YES(O(1),O(n^1)) 0.00/0.73 YES(O(1),O(n^1)) 0.00/0.73 0.00/0.73 We are left with following problem, upon which TcT provides the 0.00/0.73 certificate YES(O(1),O(n^1)). 0.00/0.73 0.00/0.73 Strict Trs: 0.00/0.73 { first(X1, X2) -> n__first(X1, X2) 0.00/0.73 , first(0(), X) -> nil() 0.00/0.73 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 0.00/0.73 , s(X) -> n__s(X) 0.00/0.73 , activate(X) -> X 0.00/0.73 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__from(X)) -> from(activate(X)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) 0.00/0.73 , from(X) -> cons(X, n__from(n__s(X))) 0.00/0.73 , from(X) -> n__from(X) } 0.00/0.73 Obligation: 0.00/0.73 innermost runtime complexity 0.00/0.73 Answer: 0.00/0.73 YES(O(1),O(n^1)) 0.00/0.73 0.00/0.73 Arguments of following rules are not normal-forms: 0.00/0.73 0.00/0.73 { first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) } 0.00/0.73 0.00/0.73 All above mentioned rules can be savely removed. 0.00/0.73 0.00/0.73 We are left with following problem, upon which TcT provides the 0.00/0.73 certificate YES(O(1),O(n^1)). 0.00/0.73 0.00/0.73 Strict Trs: 0.00/0.73 { first(X1, X2) -> n__first(X1, X2) 0.00/0.73 , first(0(), X) -> nil() 0.00/0.73 , s(X) -> n__s(X) 0.00/0.73 , activate(X) -> X 0.00/0.73 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__from(X)) -> from(activate(X)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) 0.00/0.73 , from(X) -> cons(X, n__from(n__s(X))) 0.00/0.73 , from(X) -> n__from(X) } 0.00/0.73 Obligation: 0.00/0.73 innermost runtime complexity 0.00/0.73 Answer: 0.00/0.73 YES(O(1),O(n^1)) 0.00/0.73 0.00/0.73 The weightgap principle applies (using the following nonconstant 0.00/0.73 growth matrix-interpretation) 0.00/0.73 0.00/0.73 The following argument positions are usable: 0.00/0.73 Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} 0.00/0.73 0.00/0.73 TcT has computed the following matrix interpretation satisfying 0.00/0.73 not(EDA) and not(IDA(1)). 0.00/0.73 0.00/0.73 [first](x1, x2) = [1] x1 + [1] x2 + [1] 0.00/0.73 0.00/0.73 [0] = [7] 0.00/0.73 0.00/0.73 [nil] = [7] 0.00/0.73 0.00/0.73 [s](x1) = [1] x1 + [7] 0.00/0.73 0.00/0.73 [cons](x1, x2) = [1] x2 + [0] 0.00/0.73 0.00/0.73 [n__first](x1, x2) = [1] x1 + [1] x2 + [7] 0.00/0.73 0.00/0.73 [activate](x1) = [1] x1 + [7] 0.00/0.73 0.00/0.73 [from](x1) = [1] x1 + [7] 0.00/0.73 0.00/0.73 [n__from](x1) = [1] x1 + [7] 0.00/0.73 0.00/0.73 [n__s](x1) = [1] x1 + [7] 0.00/0.73 0.00/0.73 The order satisfies the following ordering constraints: 0.00/0.73 0.00/0.73 [first(X1, X2)] = [1] X1 + [1] X2 + [1] 0.00/0.73 ? [1] X1 + [1] X2 + [7] 0.00/0.73 = [n__first(X1, X2)] 0.00/0.73 0.00/0.73 [first(0(), X)] = [1] X + [8] 0.00/0.73 > [7] 0.00/0.73 = [nil()] 0.00/0.73 0.00/0.73 [s(X)] = [1] X + [7] 0.00/0.73 >= [1] X + [7] 0.00/0.73 = [n__s(X)] 0.00/0.73 0.00/0.73 [activate(X)] = [1] X + [7] 0.00/0.73 > [1] X + [0] 0.00/0.73 = [X] 0.00/0.73 0.00/0.73 [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [14] 0.00/0.73 ? [1] X1 + [1] X2 + [15] 0.00/0.73 = [first(activate(X1), activate(X2))] 0.00/0.73 0.00/0.73 [activate(n__from(X))] = [1] X + [14] 0.00/0.73 >= [1] X + [14] 0.00/0.73 = [from(activate(X))] 0.00/0.73 0.00/0.73 [activate(n__s(X))] = [1] X + [14] 0.00/0.73 >= [1] X + [14] 0.00/0.73 = [s(activate(X))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [7] 0.00/0.73 ? [1] X + [14] 0.00/0.73 = [cons(X, n__from(n__s(X)))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [7] 0.00/0.73 >= [1] X + [7] 0.00/0.73 = [n__from(X)] 0.00/0.73 0.00/0.73 0.00/0.73 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.73 0.00/0.73 We are left with following problem, upon which TcT provides the 0.00/0.73 certificate YES(O(1),O(n^1)). 