YES(O(1),O(n^2)) 38.96/19.26 YES(O(1),O(n^2)) 38.96/19.26 38.96/19.26 We are left with following problem, upon which TcT provides the 38.96/19.26 certificate YES(O(1),O(n^2)). 38.96/19.26 38.96/19.26 Strict Trs: 38.96/19.26 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.26 , a__fst(0(), Z) -> nil() 38.96/19.26 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.26 , mark(0()) -> 0() 38.96/19.26 , mark(nil()) -> nil() 38.96/19.26 , mark(s(X)) -> s(X) 38.96/19.26 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.26 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.26 , mark(from(X)) -> a__from(mark(X)) 38.96/19.26 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.26 , mark(len(X)) -> a__len(mark(X)) 38.96/19.26 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.26 , a__from(X) -> from(X) 38.96/19.26 , a__add(X1, X2) -> add(X1, X2) 38.96/19.26 , a__add(0(), X) -> mark(X) 38.96/19.26 , a__add(s(X), Y) -> s(add(X, Y)) 38.96/19.26 , a__len(X) -> len(X) 38.96/19.26 , a__len(nil()) -> 0() 38.96/19.26 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.26 Obligation: 38.96/19.26 innermost runtime complexity 38.96/19.26 Answer: 38.96/19.26 YES(O(1),O(n^2)) 38.96/19.26 38.96/19.26 The weightgap principle applies (using the following nonconstant 38.96/19.26 growth matrix-interpretation) 38.96/19.26 38.96/19.26 The following argument positions are usable: 38.96/19.26 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.27 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.27 38.96/19.27 TcT has computed the following matrix interpretation satisfying 38.96/19.27 not(EDA) and not(IDA(1)). 38.96/19.27 38.96/19.27 [a__fst](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [0] = [0] 38.96/19.27 38.96/19.27 [nil] = [0] 38.96/19.27 38.96/19.27 [s](x1) = [0] 38.96/19.27 38.96/19.27 [cons](x1, x2) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [mark](x1) = [0] 38.96/19.27 38.96/19.27 [fst](x1, x2) = [1] x1 + [1] x2 + [0] 38.96/19.27 38.96/19.27 [a__from](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [from](x1) = [1] x1 + [7] 38.96/19.27 38.96/19.27 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 38.96/19.27 38.96/19.27 [add](x1, x2) = [1] x1 + [1] x2 + [7] 38.96/19.27 38.96/19.27 [a__len](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [len](x1) = [1] x1 + [7] 38.96/19.27 38.96/19.27 The order satisfies the following ordering constraints: 38.96/19.27 38.96/19.27 [a__fst(X1, X2)] = [1] X1 + [1] X2 + [1] 38.96/19.27 > [1] X1 + [1] X2 + [0] 38.96/19.27 = [fst(X1, X2)] 38.96/19.27 38.96/19.27 [a__fst(0(), Z)] = [1] Z + [1] 38.96/19.27 > [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [a__fst(s(X), cons(Y, Z))] = [1] Y + [1] 38.96/19.27 > [0] 38.96/19.27 = [cons(mark(Y), fst(X, Z))] 38.96/19.27 38.96/19.27 [mark(0())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [mark(nil())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [mark(s(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(X)] 38.96/19.27 38.96/19.27 [mark(cons(X1, X2))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [cons(mark(X1), X2)] 38.96/19.27 38.96/19.27 [mark(fst(X1, X2))] = [0] 38.96/19.27 ? [1] 38.96/19.27 = [a__fst(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(from(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [a__from(mark(X))] 38.96/19.27 38.96/19.27 [mark(add(X1, X2))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [a__add(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(len(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [a__len(mark(X))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [0] 38.96/19.27 >= [0] 38.96/19.27 = [cons(mark(X), from(s(X)))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [0] 38.96/19.27 ? [1] X + [7] 38.96/19.27 = [from(X)] 38.96/19.27 38.96/19.27 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 38.96/19.27 ? [1] X1 + [1] X2 + [7] 38.96/19.27 = [add(X1, X2)] 38.96/19.27 38.96/19.27 [a__add(0(), X)] = [1] X + [0] 38.96/19.27 >= [0] 38.96/19.27 = [mark(X)] 38.96/19.27 38.96/19.27 [a__add(s(X), Y)] = [1] Y + [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(add(X, Y))] 38.96/19.27 38.96/19.27 [a__len(X)] = [1] X + [0] 38.96/19.27 ? [1] X + [7] 38.96/19.27 = [len(X)] 38.96/19.27 38.96/19.27 [a__len(nil())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [a__len(cons(X, Z))] = [1] X + [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(len(Z))] 38.96/19.27 38.96/19.27 38.96/19.27 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 38.96/19.27 38.96/19.27 We are left with following problem, upon which TcT provides the 38.96/19.27 certificate YES(O(1),O(n^2)). 38.96/19.27 38.96/19.27 Strict Trs: 38.96/19.27 { mark(0()) -> 0() 38.96/19.27 , mark(nil()) -> nil() 38.96/19.27 , mark(s(X)) -> s(X) 38.96/19.27 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.27 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.27 , mark(from(X)) -> a__from(mark(X)) 38.96/19.27 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.27 , mark(len(X)) -> a__len(mark(X)) 38.96/19.27 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.27 , a__from(X) -> from(X) 38.96/19.27 , a__add(X1, X2) -> add(X1, X2) 38.96/19.27 , a__add(0(), X) -> mark(X) 38.96/19.27 , a__add(s(X), Y) -> s(add(X, Y)) 38.96/19.27 , a__len(X) -> len(X) 38.96/19.27 , a__len(nil()) -> 0() 38.96/19.27 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.27 Weak Trs: 38.96/19.27 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.27 , a__fst(0(), Z) -> nil() 38.96/19.27 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) } 38.96/19.27 Obligation: 38.96/19.27 innermost runtime complexity 38.96/19.27 Answer: 38.96/19.27 YES(O(1),O(n^2)) 38.96/19.27 38.96/19.27 The weightgap principle applies (using the following nonconstant 38.96/19.27 growth matrix-interpretation) 38.96/19.27 38.96/19.27 The following argument positions are usable: 38.96/19.27 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.27 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.27 38.96/19.27 TcT has computed the following matrix interpretation satisfying 38.96/19.27 not(EDA) and not(IDA(1)). 