YES(O(1),O(n^1)) 0.00/0.67 YES(O(1),O(n^1)) 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(n^1)). 0.00/0.67 0.00/0.67 Strict Trs: 0.00/0.67 { prime(0()) -> false() 0.00/0.67 , prime(s(0())) -> false() 0.00/0.67 , prime(s(s(x))) -> prime1(s(s(x)), s(x)) 0.00/0.67 , prime1(x, 0()) -> false() 0.00/0.67 , prime1(x, s(0())) -> true() 0.00/0.67 , prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) 0.00/0.67 , divp(x, y) -> =(rem(x, y), 0()) } 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(n^1)) 0.00/0.67 0.00/0.67 We add the following weak dependency pairs: 0.00/0.67 0.00/0.67 Strict DPs: 0.00/0.67 { prime^#(0()) -> c_1() 0.00/0.67 , prime^#(s(0())) -> c_2() 0.00/0.67 , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) 0.00/0.67 , prime1^#(x, 0()) -> c_4() 0.00/0.67 , prime1^#(x, s(0())) -> c_5() 0.00/0.67 , prime1^#(x, s(s(y))) -> 0.00/0.67 c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) 0.00/0.67 , divp^#(x, y) -> c_7() } 0.00/0.67 0.00/0.67 and mark the set of starting terms. 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(n^1)). 0.00/0.67 0.00/0.67 Strict DPs: 0.00/0.67 { prime^#(0()) -> c_1() 0.00/0.67 , prime^#(s(0())) -> c_2() 0.00/0.67 , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) 0.00/0.67 , prime1^#(x, 0()) -> c_4() 0.00/0.67 , prime1^#(x, s(0())) -> c_5() 0.00/0.67 , prime1^#(x, s(s(y))) -> 0.00/0.67 c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) 0.00/0.67 , divp^#(x, y) -> c_7() } 0.00/0.67 Strict Trs: 0.00/0.67 { prime(0()) -> false() 0.00/0.67 , prime(s(0())) -> false() 0.00/0.67 , prime(s(s(x))) -> prime1(s(s(x)), s(x)) 0.00/0.67 , prime1(x, 0()) -> false() 0.00/0.67 , prime1(x, s(0())) -> true() 0.00/0.67 , prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) 0.00/0.67 , divp(x, y) -> =(rem(x, y), 0()) } 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(n^1)) 0.00/0.67 0.00/0.67 No rule is usable, rules are removed from the input problem. 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(n^1)). 0.00/0.67 0.00/0.67 Strict DPs: 0.00/0.67 { prime^#(0()) -> c_1() 0.00/0.67 , prime^#(s(0())) -> c_2() 0.00/0.67 , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) 0.00/0.67 , prime1^#(x, 0()) -> c_4() 0.00/0.67 , prime1^#(x, s(0())) -> c_5() 0.00/0.67 , prime1^#(x, s(s(y))) -> 0.00/0.67 c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) 0.00/0.67 , divp^#(x, y) -> c_7() } 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(n^1)) 0.00/0.67 0.00/0.67 The weightgap principle applies (using the following constant 0.00/0.67 growth matrix-interpretation) 0.00/0.67 0.00/0.67 The following argument positions are usable: 0.00/0.67 Uargs(c_3) = {1}, Uargs(c_6) = {1, 2} 0.00/0.67 0.00/0.67 TcT has computed the following constructor-restricted matrix 0.00/0.67 interpretation. 