0.00/0.73 0.00/0.73 Strict Trs: 0.00/0.73 { first(X1, X2) -> n__first(X1, X2) 0.00/0.73 , s(X) -> n__s(X) 0.00/0.73 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__from(X)) -> from(activate(X)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) 0.00/0.73 , from(X) -> cons(X, n__from(n__s(X))) 0.00/0.73 , from(X) -> n__from(X) } 0.00/0.73 Weak Trs: 0.00/0.73 { first(0(), X) -> nil() 0.00/0.73 , activate(X) -> X } 0.00/0.73 Obligation: 0.00/0.73 innermost runtime complexity 0.00/0.73 Answer: 0.00/0.73 YES(O(1),O(n^1)) 0.00/0.73 0.00/0.73 The weightgap principle applies (using the following nonconstant 0.00/0.73 growth matrix-interpretation) 0.00/0.73 0.00/0.73 The following argument positions are usable: 0.00/0.73 Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} 0.00/0.73 0.00/0.73 TcT has computed the following matrix interpretation satisfying 0.00/0.73 not(EDA) and not(IDA(1)). 0.00/0.73 0.00/0.73 [first](x1, x2) = [1] x1 + [1] x2 + [1] 0.00/0.73 0.00/0.73 [0] = [7] 0.00/0.73 0.00/0.73 [nil] = [0] 0.00/0.73 0.00/0.73 [s](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [cons](x1, x2) = [1] x1 + [7] 0.00/0.73 0.00/0.73 [n__first](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.73 0.00/0.73 [activate](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [from](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [n__from](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [n__s](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 The order satisfies the following ordering constraints: 0.00/0.73 0.00/0.73 [first(X1, X2)] = [1] X1 + [1] X2 + [1] 0.00/0.73 > [1] X1 + [1] X2 + [0] 0.00/0.73 = [n__first(X1, X2)] 0.00/0.73 0.00/0.73 [first(0(), X)] = [1] X + [8] 0.00/0.73 > [0] 0.00/0.73 = [nil()] 0.00/0.73 0.00/0.73 [s(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [n__s(X)] 0.00/0.73 0.00/0.73 [activate(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [X] 0.00/0.73 0.00/0.73 [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [0] 0.00/0.73 ? [1] X1 + [1] X2 + [1] 0.00/0.73 = [first(activate(X1), activate(X2))] 0.00/0.73 0.00/0.73 [activate(n__from(X))] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [from(activate(X))] 0.00/0.73 0.00/0.73 [activate(n__s(X))] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [s(activate(X))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [0] 0.00/0.73 ? [1] X + [7] 0.00/0.73 = [cons(X, n__from(n__s(X)))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [n__from(X)] 0.00/0.73 0.00/0.73 0.00/0.73 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.73 0.00/0.73 We are left with following problem, upon which TcT provides the 0.00/0.73 certificate YES(O(1),O(n^1)). 0.00/0.73 0.00/0.73 Strict Trs: 0.00/0.73 { s(X) -> n__s(X) 0.00/0.73 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__from(X)) -> from(activate(X)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) 0.00/0.73 , from(X) -> cons(X, n__from(n__s(X))) 0.00/0.73 , from(X) -> n__from(X) } 0.00/0.73 Weak Trs: 0.00/0.73 { first(X1, X2) -> n__first(X1, X2) 0.00/0.73 , first(0(), X) -> nil() 0.00/0.73 , activate(X) -> X } 0.00/0.73 Obligation: 0.00/0.73 innermost runtime complexity 0.00/0.73 Answer: 0.00/0.73 YES(O(1),O(n^1)) 0.00/0.73 0.00/0.73 The weightgap principle applies (using the following nonconstant 0.00/0.73 growth matrix-interpretation) 0.00/0.73 0.00/0.73 The following argument positions are usable: 0.00/0.73 Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} 0.00/0.73 0.00/0.73 TcT has computed the following matrix interpretation satisfying 0.00/0.73 not(EDA) and not(IDA(1)). 