YES(O(1),O(n^4)) 772.02/297.03 YES(O(1),O(n^4)) 772.02/297.03 772.02/297.03 We are left with following problem, upon which TcT provides the 772.02/297.03 certificate YES(O(1),O(n^4)). 772.02/297.03 772.02/297.03 Strict Trs: 772.02/297.03 { a__2nd(X) -> 2nd(X) 772.02/297.03 , a__2nd(cons(X, cons(Y, Z))) -> mark(Y) 772.02/297.03 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 772.02/297.03 , mark(from(X)) -> a__from(mark(X)) 772.02/297.03 , mark(s(X)) -> s(mark(X)) 772.02/297.03 , mark(2nd(X)) -> a__2nd(mark(X)) 772.02/297.03 , a__from(X) -> cons(mark(X), from(s(X))) 772.02/297.03 , a__from(X) -> from(X) } 772.02/297.03 Obligation: 772.02/297.03 innermost runtime complexity 772.02/297.03 Answer: 772.02/297.03 YES(O(1),O(n^4)) 772.02/297.03 772.02/297.03 The weightgap principle applies (using the following nonconstant 772.02/297.03 growth matrix-interpretation) 772.02/297.03 772.02/297.03 The following argument positions are usable: 772.02/297.03 Uargs(a__2nd) = {1}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 772.02/297.03 Uargs(s) = {1} 772.02/297.03 772.02/297.03 TcT has computed the following matrix interpretation satisfying 772.02/297.03 not(EDA) and not(IDA(1)). 772.02/297.03 772.02/297.03 [a__2nd](x1) = [1] x1 + [7] 772.02/297.03 772.02/297.03 [cons](x1, x2) = [1] x1 + [7] 772.02/297.03 772.02/297.03 [mark](x1) = [7] 772.02/297.03 772.02/297.03 [a__from](x1) = [1] x1 + [7] 772.02/297.03 772.02/297.03 [from](x1) = [1] x1 + [6] 772.02/297.03 772.02/297.03 [s](x1) = [1] x1 + [7] 772.02/297.03 772.02/297.03 [2nd](x1) = [1] x1 + [6] 772.02/297.03 772.02/297.03 The order satisfies the following ordering constraints: 772.02/297.03 772.02/297.03 [a__2nd(X)] = [1] X + [7] 772.02/297.03 > [1] X + [6] 772.02/297.03 = [2nd(X)] 772.02/297.03 772.02/297.03 [a__2nd(cons(X, cons(Y, Z)))] = [1] X + [14] 772.02/297.03 > [7] 772.02/297.03 = [mark(Y)] 772.02/297.03 772.02/297.03 [mark(cons(X1, X2))] = [7] 772.02/297.03 ? [14] 772.02/297.03 = [cons(mark(X1), X2)] 772.02/297.03 772.02/297.03 [mark(from(X))] = [7] 772.02/297.03 ? [14] 772.02/297.03 = [a__from(mark(X))] 772.02/297.03 772.02/297.03 [mark(s(X))] = [7] 772.02/297.03 ? [14] 772.02/297.03 = [s(mark(X))] 772.02/297.03 772.02/297.03 [mark(2nd(X))] = [7] 772.02/297.03 ? [14] 772.02/297.03 = [a__2nd(mark(X))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1] X + [7] 772.02/297.03 ? [14] 772.02/297.03 = [cons(mark(X), from(s(X)))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1] X + [7] 772.02/297.03 > [1] X + [6] 772.02/297.03 = [from(X)] 772.02/297.03 772.02/297.03 772.02/297.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 772.02/297.03 772.02/297.03 We are left with following problem, upon which TcT provides the 772.02/297.03 certificate YES(O(1),O(n^4)). 772.02/297.03 772.02/297.03 Strict Trs: 772.02/297.03 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 772.02/297.03 , mark(from(X)) -> a__from(mark(X)) 772.02/297.03 , mark(s(X)) -> s(mark(X)) 772.02/297.03 , mark(2nd(X)) -> a__2nd(mark(X)) 772.02/297.03 , a__from(X) -> cons(mark(X), from(s(X))) } 772.02/297.03 Weak Trs: 772.02/297.03 { a__2nd(X) -> 2nd(X) 772.02/297.03 , a__2nd(cons(X, cons(Y, Z))) -> mark(Y) 772.02/297.03 , a__from(X) -> from(X) } 772.02/297.03 Obligation: 772.02/297.03 innermost runtime complexity 772.02/297.03 Answer: 772.02/297.03 YES(O(1),O(n^4)) 772.02/297.03 772.02/297.03 The weightgap principle applies (using the following nonconstant 772.02/297.03 growth matrix-interpretation) 772.02/297.03 772.02/297.03 The following argument positions are usable: 772.02/297.03 Uargs(a__2nd) = {1}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 772.02/297.03 Uargs(s) = {1} 772.02/297.03 772.02/297.03 TcT has computed the following matrix interpretation satisfying 772.02/297.03 not(EDA) and not(IDA(1)). 