38.96/19.27 38.96/19.27 [a__fst](x1, x2) = [1] x1 + [1] x2 + [4] 38.96/19.27 38.96/19.27 [0] = [0] 38.96/19.27 38.96/19.27 [nil] = [0] 38.96/19.27 38.96/19.27 [s](x1) = [4] 38.96/19.27 38.96/19.27 [cons](x1, x2) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [mark](x1) = [0] 38.96/19.27 38.96/19.27 [fst](x1, x2) = [1] x1 + [1] x2 + [3] 38.96/19.27 38.96/19.27 [a__from](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [from](x1) = [1] x1 + [7] 38.96/19.27 38.96/19.27 [a__add](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [add](x1, x2) = [1] x1 + [1] x2 + [6] 38.96/19.27 38.96/19.27 [a__len](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [len](x1) = [1] x1 + [7] 38.96/19.27 38.96/19.27 The order satisfies the following ordering constraints: 38.96/19.27 38.96/19.27 [a__fst(X1, X2)] = [1] X1 + [1] X2 + [4] 38.96/19.27 > [1] X1 + [1] X2 + [3] 38.96/19.27 = [fst(X1, X2)] 38.96/19.27 38.96/19.27 [a__fst(0(), Z)] = [1] Z + [4] 38.96/19.27 > [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [a__fst(s(X), cons(Y, Z))] = [1] Y + [8] 38.96/19.27 > [0] 38.96/19.27 = [cons(mark(Y), fst(X, Z))] 38.96/19.27 38.96/19.27 [mark(0())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [mark(nil())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [mark(s(X))] = [0] 38.96/19.27 ? [4] 38.96/19.27 = [s(X)] 38.96/19.27 38.96/19.27 [mark(cons(X1, X2))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [cons(mark(X1), X2)] 38.96/19.27 38.96/19.27 [mark(fst(X1, X2))] = [0] 38.96/19.27 ? [4] 38.96/19.27 = [a__fst(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(from(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [a__from(mark(X))] 38.96/19.27 38.96/19.27 [mark(add(X1, X2))] = [0] 38.96/19.27 ? [1] 38.96/19.27 = [a__add(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(len(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [a__len(mark(X))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [0] 38.96/19.27 >= [0] 38.96/19.27 = [cons(mark(X), from(s(X)))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [0] 38.96/19.27 ? [1] X + [7] 38.96/19.27 = [from(X)] 38.96/19.27 38.96/19.27 [a__add(X1, X2)] = [1] X1 + [1] X2 + [1] 38.96/19.27 ? [1] X1 + [1] X2 + [6] 38.96/19.27 = [add(X1, X2)] 38.96/19.27 38.96/19.27 [a__add(0(), X)] = [1] X + [1] 38.96/19.27 > [0] 38.96/19.27 = [mark(X)] 38.96/19.27 38.96/19.27 [a__add(s(X), Y)] = [1] Y + [5] 38.96/19.27 > [4] 38.96/19.27 = [s(add(X, Y))] 38.96/19.27 38.96/19.27 [a__len(X)] = [1] X + [0] 38.96/19.27 ? [1] X + [7] 38.96/19.27 = [len(X)] 38.96/19.27 38.96/19.27 [a__len(nil())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [a__len(cons(X, Z))] = [1] X + [0] 38.96/19.27 ? [4] 38.96/19.27 = [s(len(Z))] 38.96/19.27 38.96/19.27 38.96/19.27 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 38.96/19.27 38.96/19.27 We are left with following problem, upon which TcT provides the 38.96/19.27 certificate YES(O(1),O(n^2)). 38.96/19.27 38.96/19.27 Strict Trs: 38.96/19.27 { mark(0()) -> 0() 38.96/19.27 , mark(nil()) -> nil() 38.96/19.27 , mark(s(X)) -> s(X) 38.96/19.27 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.27 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.27 , mark(from(X)) -> a__from(mark(X)) 38.96/19.27 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.27 , mark(len(X)) -> a__len(mark(X)) 38.96/19.27 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.27 , a__from(X) -> from(X) 38.96/19.27 , a__add(X1, X2) -> add(X1, X2) 38.96/19.27 , a__len(X) -> len(X) 38.96/19.27 , a__len(nil()) -> 0() 38.96/19.27 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.27 Weak Trs: 38.96/19.27 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.27 , a__fst(0(), Z) -> nil() 38.96/19.27 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.27 , a__add(0(), X) -> mark(X) 38.96/19.27 , a__add(s(X), Y) -> s(add(X, Y)) } 38.96/19.27 Obligation: 38.96/19.27 innermost runtime complexity 38.96/19.27 Answer: 38.96/19.27 YES(O(1),O(n^2)) 38.96/19.27 38.96/19.27 The weightgap principle applies (using the following nonconstant 38.96/19.27 growth matrix-interpretation) 38.96/19.27 38.96/19.27 The following argument positions are usable: 38.96/19.27 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.27 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.27 38.96/19.27 TcT has computed the following matrix interpretation satisfying 38.96/19.27 not(EDA) and not(IDA(1)). 38.96/19.27 38.96/19.27 [a__fst](x1, x2) = [1] x1 + [1] x2 + [4] 38.96/19.27 38.96/19.27 [0] = [0] 38.96/19.27 38.96/19.27 [nil] = [0] 38.96/19.27 38.96/19.27 [s](x1) = [0] 38.96/19.27 38.96/19.27 [cons](x1, x2) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [mark](x1) = [0] 38.96/19.27 38.96/19.27 [fst](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [a__from](x1) = [1] x1 + [1] 38.96/19.27 38.96/19.27 [from](x1) = [5] 38.96/19.27 38.96/19.27 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 38.96/19.27 38.96/19.27 [add](x1, x2) = [1] x1 + [1] x2 + [5] 38.96/19.27 38.96/19.27 [a__len](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [len](x1) = [1] x1 + [5] 38.96/19.27 38.96/19.27 The order satisfies the following ordering constraints: 38.96/19.27 38.96/19.27 [a__fst(X1, X2)] = [1] X1 + [1] X2 + [4] 38.96/19.27 > [1] X1 + [1] X2 + [1] 38.96/19.27 = [fst(X1, X2)] 38.96/19.27 38.96/19.27 [a__fst(0(), Z)] = [1] Z + [4] 38.96/19.27 > [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [a__fst(s(X), cons(Y, Z))] = [1] Y + [4] 38.96/19.27 > [0] 38.96/19.27 = [cons(mark(Y), fst(X, Z))] 38.96/19.27 38.96/19.27 [mark(0())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [mark(nil())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [mark(s(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(X)] 38.96/19.27 38.96/19.27 [mark(cons(X1, X2))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [cons(mark(X1), X2)] 38.96/19.27 38.96/19.27 [mark(fst(X1, X2))] = [0] 38.96/19.27 ? [4] 38.96/19.27 = [a__fst(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(from(X))] = [0] 38.96/19.27 ? [1] 38.96/19.27 = [a__from(mark(X))] 38.96/19.27 38.96/19.27 [mark(add(X1, X2))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [a__add(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(len(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [a__len(mark(X))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [1] 38.96/19.27 > [0] 38.96/19.27 = [cons(mark(X), from(s(X)))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [1] 38.96/19.27 ? [5] 38.96/19.27 = [from(X)] 38.96/19.27 38.96/19.27 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 38.96/19.27 ? [1] X1 + [1] X2 + [5] 38.96/19.27 = [add(X1, X2)] 38.96/19.27 38.96/19.27 [a__add(0(), X)] = [1] X + [0] 38.96/19.27 >= [0] 38.96/19.27 = [mark(X)] 38.96/19.27 38.96/19.27 [a__add(s(X), Y)] = [1] Y + [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(add(X, Y))] 38.96/19.27 38.96/19.27 [a__len(X)] = [1] X + [0] 38.96/19.27 ? [1] X + [5] 38.96/19.27 = [len(X)] 38.96/19.27 38.96/19.27 [a__len(nil())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [a__len(cons(X, Z))] = [1] X + [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(len(Z))] 38.96/19.27 38.96/19.27 38.96/19.27 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 38.96/19.27 38.96/19.27 We are left with following problem, upon which TcT provides the 38.96/19.27 certificate YES(O(1),O(n^2)). 