YES(O(1),O(n^1)) 91.34/77.16 YES(O(1),O(n^1)) 91.34/77.16 91.34/77.16 We are left with following problem, upon which TcT provides the 91.34/77.16 certificate YES(O(1),O(n^1)). 91.34/77.16 91.34/77.16 Strict Trs: 91.34/77.16 { half(0()) -> 0() 91.34/77.16 , half(s(0())) -> 0() 91.34/77.16 , half(s(s(x))) -> s(half(x)) 91.34/77.16 , lastbit(0()) -> 0() 91.34/77.16 , lastbit(s(0())) -> s(0()) 91.34/77.16 , lastbit(s(s(x))) -> lastbit(x) 91.34/77.16 , conv(0()) -> cons(nil(), 0()) 91.34/77.16 , conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) } 91.34/77.16 Obligation: 91.34/77.16 innermost runtime complexity 91.34/77.16 Answer: 91.34/77.16 YES(O(1),O(n^1)) 91.34/77.16 91.34/77.16 The weightgap principle applies (using the following nonconstant 91.34/77.16 growth matrix-interpretation) 91.34/77.16 91.34/77.16 The following argument positions are usable: 91.34/77.16 Uargs(s) = {1}, Uargs(conv) = {1}, Uargs(cons) = {1, 2} 91.34/77.16 91.34/77.16 TcT has computed the following matrix interpretation satisfying 91.34/77.16 not(EDA) and not(IDA(1)). 91.34/77.16 91.34/77.16 [half](x1) = [0] 91.34/77.16 91.34/77.16 [0] = [0] 91.34/77.16 91.34/77.16 [s](x1) = [1] x1 + [0] 91.34/77.16 91.34/77.16 [lastbit](x1) = [4] 91.34/77.16 91.34/77.16 [conv](x1) = [1] x1 + [0] 91.34/77.16 91.34/77.16 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 91.34/77.16 91.34/77.16 [nil] = [0] 91.34/77.16 91.34/77.16 The order satisfies the following ordering constraints: 91.34/77.16 91.34/77.16 [half(0())] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [half(s(0()))] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [half(s(s(x)))] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [s(half(x))] 91.34/77.16 91.34/77.16 [lastbit(0())] = [4] 91.34/77.16 > [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [lastbit(s(0()))] = [4] 91.34/77.16 > [0] 91.34/77.16 = [s(0())] 91.34/77.16 91.34/77.16 [lastbit(s(s(x)))] = [4] 91.34/77.16 >= [4] 91.34/77.16 = [lastbit(x)] 91.34/77.16 91.34/77.16 [conv(0())] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [cons(nil(), 0())] 91.34/77.16 91.34/77.16 [conv(s(x))] = [1] x + [0] 91.34/77.16 ? [4] 91.34/77.16 = [cons(conv(half(s(x))), lastbit(s(x)))] 91.34/77.16 91.34/77.16 91.34/77.16 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 91.34/77.16 91.34/77.16 We are left with following problem, upon which TcT provides the 91.34/77.16 certificate YES(O(1),O(n^1)). 91.34/77.16 91.34/77.16 Strict Trs: 91.34/77.16 { half(0()) -> 0() 91.34/77.16 , half(s(0())) -> 0() 91.34/77.16 , half(s(s(x))) -> s(half(x)) 91.34/77.16 , lastbit(s(s(x))) -> lastbit(x) 91.34/77.16 , conv(0()) -> cons(nil(), 0()) 91.34/77.16 , conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) } 91.34/77.16 Weak Trs: 91.34/77.16 { lastbit(0()) -> 0() 91.34/77.16 , lastbit(s(0())) -> s(0()) } 91.34/77.16 Obligation: 91.34/77.16 innermost runtime complexity 91.34/77.16 Answer: 91.34/77.16 YES(O(1),O(n^1)) 91.34/77.16 91.34/77.16 The weightgap principle applies (using the following nonconstant 91.34/77.16 growth matrix-interpretation) 91.34/77.16 91.34/77.16 The following argument positions are usable: 91.34/77.16 Uargs(s) = {1}, Uargs(conv) = {1}, Uargs(cons) = {1, 2} 91.34/77.16 91.34/77.16 TcT has computed the following matrix interpretation satisfying 91.34/77.16 not(EDA) and not(IDA(1)). 