rules) is
31.51/18.78 YES(?,O(n^2)) . These rules are moved into the corresponding weak
31.51/18.78 component(s).
31.51/18.78
31.51/18.78 Sub-proof:
31.51/18.78 ----------
31.51/18.78 The following argument positions are usable:
31.51/18.78 Uargs(a__first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {1},
31.51/18.78 Uargs(a__from) = {1}
31.51/18.78
31.51/18.78 TcT has computed the following constructor-based matrix
31.51/18.78 interpretation satisfying not(EDA).
31.51/18.78
31.51/18.78 [a__first](x1, x2) = [1 0] x1 + [1 4] x2 + [0]
31.51/18.78 [0 1] [0 1] [0]
31.51/18.78
31.51/18.78 [0] = [0]
31.51/18.78 [0]
31.51/18.78
31.51/18.78 [nil] = [0]
31.51/18.78 [0]
31.51/18.78
31.51/18.78 [s](x1) = [1 0] x1 + [0]
31.51/18.78 [0 1] [2]
31.51/18.78
31.51/18.78 [cons](x1, x2) = [1 0] x1 + [4]
31.51/18.78 [0 1] [2]
31.51/18.78
31.51/18.78 [mark](x1) = [1 2] x1 + [0]
31.51/18.78 [0 1] [0]
31.51/18.78
31.51/18.78 [first](x1, x2) = [1 0] x1 + [1 4] x2 + [0]
31.51/18.78 [0 1] [0 1] [0]
31.51/18.78
31.51/18.78 [a__from](x1) = [1 2] x1 + [7]
31.51/18.78 [0 1] [4]
31.51/18.78
31.51/18.78 [from](x1) = [1 2] x1 + [0]
31.51/18.78 [0 1] [4]
31.51/18.78
31.51/18.78 The order satisfies the following ordering constraints:
31.51/18.78
31.51/18.78 [a__first(X1, X2)] = [1 0] X1 + [1 4] X2 + [0]
31.51/18.78 [0 1] [0 1] [0]
31.51/18.78 >= [1 0] X1 + [1 4] X2 + [0]
31.51/18.78 [0 1] [0 1] [0]
31.51/18.78 = [first(X1, X2)]
31.51/18.78
31.51/18.78 [a__first(0(), X)] = [1 4] X + [0]
31.51/18.78 [0 1] [0]
31.51/18.78 >= [0]
31.51/18.78 [0]
31.51/18.78 = [nil()]
31.51/18.78
31.51/18.78 [a__first(s(X), cons(Y, Z))] = [1 0] X + [1 4] Y + [12]
31.51/18.78 [0 1] [0 1] [4]
31.51/18.78 > [1 2] Y + [4]
31.51/18.78 [0 1] [2]
31.51/18.78 = [cons(mark(Y), first(X, Z))]
31.51/18.78
31.51/18.78 [mark(0())] = [0]
31.51/18.78 [0]
31.51/18.78 >= [0]
31.51/18.78 [0]
31.51/18.78 = [0()]
31.51/18.78
31.51/18.78 [mark(nil())] = [0]
31.51/18.78 [0]
31.51/18.78 >= [0]
31.51/18.78 [0]
31.51/18.78 = [nil()]
31.51/18.78
31.51/18.78 [mark(s(X))] = [1 2] X + [4]
31.51/18.78 [0 1] [2]
31.51/18.78 > [1 2] X + [0]
31.51/18.78 [0 1] [2]
31.51/18.78 = [s(mark(X))]
31.51/18.78
31.51/18.78 [mark(cons(X1, X2))] = [1 2] X1 + [8]
31.51/18.78 [0 1] [2]
31.51/18.78 > [1 2] X1 + [4]
31.51/18.78 [0 1] [2]
31.51/18.78 = [cons(mark(X1), X2)]
31.51/18.78
31.51/18.78 [mark(first(X1, X2))] = [1 2] X1 + [1 6] X2 + [0]
31.51/18.78 [0 1] [0 1] [0]
31.51/18.78 >= [1 2] X1 + [1 6] X2 + [0]
31.51/18.78 [0 1] [0 1] [0]
31.51/18.78 = [a__first(mark(X1), mark(X2))]
31.51/18.78
31.51/18.78 [mark(from(X))] = [1 4] X + [8]
31.51/18.78 [0 1] [4]
31.51/18.78 > [1 4] X + [7]
31.51/18.78 [0 1] [4]
31.51/18.78 = [a__from(mark(X))]
31.51/18.78
31.51/18.78 [a__from(X)] = [1 2] X + [7]
31.51/18.78 [0 1] [4]
31.51/18.78 > [1 2] X + [4]
31.51/18.78 [0 1] [2]
31.51/18.78 = [cons(mark(X), from(s(X)))]
31.51/18.78
31.51/18.78 [a__from(X)] = [1 2] X + [7]
31.51/18.78 [0 1] [4]
31.51/18.78 > [1 2] X + [0]
31.51/18.78 [0 1] [4]
31.51/18.78 = [from(X)]
31.51/18.78
31.51/18.78
31.51/18.78 We return to the main proof.
31.51/18.78
31.51/18.78 We are left with following problem, upon which TcT provides the
31.51/18.78 certificate YES(O(1),O(1)).
31.51/18.78
31.51/18.78 Weak Trs:
31.51/18.78 { a__first(X1, X2) -> first(X1, X2)
31.51/18.78 , a__first(0(), X) -> nil()
31.51/18.78 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
31.51/18.78 , mark(0()) -> 0()
31.51/18.78 , mark(nil()) -> nil()
31.51/18.78 , mark(s(X)) -> s(mark(X))
31.51/18.78 , mark(cons(X1, X2)) -> cons(mark(X1), X2)
31.51/18.78 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
31.51/18.78 , mark(from(X)) -> a__from(mark(X))
31.51/18.78 , a__from(X) -> cons(mark(X), from(s(X)))
31.51/18.78 , a__from(X) -> from(X) }
31.51/18.78 Obligation:
31.51/18.78 innermost runtime complexity
31.51/18.78 Answer:
31.51/18.78 YES(O(1),O(1))
31.51/18.78
31.51/18.78 Empty rules are trivially bounded
31.51/18.78
31.51/18.78 Hurray, we answered YES(O(1),O(n^2))
31.51/18.78 EOF