38.96/19.27 38.96/19.27 Strict Trs: 38.96/19.27 { mark(0()) -> 0() 38.96/19.27 , mark(nil()) -> nil() 38.96/19.27 , mark(s(X)) -> s(X) 38.96/19.27 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.27 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.27 , mark(from(X)) -> a__from(mark(X)) 38.96/19.27 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.27 , mark(len(X)) -> a__len(mark(X)) 38.96/19.27 , a__from(X) -> from(X) 38.96/19.27 , a__add(X1, X2) -> add(X1, X2) 38.96/19.27 , a__len(X) -> len(X) 38.96/19.27 , a__len(nil()) -> 0() 38.96/19.27 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.27 Weak Trs: 38.96/19.27 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.27 , a__fst(0(), Z) -> nil() 38.96/19.27 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.27 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.27 , a__add(0(), X) -> mark(X) 38.96/19.27 , a__add(s(X), Y) -> s(add(X, Y)) } 38.96/19.27 Obligation: 38.96/19.27 innermost runtime complexity 38.96/19.27 Answer: 38.96/19.27 YES(O(1),O(n^2)) 38.96/19.27 38.96/19.27 The weightgap principle applies (using the following nonconstant 38.96/19.27 growth matrix-interpretation) 38.96/19.27 38.96/19.27 The following argument positions are usable: 38.96/19.27 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.27 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.27 38.96/19.27 TcT has computed the following matrix interpretation satisfying 38.96/19.27 not(EDA) and not(IDA(1)). 38.96/19.27 38.96/19.27 [a__fst](x1, x2) = [1] x1 + [1] x2 + [7] 38.96/19.27 38.96/19.27 [0] = [2] 38.96/19.27 38.96/19.27 [nil] = [0] 38.96/19.27 38.96/19.27 [s](x1) = [1] 38.96/19.27 38.96/19.27 [cons](x1, x2) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [mark](x1) = [1] 38.96/19.27 38.96/19.27 [fst](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [a__from](x1) = [1] x1 + [7] 38.96/19.27 38.96/19.27 [from](x1) = [5] 38.96/19.27 38.96/19.27 [a__add](x1, x2) = [1] x1 + [1] x2 + [3] 38.96/19.27 38.96/19.27 [add](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [a__len](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [len](x1) = [1] x1 + [5] 38.96/19.27 38.96/19.27 The order satisfies the following ordering constraints: 38.96/19.27 38.96/19.27 [a__fst(X1, X2)] = [1] X1 + [1] X2 + [7] 38.96/19.27 > [1] X1 + [1] X2 + [1] 38.96/19.27 = [fst(X1, X2)] 38.96/19.27 38.96/19.27 [a__fst(0(), Z)] = [1] Z + [9] 38.96/19.27 > [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [a__fst(s(X), cons(Y, Z))] = [1] Y + [8] 38.96/19.27 > [1] 38.96/19.27 = [cons(mark(Y), fst(X, Z))] 38.96/19.27 38.96/19.27 [mark(0())] = [1] 38.96/19.27 ? [2] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [mark(nil())] = [1] 38.96/19.27 > [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [mark(s(X))] = [1] 38.96/19.27 >= [1] 38.96/19.27 = [s(X)] 38.96/19.27 38.96/19.27 [mark(cons(X1, X2))] = [1] 38.96/19.27 >= [1] 38.96/19.27 = [cons(mark(X1), X2)] 38.96/19.27 38.96/19.27 [mark(fst(X1, X2))] = [1] 38.96/19.27 ? [9] 38.96/19.27 = [a__fst(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(from(X))] = [1] 38.96/19.27 ? [8] 38.96/19.27 = [a__from(mark(X))] 38.96/19.27 38.96/19.27 [mark(add(X1, X2))] = [1] 38.96/19.27 ? [5] 38.96/19.27 = [a__add(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(len(X))] = [1] 38.96/19.27 >= [1] 38.96/19.27 = [a__len(mark(X))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [7] 38.96/19.27 > [1] 38.96/19.27 = [cons(mark(X), from(s(X)))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [7] 38.96/19.27 > [5] 38.96/19.27 = [from(X)] 38.96/19.27 38.96/19.27 [a__add(X1, X2)] = [1] X1 + [1] X2 + [3] 38.96/19.27 > [1] X1 + [1] X2 + [1] 38.96/19.27 = [add(X1, X2)] 38.96/19.27 38.96/19.27 [a__add(0(), X)] = [1] X + [5] 38.96/19.27 > [1] 38.96/19.27 = [mark(X)] 38.96/19.27 38.96/19.27 [a__add(s(X), Y)] = [1] Y + [4] 38.96/19.27 > [1] 38.96/19.27 = [s(add(X, Y))] 38.96/19.27 38.96/19.27 [a__len(X)] = [1] X + [0] 38.96/19.27 ? [1] X + [5] 38.96/19.27 = [len(X)] 38.96/19.27 38.96/19.27 [a__len(nil())] = [0] 38.96/19.27 ? [2] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [a__len(cons(X, Z))] = [1] X + [0] 38.96/19.27 ? [1] 38.96/19.27 = [s(len(Z))] 38.96/19.27 38.96/19.27 38.96/19.27 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 38.96/19.27 38.96/19.27 We are left with following problem, upon which TcT provides the 38.96/19.27 certificate YES(O(1),O(n^2)). 38.96/19.27 38.96/19.27 Strict Trs: 38.96/19.27 { mark(0()) -> 0() 38.96/19.27 , mark(s(X)) -> s(X) 38.96/19.27 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.27 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.27 , mark(from(X)) -> a__from(mark(X)) 38.96/19.27 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.27 , mark(len(X)) -> a__len(mark(X)) 38.96/19.27 , a__len(X) -> len(X) 38.96/19.27 , a__len(nil()) -> 0() 38.96/19.27 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.27 Weak Trs: 38.96/19.27 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.27 , a__fst(0(), Z) -> nil() 38.96/19.27 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.27 , mark(nil()) -> nil() 38.96/19.27 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.27 , a__from(X) -> from(X) 38.96/19.27 , a__add(X1, X2) -> add(X1, X2) 38.96/19.27 , a__add(0(), X) -> mark(X) 38.96/19.27 , a__add(s(X), Y) -> s(add(X, Y)) } 38.96/19.27 Obligation: 38.96/19.27 innermost runtime complexity 38.96/19.27 Answer: 38.96/19.27 YES(O(1),O(n^2)) 38.96/19.27 38.96/19.27 The weightgap principle applies (using the following nonconstant 38.96/19.27 growth matrix-interpretation) 38.96/19.27 38.96/19.27 The following argument positions are usable: 38.96/19.27 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.27 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.27 38.96/19.27 TcT has computed the following matrix interpretation satisfying 38.96/19.27 not(EDA) and not(IDA(1)). 