91.34/77.16 91.34/77.16 [half](x1) = [0] 91.34/77.16 91.34/77.16 [0] = [0] 91.34/77.16 91.34/77.16 [s](x1) = [1] x1 + [1] 91.34/77.16 91.34/77.16 [lastbit](x1) = [1] x1 + [0] 91.34/77.16 91.34/77.16 [conv](x1) = [1] x1 + [0] 91.34/77.16 91.34/77.16 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 91.34/77.16 91.34/77.16 [nil] = [0] 91.34/77.16 91.34/77.16 The order satisfies the following ordering constraints: 91.34/77.16 91.34/77.16 [half(0())] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [half(s(0()))] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [half(s(s(x)))] = [0] 91.34/77.16 ? [1] 91.34/77.16 = [s(half(x))] 91.34/77.16 91.34/77.16 [lastbit(0())] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [lastbit(s(0()))] = [1] 91.34/77.16 >= [1] 91.34/77.16 = [s(0())] 91.34/77.16 91.34/77.16 [lastbit(s(s(x)))] = [1] x + [2] 91.34/77.16 > [1] x + [0] 91.34/77.16 = [lastbit(x)] 91.34/77.16 91.34/77.16 [conv(0())] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [cons(nil(), 0())] 91.34/77.16 91.34/77.16 [conv(s(x))] = [1] x + [1] 91.34/77.16 >= [1] x + [1] 91.34/77.16 = [cons(conv(half(s(x))), lastbit(s(x)))] 91.34/77.16 91.34/77.16 91.34/77.16 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 91.34/77.16 91.34/77.16 We are left with following problem, upon which TcT provides the 91.34/77.16 certificate YES(O(1),O(n^1)). 91.34/77.16 91.34/77.16 Strict Trs: 91.34/77.16 { half(0()) -> 0() 91.34/77.16 , half(s(0())) -> 0() 91.34/77.16 , half(s(s(x))) -> s(half(x)) 91.34/77.16 , conv(0()) -> cons(nil(), 0()) 91.34/77.16 , conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) } 91.34/77.16 Weak Trs: 91.34/77.16 { lastbit(0()) -> 0() 91.34/77.16 , lastbit(s(0())) -> s(0()) 91.34/77.16 , lastbit(s(s(x))) -> lastbit(x) } 91.34/77.16 Obligation: 91.34/77.16 innermost runtime complexity 91.34/77.16 Answer: 91.34/77.16 YES(O(1),O(n^1)) 91.34/77.16 91.34/77.16 The weightgap principle applies (using the following nonconstant 91.34/77.16 growth matrix-interpretation) 91.34/77.16 91.34/77.16 The following argument positions are usable: 91.34/77.16 Uargs(s) = {1}, Uargs(conv) = {1}, Uargs(cons) = {1, 2} 91.34/77.16 91.34/77.16 TcT has computed the following matrix interpretation satisfying 91.34/77.16 not(EDA) and not(IDA(1)). 91.34/77.16 91.34/77.16 [half](x1) = [1] 91.34/77.16 91.34/77.16 [0] = [0] 91.34/77.16 91.34/77.16 [s](x1) = [1] x1 + [0] 91.34/77.16 91.34/77.16 [lastbit](x1) = [0] 91.34/77.16 91.34/77.16 [conv](x1) = [1] x1 + [0] 91.34/77.16 91.34/77.16 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 91.34/77.16 91.34/77.16 [nil] = [0] 91.34/77.16 91.34/77.16 The order satisfies the following ordering constraints: 91.34/77.16 91.34/77.16 [half(0())] = [1] 91.34/77.16 > [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [half(s(0()))] = [1] 