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