38.96/19.27 38.96/19.27 [a__fst](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [0] = [4] 38.96/19.27 38.96/19.27 [nil] = [0] 38.96/19.27 38.96/19.27 [s](x1) = [7] 38.96/19.27 38.96/19.27 [cons](x1, x2) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [mark](x1) = [0] 38.96/19.27 38.96/19.27 [fst](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [a__from](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [from](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [a__add](x1, x2) = [1] x1 + [1] x2 + [4] 38.96/19.27 38.96/19.27 [add](x1, x2) = [1] x1 + [1] x2 + [3] 38.96/19.27 38.96/19.27 [a__len](x1) = [1] x1 + [1] 38.96/19.27 38.96/19.27 [len](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 The order satisfies the following ordering constraints: 38.96/19.27 38.96/19.27 [a__fst(X1, X2)] = [1] X1 + [1] X2 + [1] 38.96/19.27 >= [1] X1 + [1] X2 + [1] 38.96/19.27 = [fst(X1, X2)] 38.96/19.27 38.96/19.27 [a__fst(0(), Z)] = [1] Z + [5] 38.96/19.27 > [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [a__fst(s(X), cons(Y, Z))] = [1] Y + [8] 38.96/19.27 > [0] 38.96/19.27 = [cons(mark(Y), fst(X, Z))] 38.96/19.27 38.96/19.27 [mark(0())] = [0] 38.96/19.27 ? [4] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [mark(nil())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [mark(s(X))] = [0] 38.96/19.27 ? [7] 38.96/19.27 = [s(X)] 38.96/19.27 38.96/19.27 [mark(cons(X1, X2))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [cons(mark(X1), X2)] 38.96/19.27 38.96/19.27 [mark(fst(X1, X2))] = [0] 38.96/19.27 ? [1] 38.96/19.27 = [a__fst(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(from(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [a__from(mark(X))] 38.96/19.27 38.96/19.27 [mark(add(X1, X2))] = [0] 38.96/19.27 ? [4] 38.96/19.27 = [a__add(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(len(X))] = [0] 38.96/19.27 ? [1] 38.96/19.27 = [a__len(mark(X))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [0] 38.96/19.27 >= [0] 38.96/19.27 = [cons(mark(X), from(s(X)))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [0] 38.96/19.27 >= [1] X + [0] 38.96/19.27 = [from(X)] 38.96/19.27 38.96/19.27 [a__add(X1, X2)] = [1] X1 + [1] X2 + [4] 38.96/19.27 > [1] X1 + [1] X2 + [3] 38.96/19.27 = [add(X1, X2)] 38.96/19.27 38.96/19.27 [a__add(0(), X)] = [1] X + [8] 38.96/19.27 > [0] 38.96/19.27 = [mark(X)] 38.96/19.27 38.96/19.27 [a__add(s(X), Y)] = [1] Y + [11] 38.96/19.27 > [7] 38.96/19.27 = [s(add(X, Y))] 38.96/19.27 38.96/19.27 [a__len(X)] = [1] X + [1] 38.96/19.27 > [1] X + [0] 38.96/19.27 = [len(X)] 38.96/19.27 38.96/19.27 [a__len(nil())] = [1] 38.96/19.27 ? [4] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [a__len(cons(X, Z))] = [1] X + [1] 38.96/19.27 ? [7] 38.96/19.27 = [s(len(Z))] 38.96/19.27 38.96/19.27 38.96/19.27 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 38.96/19.27 38.96/19.27 We are left with following problem, upon which TcT provides the 38.96/19.27 certificate YES(O(1),O(n^2)). 38.96/19.27 38.96/19.27 Strict Trs: 38.96/19.27 { mark(0()) -> 0() 38.96/19.27 , mark(s(X)) -> s(X) 38.96/19.27 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.27 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.27 , mark(from(X)) -> a__from(mark(X)) 38.96/19.27 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.27 , mark(len(X)) -> a__len(mark(X)) 38.96/19.27 , a__len(nil()) -> 0() 38.96/19.27 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.27 Weak Trs: 38.96/19.27 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.27 , a__fst(0(), Z) -> nil() 38.96/19.27 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.27 , mark(nil()) -> nil() 38.96/19.27 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.27 , a__from(X) -> from(X) 38.96/19.27 , a__add(X1, X2) -> add(X1, X2) 38.96/19.27 , a__add(0(), X) -> mark(X) 38.96/19.27 , a__add(s(X), Y) -> s(add(X, Y)) 38.96/19.27 , a__len(X) -> len(X) } 38.96/19.27 Obligation: 38.96/19.27 innermost runtime complexity 38.96/19.27 Answer: 38.96/19.27 YES(O(1),O(n^2)) 38.96/19.27 38.96/19.27 The weightgap principle applies (using the following nonconstant 38.96/19.27 growth matrix-interpretation) 38.96/19.27 38.96/19.27 The following argument positions are usable: 38.96/19.27 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.27 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.27 38.96/19.27 TcT has computed the following matrix interpretation satisfying 38.96/19.27 not(EDA) and not(IDA(1)). 38.96/19.27 38.96/19.27 [a__fst](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [0] = [0] 38.96/19.27 38.96/19.27 [nil] = [1] 38.96/19.27 38.96/19.27 [s](x1) = [0] 38.96/19.27 38.96/19.27 [cons](x1, x2) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [mark](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [fst](x1, x2) = [1] x1 + [1] x2 + [0] 38.96/19.27 38.96/19.27 [a__from](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [from](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 38.96/19.27 38.96/19.27 [add](x1, x2) = [1] x1 + [1] x2 + [0] 38.96/19.27 38.96/19.27 [a__len](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [len](x1) = [1] x1 + [0] 38.96/19.27 38.96/19.27 The order satisfies the following ordering constraints: 38.96/19.27 38.96/19.27 [a__fst(X1, X2)] = [1] X1 + [1] X2 + [1] 38.96/19.27 > [1] X1 + [1] X2 + [0] 38.96/19.27 = [fst(X1, X2)] 38.96/19.27 38.96/19.27 [a__fst(0(), Z)] = [1] Z + [1] 38.96/19.27 >= [1] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [a__fst(s(X), cons(Y, Z))] = [1] Y + [1] 38.96/19.27 > [1] Y + [0] 38.96/19.27 = [cons(mark(Y), fst(X, Z))] 38.96/19.27 38.96/19.27 [mark(0())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [mark(nil())] = [1] 38.96/19.27 >= [1] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [mark(s(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(X)] 38.96/19.27 38.96/19.27 [mark(cons(X1, X2))] = [1] X1 + [0] 38.96/19.27 >= [1] X1 + [0] 38.96/19.27 = [cons(mark(X1), X2)] 38.96/19.27 38.96/19.27 [mark(fst(X1, X2))] = [1] X1 + [1] X2 + [0] 38.96/19.27 ? [1] X1 + [1] X2 + [1] 38.96/19.27 = [a__fst(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(from(X))] = [1] X + [0] 38.96/19.27 >= [1] X + [0] 38.96/19.27 = [a__from(mark(X))] 38.96/19.27 38.96/19.27 [mark(add(X1, X2))] = [1] X1 + [1] X2 + [0] 38.96/19.27 >= [1] X1 + [1] X2 + [0] 38.96/19.27 = [a__add(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(len(X))] = [1] X + [0] 38.96/19.27 >= [1] X + [0] 38.96/19.27 = [a__len(mark(X))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [0] 38.96/19.27 >= [1] X + [0] 38.96/19.27 = [cons(mark(X), from(s(X)))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [0] 38.96/19.27 >= [1] X + [0] 38.96/19.27 = [from(X)] 38.96/19.27 38.96/19.27 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 38.96/19.27 >= [1] X1 + [1] X2 + [0] 38.96/19.27 = [add(X1, X2)] 38.96/19.27 38.96/19.27 [a__add(0(), X)] = [1] X + [0] 38.96/19.27 >= [1] X + [0] 38.96/19.27 = [mark(X)] 38.96/19.27 38.96/19.27 [a__add(s(X), Y)] = [1] Y + [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(add(X, Y))] 38.96/19.27 38.96/19.27 [a__len(X)] = [1] X + [0] 38.96/19.27 >= [1] X + [0] 38.96/19.27 = [len(X)] 38.96/19.27 38.96/19.27 [a__len(nil())] = [1] 38.96/19.27 > [0] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [a__len(cons(X, Z))] = [1] X + [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(len(Z))] 38.96/19.27 38.96/19.27 38.96/19.27 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 38.96/19.27 38.96/19.27 We are left with following problem, upon which TcT provides the 38.96/19.27 certificate YES(O(1),O(n^2)). 