YES(O(1),O(n^1)) 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict Trs: 0.00/0.86 { 2nd(cons1(X, cons(Y, Z))) -> Y 0.00/0.86 , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1))) 0.00/0.86 , activate(X) -> X 0.00/0.86 , activate(n__from(X)) -> from(X) 0.00/0.86 , from(X) -> cons(X, n__from(s(X))) 0.00/0.86 , from(X) -> n__from(X) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 The weightgap principle applies (using the following nonconstant 0.00/0.86 growth matrix-interpretation) 0.00/0.86 0.00/0.86 The following argument positions are usable: 0.00/0.86 Uargs(2nd) = {1}, Uargs(cons1) = {2} 0.00/0.86 0.00/0.86 TcT has computed the following matrix interpretation satisfying 0.00/0.86 not(EDA) and not(IDA(1)). 0.00/0.86 0.00/0.86 [2nd](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [cons1](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.86 0.00/0.86 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.86 0.00/0.86 [activate](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [from](x1) = [1] x1 + [5] 0.00/0.86 0.00/0.86 [n__from](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [s](x1) = [0] 0.00/0.86 0.00/0.86 The order satisfies the following ordering constraints: 0.00/0.86 0.00/0.86 [2nd(cons1(X, cons(Y, Z)))] = [1] X + [1] Y + [1] Z + [0] 0.00/0.86 >= [1] Y + [0] 0.00/0.86 = [Y] 0.00/0.86 0.00/0.86 [2nd(cons(X, X1))] = [1] X + [1] X1 + [0] 0.00/0.86 >= [1] X + [1] X1 + [0] 0.00/0.86 = [2nd(cons1(X, activate(X1)))] 0.00/0.86 0.00/0.86 [activate(X)] = [1] X + [0] 0.00/0.86 >= [1] X + [0] 0.00/0.86 = [X] 0.00/0.86 0.00/0.86 [activate(n__from(X))] = [1] X + [0] 0.00/0.86 ? [1] X + [5] 0.00/0.86 = [from(X)] 0.00/0.86 0.00/0.86 [from(X)] = [1] X + [5] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [cons(X, n__from(s(X)))] 0.00/0.86 0.00/0.86 [from(X)] = [1] X + [5] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [n__from(X)] 0.00/0.86 0.00/0.86 0.00/0.86 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict Trs: 0.00/0.86 { 2nd(cons1(X, cons(Y, Z))) -> Y 0.00/0.86 , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1))) 0.00/0.86 , activate(X) -> X 0.00/0.86 , activate(n__from(X)) -> from(X) } 0.00/0.86 Weak Trs: 0.00/0.86 { from(X) -> cons(X, n__from(s(X))) 0.00/0.86 , from(X) -> n__from(X) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 The weightgap principle applies (using the following nonconstant 0.00/0.86 growth matrix-interpretation) 0.00/0.86 0.00/0.86 The following argument positions are usable: 0.00/0.86 Uargs(2nd) = {1}, Uargs(cons1) = {2} 0.00/0.86 0.00/0.86 TcT has computed the following matrix interpretation satisfying 0.00/0.86 not(EDA) and not(IDA(1)). 0.00/0.86 0.00/0.86 [2nd](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [cons1](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.86 0.00/0.86 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.86 0.00/0.86 [activate](x1) = [1] x1 + [4] 0.00/0.86 0.00/0.86 [from](x1) = [1] x1 + [4] 0.00/0.86 0.00/0.86 [n__from](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [s](x1) = [0] 0.00/0.86 0.00/0.86 The order satisfies the following ordering constraints: 0.00/0.86 0.00/0.86 [2nd(cons1(X, cons(Y, Z)))] = [1] X + [1] Y + [1] Z + [0] 0.00/0.86 >= [1] Y + [0] 0.00/0.86 = [Y] 0.00/0.86 0.00/0.86 [2nd(cons(X, X1))] = [1] X + [1] X1 + [0] 0.00/0.86 ? [1] X + [1] X1 + [4] 0.00/0.86 = [2nd(cons1(X, activate(X1)))] 0.00/0.86 0.00/0.86 [activate(X)] = [1] X + [4] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [X] 0.00/0.86 0.00/0.86 [activate(n__from(X))] = [1] X + [4] 0.00/0.86 >= [1] X + [4] 0.00/0.86 = [from(X)] 0.00/0.86 0.00/0.86 [from(X)] = [1] X + [4] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [cons(X, n__from(s(X)))] 0.00/0.86 0.00/0.86 [from(X)] = [1] X + [4] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [n__from(X)] 0.00/0.86 0.00/0.86 0.00/0.86 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict Trs: 0.00/0.86 { 2nd(cons1(X, cons(Y, Z))) -> Y 0.00/0.86 , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1))) 0.00/0.86 , activate(n__from(X)) -> from(X) } 0.00/0.86 Weak Trs: 0.00/0.86 { activate(X) -> X 0.00/0.86 , from(X) -> cons(X, n__from(s(X))) 0.00/0.86 , from(X) -> n__from(X) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 The weightgap principle applies (using the following nonconstant 0.00/0.86 growth matrix-interpretation) 0.00/0.86 0.00/0.86 The following argument positions are usable: 0.00/0.86 Uargs(2nd) = {1}, Uargs(cons1) = {2} 0.00/0.86 0.00/0.86 TcT has computed the following matrix interpretation satisfying 0.00/0.86 not(EDA) and not(IDA(1)). 0.00/0.86 0.00/0.86 [2nd](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [cons1](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.86 0.00/0.86 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.86 0.00/0.86 [activate](x1) = [1] x1 + [1] 0.00/0.86 0.00/0.86 [from](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [n__from](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [s](x1) = [0] 0.00/0.86 0.00/0.86 The order satisfies the following ordering constraints: 0.00/0.86 0.00/0.86 [2nd(cons1(X, cons(Y, Z)))] = [1] X + [1] Y + [1] Z + [0] 0.00/0.86 >= [1] Y + [0] 0.00/0.86 = [Y] 0.00/0.86 0.00/0.86 [2nd(cons(X, X1))] = [1] X + [1] X1 + [0] 0.00/0.86 ? [1] X + [1] X1 + [1] 0.00/0.86 = [2nd(cons1(X, activate(X1)))] 0.00/0.86 0.00/0.86 [activate(X)] = [1] X + [1] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [X] 0.00/0.86 0.00/0.86 [activate(n__from(X))] = [1] X + [1] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [from(X)] 0.00/0.86 0.00/0.86 [from(X)] = [1] X + [0] 0.00/0.86 >= [1] X + [0] 0.00/0.86 = [cons(X, n__from(s(X)))] 0.00/0.86 0.00/0.86 [from(X)] = [1] X + [0] 0.00/0.86 >= [1] X + [0] 0.00/0.86 = [n__from(X)] 0.00/0.86 0.00/0.86 0.00/0.86 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict Trs: 0.00/0.86 { 2nd(cons1(X, cons(Y, Z))) -> Y 0.00/0.86 , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1))) } 0.00/0.86 Weak Trs: 0.00/0.86 { activate(X) -> X 0.00/0.86 , activate(n__from(X)) -> from(X) 0.00/0.86 , from(X) -> cons(X, n__from(s(X))) 0.00/0.86 , from(X) -> n__from(X) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 The weightgap principle applies (using the following nonconstant 0.00/0.86 growth matrix-interpretation) 0.00/0.86 0.00/0.86 The following argument positions are usable: 0.00/0.86 Uargs(2nd) = {1}, Uargs(cons1) = {2} 0.00/0.86 0.00/0.86 TcT has computed the following matrix interpretation satisfying 0.00/0.86 not(EDA) and not(IDA(1)). 0.00/0.86 0.00/0.86 [2nd](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [cons1](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.86 0.00/0.86 [cons](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.86 0.00/0.86 [activate](x1) = [1] x1 + [4] 0.00/0.86 0.00/0.86 [from](x1) = [1] x1 + [4] 0.00/0.86 0.00/0.86 [n__from](x1) = [1] x1 + [0] 0.00/0.86 0.00/0.86 [s](x1) = [0] 0.00/0.86 0.00/0.86 The order satisfies the following ordering constraints: 0.00/0.86 0.00/0.86 [2nd(cons1(X, cons(Y, Z)))] = [1] X + [1] Y + [1] Z + [8] 0.00/0.86 > [1] Y + [0] 0.00/0.86 = [Y] 0.00/0.86 0.00/0.86 [2nd(cons(X, X1))] = [1] X + [1] X1 + [4] 0.00/0.86 ? [1] X + [1] X1 + [8] 0.00/0.86 = [2nd(cons1(X, activate(X1)))] 0.00/0.86 0.00/0.86 [activate(X)] = [1] X + [4] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [X] 0.00/0.86 0.00/0.86 [activate(n__from(X))] = [1] X + [4] 0.00/0.86 >= [1] X + [4] 0.00/0.86 = [from(X)] 0.00/0.86 0.00/0.86 [from(X)] = [1] X + [4] 0.00/0.86 >= [1] X + [4] 0.00/0.86 = [cons(X, n__from(s(X)))] 0.00/0.86 0.00/0.86 [from(X)] = [1] X + [4] 0.00/0.86 > [1] X + [0] 0.00/0.86 = [n__from(X)] 0.00/0.86 0.00/0.86 0.00/0.86 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(n^1)). 