YES(O(1),O(n^1)) 69.51/24.03 YES(O(1),O(n^1)) 69.51/24.03 69.51/24.03 We are left with following problem, upon which TcT provides the 69.51/24.03 certificate YES(O(1),O(n^1)). 69.51/24.03 69.51/24.03 Strict Trs: 69.51/24.03 { half(0()) -> 0() 69.51/24.03 , half(s(s(x))) -> s(half(x)) 69.51/24.03 , log(s(0())) -> 0() 69.51/24.03 , log(s(s(x))) -> s(log(s(half(x)))) } 69.51/24.03 Obligation: 69.51/24.03 runtime complexity 69.51/24.03 Answer: 69.51/24.03 YES(O(1),O(n^1)) 69.51/24.03 69.51/24.03 The input is overlay and right-linear. Switching to innermost 69.51/24.03 rewriting. 69.51/24.03 69.51/24.03 We are left with following problem, upon which TcT provides the 69.51/24.03 certificate YES(O(1),O(n^1)). 69.51/24.03 69.51/24.03 Strict Trs: 69.51/24.03 { half(0()) -> 0() 69.51/24.03 , half(s(s(x))) -> s(half(x)) 69.51/24.03 , log(s(0())) -> 0() 69.51/24.03 , log(s(s(x))) -> s(log(s(half(x)))) } 69.51/24.03 Obligation: 69.51/24.03 innermost runtime complexity 69.51/24.03 Answer: 69.51/24.03 YES(O(1),O(n^1)) 69.51/24.03 69.51/24.03 The weightgap principle applies (using the following nonconstant 69.51/24.03 growth matrix-interpretation) 69.51/24.03 69.51/24.03 The following argument positions are usable: 69.51/24.03 Uargs(s) = {1}, Uargs(log) = {1} 69.51/24.03 69.51/24.03 TcT has computed the following matrix interpretation satisfying 69.51/24.03 not(EDA) and not(IDA(1)). 69.51/24.03 69.51/24.03 [half](x1) = [0] 69.51/24.03 69.51/24.03 [0] = [4] 69.51/24.03 69.51/24.03 [s](x1) = [1] x1 + [4] 69.51/24.03 69.51/24.03 [log](x1) = [1] x1 + [0] 69.51/24.03 69.51/24.03 The order satisfies the following ordering constraints: 69.51/24.03 69.51/24.03 [half(0())] = [0] 69.51/24.03 ? [4] 69.51/24.03 = [0()] 69.51/24.03 69.51/24.03 [half(s(s(x)))] = [0] 69.51/24.03 ? [4] 69.51/24.03 = [s(half(x))] 69.51/24.03 69.51/24.03 [log(s(0()))] = [8] 69.51/24.03 > [4] 69.51/24.03 = [0()] 69.51/24.03 69.51/24.03 [log(s(s(x)))] = [1] x + [8] 69.51/24.03 >= [8] 69.51/24.03 = [s(log(s(half(x))))] 69.51/24.03 69.51/24.03 69.51/24.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 69.51/24.03 69.51/24.03 We are left with following problem, upon which TcT provides the 69.51/24.03 certificate YES(O(1),O(n^1)). 69.51/24.03 69.51/24.03 Strict Trs: 69.51/24.03 { half(0()) -> 0() 69.51/24.03 , half(s(s(x))) -> s(half(x)) 69.51/24.03 , log(s(s(x))) -> s(log(s(half(x)))) } 69.51/24.03 Weak Trs: { log(s(0())) -> 0() } 69.51/24.03 Obligation: 69.51/24.03 innermost runtime complexity 69.51/24.03 Answer: 69.51/24.03 YES(O(1),O(n^1)) 69.51/24.03 69.51/24.03 We use the processor 'matrix interpretation of dimension 1' to 69.51/24.03 orient following rules strictly. 69.51/24.03 69.51/24.03 Trs: { half(s(s(x))) -> s(half(x)) } 69.51/24.03 69.51/24.03 The induced complexity on above rules (modulo remaining rules) is 69.51/24.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak 69.51/24.03 component(s). 69.51/24.03 69.51/24.03 Sub-proof: 69.51/24.03 ---------- 69.51/24.03 The following argument positions are usable: 69.51/24.03 Uargs(s) = {1}, Uargs(log) = {1} 69.51/24.03 69.51/24.03 TcT has computed the following constructor-based matrix 69.51/24.03 interpretation satisfying not(EDA). 69.51/24.03 69.51/24.03 [half](x1) = [1] x1 + [0] 69.51/24.03 69.51/24.03 [0] = [0] 69.51/24.03 69.51/24.03 [s](x1) = [1] x1 + [4] 69.51/24.03 69.51/24.03 [log](x1) = [1] x1 + [0] 69.51/24.03 69.51/24.03 The order satisfies the following ordering constraints: 69.51/24.03 69.51/24.03 [half(0())] = [0] 69.51/24.03 >= [0] 69.51/24.03 = [0()] 69.51/24.03 69.51/24.03 [half(s(s(x)))] = [1] x + [8] 69.51/24.03 > [1] x + [4] 69.51/24.03 = [s(half(x))] 69.51/24.03 69.51/24.03 [log(s(0()))] = [4] 69.51/24.03 > [0] 69.51/24.03 = [0()] 69.51/24.03 69.51/24.03 [log(s(s(x)))] = [1] x + [8] 69.51/24.03 >= [1] x + [8] 69.51/24.03 = [s(log(s(half(x))))] 69.51/24.03 69.51/24.03 69.51/24.03 We return to the main proof. 