91.34/77.16 > [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [half(s(s(x)))] = [1] 91.34/77.16 >= [1] 91.34/77.16 = [s(half(x))] 91.34/77.16 91.34/77.16 [lastbit(0())] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [lastbit(s(0()))] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [s(0())] 91.34/77.16 91.34/77.16 [lastbit(s(s(x)))] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [lastbit(x)] 91.34/77.16 91.34/77.16 [conv(0())] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [cons(nil(), 0())] 91.34/77.16 91.34/77.16 [conv(s(x))] = [1] x + [0] 91.34/77.16 ? [1] 91.34/77.16 = [cons(conv(half(s(x))), lastbit(s(x)))] 91.34/77.16 91.34/77.16 91.34/77.16 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 91.34/77.16 91.34/77.16 We are left with following problem, upon which TcT provides the 91.34/77.16 certificate YES(O(1),O(n^1)). 91.34/77.16 91.34/77.16 Strict Trs: 91.34/77.16 { half(s(s(x))) -> s(half(x)) 91.34/77.16 , conv(0()) -> cons(nil(), 0()) 91.34/77.16 , conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) } 91.34/77.16 Weak Trs: 91.34/77.16 { half(0()) -> 0() 91.34/77.16 , half(s(0())) -> 0() 91.34/77.16 , lastbit(0()) -> 0() 91.34/77.16 , lastbit(s(0())) -> s(0()) 91.34/77.16 , lastbit(s(s(x))) -> lastbit(x) } 91.34/77.16 Obligation: 91.34/77.16 innermost runtime complexity 91.34/77.16 Answer: 91.34/77.16 YES(O(1),O(n^1)) 91.34/77.16 91.34/77.16 The weightgap principle applies (using the following nonconstant 91.34/77.16 growth matrix-interpretation) 91.34/77.16 91.34/77.16 The following argument positions are usable: 91.34/77.16 Uargs(s) = {1}, Uargs(conv) = {1}, Uargs(cons) = {1, 2} 91.34/77.16 91.34/77.16 TcT has computed the following matrix interpretation satisfying 91.34/77.16 not(EDA) and not(IDA(1)). 91.34/77.16 91.34/77.16 [half](x1) = [0] 91.34/77.16 91.34/77.16 [0] = [0] 91.34/77.16 91.34/77.16 [s](x1) = [1] x1 + [0] 91.34/77.16 91.34/77.16 [lastbit](x1) = [0] 91.34/77.16 91.34/77.16 [conv](x1) = [1] x1 + [1] 91.34/77.16 91.34/77.16 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 91.34/77.16 91.34/77.16 [nil] = [0] 91.34/77.16 91.34/77.16 The order satisfies the following ordering constraints: 91.34/77.16 91.34/77.16 [half(0())] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [half(s(0()))] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [half(s(s(x)))] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [s(half(x))] 91.34/77.16 91.34/77.16 [lastbit(0())] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [0()] 91.34/77.16 91.34/77.16 [lastbit(s(0()))] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [s(0())] 91.34/77.16 91.34/77.16 [lastbit(s(s(x)))] = [0] 91.34/77.16 >= [0] 91.34/77.16 = [lastbit(x)] 91.34/77.16 91.34/77.16 [conv(0())] = [1] 91.34/77.16 > [0] 91.34/77.16 = [cons(nil(), 0())] 91.34/77.16 91.34/77.16 [conv(s(x))] = [1] x + [1] 91.34/77.16 >= [1] 91.34/77.16 = [cons(conv(half(s(x))), lastbit(s(x)))] 91.34/77.16 91.34/77.16 91.34/77.16 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 91.34/77.16 91.34/77.16 We are left with following problem, upon which TcT provides the 91.34/77.16 certificate YES(O(1),O(n^1)). 