YES(O(1),O(n^1)) 69.68/24.02 YES(O(1),O(n^1)) 69.68/24.02 69.68/24.02 We are left with following problem, upon which TcT provides the 69.68/24.02 certificate YES(O(1),O(n^1)). 69.68/24.02 69.68/24.02 Strict Trs: 69.68/24.02 { half(0()) -> 0() 69.68/24.02 , half(s(s(x))) -> s(half(x)) 69.68/24.02 , log(s(0())) -> 0() 69.68/24.02 , log(s(s(x))) -> s(log(s(half(x)))) } 69.68/24.02 Obligation: 69.68/24.02 innermost runtime complexity 69.68/24.02 Answer: 69.68/24.02 YES(O(1),O(n^1)) 69.68/24.02 69.68/24.02 The weightgap principle applies (using the following nonconstant 69.68/24.02 growth matrix-interpretation) 69.68/24.02 69.68/24.02 The following argument positions are usable: 69.68/24.02 Uargs(s) = {1}, Uargs(log) = {1} 69.68/24.02 69.68/24.02 TcT has computed the following matrix interpretation satisfying 69.68/24.02 not(EDA) and not(IDA(1)). 69.68/24.02 69.68/24.02 [half](x1) = [0] 69.68/24.02 69.68/24.02 [0] = [4] 69.68/24.02 69.68/24.02 [s](x1) = [1] x1 + [4] 69.68/24.02 69.68/24.02 [log](x1) = [1] x1 + [0] 69.68/24.02 69.68/24.02 The order satisfies the following ordering constraints: 69.68/24.02 69.68/24.02 [half(0())] = [0] 69.68/24.02 ? [4] 69.68/24.02 = [0()] 69.68/24.02 69.68/24.02 [half(s(s(x)))] = [0] 69.68/24.02 ? [4] 69.68/24.02 = [s(half(x))] 69.68/24.02 69.68/24.02 [log(s(0()))] = [8] 69.68/24.02 > [4] 69.68/24.02 = [0()] 69.68/24.02 69.68/24.02 [log(s(s(x)))] = [1] x + [8] 69.68/24.02 >= [8] 69.68/24.02 = [s(log(s(half(x))))] 69.68/24.02 69.68/24.02 69.68/24.02 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 69.68/24.02 69.68/24.02 We are left with following problem, upon which TcT provides the 69.68/24.02 certificate YES(O(1),O(n^1)). 69.68/24.02 69.68/24.02 Strict Trs: 69.68/24.02 { half(0()) -> 0() 69.68/24.02 , half(s(s(x))) -> s(half(x)) 69.68/24.02 , log(s(s(x))) -> s(log(s(half(x)))) } 69.68/24.02 Weak Trs: { log(s(0())) -> 0() } 69.68/24.02 Obligation: 69.68/24.02 innermost runtime complexity 69.68/24.02 Answer: 69.68/24.02 YES(O(1),O(n^1)) 69.68/24.02 69.68/24.02 We use the processor 'matrix interpretation of dimension 1' to 69.68/24.02 orient following rules strictly. 69.68/24.02 69.68/24.02 Trs: { half(s(s(x))) -> s(half(x)) } 69.68/24.02 69.68/24.02 The induced complexity on above rules (modulo remaining rules) is 69.68/24.02 YES(?,O(n^1)) . These rules are moved into the corresponding weak 69.68/24.02 component(s). 69.68/24.02 69.68/24.02 Sub-proof: 69.68/24.02 ---------- 69.68/24.02 The following argument positions are usable: 69.68/24.02 Uargs(s) = {1}, Uargs(log) = {1} 69.68/24.02 69.68/24.02 TcT has computed the following constructor-based matrix 69.68/24.02 interpretation satisfying not(EDA). 69.68/24.02 69.68/24.02 [half](x1) = [1] x1 + [0] 69.68/24.02 69.68/24.02 [0] = [0] 69.68/24.02 69.68/24.02 [s](x1) = [1] x1 + [4] 69.68/24.02 69.68/24.02 [log](x1) = [1] x1 + [0] 69.68/24.02 69.68/24.02 The order satisfies the following ordering constraints: 69.68/24.02 69.68/24.02 [half(0())] = [0] 69.68/24.02 >= [0] 69.68/24.02 = [0()] 69.68/24.02 69.68/24.02 [half(s(s(x)))] = [1] x + [8] 69.68/24.02 > [1] x + [4] 69.68/24.02 = [s(half(x))] 69.68/24.02 69.68/24.02 [log(s(0()))] = [4] 69.68/24.02 > [0] 69.68/24.02 = [0()] 69.68/24.02 69.68/24.02 [log(s(s(x)))] = [1] x + [8] 69.68/24.02 >= [1] x + [8] 69.68/24.02 = [s(log(s(half(x))))] 69.68/24.02 69.68/24.02 69.68/24.02 We return to the main proof. 69.68/24.02 69.68/24.02 We are left with following problem, upon which TcT provides the 69.68/24.02 certificate YES(O(1),O(n^1)). 