0.00/0.86 0.00/0.86 Strict Trs: { 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1))) } 0.00/0.86 Weak Trs: 0.00/0.86 { 2nd(cons1(X, cons(Y, Z))) -> Y 0.00/0.86 , activate(X) -> X 0.00/0.86 , activate(n__from(X)) -> from(X) 0.00/0.86 , from(X) -> cons(X, n__from(s(X))) 0.00/0.86 , from(X) -> n__from(X) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(n^1)) 0.00/0.86 0.00/0.86 We use the processor 'matrix interpretation of dimension 2' to 0.00/0.86 orient following rules strictly. 0.00/0.86 0.00/0.86 Trs: { 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1))) } 0.00/0.86 0.00/0.86 The induced complexity on above rules (modulo remaining rules) is 0.00/0.86 YES(?,O(n^1)) . These rules are moved into the corresponding weak 0.00/0.86 component(s). 0.00/0.86 0.00/0.86 Sub-proof: 0.00/0.86 ---------- 0.00/0.86 The following argument positions are usable: 0.00/0.86 Uargs(2nd) = {1}, Uargs(cons1) = {2} 0.00/0.86 0.00/0.86 TcT has computed the following constructor-based matrix 0.00/0.86 interpretation satisfying not(EDA) and not(IDA(1)). 0.00/0.86 0.00/0.86 [2nd](x1) = [2 1] x1 + [0] 0.00/0.86 [1 0] [0] 0.00/0.86 0.00/0.86 [cons1](x1, x2) = [1 0] x2 + [0] 0.00/0.86 [0 0] [1] 0.00/0.86 0.00/0.86 [cons](x1, x2) = [1 1] x1 + [1 0] x2 + [0] 0.00/0.86 [0 0] [0 0] [4] 0.00/0.86 0.00/0.86 [activate](x1) = [1 0] x1 + [0] 0.00/0.86 [0 1] [4] 0.00/0.86 0.00/0.86 [from](x1) = [1 1] x1 + [0] 0.00/0.86 [0 0] [4] 0.00/0.86 0.00/0.86 [n__from](x1) = [1 1] x1 + [0] 0.00/0.86 [0 0] [0] 0.00/0.86 0.00/0.86 [s](x1) = [0] 0.00/0.86 [0] 0.00/0.86 0.00/0.86 The order satisfies the following ordering constraints: 0.00/0.86 0.00/0.86 [2nd(cons1(X, cons(Y, Z)))] = [2 2] Y + [2 0] Z + [1] 0.00/0.86 [1 1] [1 0] [0] 0.00/0.86 > [1 0] Y + [0] 0.00/0.86 [0 1] [0] 0.00/0.86 = [Y] 0.00/0.86 0.00/0.86 [2nd(cons(X, X1))] = [2 2] X + [2 0] X1 + [4] 0.00/0.86 [1 1] [1 0] [0] 0.00/0.86 > [2 0] X1 + [1] 0.00/0.86 [1 0] [0] 0.00/0.86 = [2nd(cons1(X, activate(X1)))] 0.00/0.86 0.00/0.86 [activate(X)] = [1 0] X + [0] 0.00/0.86 [0 1] [4] 0.00/0.86 >= [1 0] X + [0] 0.00/0.86 [0 1] [0] 0.00/0.86 = [X] 0.00/0.86 0.00/0.86 [activate(n__from(X))] = [1 1] X + [0] 0.00/0.86 [0 0] [4] 0.00/0.86 >= [1 1] X + [0] 0.00/0.86 [0 0] [4] 0.00/0.86 = [from(X)] 0.00/0.86 0.00/0.86 [from(X)] = [1 1] X + [0] 0.00/0.86 [0 0] [4] 0.00/0.86 >= [1 1] X + [0] 0.00/0.86 [0 0] [4] 0.00/0.86 = [cons(X, n__from(s(X)))] 0.00/0.86 0.00/0.86 [from(X)] = [1 1] X + [0] 0.00/0.86 [0 0] [4] 0.00/0.86 >= [1 1] X + [0] 0.00/0.86 [0 0] [0] 0.00/0.86 = [n__from(X)] 0.00/0.86 0.00/0.86 0.00/0.86 We return to the main proof. 0.00/0.86 0.00/0.86 We are left with following problem, upon which TcT provides the 0.00/0.86 certificate YES(O(1),O(1)). 0.00/0.86 0.00/0.86 Weak Trs: 0.00/0.86 { 2nd(cons1(X, cons(Y, Z))) -> Y 0.00/0.86 , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1))) 0.00/0.86 , activate(X) -> X 0.00/0.86 , activate(n__from(X)) -> from(X) 0.00/0.86 , from(X) -> cons(X, n__from(s(X))) 0.00/0.86 , from(X) -> n__from(X) } 0.00/0.86 Obligation: 0.00/0.86 innermost runtime complexity 0.00/0.86 Answer: 0.00/0.86 YES(O(1),O(1)) 0.00/0.86 0.00/0.86 Empty rules are trivially bounded 0.00/0.87 0.00/0.87 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.87 EOF