38.96/19.27 38.96/19.27 Strict Trs: 38.96/19.27 { mark(0()) -> 0() 38.96/19.27 , mark(s(X)) -> s(X) 38.96/19.27 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.27 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.27 , mark(from(X)) -> a__from(mark(X)) 38.96/19.27 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.27 , mark(len(X)) -> a__len(mark(X)) 38.96/19.27 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.27 Weak Trs: 38.96/19.27 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.27 , a__fst(0(), Z) -> nil() 38.96/19.27 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.27 , mark(nil()) -> nil() 38.96/19.27 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.27 , a__from(X) -> from(X) 38.96/19.27 , a__add(X1, X2) -> add(X1, X2) 38.96/19.27 , a__add(0(), X) -> mark(X) 38.96/19.27 , a__add(s(X), Y) -> s(add(X, Y)) 38.96/19.27 , a__len(X) -> len(X) 38.96/19.27 , a__len(nil()) -> 0() } 38.96/19.27 Obligation: 38.96/19.27 innermost runtime complexity 38.96/19.27 Answer: 38.96/19.27 YES(O(1),O(n^2)) 38.96/19.27 38.96/19.27 The weightgap principle applies (using the following nonconstant 38.96/19.27 growth matrix-interpretation) 38.96/19.27 38.96/19.27 The following argument positions are usable: 38.96/19.27 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.27 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.27 38.96/19.27 TcT has computed the following matrix interpretation satisfying 38.96/19.27 not(EDA) and not(IDA(1)). 38.96/19.27 38.96/19.27 [a__fst](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [0] = [3] 38.96/19.27 38.96/19.27 [nil] = [0] 38.96/19.27 38.96/19.27 [s](x1) = [0] 38.96/19.27 38.96/19.27 [cons](x1, x2) = [1] x1 + [0] 38.96/19.27 38.96/19.27 [mark](x1) = [0] 38.96/19.27 38.96/19.27 [fst](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.27 38.96/19.27 [a__from](x1) = [1] x1 + [4] 38.96/19.27 38.96/19.27 [from](x1) = [1] 38.96/19.27 38.96/19.27 [a__add](x1, x2) = [1] x1 + [1] x2 + [5] 38.96/19.27 38.96/19.27 [add](x1, x2) = [1] x1 + [1] x2 + [2] 38.96/19.27 38.96/19.27 [a__len](x1) = [1] x1 + [4] 38.96/19.27 38.96/19.27 [len](x1) = [1] x1 + [2] 38.96/19.27 38.96/19.27 The order satisfies the following ordering constraints: 38.96/19.27 38.96/19.27 [a__fst(X1, X2)] = [1] X1 + [1] X2 + [1] 38.96/19.27 >= [1] X1 + [1] X2 + [1] 38.96/19.27 = [fst(X1, X2)] 38.96/19.27 38.96/19.27 [a__fst(0(), Z)] = [1] Z + [4] 38.96/19.27 > [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [a__fst(s(X), cons(Y, Z))] = [1] Y + [1] 38.96/19.27 > [0] 38.96/19.27 = [cons(mark(Y), fst(X, Z))] 38.96/19.27 38.96/19.27 [mark(0())] = [0] 38.96/19.27 ? [3] 38.96/19.27 = [0()] 38.96/19.27 38.96/19.27 [mark(nil())] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [nil()] 38.96/19.27 38.96/19.27 [mark(s(X))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [s(X)] 38.96/19.27 38.96/19.27 [mark(cons(X1, X2))] = [0] 38.96/19.27 >= [0] 38.96/19.27 = [cons(mark(X1), X2)] 38.96/19.27 38.96/19.27 [mark(fst(X1, X2))] = [0] 38.96/19.27 ? [1] 38.96/19.27 = [a__fst(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(from(X))] = [0] 38.96/19.27 ? [4] 38.96/19.27 = [a__from(mark(X))] 38.96/19.27 38.96/19.27 [mark(add(X1, X2))] = [0] 38.96/19.27 ? [5] 38.96/19.27 = [a__add(mark(X1), mark(X2))] 38.96/19.27 38.96/19.27 [mark(len(X))] = [0] 38.96/19.27 ? [4] 38.96/19.27 = [a__len(mark(X))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [4] 38.96/19.27 > [0] 38.96/19.27 = [cons(mark(X), from(s(X)))] 38.96/19.27 38.96/19.27 [a__from(X)] = [1] X + [4] 38.96/19.27 > [1] 38.96/19.27 = [from(X)] 38.96/19.27 38.96/19.27 [a__add(X1, X2)] = [1] X1 + [1] X2 + [5] 38.96/19.27 > [1] X1 + [1] X2 + [2] 38.96/19.27 = [add(X1, X2)] 38.96/19.27 38.96/19.27 [a__add(0(), X)] = [1] X + [8] 38.96/19.27 > [0] 38.96/19.27 = [mark(X)] 38.96/19.27 38.96/19.27 [a__add(s(X), Y)] = [1] Y + [5] 38.96/19.27 > [0] 38.96/19.27 = [s(add(X, Y))] 38.96/19.28 38.96/19.28 [a__len(X)] = [1] X + [4] 38.96/19.28 > [1] X + [2] 38.96/19.28 = [len(X)] 38.96/19.28 38.96/19.28 [a__len(nil())] = [4] 38.96/19.28 > [3] 38.96/19.28 = [0()] 38.96/19.28 38.96/19.28 [a__len(cons(X, Z))] = [1] X + [4] 38.96/19.28 > [0] 38.96/19.28 = [s(len(Z))] 38.96/19.28 38.96/19.28 38.96/19.28 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 38.96/19.28 38.96/19.28 We are left with following problem, upon which TcT provides the 38.96/19.28 certificate YES(O(1),O(n^2)). 38.96/19.28 38.96/19.28 Strict Trs: 38.96/19.28 { mark(0()) -> 0() 38.96/19.28 , mark(s(X)) -> s(X) 38.96/19.28 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.28 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.28 , mark(from(X)) -> a__from(mark(X)) 38.96/19.28 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.28 , mark(len(X)) -> a__len(mark(X)) } 38.96/19.28 Weak Trs: 38.96/19.28 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.28 , a__fst(0(), Z) -> nil() 38.96/19.28 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.28 , mark(nil()) -> nil() 38.96/19.28 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.28 , a__from(X) -> from(X) 38.96/19.28 , a__add(X1, X2) -> add(X1, X2) 38.96/19.28 , a__add(0(), X) -> mark(X) 38.96/19.28 , a__add(s(X), Y) -> s(add(X, Y)) 38.96/19.28 , a__len(X) -> len(X) 38.96/19.28 , a__len(nil()) -> 0() 38.96/19.28 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.28 Obligation: 38.96/19.28 innermost runtime complexity 38.96/19.28 Answer: 38.96/19.28 YES(O(1),O(n^2)) 38.96/19.28 38.96/19.28 The weightgap principle applies (using the following nonconstant 38.96/19.28 growth matrix-interpretation) 38.96/19.28 38.96/19.28 The following argument positions are usable: 38.96/19.28 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.28 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.28 38.96/19.28 TcT has computed the following matrix interpretation satisfying 38.96/19.28 not(EDA) and not(IDA(1)). 