91.34/77.16 91.34/77.16 Strict Trs: 91.34/77.16 { half(s(s(x))) -> s(half(x)) 91.34/77.16 , conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) } 91.34/77.16 Weak Trs: 91.34/77.16 { half(0()) -> 0() 91.34/77.16 , half(s(0())) -> 0() 91.34/77.16 , lastbit(0()) -> 0() 91.34/77.16 , lastbit(s(0())) -> s(0()) 91.34/77.16 , lastbit(s(s(x))) -> lastbit(x) 91.34/77.17 , conv(0()) -> cons(nil(), 0()) } 91.34/77.17 Obligation: 91.34/77.17 innermost runtime complexity 91.34/77.17 Answer: 91.34/77.17 YES(O(1),O(n^1)) 91.34/77.17 91.34/77.17 The weightgap principle applies (using the following nonconstant 91.34/77.17 growth matrix-interpretation) 91.34/77.17 91.34/77.17 The following argument positions are usable: 91.34/77.17 Uargs(s) = {1}, Uargs(conv) = {1}, Uargs(cons) = {1, 2} 91.34/77.17 91.34/77.17 TcT has computed the following matrix interpretation satisfying 91.34/77.17 not(EDA) and not(IDA(1)). 91.34/77.17 91.34/77.17 [half](x1) = [1] x1 + [0] 91.34/77.17 91.34/77.17 [0] = [0] 91.34/77.17 91.34/77.17 [s](x1) = [1] x1 + [4] 91.34/77.17 91.34/77.17 [lastbit](x1) = [4] 91.34/77.17 91.34/77.17 [conv](x1) = [1] x1 + [0] 91.34/77.17 91.34/77.17 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 91.34/77.17 91.34/77.17 [nil] = [0] 91.34/77.17 91.34/77.17 The order satisfies the following ordering constraints: 91.34/77.17 91.34/77.17 [half(0())] = [0] 91.34/77.17 >= [0] 91.34/77.17 = [0()] 91.34/77.17 91.34/77.17 [half(s(0()))] = [4] 91.34/77.17 > [0] 91.34/77.17 = [0()] 91.34/77.17 91.34/77.17 [half(s(s(x)))] = [1] x + [8] 91.34/77.17 > [1] x + [4] 91.34/77.17 = [s(half(x))] 91.34/77.17 91.34/77.17 [lastbit(0())] = [4] 91.34/77.17 > [0] 91.34/77.17 = [0()] 91.34/77.17 91.34/77.17 [lastbit(s(0()))] = [4] 91.34/77.17 >= [4] 91.34/77.17 = [s(0())] 91.34/77.17 91.34/77.17 [lastbit(s(s(x)))] = [4] 91.34/77.17 >= [4] 91.34/77.17 = [lastbit(x)] 91.34/77.17 91.34/77.17 [conv(0())] = [0] 91.34/77.17 >= [0] 91.34/77.17 = [cons(nil(), 0())] 91.34/77.17 91.34/77.17 [conv(s(x))] = [1] x + [4] 91.34/77.17 ? [1] x + [8] 91.34/77.17 = [cons(conv(half(s(x))), lastbit(s(x)))] 91.34/77.17 91.34/77.17 91.34/77.17 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 91.34/77.17 91.34/77.17 We are left with following problem, upon which TcT provides the 91.34/77.17 certificate YES(O(1),O(n^1)). 91.34/77.17 91.34/77.17 Strict Trs: { conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) } 91.34/77.17 Weak Trs: 91.34/77.17 { half(0()) -> 0() 91.34/77.17 , half(s(0())) -> 0() 91.34/77.17 , half(s(s(x))) -> s(half(x)) 91.34/77.17 , lastbit(0()) -> 0() 91.34/77.17 , lastbit(s(0())) -> s(0()) 91.34/77.17 , lastbit(s(s(x))) -> lastbit(x) 91.34/77.17 , conv(0()) -> cons(nil(), 0()) } 91.34/77.17 Obligation: 91.34/77.17 innermost runtime complexity 91.34/77.17 Answer: 91.34/77.17 YES(O(1),O(n^1)) 91.34/77.17 91.34/77.17 We use the processor 'matrix interpretation of dimension 2' to 91.34/77.17 orient following rules strictly. 