772.02/297.03 772.02/297.03 [a__2nd](x1) = [1] x1 + [7] 772.02/297.03 772.02/297.03 [cons](x1, x2) = [1] x1 + [0] 772.02/297.03 772.02/297.03 [mark](x1) = [0] 772.02/297.03 772.02/297.03 [a__from](x1) = [1] x1 + [4] 772.02/297.03 772.02/297.03 [from](x1) = [1] x1 + [4] 772.02/297.03 772.02/297.03 [s](x1) = [1] x1 + [7] 772.02/297.03 772.02/297.03 [2nd](x1) = [1] x1 + [7] 772.02/297.03 772.02/297.03 The order satisfies the following ordering constraints: 772.02/297.03 772.02/297.03 [a__2nd(X)] = [1] X + [7] 772.02/297.03 >= [1] X + [7] 772.02/297.03 = [2nd(X)] 772.02/297.03 772.02/297.03 [a__2nd(cons(X, cons(Y, Z)))] = [1] X + [7] 772.02/297.03 > [0] 772.02/297.03 = [mark(Y)] 772.02/297.03 772.02/297.03 [mark(cons(X1, X2))] = [0] 772.02/297.03 >= [0] 772.02/297.03 = [cons(mark(X1), X2)] 772.02/297.03 772.02/297.03 [mark(from(X))] = [0] 772.02/297.03 ? [4] 772.02/297.03 = [a__from(mark(X))] 772.02/297.03 772.02/297.03 [mark(s(X))] = [0] 772.02/297.03 ? [7] 772.02/297.03 = [s(mark(X))] 772.02/297.03 772.02/297.03 [mark(2nd(X))] = [0] 772.02/297.03 ? [7] 772.02/297.03 = [a__2nd(mark(X))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1] X + [4] 772.02/297.03 > [0] 772.02/297.03 = [cons(mark(X), from(s(X)))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1] X + [4] 772.02/297.03 >= [1] X + [4] 772.02/297.03 = [from(X)] 772.02/297.03 772.02/297.03 772.02/297.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 772.02/297.03 772.02/297.03 We are left with following problem, upon which TcT provides the 772.02/297.03 certificate YES(O(1),O(n^4)). 772.02/297.03 772.02/297.03 Strict Trs: 772.02/297.03 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 772.02/297.03 , mark(from(X)) -> a__from(mark(X)) 772.02/297.03 , mark(s(X)) -> s(mark(X)) 772.02/297.03 , mark(2nd(X)) -> a__2nd(mark(X)) } 772.02/297.03 Weak Trs: 772.02/297.03 { a__2nd(X) -> 2nd(X) 772.02/297.03 , a__2nd(cons(X, cons(Y, Z))) -> mark(Y) 772.02/297.03 , a__from(X) -> cons(mark(X), from(s(X))) 772.02/297.03 , a__from(X) -> from(X) } 772.02/297.03 Obligation: 772.02/297.03 innermost runtime complexity 772.02/297.03 Answer: 772.02/297.03 YES(O(1),O(n^4)) 772.02/297.03 772.02/297.03 We use the processor 'matrix interpretation of dimension 4' to 772.02/297.03 orient following rules strictly. 772.02/297.03 772.02/297.03 Trs: 772.02/297.03 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 772.02/297.03 , mark(from(X)) -> a__from(mark(X)) } 772.02/297.03 772.02/297.03 The induced complexity on above rules (modulo remaining rules) is 772.02/297.03 YES(?,O(n^4)) . These rules are moved into the corresponding weak 772.02/297.03 component(s). 772.02/297.03 772.02/297.03 Sub-proof: 772.02/297.03 ---------- 772.02/297.03 The following argument positions are usable: 772.02/297.03 Uargs(a__2nd) = {1}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 772.02/297.03 Uargs(s) = {1} 772.02/297.03 772.02/297.03 TcT has computed the following constructor-based matrix 772.02/297.03 interpretation satisfying not(EDA). 