69.68/24.02 69.68/24.02 Strict Trs: 69.68/24.02 { half(0()) -> 0() 69.68/24.02 , log(s(s(x))) -> s(log(s(half(x)))) } 69.68/24.02 Weak Trs: 69.68/24.02 { half(s(s(x))) -> s(half(x)) 69.68/24.02 , log(s(0())) -> 0() } 69.68/24.02 Obligation: 69.68/24.02 innermost runtime complexity 69.68/24.02 Answer: 69.68/24.02 YES(O(1),O(n^1)) 69.68/24.02 69.68/24.02 The weightgap principle applies (using the following nonconstant 69.68/24.02 growth matrix-interpretation) 69.68/24.02 69.68/24.02 The following argument positions are usable: 69.68/24.02 Uargs(s) = {1}, Uargs(log) = {1} 69.68/24.02 69.68/24.02 TcT has computed the following matrix interpretation satisfying 69.68/24.02 not(EDA) and not(IDA(1)). 69.68/24.02 69.68/24.02 [half](x1) = [1] x1 + [4] 69.68/24.02 69.68/24.02 [0] = [4] 69.68/24.02 69.68/24.02 [s](x1) = [1] x1 + [4] 69.68/24.02 69.68/24.02 [log](x1) = [1] x1 + [0] 69.68/24.02 69.68/24.02 The order satisfies the following ordering constraints: 69.68/24.02 69.68/24.02 [half(0())] = [8] 69.68/24.02 > [4] 69.68/24.02 = [0()] 69.68/24.02 69.68/24.02 [half(s(s(x)))] = [1] x + [12] 69.68/24.02 > [1] x + [8] 69.68/24.02 = [s(half(x))] 69.68/24.02 69.68/24.02 [log(s(0()))] = [8] 69.68/24.02 > [4] 69.68/24.02 = [0()] 69.68/24.02 69.68/24.02 [log(s(s(x)))] = [1] x + [8] 69.68/24.02 ? [1] x + [12] 69.68/24.02 = [s(log(s(half(x))))] 69.68/24.02 69.68/24.02 69.68/24.02 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 69.68/24.02 69.68/24.02 We are left with following problem, upon which TcT provides the 69.68/24.02 certificate YES(O(1),O(n^1)). 69.68/24.02 69.68/24.02 Strict Trs: { log(s(s(x))) -> s(log(s(half(x)))) } 69.68/24.02 Weak Trs: 69.68/24.02 { half(0()) -> 0() 69.68/24.02 , half(s(s(x))) -> s(half(x)) 69.68/24.02 , log(s(0())) -> 0() } 69.68/24.02 Obligation: 69.68/24.02 innermost runtime complexity 69.68/24.02 Answer: 69.68/24.02 YES(O(1),O(n^1)) 69.68/24.02 69.68/24.02 We use the processor 'matrix interpretation of dimension 1' to 69.68/24.02 orient following rules strictly. 69.68/24.02 69.68/24.02 Trs: { log(s(s(x))) -> s(log(s(half(x)))) } 69.68/24.02 69.68/24.02 The induced complexity on above rules (modulo remaining rules) is 69.68/24.02 YES(?,O(n^1)) . These rules are moved into the corresponding weak 69.68/24.02 component(s). 69.68/24.02 69.68/24.02 Sub-proof: 69.68/24.02 ---------- 69.68/24.02 The following argument positions are usable: 69.68/24.02 Uargs(s) = {1}, Uargs(log) = {1} 69.68/24.02 69.68/24.02 TcT has computed the following constructor-based matrix 69.68/24.02 interpretation satisfying not(EDA). 69.68/24.02 69.68/24.02 [half](x1) = [1] x1 + [0] 69.68/24.02 69.68/24.02 [0] = [2] 69.68/24.02 69.68/24.02 [s](x1) = [1] x1 + [2] 69.68/24.02 69.68/24.02 [log](x1) = [2] x1 + [0] 69.68/24.02 69.68/24.02 The order satisfies the following ordering constraints: 69.68/24.02 69.68/24.02 [half(0())] = [2] 69.68/24.02 >= [2] 69.68/24.02 = [0()] 69.68/24.02 69.68/24.02 [half(s(s(x)))] = [1] x + [4] 69.68/24.02 > [1] x + [2] 69.68/24.02 = [s(half(x))] 69.68/24.02 69.68/24.02 [log(s(0()))] = [8] 69.68/24.02 > [2] 69.68/24.02 = [0()] 69.68/24.02 69.68/24.02 [log(s(s(x)))] = [2] x + [8] 69.68/24.02 > [2] x + [6] 69.68/24.02 = [s(log(s(half(x))))] 69.68/24.02 69.68/24.02 69.68/24.02 We return to the main proof. 69.68/24.02 69.68/24.02 We are left with following problem, upon which TcT provides the 69.68/24.02 certificate YES(O(1),O(1)). 69.68/24.02 69.68/24.02 Weak Trs: 69.68/24.02 { half(0()) -> 0() 69.68/24.02 , half(s(s(x))) -> s(half(x)) 69.68/24.02 , log(s(0())) -> 0() 69.68/24.02 , log(s(s(x))) -> s(log(s(half(x)))) } 69.68/24.02 Obligation: 69.68/24.02 innermost runtime complexity 69.68/24.02 Answer: 69.68/24.02 YES(O(1),O(1)) 69.68/24.02 69.68/24.02 Empty rules are trivially bounded 69.68/24.02 69.68/24.02 Hurray, we answered YES(O(1),O(n^1)) 69.68/24.02 EOF