91.34/77.17 91.34/77.17 Trs: { conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) } 91.34/77.17 91.34/77.17 The induced complexity on above rules (modulo remaining rules) is 91.34/77.17 YES(?,O(n^1)) . These rules are moved into the corresponding weak 91.34/77.17 component(s). 91.34/77.17 91.34/77.17 Sub-proof: 91.34/77.17 ---------- 91.34/77.17 The following argument positions are usable: 91.34/77.17 Uargs(s) = {1}, Uargs(conv) = {1}, Uargs(cons) = {1, 2} 91.34/77.17 91.34/77.17 TcT has computed the following constructor-based matrix 91.34/77.17 interpretation satisfying not(EDA) and not(IDA(1)). 91.34/77.17 91.34/77.17 [half](x1) = [1 0] x1 + [0] 91.34/77.17 [1 0] [2] 91.34/77.17 91.34/77.17 [0] = [0] 91.34/77.17 [0] 91.34/77.17 91.34/77.17 [s](x1) = [1 0] x1 + [1] 91.34/77.17 [1 0] [4] 91.34/77.17 91.34/77.17 [lastbit](x1) = [1] 91.34/77.17 [4] 91.34/77.17 91.34/77.17 [conv](x1) = [1 2] x1 + [0] 91.34/77.17 [4 0] [0] 91.34/77.17 91.34/77.17 [cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 91.34/77.17 [0 0] [0 0] [0] 91.34/77.17 91.34/77.17 [nil] = [0] 91.34/77.17 [0] 91.34/77.17 91.34/77.17 The order satisfies the following ordering constraints: 91.34/77.17 91.34/77.17 [half(0())] = [0] 91.34/77.17 [2] 91.34/77.17 >= [0] 91.34/77.17 [0] 91.34/77.17 = [0()] 91.34/77.17 91.34/77.17 [half(s(0()))] = [1] 91.34/77.17 [3] 91.34/77.17 > [0] 91.34/77.17 [0] 91.34/77.17 = [0()] 91.34/77.17 91.34/77.17 [half(s(s(x)))] = [1 0] x + [2] 91.34/77.17 [1 0] [4] 91.34/77.17 > [1 0] x + [1] 91.34/77.17 [1 0] [4] 91.34/77.17 = [s(half(x))] 91.34/77.17 91.34/77.17 [lastbit(0())] = [1] 91.34/77.17 [4] 91.34/77.17 > [0] 91.34/77.17 [0] 91.34/77.17 = [0()] 91.34/77.17 91.34/77.17 [lastbit(s(0()))] = [1] 91.34/77.17 [4] 91.34/77.17 >= [1] 91.34/77.17 [4] 91.34/77.17 = [s(0())] 91.34/77.17 91.34/77.17 [lastbit(s(s(x)))] = [1] 91.34/77.17 [4] 91.34/77.17 >= [1] 91.34/77.17 [4] 91.34/77.17 = [lastbit(x)] 91.34/77.17 91.34/77.17 [conv(0())] = [0] 91.34/77.17 [0] 91.34/77.17 >= [0] 91.34/77.17 [0] 91.34/77.17 = [cons(nil(), 0())] 91.34/77.17 91.34/77.17 [conv(s(x))] = [3 0] x + [9] 91.34/77.17 [4 0] [4] 91.34/77.17 > [3 0] x + [8] 91.34/77.17 [0 0] [0] 91.34/77.17 = [cons(conv(half(s(x))), lastbit(s(x)))] 91.34/77.17 91.34/77.17 91.34/77.17 We return to the main proof. 91.34/77.17 91.34/77.17 We are left with following problem, upon which TcT provides the 91.34/77.17 certificate YES(O(1),O(1)). 91.34/77.17 91.34/77.17 Weak Trs: 91.34/77.17 { half(0()) -> 0() 91.34/77.17 , half(s(0())) -> 0() 91.34/77.17 , half(s(s(x))) -> s(half(x)) 91.34/77.17 , lastbit(0()) -> 0() 91.34/77.17 , lastbit(s(0())) -> s(0()) 91.34/77.17 , lastbit(s(s(x))) -> lastbit(x) 91.34/77.17 , conv(0()) -> cons(nil(), 0()) 91.34/77.17 , conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) } 91.34/77.17 Obligation: 91.34/77.17 innermost runtime complexity 91.34/77.17 Answer: 91.34/77.17 YES(O(1),O(1)) 91.34/77.17 91.34/77.17 Empty rules are trivially bounded 91.34/77.17 91.34/77.17 Hurray, we answered YES(O(1),O(n^1)) 91.34/77.18 EOF