0.00/0.73 0.00/0.73 [first](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.73 0.00/0.73 [0] = [3] 0.00/0.73 0.00/0.73 [nil] = [7] 0.00/0.73 0.00/0.73 [s](x1) = [1] x1 + [1] 0.00/0.73 0.00/0.73 [cons](x1, x2) = [1] x1 + [7] 0.00/0.73 0.00/0.73 [n__first](x1, x2) = [1] x1 + [1] x2 + [1] 0.00/0.73 0.00/0.73 [activate](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [from](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [n__from](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [n__s](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 The order satisfies the following ordering constraints: 0.00/0.73 0.00/0.73 [first(X1, X2)] = [1] X1 + [1] X2 + [4] 0.00/0.73 > [1] X1 + [1] X2 + [1] 0.00/0.73 = [n__first(X1, X2)] 0.00/0.73 0.00/0.73 [first(0(), X)] = [1] X + [7] 0.00/0.73 >= [7] 0.00/0.73 = [nil()] 0.00/0.73 0.00/0.73 [s(X)] = [1] X + [1] 0.00/0.73 > [1] X + [0] 0.00/0.73 = [n__s(X)] 0.00/0.73 0.00/0.73 [activate(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [X] 0.00/0.73 0.00/0.73 [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [1] 0.00/0.73 ? [1] X1 + [1] X2 + [4] 0.00/0.73 = [first(activate(X1), activate(X2))] 0.00/0.73 0.00/0.73 [activate(n__from(X))] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [from(activate(X))] 0.00/0.73 0.00/0.73 [activate(n__s(X))] = [1] X + [0] 0.00/0.73 ? [1] X + [1] 0.00/0.73 = [s(activate(X))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [0] 0.00/0.73 ? [1] X + [7] 0.00/0.73 = [cons(X, n__from(n__s(X)))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [n__from(X)] 0.00/0.73 0.00/0.73 0.00/0.73 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.73 0.00/0.73 We are left with following problem, upon which TcT provides the 0.00/0.73 certificate YES(O(1),O(n^1)). 0.00/0.73 0.00/0.73 Strict Trs: 0.00/0.73 { activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__from(X)) -> from(activate(X)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) 0.00/0.73 , from(X) -> cons(X, n__from(n__s(X))) 0.00/0.73 , from(X) -> n__from(X) } 0.00/0.73 Weak Trs: 0.00/0.73 { first(X1, X2) -> n__first(X1, X2) 0.00/0.73 , first(0(), X) -> nil() 0.00/0.73 , s(X) -> n__s(X) 0.00/0.73 , activate(X) -> X } 0.00/0.73 Obligation: 0.00/0.73 innermost runtime complexity 0.00/0.73 Answer: 0.00/0.73 YES(O(1),O(n^1)) 0.00/0.73 0.00/0.73 The weightgap principle applies (using the following nonconstant 0.00/0.73 growth matrix-interpretation) 0.00/0.73 0.00/0.73 The following argument positions are usable: 0.00/0.73 Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} 0.00/0.73 0.00/0.73 TcT has computed the following matrix interpretation satisfying 0.00/0.73 not(EDA) and not(IDA(1)). 0.00/0.73 0.00/0.73 [first](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.73 0.00/0.73 [0] = [7] 0.00/0.73 0.00/0.73 [nil] = [7] 0.00/0.73 0.00/0.73 [s](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [cons](x1, x2) = [1] x1 + [7] 0.00/0.73 0.00/0.73 [n__first](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.73 0.00/0.73 [activate](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [from](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [n__from](x1) = [1] x1 + [1] 0.00/0.73 0.00/0.73 [n__s](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 The order satisfies the following ordering constraints: 0.00/0.73 0.00/0.73 [first(X1, X2)] = [1] X1 + [1] X2 + [0] 0.00/0.73 >= [1] X1 + [1] X2 + [0] 0.00/0.73 = [n__first(X1, X2)] 0.00/0.73 0.00/0.73 [first(0(), X)] = [1] X + [7] 0.00/0.73 >= [7] 0.00/0.73 = [nil()] 0.00/0.73 0.00/0.73 [s(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [n__s(X)] 0.00/0.73 0.00/0.73 [activate(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [X] 0.00/0.73 0.00/0.73 [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [0] 0.00/0.73 >= [1] X1 + [1] X2 + [0] 0.00/0.73 = [first(activate(X1), activate(X2))] 0.00/0.73 0.00/0.73 [activate(n__from(X))] = [1] X + [1] 0.00/0.73 > [1] X + [0] 0.00/0.73 = [from(activate(X))] 0.00/0.73 0.00/0.73 [activate(n__s(X))] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [s(activate(X))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [0] 0.00/0.73 ? [1] X + [7] 0.00/0.73 = [cons(X, n__from(n__s(X)))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [0] 0.00/0.73 ? [1] X + [1] 0.00/0.73 = [n__from(X)] 0.00/0.73 0.00/0.73 0.00/0.73 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.73 0.00/0.73 We are left with following problem, upon which TcT provides the 0.00/0.73 certificate YES(O(1),O(n^1)). 0.00/0.73 0.00/0.73 Strict Trs: 0.00/0.73 { activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) 0.00/0.73 , from(X) -> cons(X, n__from(n__s(X))) 0.00/0.73 , from(X) -> n__from(X) } 0.00/0.73 Weak Trs: 0.00/0.73 { first(X1, X2) -> n__first(X1, X2) 0.00/0.73 , first(0(), X) -> nil() 0.00/0.73 , s(X) -> n__s(X) 0.00/0.73 , activate(X) -> X 0.00/0.73 , activate(n__from(X)) -> from(activate(X)) } 0.00/0.73 Obligation: 0.00/0.73 innermost runtime complexity 0.00/0.73 Answer: 0.00/0.73 YES(O(1),O(n^1)) 0.00/0.73 0.00/0.73 The weightgap principle applies (using the following nonconstant 0.00/0.73 growth matrix-interpretation) 0.00/0.73 0.00/0.73 The following argument positions are usable: 0.00/0.73 Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} 0.00/0.73 0.00/0.73 TcT has computed the following matrix interpretation satisfying 0.00/0.73 not(EDA) and not(IDA(1)). 0.00/0.73 0.00/0.73 [first](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.73 0.00/0.73 [0] = [7] 0.00/0.73 0.00/0.73 [nil] = [7] 0.00/0.73 0.00/0.73 [s](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [cons](x1, x2) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [n__first](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.73 0.00/0.73 [activate](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [from](x1) = [1] x1 + [1] 0.00/0.73 0.00/0.73 [n__from](x1) = [1] x1 + [1] 0.00/0.73 0.00/0.73 [n__s](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 The order satisfies the following ordering constraints: 0.00/0.73 0.00/0.73 [first(X1, X2)] = [1] X1 + [1] X2 + [0] 0.00/0.73 >= [1] X1 + [1] X2 + [0] 0.00/0.73 = [n__first(X1, X2)] 0.00/0.73 0.00/0.73 [first(0(), X)] = [1] X + [7] 0.00/0.73 >= [7] 0.00/0.73 = [nil()] 0.00/0.73 0.00/0.73 [s(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [n__s(X)] 0.00/0.73 0.00/0.73 [activate(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [X] 0.00/0.73 0.00/0.73 [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [0] 0.00/0.73 >= [1] X1 + [1] X2 + [0] 0.00/0.73 = [first(activate(X1), activate(X2))] 0.00/0.73 0.00/0.73 [activate(n__from(X))] = [1] X + [1] 0.00/0.73 >= [1] X + [1] 0.00/0.73 = [from(activate(X))] 0.00/0.73 0.00/0.73 [activate(n__s(X))] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [s(activate(X))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [1] 0.00/0.73 > [1] X + [0] 0.00/0.73 = [cons(X, n__from(n__s(X)))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [1] 0.00/0.73 >= [1] X + [1] 0.00/0.73 = [n__from(X)] 0.00/0.73 0.00/0.73 0.00/0.73 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.73 0.00/0.73 We are left with following problem, upon which TcT provides the 0.00/0.73 certificate YES(O(1),O(n^1)). 