69.51/24.03 69.51/24.03 We are left with following problem, upon which TcT provides the 69.51/24.03 certificate YES(O(1),O(n^1)). 69.51/24.03 69.51/24.03 Strict Trs: 69.51/24.03 { half(0()) -> 0() 69.51/24.03 , log(s(s(x))) -> s(log(s(half(x)))) } 69.51/24.03 Weak Trs: 69.51/24.03 { half(s(s(x))) -> s(half(x)) 69.51/24.03 , log(s(0())) -> 0() } 69.51/24.03 Obligation: 69.51/24.03 innermost runtime complexity 69.51/24.03 Answer: 69.51/24.03 YES(O(1),O(n^1)) 69.51/24.03 69.51/24.03 The weightgap principle applies (using the following nonconstant 69.51/24.03 growth matrix-interpretation) 69.51/24.03 69.51/24.03 The following argument positions are usable: 69.51/24.03 Uargs(s) = {1}, Uargs(log) = {1} 69.51/24.03 69.51/24.03 TcT has computed the following matrix interpretation satisfying 69.51/24.03 not(EDA) and not(IDA(1)). 69.51/24.03 69.51/24.03 [half](x1) = [1] x1 + [4] 69.51/24.03 69.51/24.03 [0] = [4] 69.51/24.03 69.51/24.03 [s](x1) = [1] x1 + [4] 69.51/24.03 69.51/24.03 [log](x1) = [1] x1 + [0] 69.51/24.03 69.51/24.03 The order satisfies the following ordering constraints: 69.51/24.03 69.51/24.03 [half(0())] = [8] 69.51/24.03 > [4] 69.51/24.03 = [0()] 69.51/24.03 69.51/24.03 [half(s(s(x)))] = [1] x + [12] 69.51/24.03 > [1] x + [8] 69.51/24.03 = [s(half(x))] 69.51/24.03 69.51/24.03 [log(s(0()))] = [8] 69.51/24.03 > [4] 69.51/24.03 = [0()] 69.51/24.03 69.51/24.03 [log(s(s(x)))] = [1] x + [8] 69.51/24.03 ? [1] x + [12] 69.51/24.03 = [s(log(s(half(x))))] 69.51/24.03 69.51/24.03 69.51/24.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 69.51/24.03 69.51/24.03 We are left with following problem, upon which TcT provides the 69.51/24.03 certificate YES(O(1),O(n^1)). 69.51/24.03 69.51/24.03 Strict Trs: { log(s(s(x))) -> s(log(s(half(x)))) } 69.51/24.03 Weak Trs: 69.51/24.03 { half(0()) -> 0() 69.51/24.03 , half(s(s(x))) -> s(half(x)) 69.51/24.03 , log(s(0())) -> 0() } 69.51/24.03 Obligation: 69.51/24.03 innermost runtime complexity 69.51/24.03 Answer: 69.51/24.03 YES(O(1),O(n^1)) 69.51/24.03 69.51/24.03 We use the processor 'matrix interpretation of dimension 1' to 69.51/24.03 orient following rules strictly. 69.51/24.03 69.51/24.03 Trs: { log(s(s(x))) -> s(log(s(half(x)))) } 69.51/24.03 69.51/24.03 The induced complexity on above rules (modulo remaining rules) is 69.51/24.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak 69.51/24.03 component(s). 69.51/24.03 69.51/24.03 Sub-proof: 69.51/24.03 ---------- 69.51/24.03 The following argument positions are usable: 69.51/24.03 Uargs(s) = {1}, Uargs(log) = {1} 69.51/24.03 69.51/24.03 TcT has computed the following constructor-based matrix 69.51/24.03 interpretation satisfying not(EDA). 69.51/24.03 69.51/24.03 [half](x1) = [1] x1 + [0] 69.51/24.03 69.51/24.03 [0] = [2] 69.51/24.03 69.51/24.03 [s](x1) = [1] x1 + [2] 69.51/24.03 69.51/24.03 [log](x1) = [2] x1 + [0] 69.51/24.03 69.51/24.03 The order satisfies the following ordering constraints: 69.51/24.03 69.51/24.03 [half(0())] = [2] 69.51/24.03 >= [2] 69.51/24.03 = [0()] 69.51/24.03 69.51/24.03 [half(s(s(x)))] = [1] x + [4] 69.51/24.03 > [1] x + [2] 69.51/24.03 = [s(half(x))] 69.51/24.03 69.51/24.03 [log(s(0()))] = [8] 69.51/24.03 > [2] 69.51/24.03 = [0()] 69.51/24.03 69.51/24.03 [log(s(s(x)))] = [2] x + [8] 69.51/24.03 > [2] x + [6] 69.51/24.03 = [s(log(s(half(x))))] 69.51/24.03 69.51/24.03 69.51/24.03 We return to the main proof. 69.51/24.03 69.51/24.03 We are left with following problem, upon which TcT provides the 69.51/24.03 certificate YES(O(1),O(1)). 69.51/24.03 69.51/24.03 Weak Trs: 69.51/24.03 { half(0()) -> 0() 69.51/24.03 , half(s(s(x))) -> s(half(x)) 69.51/24.03 , log(s(0())) -> 0() 69.51/24.03 , log(s(s(x))) -> s(log(s(half(x)))) } 69.51/24.03 Obligation: 69.51/24.03 innermost runtime complexity 69.51/24.03 Answer: 69.51/24.03 YES(O(1),O(1)) 69.51/24.03 69.51/24.03 Empty rules are trivially bounded 69.51/24.03 69.51/24.03 Hurray, we answered YES(O(1),O(n^1)) 69.51/24.04 EOF