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [a__2nd](x1) = [1 0 0 1] x1 + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 772.02/297.03 [1 0 0 0] [0 1 0 0] [0] 772.02/297.03 [cons](x1, x2) = [1 0 0 0] x1 + [0 0 0 0] x2 + [0] 772.02/297.03 [0 0 0 1] [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0 0 1 0] [1] 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [mark](x1) = [1 0 0 1] x1 + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [a__from](x1) = [1 0 0 1] x1 + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [from](x1) = [0 0 0 0] x1 + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 772.02/297.03 [1 0 0 0] [0] 772.02/297.03 [s](x1) = [0 0 0 0] x1 + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [2nd](x1) = [0 0 0 0] x1 + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 772.02/297.03 The order satisfies the following ordering constraints: 772.02/297.03 772.02/297.03 [a__2nd(X)] = [1 0 0 1] [0] 772.02/297.03 [1 0 0 1] X + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 >= [1 0 0 1] [0] 772.02/297.03 [0 0 0 0] X + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [2nd(X)] 772.02/297.03 772.02/297.03 [a__2nd(cons(X, cons(Y, Z)))] = [1 0 0 1] [1 0 0 1] [1] 772.02/297.03 [1 0 0 1] X + [1 0 0 1] Y + [1] 772.02/297.03 [0 0 0 1] [0 0 0 1] [1] 772.02/297.03 [0 0 0 1] [0 0 0 1] [1] 772.02/297.03 > [1 0 0 1] [0] 772.02/297.03 [1 0 0 1] Y + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [mark(Y)] 772.02/297.03 772.02/297.03 [mark(cons(X1, X2))] = [1 0 0 1] [0 1 1 0] [1] 772.02/297.03 [1 0 0 1] X1 + [0 1 1 0] X2 + [1] 772.02/297.03 [0 0 0 1] [0 0 1 0] [1] 772.02/297.03 [0 0 0 1] [0 0 1 0] [1] 772.02/297.03 > [1 0 0 1] [0 1 0 0] [0] 772.02/297.03 [1 0 0 1] X1 + [0 0 0 0] X2 + [0] 772.02/297.03 [0 0 0 1] [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0 0 1 0] [1] 772.02/297.03 = [cons(mark(X1), X2)] 772.02/297.03 772.02/297.03 [mark(from(X))] = [1 0 0 2] [1] 772.02/297.03 [1 0 0 2] X + [1] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 > [1 0 0 2] [0] 772.02/297.03 [1 0 0 2] X + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 = [a__from(mark(X))] 772.02/297.03 772.02/297.03 [mark(s(X))] = [1 0 0 1] [0] 772.02/297.03 [1 0 0 1] X + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 >= [1 0 0 1] [0] 772.02/297.03 [0 0 0 0] X + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [s(mark(X))] 772.02/297.03 772.02/297.03 [mark(2nd(X))] = [1 0 0 2] [0] 772.02/297.03 [1 0 0 2] X + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 >= [1 0 0 2] [0] 772.02/297.03 [1 0 0 2] X + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [a__2nd(mark(X))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1 0 0 1] [0] 772.02/297.03 [1 0 0 1] X + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 >= [1 0 0 1] [0] 772.02/297.03 [1 0 0 1] X + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 = [cons(mark(X), from(s(X)))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1 0 0 1] [0] 772.02/297.03 [1 0 0 1] X + [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 >= [1 0 0 1] [0] 772.02/297.03 [0 0 0 0] X + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 = [from(X)] 772.02/297.03 772.02/297.03 772.02/297.03 We return to the main proof. 772.02/297.03 772.02/297.03 We are left with following problem, upon which TcT provides the 772.02/297.03 certificate YES(O(1),O(n^4)). 