38.96/19.28 38.96/19.28 [a__fst](x1, x2) = [1] x1 + [1] x2 + [2] 38.96/19.28 38.96/19.28 [0] = [2] 38.96/19.28 38.96/19.28 [nil] = [1] 38.96/19.28 38.96/19.28 [s](x1) = [0] 38.96/19.28 38.96/19.28 [cons](x1, x2) = [1] x1 + [0] 38.96/19.28 38.96/19.28 [mark](x1) = [1] 38.96/19.28 38.96/19.28 [fst](x1, x2) = [1] x1 + [1] x2 + [2] 38.96/19.28 38.96/19.28 [a__from](x1) = [1] x1 + [7] 38.96/19.28 38.96/19.28 [from](x1) = [1] 38.96/19.28 38.96/19.28 [a__add](x1, x2) = [1] x1 + [1] x2 + [2] 38.96/19.28 38.96/19.28 [add](x1, x2) = [1] x1 + [1] x2 + [2] 38.96/19.28 38.96/19.28 [a__len](x1) = [1] x1 + [7] 38.96/19.28 38.96/19.28 [len](x1) = [1] x1 + [1] 38.96/19.28 38.96/19.28 The order satisfies the following ordering constraints: 38.96/19.28 38.96/19.28 [a__fst(X1, X2)] = [1] X1 + [1] X2 + [2] 38.96/19.28 >= [1] X1 + [1] X2 + [2] 38.96/19.28 = [fst(X1, X2)] 38.96/19.28 38.96/19.28 [a__fst(0(), Z)] = [1] Z + [4] 38.96/19.28 > [1] 38.96/19.28 = [nil()] 38.96/19.28 38.96/19.28 [a__fst(s(X), cons(Y, Z))] = [1] Y + [2] 38.96/19.28 > [1] 38.96/19.28 = [cons(mark(Y), fst(X, Z))] 38.96/19.28 38.96/19.28 [mark(0())] = [1] 38.96/19.28 ? [2] 38.96/19.28 = [0()] 38.96/19.28 38.96/19.28 [mark(nil())] = [1] 38.96/19.28 >= [1] 38.96/19.28 = [nil()] 38.96/19.28 38.96/19.28 [mark(s(X))] = [1] 38.96/19.28 > [0] 38.96/19.28 = [s(X)] 38.96/19.28 38.96/19.28 [mark(cons(X1, X2))] = [1] 38.96/19.28 >= [1] 38.96/19.28 = [cons(mark(X1), X2)] 38.96/19.28 38.96/19.28 [mark(fst(X1, X2))] = [1] 38.96/19.28 ? [4] 38.96/19.28 = [a__fst(mark(X1), mark(X2))] 38.96/19.28 38.96/19.28 [mark(from(X))] = [1] 38.96/19.28 ? [8] 38.96/19.28 = [a__from(mark(X))] 38.96/19.28 38.96/19.28 [mark(add(X1, X2))] = [1] 38.96/19.28 ? [4] 38.96/19.28 = [a__add(mark(X1), mark(X2))] 38.96/19.28 38.96/19.28 [mark(len(X))] = [1] 38.96/19.28 ? [8] 38.96/19.28 = [a__len(mark(X))] 38.96/19.28 38.96/19.28 [a__from(X)] = [1] X + [7] 38.96/19.28 > [1] 38.96/19.28 = [cons(mark(X), from(s(X)))] 38.96/19.28 38.96/19.28 [a__from(X)] = [1] X + [7] 38.96/19.28 > [1] 38.96/19.28 = [from(X)] 38.96/19.28 38.96/19.28 [a__add(X1, X2)] = [1] X1 + [1] X2 + [2] 38.96/19.28 >= [1] X1 + [1] X2 + [2] 38.96/19.28 = [add(X1, X2)] 38.96/19.28 38.96/19.28 [a__add(0(), X)] = [1] X + [4] 38.96/19.28 > [1] 38.96/19.28 = [mark(X)] 38.96/19.28 38.96/19.28 [a__add(s(X), Y)] = [1] Y + [2] 38.96/19.28 > [0] 38.96/19.28 = [s(add(X, Y))] 38.96/19.28 38.96/19.28 [a__len(X)] = [1] X + [7] 38.96/19.28 > [1] X + [1] 38.96/19.28 = [len(X)] 38.96/19.28 38.96/19.28 [a__len(nil())] = [8] 38.96/19.28 > [2] 38.96/19.28 = [0()] 38.96/19.28 38.96/19.28 [a__len(cons(X, Z))] = [1] X + [7] 38.96/19.28 > [0] 38.96/19.28 = [s(len(Z))] 38.96/19.28 38.96/19.28 38.96/19.28 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 38.96/19.28 38.96/19.28 We are left with following problem, upon which TcT provides the 38.96/19.28 certificate YES(O(1),O(n^2)). 38.96/19.28 38.96/19.28 Strict Trs: 38.96/19.28 { mark(0()) -> 0() 38.96/19.28 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.28 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.28 , mark(from(X)) -> a__from(mark(X)) 38.96/19.28 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.28 , mark(len(X)) -> a__len(mark(X)) } 38.96/19.28 Weak Trs: 38.96/19.28 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.28 , a__fst(0(), Z) -> nil() 38.96/19.28 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.28 , mark(nil()) -> nil() 38.96/19.28 , mark(s(X)) -> s(X) 38.96/19.28 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.28 , a__from(X) -> from(X) 38.96/19.28 , a__add(X1, X2) -> add(X1, X2) 38.96/19.28 , a__add(0(), X) -> mark(X) 38.96/19.28 , a__add(s(X), Y) -> s(add(X, Y)) 38.96/19.28 , a__len(X) -> len(X) 38.96/19.28 , a__len(nil()) -> 0() 38.96/19.28 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.28 Obligation: 38.96/19.28 innermost runtime complexity 38.96/19.28 Answer: 38.96/19.28 YES(O(1),O(n^2)) 38.96/19.28 38.96/19.28 The weightgap principle applies (using the following nonconstant 38.96/19.28 growth matrix-interpretation) 38.96/19.28 38.96/19.28 The following argument positions are usable: 38.96/19.28 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.28 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.28 38.96/19.28 TcT has computed the following matrix interpretation satisfying 38.96/19.28 not(EDA) and not(IDA(1)). 38.96/19.28 38.96/19.28 [a__fst](x1, x2) = [1] x1 + [1] x2 + [2] 38.96/19.28 38.96/19.28 [0] = [0] 38.96/19.28 38.96/19.28 [nil] = [0] 38.96/19.28 38.96/19.28 [s](x1) = [0] 38.96/19.28 38.96/19.28 [cons](x1, x2) = [1] x1 + [0] 38.96/19.28 38.96/19.28 [mark](x1) = [1] 38.96/19.28 38.96/19.28 [fst](x1, x2) = [1] x1 + [1] x2 + [1] 38.96/19.28 38.96/19.28 [a__from](x1) = [1] x1 + [7] 38.96/19.28 38.96/19.28 [from](x1) = [1] x1 + [3] 38.96/19.28 38.96/19.28 [a__add](x1, x2) = [1] x1 + [1] x2 + [6] 38.96/19.28 38.96/19.28 [add](x1, x2) = [1] x1 + [1] x2 + [3] 38.96/19.28 38.96/19.28 [a__len](x1) = [1] x1 + [0] 38.96/19.28 38.96/19.28 [len](x1) = [1] x1 + [0] 38.96/19.28 38.96/19.28 The order satisfies the following ordering constraints: 38.96/19.28 38.96/19.28 [a__fst(X1, X2)] = [1] X1 + [1] X2 + [2] 38.96/19.28 > [1] X1 + [1] X2 + [1] 38.96/19.28 = [fst(X1, X2)] 38.96/19.28 38.96/19.28 [a__fst(0(), Z)] = [1] Z + [2] 38.96/19.28 > [0] 38.96/19.28 = [nil()] 38.96/19.28 38.96/19.28 [a__fst(s(X), cons(Y, Z))] = [1] Y + [2] 38.96/19.28 > [1] 38.96/19.28 = [cons(mark(Y), fst(X, Z))] 38.96/19.28 38.96/19.28 [mark(0())] = [1] 38.96/19.28 > [0] 38.96/19.28 = [0()] 38.96/19.28 38.96/19.28 [mark(nil())] = [1] 38.96/19.28 > [0] 38.96/19.28 = [nil()] 38.96/19.28 38.96/19.28 [mark(s(X))] = [1] 38.96/19.28 > [0] 38.96/19.28 = [s(X)] 38.96/19.28 38.96/19.28 [mark(cons(X1, X2))] = [1] 38.96/19.28 >= [1] 38.96/19.28 = [cons(mark(X1), X2)] 38.96/19.28 38.96/19.28 [mark(fst(X1, X2))] = [1] 38.96/19.28 ? [4] 38.96/19.28 = [a__fst(mark(X1), mark(X2))] 38.96/19.28 38.96/19.28 [mark(from(X))] = [1] 38.96/19.28 ? [8] 38.96/19.28 = [a__from(mark(X))] 38.96/19.28 38.96/19.28 [mark(add(X1, X2))] = [1] 38.96/19.28 ? [8] 38.96/19.28 = [a__add(mark(X1), mark(X2))] 38.96/19.28 38.96/19.28 [mark(len(X))] = [1] 38.96/19.28 >= [1] 38.96/19.28 = [a__len(mark(X))] 38.96/19.28 38.96/19.28 [a__from(X)] = [1] X + [7] 38.96/19.28 > [1] 38.96/19.28 = [cons(mark(X), from(s(X)))] 38.96/19.28 38.96/19.28 [a__from(X)] = [1] X + [7] 38.96/19.28 > [1] X + [3] 38.96/19.28 = [from(X)] 38.96/19.28 38.96/19.28 [a__add(X1, X2)] = [1] X1 + [1] X2 + [6] 38.96/19.28 > [1] X1 + [1] X2 + [3] 38.96/19.28 = [add(X1, X2)] 38.96/19.28 38.96/19.28 [a__add(0(), X)] = [1] X + [6] 38.96/19.28 > [1] 38.96/19.28 = [mark(X)] 38.96/19.28 38.96/19.28 [a__add(s(X), Y)] = [1] Y + [6] 38.96/19.28 > [0] 38.96/19.28 = [s(add(X, Y))] 38.96/19.28 38.96/19.28 [a__len(X)] = [1] X + [0] 38.96/19.28 >= [1] X + [0] 38.96/19.28 = [len(X)] 38.96/19.28 38.96/19.28 [a__len(nil())] = [0] 38.96/19.28 >= [0] 38.96/19.28 = [0()] 38.96/19.28 38.96/19.28 [a__len(cons(X, Z))] = [1] X + [0] 38.96/19.28 >= [0] 38.96/19.28 = [s(len(Z))] 38.96/19.28 38.96/19.28 38.96/19.28 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 38.96/19.28 38.96/19.28 We are left with following problem, upon which TcT provides the 38.96/19.28 certificate YES(O(1),O(n^2)). 