0.00/0.73 0.00/0.73 Strict Trs: 0.00/0.73 { activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) 0.00/0.73 , from(X) -> n__from(X) } 0.00/0.73 Weak Trs: 0.00/0.73 { first(X1, X2) -> n__first(X1, X2) 0.00/0.73 , first(0(), X) -> nil() 0.00/0.73 , s(X) -> n__s(X) 0.00/0.73 , activate(X) -> X 0.00/0.73 , activate(n__from(X)) -> from(activate(X)) 0.00/0.73 , from(X) -> cons(X, n__from(n__s(X))) } 0.00/0.73 Obligation: 0.00/0.73 innermost runtime complexity 0.00/0.73 Answer: 0.00/0.73 YES(O(1),O(n^1)) 0.00/0.73 0.00/0.73 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.73 orient following rules strictly. 0.00/0.73 0.00/0.73 Trs: 0.00/0.73 { activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) } 0.00/0.73 0.00/0.73 The induced complexity on above rules (modulo remaining rules) is 0.00/0.73 YES(?,O(n^1)) . These rules are moved into the corresponding weak 0.00/0.73 component(s). 0.00/0.73 0.00/0.73 Sub-proof: 0.00/0.73 ---------- 0.00/0.73 The following argument positions are usable: 0.00/0.73 Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} 0.00/0.73 0.00/0.73 TcT has computed the following constructor-based matrix 0.00/0.73 interpretation satisfying not(EDA). 0.00/0.73 0.00/0.73 [first](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.73 0.00/0.73 [0] = [1] 0.00/0.73 0.00/0.73 [nil] = [3] 0.00/0.73 0.00/0.73 [s](x1) = [1] x1 + [4] 0.00/0.73 0.00/0.73 [cons](x1, x2) = [1] x1 + [2] 0.00/0.73 0.00/0.73 [n__first](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.73 0.00/0.73 [activate](x1) = [2] x1 + [0] 0.00/0.73 0.00/0.73 [from](x1) = [1] x1 + [2] 0.00/0.73 0.00/0.73 [n__from](x1) = [1] x1 + [2] 0.00/0.73 0.00/0.73 [n__s](x1) = [1] x1 + [4] 0.00/0.73 0.00/0.73 The order satisfies the following ordering constraints: 0.00/0.73 0.00/0.73 [first(X1, X2)] = [1] X1 + [1] X2 + [4] 0.00/0.73 >= [1] X1 + [1] X2 + [4] 0.00/0.73 = [n__first(X1, X2)] 0.00/0.73 0.00/0.73 [first(0(), X)] = [1] X + [5] 0.00/0.73 > [3] 0.00/0.73 = [nil()] 0.00/0.73 0.00/0.73 [s(X)] = [1] X + [4] 0.00/0.73 >= [1] X + [4] 0.00/0.73 = [n__s(X)] 0.00/0.73 0.00/0.73 [activate(X)] = [2] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [X] 0.00/0.73 0.00/0.73 [activate(n__first(X1, X2))] = [2] X1 + [2] X2 + [8] 0.00/0.73 > [2] X1 + [2] X2 + [4] 0.00/0.73 = [first(activate(X1), activate(X2))] 0.00/0.73 0.00/0.73 [activate(n__from(X))] = [2] X + [4] 0.00/0.73 > [2] X + [2] 0.00/0.73 = [from(activate(X))] 0.00/0.73 0.00/0.73 [activate(n__s(X))] = [2] X + [8] 0.00/0.73 > [2] X + [4] 0.00/0.73 = [s(activate(X))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [2] 0.00/0.73 >= [1] X + [2] 0.00/0.73 = [cons(X, n__from(n__s(X)))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [2] 0.00/0.73 >= [1] X + [2] 0.00/0.73 = [n__from(X)] 0.00/0.73 0.00/0.73 0.00/0.73 We return to the main proof. 0.00/0.73 0.00/0.73 We are left with following problem, upon which TcT provides the 0.00/0.73 certificate YES(O(1),O(n^1)). 