772.02/297.03 772.02/297.03 Strict Trs: 772.02/297.03 { mark(s(X)) -> s(mark(X)) 772.02/297.03 , mark(2nd(X)) -> a__2nd(mark(X)) } 772.02/297.03 Weak Trs: 772.02/297.03 { a__2nd(X) -> 2nd(X) 772.02/297.03 , a__2nd(cons(X, cons(Y, Z))) -> mark(Y) 772.02/297.03 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 772.02/297.03 , mark(from(X)) -> a__from(mark(X)) 772.02/297.03 , a__from(X) -> cons(mark(X), from(s(X))) 772.02/297.03 , a__from(X) -> from(X) } 772.02/297.03 Obligation: 772.02/297.03 innermost runtime complexity 772.02/297.03 Answer: 772.02/297.03 YES(O(1),O(n^4)) 772.02/297.03 772.02/297.03 We use the processor 'matrix interpretation of dimension 4' to 772.02/297.03 orient following rules strictly. 772.02/297.03 772.02/297.03 Trs: { mark(s(X)) -> s(mark(X)) } 772.02/297.03 772.02/297.03 The induced complexity on above rules (modulo remaining rules) is 772.02/297.03 YES(?,O(n^4)) . These rules are moved into the corresponding weak 772.02/297.03 component(s). 772.02/297.03 772.02/297.03 Sub-proof: 772.02/297.03 ---------- 772.02/297.03 The following argument positions are usable: 772.02/297.03 Uargs(a__2nd) = {1}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 772.02/297.03 Uargs(s) = {1} 772.02/297.03 772.02/297.03 TcT has computed the following constructor-based matrix 772.02/297.03 interpretation satisfying not(EDA). 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [a__2nd](x1) = [0 0 0 1] x1 + [0] 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 772.02/297.03 [1 0 0 0] [0 0 1 0] [0] 772.02/297.03 [cons](x1, x2) = [0 0 0 1] x1 + [0 0 0 0] x2 + [0] 772.02/297.03 [1 0 0 0] [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0 1 0 0] [0] 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [mark](x1) = [0 0 0 1] x1 + [0] 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [a__from](x1) = [0 0 0 1] x1 + [0] 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [from](x1) = [0 0 0 0] x1 + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 772.02/297.03 [1 0 0 0] [0] 772.02/297.03 [s](x1) = [0 0 0 0] x1 + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [2nd](x1) = [0 0 0 0] x1 + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 772.02/297.03 The order satisfies the following ordering constraints: 772.02/297.03 772.02/297.03 [a__2nd(X)] = [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] X + [0] 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 >= [1 0 0 1] [0] 772.02/297.03 [0 0 0 0] X + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [2nd(X)] 772.02/297.03 772.02/297.03 [a__2nd(cons(X, cons(Y, Z)))] = [1 0 0 1] [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] X + [0 0 0 1] Y + [0] 772.02/297.03 [1 0 0 1] [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0 0 0 1] [0] 772.02/297.03 >= [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] Y + [0] 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [mark(Y)] 772.02/297.03 772.02/297.03 [mark(cons(X1, X2))] = [1 0 0 1] [0 1 1 0] [0] 772.02/297.03 [0 0 0 1] X1 + [0 1 0 0] X2 + [0] 772.02/297.03 [1 0 0 1] [0 1 1 0] [0] 772.02/297.03 [0 0 0 1] [0 1 0 0] [0] 772.02/297.03 >= [1 0 0 1] [0 0 1 0] [0] 772.02/297.03 [0 0 0 1] X1 + [0 0 0 0] X2 + [0] 772.02/297.03 [1 0 0 1] [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0 1 0 0] [0] 772.02/297.03 = [cons(mark(X1), X2)] 772.02/297.03 772.02/297.03 [mark(from(X))] = [1 0 0 2] [0] 772.02/297.03 [0 0 0 1] X + [0] 772.02/297.03 [1 0 0 2] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 >= [1 0 0 2] [0] 772.02/297.03 [0 0 0 1] X + [0] 772.02/297.03 [1 0 0 2] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [a__from(mark(X))] 772.02/297.03 772.02/297.03 [mark(s(X))] = [1 0 0 1] [1] 772.02/297.03 [0 0 0 1] X + [1] 772.02/297.03 [1 0 0 1] [1] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 > [1 0 0 1] [0] 772.02/297.03 [0 0 0 0] X + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [1] 772.02/297.03 = [s(mark(X))] 772.02/297.03 772.02/297.03 [mark(2nd(X))] = [1 0 0 2] [0] 772.02/297.03 [0 0 0 1] X + [0] 772.02/297.03 [1 0 0 2] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 >= [1 0 0 2] [0] 772.02/297.03 [0 0 0 1] X + [0] 772.02/297.03 [1 0 0 2] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [a__2nd(mark(X))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] X + [0] 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 >= [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] X + [0] 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [cons(mark(X), from(s(X)))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] X + [0] 772.02/297.03 [1 0 0 1] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 >= [1 0 0 1] [0] 772.02/297.03 [0 0 0 0] X + [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 [0 0 0 1] [0] 772.02/297.03 = [from(X)] 772.02/297.03 772.02/297.03 772.02/297.03 We return to the main proof. 772.02/297.03 772.02/297.03 We are left with following problem, upon which TcT provides the 772.02/297.03 certificate YES(O(1),O(n^4)). 772.02/297.03 772.02/297.03 Strict Trs: { mark(2nd(X)) -> a__2nd(mark(X)) } 772.02/297.03 Weak Trs: 772.02/297.03 { a__2nd(X) -> 2nd(X) 772.02/297.03 , a__2nd(cons(X, cons(Y, Z))) -> mark(Y) 772.02/297.03 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 772.02/297.03 , mark(from(X)) -> a__from(mark(X)) 772.02/297.03 , mark(s(X)) -> s(mark(X)) 772.02/297.03 , a__from(X) -> cons(mark(X), from(s(X))) 772.02/297.03 , a__from(X) -> from(X) } 772.02/297.03 Obligation: 772.02/297.03 innermost runtime complexity 772.02/297.03 Answer: 772.02/297.03 YES(O(1),O(n^4)) 772.02/297.03 772.02/297.03 We use the processor 'matrix interpretation of dimension 4' to 772.02/297.03 orient following rules strictly. 772.02/297.03 772.02/297.03 Trs: { mark(2nd(X)) -> a__2nd(mark(X)) } 772.02/297.03 772.02/297.03 The induced complexity on above rules (modulo remaining rules) is 772.02/297.03 YES(?,O(n^4)) . These rules are moved into the corresponding weak 772.02/297.03 component(s). 772.02/297.03 772.02/297.03 Sub-proof: 772.02/297.03 ---------- 772.02/297.03 The following argument positions are usable: 772.02/297.03 Uargs(a__2nd) = {1}, Uargs(cons) = {1}, Uargs(a__from) = {1}, 772.02/297.03 Uargs(s) = {1} 772.02/297.03 772.02/297.03 TcT has computed the following constructor-based matrix 772.02/297.03 interpretation satisfying not(EDA). 772.02/297.03 772.02/297.03 [1 0 0 0] [1] 772.02/297.03 [a__2nd](x1) = [1 0 0 0] x1 + [0] 772.02/297.03 [0 0 1 0] [1] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 772.02/297.03 [1 0 0 0] [0 1 0 1] [0] 772.02/297.03 [cons](x1, x2) = [1 0 0 0] x1 + [0 0 0 0] x2 + [0] 772.02/297.03 [0 0 1 0] [0 0 0 1] [0] 772.02/297.03 [0 0 1 0] [0 0 0 0] [0] 772.02/297.03 772.02/297.03 [1 0 1 0] [0] 772.02/297.03 [mark](x1) = [1 0 1 0] x1 + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 772.02/297.03 [1 0 1 0] [0] 772.02/297.03 [a__from](x1) = [1 0 1 0] x1 + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 772.02/297.03 [1 0 1 0] [0] 772.02/297.03 [from](x1) = [0 0 0 0] x1 + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 772.02/297.03 [1 0 0 0] [0] 772.02/297.03 [s](x1) = [0 0 0 0] x1 + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 772.02/297.03 [1 0 0 0] [1] 772.02/297.03 [2nd](x1) = [0 0 0 0] x1 + [0] 772.02/297.03 [0 0 1 0] [1] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 772.02/297.03 The order satisfies the following ordering constraints: 772.02/297.03 772.02/297.03 [a__2nd(X)] = [1 0 0 0] [1] 772.02/297.03 [1 0 0 0] X + [0] 772.02/297.03 [0 0 1 0] [1] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 >= [1 0 0 0] [1] 772.02/297.03 [0 0 0 0] X + [0] 772.02/297.03 [0 0 1 0] [1] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 = [2nd(X)] 772.02/297.03 772.02/297.03 [a__2nd(cons(X, cons(Y, Z)))] = [1 0 0 0] [1 0 1 0] [1] 772.02/297.03 [1 0 0 0] X + [1 0 1 0] Y + [0] 772.02/297.03 [0 0 1 0] [0 0 1 0] [1] 772.02/297.03 [0 0 1 0] [0 0 1 0] [0] 772.02/297.03 > [1 0 1 0] [0] 772.02/297.03 [1 0 1 0] Y + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 = [mark(Y)] 772.02/297.03 772.02/297.03 [mark(cons(X1, X2))] = [1 0 1 0] [0 1 0 2] [0] 772.02/297.03 [1 0 1 0] X1 + [0 1 0 2] X2 + [0] 772.02/297.03 [0 0 1 0] [0 0 0 1] [0] 772.02/297.03 [0 0 1 0] [0 0 0 1] [0] 772.02/297.03 >= [1 0 1 0] [0 1 0 1] [0] 772.02/297.03 [1 0 1 0] X1 + [0 0 0 0] X2 + [0] 772.02/297.03 [0 0 1 0] [0 0 0 1] [0] 772.02/297.03 [0 0 1 0] [0 0 0 0] [0] 772.02/297.03 = [cons(mark(X1), X2)] 772.02/297.03 772.02/297.03 [mark(from(X))] = [1 0 2 0] [0] 772.02/297.03 [1 0 2 0] X + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 >= [1 0 2 0] [0] 772.02/297.03 [1 0 2 0] X + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 = [a__from(mark(X))] 772.02/297.03 772.02/297.03 [mark(s(X))] = [1 0 1 0] [0] 772.02/297.03 [1 0 1 0] X + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 >= [1 0 1 0] [0] 772.02/297.03 [0 0 0 0] X + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 = [s(mark(X))] 772.02/297.03 772.02/297.03 [mark(2nd(X))] = [1 0 1 0] [2] 772.02/297.03 [1 0 1 0] X + [2] 772.02/297.03 [0 0 1 0] [1] 772.02/297.03 [0 0 1 0] [1] 772.02/297.03 > [1 0 1 0] [1] 772.02/297.03 [1 0 1 0] X + [0] 772.02/297.03 [0 0 1 0] [1] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 = [a__2nd(mark(X))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1 0 1 0] [0] 772.02/297.03 [1 0 1 0] X + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 >= [1 0 1 0] [0] 772.02/297.03 [1 0 1 0] X + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 = [cons(mark(X), from(s(X)))] 772.02/297.03 772.02/297.03 [a__from(X)] = [1 0 1 0] [0] 772.02/297.03 [1 0 1 0] X + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 >= [1 0 1 0] [0] 772.02/297.03 [0 0 0 0] X + [0] 772.02/297.03 [0 0 1 0] [0] 772.02/297.03 [0 0 0 0] [0] 772.02/297.03 = [from(X)] 772.02/297.03 772.02/297.03 772.02/297.03 We return to the main proof. 772.02/297.03 772.02/297.03 We are left with following problem, upon which TcT provides the 772.02/297.03 certificate YES(O(1),O(1)). 772.02/297.03 772.02/297.03 Weak Trs: 772.02/297.03 { a__2nd(X) -> 2nd(X) 772.02/297.03 , a__2nd(cons(X, cons(Y, Z))) -> mark(Y) 772.02/297.03 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 772.02/297.03 , mark(from(X)) -> a__from(mark(X)) 772.02/297.03 , mark(s(X)) -> s(mark(X)) 772.02/297.03 , mark(2nd(X)) -> a__2nd(mark(X)) 772.02/297.03 , a__from(X) -> cons(mark(X), from(s(X))) 772.02/297.03 , a__from(X) -> from(X) } 772.02/297.03 Obligation: 772.02/297.03 innermost runtime complexity 772.02/297.03 Answer: 772.02/297.03 YES(O(1),O(1)) 772.02/297.03 772.02/297.03 Empty rules are trivially bounded 772.02/297.03 772.02/297.03 Hurray, we answered YES(O(1),O(n^4)) 772.15/297.18 EOF