38.96/19.28 38.96/19.28 Strict Trs: 38.96/19.28 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.28 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.28 , mark(from(X)) -> a__from(mark(X)) 38.96/19.28 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.28 , mark(len(X)) -> a__len(mark(X)) } 38.96/19.28 Weak Trs: 38.96/19.28 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.28 , a__fst(0(), Z) -> nil() 38.96/19.28 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.28 , mark(0()) -> 0() 38.96/19.28 , mark(nil()) -> nil() 38.96/19.28 , mark(s(X)) -> s(X) 38.96/19.28 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.28 , a__from(X) -> from(X) 38.96/19.28 , a__add(X1, X2) -> add(X1, X2) 38.96/19.28 , a__add(0(), X) -> mark(X) 38.96/19.28 , a__add(s(X), Y) -> s(add(X, Y)) 38.96/19.28 , a__len(X) -> len(X) 38.96/19.28 , a__len(nil()) -> 0() 38.96/19.28 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.28 Obligation: 38.96/19.28 innermost runtime complexity 38.96/19.28 Answer: 38.96/19.28 YES(O(1),O(n^2)) 38.96/19.28 38.96/19.28 We use the processor 'matrix interpretation of dimension 2' to 38.96/19.28 orient following rules strictly. 38.96/19.28 38.96/19.28 Trs: 38.96/19.28 { mark(from(X)) -> a__from(mark(X)) 38.96/19.28 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.28 , mark(len(X)) -> a__len(mark(X)) } 38.96/19.28 38.96/19.28 The induced complexity on above rules (modulo remaining rules) is 38.96/19.28 YES(?,O(n^2)) . These rules are moved into the corresponding weak 38.96/19.28 component(s). 38.96/19.28 38.96/19.28 Sub-proof: 38.96/19.28 ---------- 38.96/19.28 The following argument positions are usable: 38.96/19.28 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.28 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.28 38.96/19.28 TcT has computed the following constructor-based matrix 38.96/19.28 interpretation satisfying not(EDA). 38.96/19.28 38.96/19.28 [a__fst](x1, x2) = [1 5] x1 + [1 6] x2 + [0] 38.96/19.28 [0 1] [0 1] [0] 38.96/19.28 38.96/19.28 [0] = [0] 38.96/19.28 [0] 38.96/19.28 38.96/19.28 [nil] = [0] 38.96/19.28 [0] 38.96/19.28 38.96/19.28 [s](x1) = [0] 38.96/19.28 [0] 38.96/19.28 38.96/19.28 [cons](x1, x2) = [1 2] x1 + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 38.96/19.28 [mark](x1) = [1 3] x1 + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 38.96/19.28 [fst](x1, x2) = [1 5] x1 + [1 6] x2 + [0] 38.96/19.28 [0 1] [0 1] [0] 38.96/19.28 38.96/19.28 [a__from](x1) = [1 6] x1 + [7] 38.96/19.28 [0 1] [1] 38.96/19.28 38.96/19.28 [from](x1) = [1 6] x1 + [6] 38.96/19.28 [0 1] [1] 38.96/19.28 38.96/19.28 [a__add](x1, x2) = [1 5] x1 + [1 5] x2 + [1] 38.96/19.28 [0 1] [0 1] [1] 38.96/19.28 38.96/19.28 [add](x1, x2) = [1 5] x1 + [1 5] x2 + [1] 38.96/19.28 [0 1] [0 1] [1] 38.96/19.28 38.96/19.28 [a__len](x1) = [1 6] x1 + [1] 38.96/19.28 [0 1] [1] 38.96/19.28 38.96/19.28 [len](x1) = [1 6] x1 + [1] 38.96/19.28 [0 1] [1] 38.96/19.28 38.96/19.28 The order satisfies the following ordering constraints: 38.96/19.28 38.96/19.28 [a__fst(X1, X2)] = [1 5] X1 + [1 6] X2 + [0] 38.96/19.28 [0 1] [0 1] [0] 38.96/19.28 >= [1 5] X1 + [1 6] X2 + [0] 38.96/19.28 [0 1] [0 1] [0] 38.96/19.28 = [fst(X1, X2)] 38.96/19.28 38.96/19.28 [a__fst(0(), Z)] = [1 6] Z + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 >= [0] 38.96/19.28 [0] 38.96/19.28 = [nil()] 38.96/19.28 38.96/19.28 [a__fst(s(X), cons(Y, Z))] = [1 8] Y + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 >= [1 5] Y + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 = [cons(mark(Y), fst(X, Z))] 38.96/19.28 38.96/19.28 [mark(0())] = [0] 38.96/19.28 [0] 38.96/19.28 >= [0] 38.96/19.28 [0] 38.96/19.28 = [0()] 38.96/19.28 38.96/19.28 [mark(nil())] = [0] 38.96/19.28 [0] 38.96/19.28 >= [0] 38.96/19.28 [0] 38.96/19.28 = [nil()] 38.96/19.28 38.96/19.28 [mark(s(X))] = [0] 38.96/19.28 [0] 38.96/19.28 >= [0] 38.96/19.28 [0] 38.96/19.28 = [s(X)] 38.96/19.28 38.96/19.28 [mark(cons(X1, X2))] = [1 5] X1 + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 >= [1 5] X1 + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 = [cons(mark(X1), X2)] 38.96/19.28 38.96/19.28 [mark(fst(X1, X2))] = [1 8] X1 + [1 9] X2 + [0] 38.96/19.28 [0 1] [0 1] [0] 38.96/19.28 >= [1 8] X1 + [1 9] X2 + [0] 38.96/19.28 [0 1] [0 1] [0] 38.96/19.28 = [a__fst(mark(X1), mark(X2))] 38.96/19.28 38.96/19.28 [mark(from(X))] = [1 9] X + [9] 38.96/19.28 [0 1] [1] 38.96/19.28 > [1 9] X + [7] 38.96/19.28 [0 1] [1] 38.96/19.28 = [a__from(mark(X))] 38.96/19.28 38.96/19.28 [mark(add(X1, X2))] = [1 8] X1 + [1 8] X2 + [4] 38.96/19.28 [0 1] [0 1] [1] 38.96/19.28 > [1 8] X1 + [1 8] X2 + [1] 38.96/19.28 [0 1] [0 1] [1] 38.96/19.28 = [a__add(mark(X1), mark(X2))] 38.96/19.28 38.96/19.28 [mark(len(X))] = [1 9] X + [4] 38.96/19.28 [0 1] [1] 38.96/19.28 > [1 9] X + [1] 38.96/19.28 [0 1] [1] 38.96/19.28 = [a__len(mark(X))] 38.96/19.28 38.96/19.28 [a__from(X)] = [1 6] X + [7] 38.96/19.28 [0 1] [1] 38.96/19.28 > [1 5] X + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 = [cons(mark(X), from(s(X)))] 38.96/19.28 38.96/19.28 [a__from(X)] = [1 6] X + [7] 38.96/19.28 [0 1] [1] 38.96/19.28 > [1 6] X + [6] 38.96/19.28 [0 1] [1] 38.96/19.28 = [from(X)] 38.96/19.28 38.96/19.28 [a__add(X1, X2)] = [1 5] X1 + [1 5] X2 + [1] 38.96/19.28 [0 1] [0 1] [1] 38.96/19.28 >= [1 5] X1 + [1 5] X2 + [1] 38.96/19.28 [0 1] [0 1] [1] 38.96/19.28 = [add(X1, X2)] 38.96/19.28 38.96/19.28 [a__add(0(), X)] = [1 5] X + [1] 38.96/19.28 [0 1] [1] 38.96/19.28 > [1 3] X + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 = [mark(X)] 38.96/19.28 38.96/19.28 [a__add(s(X), Y)] = [1 5] Y + [1] 38.96/19.28 [0 1] [1] 38.96/19.28 > [0] 38.96/19.28 [0] 38.96/19.28 = [s(add(X, Y))] 38.96/19.28 38.96/19.28 [a__len(X)] = [1 6] X + [1] 38.96/19.28 [0 1] [1] 38.96/19.28 >= [1 6] X + [1] 38.96/19.28 [0 1] [1] 38.96/19.28 = [len(X)] 38.96/19.28 38.96/19.28 [a__len(nil())] = [1] 38.96/19.28 [1] 38.96/19.28 > [0] 38.96/19.28 [0] 38.96/19.28 = [0()] 38.96/19.28 38.96/19.28 [a__len(cons(X, Z))] = [1 8] X + [1] 38.96/19.28 [0 1] [1] 38.96/19.28 > [0] 38.96/19.28 [0] 38.96/19.28 = [s(len(Z))] 38.96/19.28 38.96/19.28 38.96/19.28 We return to the main proof. 38.96/19.28 38.96/19.28 We are left with following problem, upon which TcT provides the 38.96/19.28 certificate YES(O(1),O(n^2)). 38.96/19.28 38.96/19.28 Strict Trs: 38.96/19.28 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.28 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) } 38.96/19.28 Weak Trs: 38.96/19.28 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.28 , a__fst(0(), Z) -> nil() 38.96/19.28 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.28 , mark(0()) -> 0() 38.96/19.28 , mark(nil()) -> nil() 38.96/19.28 , mark(s(X)) -> s(X) 38.96/19.28 , mark(from(X)) -> a__from(mark(X)) 38.96/19.28 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.28 , mark(len(X)) -> a__len(mark(X)) 38.96/19.28 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.28 , a__from(X) -> from(X) 38.96/19.28 , a__add(X1, X2) -> add(X1, X2) 38.96/19.28 , a__add(0(), X) -> mark(X) 38.96/19.28 , a__add(s(X), Y) -> s(add(X, Y)) 38.96/19.28 , a__len(X) -> len(X) 38.96/19.28 , a__len(nil()) -> 0() 38.96/19.28 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.28 Obligation: 38.96/19.28 innermost runtime complexity 38.96/19.28 Answer: 38.96/19.28 YES(O(1),O(n^2)) 38.96/19.28 38.96/19.28 We use the processor 'matrix interpretation of dimension 2' to 38.96/19.28 orient following rules strictly. 