0.00/0.73 0.00/0.73 Strict Trs: { from(X) -> n__from(X) } 0.00/0.73 Weak Trs: 0.00/0.73 { first(X1, X2) -> n__first(X1, X2) 0.00/0.73 , first(0(), X) -> nil() 0.00/0.73 , s(X) -> n__s(X) 0.00/0.73 , activate(X) -> X 0.00/0.73 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__from(X)) -> from(activate(X)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) 0.00/0.73 , from(X) -> cons(X, n__from(n__s(X))) } 0.00/0.73 Obligation: 0.00/0.73 innermost runtime complexity 0.00/0.73 Answer: 0.00/0.73 YES(O(1),O(n^1)) 0.00/0.73 0.00/0.73 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.73 orient following rules strictly. 0.00/0.73 0.00/0.73 Trs: { from(X) -> n__from(X) } 0.00/0.73 0.00/0.73 The induced complexity on above rules (modulo remaining rules) is 0.00/0.73 YES(?,O(n^1)) . These rules are moved into the corresponding weak 0.00/0.73 component(s). 0.00/0.73 0.00/0.73 Sub-proof: 0.00/0.73 ---------- 0.00/0.73 The following argument positions are usable: 0.00/0.73 Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} 0.00/0.73 0.00/0.73 TcT has computed the following constructor-based matrix 0.00/0.73 interpretation satisfying not(EDA). 0.00/0.73 0.00/0.73 [first](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.73 0.00/0.73 [0] = [1] 0.00/0.73 0.00/0.73 [nil] = [1] 0.00/0.73 0.00/0.73 [s](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 [cons](x1, x2) = [1] x1 + [4] 0.00/0.73 0.00/0.73 [n__first](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.73 0.00/0.73 [activate](x1) = [4] x1 + [0] 0.00/0.73 0.00/0.73 [from](x1) = [1] x1 + [4] 0.00/0.73 0.00/0.73 [n__from](x1) = [1] x1 + [1] 0.00/0.73 0.00/0.73 [n__s](x1) = [1] x1 + [0] 0.00/0.73 0.00/0.73 The order satisfies the following ordering constraints: 0.00/0.73 0.00/0.73 [first(X1, X2)] = [1] X1 + [1] X2 + [0] 0.00/0.73 >= [1] X1 + [1] X2 + [0] 0.00/0.73 = [n__first(X1, X2)] 0.00/0.73 0.00/0.73 [first(0(), X)] = [1] X + [1] 0.00/0.73 >= [1] 0.00/0.73 = [nil()] 0.00/0.73 0.00/0.73 [s(X)] = [1] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [n__s(X)] 0.00/0.73 0.00/0.73 [activate(X)] = [4] X + [0] 0.00/0.73 >= [1] X + [0] 0.00/0.73 = [X] 0.00/0.73 0.00/0.73 [activate(n__first(X1, X2))] = [4] X1 + [4] X2 + [0] 0.00/0.73 >= [4] X1 + [4] X2 + [0] 0.00/0.73 = [first(activate(X1), activate(X2))] 0.00/0.73 0.00/0.73 [activate(n__from(X))] = [4] X + [4] 0.00/0.73 >= [4] X + [4] 0.00/0.73 = [from(activate(X))] 0.00/0.73 0.00/0.73 [activate(n__s(X))] = [4] X + [0] 0.00/0.73 >= [4] X + [0] 0.00/0.73 = [s(activate(X))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [4] 0.00/0.73 >= [1] X + [4] 0.00/0.73 = [cons(X, n__from(n__s(X)))] 0.00/0.73 0.00/0.73 [from(X)] = [1] X + [4] 0.00/0.73 > [1] X + [1] 0.00/0.73 = [n__from(X)] 0.00/0.73 0.00/0.73 0.00/0.73 We return to the main proof. 0.00/0.73 0.00/0.73 We are left with following problem, upon which TcT provides the 0.00/0.73 certificate YES(O(1),O(1)). 0.00/0.73 0.00/0.73 Weak Trs: 0.00/0.73 { first(X1, X2) -> n__first(X1, X2) 0.00/0.73 , first(0(), X) -> nil() 0.00/0.73 , s(X) -> n__s(X) 0.00/0.73 , activate(X) -> X 0.00/0.73 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.73 , activate(n__from(X)) -> from(activate(X)) 0.00/0.73 , activate(n__s(X)) -> s(activate(X)) 0.00/0.73 , from(X) -> cons(X, n__from(n__s(X))) 0.00/0.73 , from(X) -> n__from(X) } 0.00/0.73 Obligation: 0.00/0.73 innermost runtime complexity 0.00/0.73 Answer: 0.00/0.73 YES(O(1),O(1)) 0.00/0.73 0.00/0.73 Empty rules are trivially bounded 0.00/0.73 0.00/0.73 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.74 EOF