38.96/19.28 38.96/19.28 Trs: 38.96/19.28 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.28 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) } 38.96/19.28 38.96/19.28 The induced complexity on above rules (modulo remaining rules) is 38.96/19.28 YES(?,O(n^2)) . These rules are moved into the corresponding weak 38.96/19.28 component(s). 38.96/19.28 38.96/19.28 Sub-proof: 38.96/19.28 ---------- 38.96/19.28 The following argument positions are usable: 38.96/19.28 Uargs(a__fst) = {1, 2}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 38.96/19.28 Uargs(a__add) = {1, 2}, Uargs(a__len) = {1} 38.96/19.28 38.96/19.28 TcT has computed the following constructor-based matrix 38.96/19.28 interpretation satisfying not(EDA). 38.96/19.28 38.96/19.28 [a__fst](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 38.96/19.28 [0 1] [0 1] [6] 38.96/19.28 38.96/19.28 [0] = [4] 38.96/19.28 [0] 38.96/19.28 38.96/19.28 [nil] = [2] 38.96/19.28 [1] 38.96/19.28 38.96/19.28 [s](x1) = [2] 38.96/19.28 [1] 38.96/19.28 38.96/19.28 [cons](x1, x2) = [1 6] x1 + [0] 38.96/19.28 [0 1] [1] 38.96/19.28 38.96/19.28 [mark](x1) = [1 1] x1 + [3] 38.96/19.28 [0 1] [0] 38.96/19.28 38.96/19.28 [fst](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 38.96/19.28 [0 1] [0 1] [6] 38.96/19.28 38.96/19.28 [a__from](x1) = [1 7] x1 + [3] 38.96/19.28 [0 1] [4] 38.96/19.28 38.96/19.28 [from](x1) = [1 7] x1 + [0] 38.96/19.28 [0 1] [4] 38.96/19.28 38.96/19.28 [a__add](x1, x2) = [1 5] x1 + [1 2] x2 + [1] 38.96/19.28 [0 1] [0 1] [4] 38.96/19.28 38.96/19.28 [add](x1, x2) = [1 5] x1 + [1 2] x2 + [1] 38.96/19.28 [0 1] [0 1] [4] 38.96/19.28 38.96/19.28 [a__len](x1) = [1 3] x1 + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 38.96/19.28 [len](x1) = [1 3] x1 + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 38.96/19.28 The order satisfies the following ordering constraints: 38.96/19.28 38.96/19.28 [a__fst(X1, X2)] = [1 0] X1 + [1 2] X2 + [0] 38.96/19.28 [0 1] [0 1] [6] 38.96/19.28 >= [1 0] X1 + [1 2] X2 + [0] 38.96/19.28 [0 1] [0 1] [6] 38.96/19.28 = [fst(X1, X2)] 38.96/19.28 38.96/19.28 [a__fst(0(), Z)] = [1 2] Z + [4] 38.96/19.28 [0 1] [6] 38.96/19.28 > [2] 38.96/19.28 [1] 38.96/19.28 = [nil()] 38.96/19.28 38.96/19.28 [a__fst(s(X), cons(Y, Z))] = [1 8] Y + [4] 38.96/19.28 [0 1] [8] 38.96/19.28 > [1 7] Y + [3] 38.96/19.28 [0 1] [1] 38.96/19.28 = [cons(mark(Y), fst(X, Z))] 38.96/19.28 38.96/19.28 [mark(0())] = [7] 38.96/19.28 [0] 38.96/19.28 > [4] 38.96/19.28 [0] 38.96/19.28 = [0()] 38.96/19.28 38.96/19.28 [mark(nil())] = [6] 38.96/19.28 [1] 38.96/19.28 > [2] 38.96/19.28 [1] 38.96/19.28 = [nil()] 38.96/19.28 38.96/19.28 [mark(s(X))] = [6] 38.96/19.28 [1] 38.96/19.28 > [2] 38.96/19.28 [1] 38.96/19.28 = [s(X)] 38.96/19.28 38.96/19.28 [mark(cons(X1, X2))] = [1 7] X1 + [4] 38.96/19.28 [0 1] [1] 38.96/19.28 > [1 7] X1 + [3] 38.96/19.28 [0 1] [1] 38.96/19.28 = [cons(mark(X1), X2)] 38.96/19.28 38.96/19.28 [mark(fst(X1, X2))] = [1 1] X1 + [1 3] X2 + [9] 38.96/19.28 [0 1] [0 1] [6] 38.96/19.28 > [1 1] X1 + [1 3] X2 + [6] 38.96/19.28 [0 1] [0 1] [6] 38.96/19.28 = [a__fst(mark(X1), mark(X2))] 38.96/19.28 38.96/19.28 [mark(from(X))] = [1 8] X + [7] 38.96/19.28 [0 1] [4] 38.96/19.28 > [1 8] X + [6] 38.96/19.28 [0 1] [4] 38.96/19.28 = [a__from(mark(X))] 38.96/19.28 38.96/19.28 [mark(add(X1, X2))] = [1 6] X1 + [1 3] X2 + [8] 38.96/19.28 [0 1] [0 1] [4] 38.96/19.28 > [1 6] X1 + [1 3] X2 + [7] 38.96/19.28 [0 1] [0 1] [4] 38.96/19.28 = [a__add(mark(X1), mark(X2))] 38.96/19.28 38.96/19.28 [mark(len(X))] = [1 4] X + [3] 38.96/19.28 [0 1] [0] 38.96/19.28 >= [1 4] X + [3] 38.96/19.28 [0 1] [0] 38.96/19.28 = [a__len(mark(X))] 38.96/19.28 38.96/19.28 [a__from(X)] = [1 7] X + [3] 38.96/19.28 [0 1] [4] 38.96/19.28 >= [1 7] X + [3] 38.96/19.28 [0 1] [1] 38.96/19.28 = [cons(mark(X), from(s(X)))] 38.96/19.28 38.96/19.28 [a__from(X)] = [1 7] X + [3] 38.96/19.28 [0 1] [4] 38.96/19.28 > [1 7] X + [0] 38.96/19.28 [0 1] [4] 38.96/19.28 = [from(X)] 38.96/19.28 38.96/19.28 [a__add(X1, X2)] = [1 5] X1 + [1 2] X2 + [1] 38.96/19.28 [0 1] [0 1] [4] 38.96/19.28 >= [1 5] X1 + [1 2] X2 + [1] 38.96/19.28 [0 1] [0 1] [4] 38.96/19.28 = [add(X1, X2)] 38.96/19.28 38.96/19.28 [a__add(0(), X)] = [1 2] X + [5] 38.96/19.28 [0 1] [4] 38.96/19.28 > [1 1] X + [3] 38.96/19.28 [0 1] [0] 38.96/19.28 = [mark(X)] 38.96/19.28 38.96/19.28 [a__add(s(X), Y)] = [1 2] Y + [8] 38.96/19.28 [0 1] [5] 38.96/19.28 > [2] 38.96/19.28 [1] 38.96/19.28 = [s(add(X, Y))] 38.96/19.28 38.96/19.28 [a__len(X)] = [1 3] X + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 >= [1 3] X + [0] 38.96/19.28 [0 1] [0] 38.96/19.28 = [len(X)] 38.96/19.28 38.96/19.28 [a__len(nil())] = [5] 38.96/19.28 [1] 38.96/19.28 > [4] 38.96/19.28 [0] 38.96/19.28 = [0()] 38.96/19.28 38.96/19.28 [a__len(cons(X, Z))] = [1 9] X + [3] 38.96/19.28 [0 1] [1] 38.96/19.28 > [2] 38.96/19.28 [1] 38.96/19.28 = [s(len(Z))] 38.96/19.28 38.96/19.28 38.96/19.28 We return to the main proof. 38.96/19.28 38.96/19.28 We are left with following problem, upon which TcT provides the 38.96/19.28 certificate YES(O(1),O(1)). 38.96/19.28 38.96/19.28 Weak Trs: 38.96/19.28 { a__fst(X1, X2) -> fst(X1, X2) 38.96/19.28 , a__fst(0(), Z) -> nil() 38.96/19.28 , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) 38.96/19.28 , mark(0()) -> 0() 38.96/19.28 , mark(nil()) -> nil() 38.96/19.28 , mark(s(X)) -> s(X) 38.96/19.28 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 38.96/19.28 , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) 38.96/19.28 , mark(from(X)) -> a__from(mark(X)) 38.96/19.28 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 38.96/19.28 , mark(len(X)) -> a__len(mark(X)) 38.96/19.28 , a__from(X) -> cons(mark(X), from(s(X))) 38.96/19.28 , a__from(X) -> from(X) 38.96/19.28 , a__add(X1, X2) -> add(X1, X2) 38.96/19.28 , a__add(0(), X) -> mark(X) 38.96/19.28 , a__add(s(X), Y) -> s(add(X, Y)) 38.96/19.28 , a__len(X) -> len(X) 38.96/19.28 , a__len(nil()) -> 0() 38.96/19.28 , a__len(cons(X, Z)) -> s(len(Z)) } 38.96/19.28 Obligation: 38.96/19.28 innermost runtime complexity 38.96/19.28 Answer: 38.96/19.28 YES(O(1),O(1)) 38.96/19.28 38.96/19.28 Empty rules are trivially bounded 38.96/19.28 38.96/19.28 Hurray, we answered YES(O(1),O(n^2)) 38.96/19.29 EOF