0.00/0.67 0.00/0.67 [0] = [0] 0.00/0.67 [0] 0.00/0.67 0.00/0.67 [s](x1) = [0] 0.00/0.67 [0] 0.00/0.67 0.00/0.67 [prime^#](x1) = [0] 0.00/0.67 [0] 0.00/0.67 0.00/0.67 [c_1] = [1] 0.00/0.67 [0] 0.00/0.67 0.00/0.67 [c_2] = [1] 0.00/0.67 [0] 0.00/0.67 0.00/0.67 [c_3](x1) = [1 0] x1 + [1] 0.00/0.67 [0 1] [0] 0.00/0.67 0.00/0.67 [prime1^#](x1, x2) = [0] 0.00/0.67 [0] 0.00/0.67 0.00/0.67 [c_4] = [1] 0.00/0.67 [0] 0.00/0.67 0.00/0.67 [c_5] = [2] 0.00/0.67 [0] 0.00/0.67 0.00/0.67 [c_6](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 0.00/0.67 [0 1] [0 1] [0] 0.00/0.67 0.00/0.67 [divp^#](x1, x2) = [0 0] x2 + [1] 0.00/0.67 [2 2] [0] 0.00/0.67 0.00/0.67 [c_7] = [0] 0.00/0.67 [0] 0.00/0.67 0.00/0.67 The order satisfies the following ordering constraints: 0.00/0.67 0.00/0.67 [prime^#(0())] = [0] 0.00/0.67 [0] 0.00/0.67 ? [1] 0.00/0.67 [0] 0.00/0.67 = [c_1()] 0.00/0.67 0.00/0.67 [prime^#(s(0()))] = [0] 0.00/0.67 [0] 0.00/0.67 ? [1] 0.00/0.67 [0] 0.00/0.67 = [c_2()] 0.00/0.67 0.00/0.67 [prime^#(s(s(x)))] = [0] 0.00/0.67 [0] 0.00/0.67 ? [1] 0.00/0.67 [0] 0.00/0.67 = [c_3(prime1^#(s(s(x)), s(x)))] 0.00/0.67 0.00/0.67 [prime1^#(x, 0())] = [0] 0.00/0.67 [0] 0.00/0.67 ? [1] 0.00/0.67 [0] 0.00/0.67 = [c_4()] 0.00/0.67 0.00/0.67 [prime1^#(x, s(0()))] = [0] 0.00/0.67 [0] 0.00/0.67 ? [2] 0.00/0.67 [0] 0.00/0.67 = [c_5()] 0.00/0.67 0.00/0.67 [prime1^#(x, s(s(y)))] = [0] 0.00/0.67 [0] 0.00/0.67 ? [0 0] x + [1] 0.00/0.67 [2 2] [0] 0.00/0.67 = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))] 0.00/0.67 0.00/0.67 [divp^#(x, y)] = [0 0] y + [1] 0.00/0.67 [2 2] [0] 0.00/0.67 > [0] 0.00/0.67 [0] 0.00/0.67 = [c_7()] 0.00/0.67 0.00/0.67 0.00/0.67 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(n^1)). 0.00/0.67 0.00/0.67 Strict DPs: 0.00/0.67 { prime^#(0()) -> c_1() 0.00/0.67 , prime^#(s(0())) -> c_2() 0.00/0.67 , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) 0.00/0.67 , prime1^#(x, 0()) -> c_4() 0.00/0.67 , prime1^#(x, s(0())) -> c_5() 0.00/0.67 , prime1^#(x, s(s(y))) -> 0.00/0.67 c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } 0.00/0.67 Weak DPs: { divp^#(x, y) -> c_7() } 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(n^1)) 0.00/0.67 0.00/0.67 We estimate the number of application of {1,2,4,5} by applications 0.00/0.67 of Pre({1,2,4,5}) = {3,6}. Here rules are labeled as follows: 0.00/0.67 0.00/0.67 DPs: 0.00/0.67 { 1: prime^#(0()) -> c_1() 0.00/0.67 , 2: prime^#(s(0())) -> c_2() 0.00/0.67 , 3: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) 0.00/0.67 , 4: prime1^#(x, 0()) -> c_4() 0.00/0.67 , 5: prime1^#(x, s(0())) -> c_5() 0.00/0.67 , 6: prime1^#(x, s(s(y))) -> 0.00/0.67 c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) 0.00/0.67 , 7: divp^#(x, y) -> c_7() } 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(n^1)). 0.00/0.67 0.00/0.67 Strict DPs: 0.00/0.67 { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) 0.00/0.67 , prime1^#(x, s(s(y))) -> 0.00/0.67 c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } 0.00/0.67 Weak DPs: 0.00/0.67 { prime^#(0()) -> c_1() 0.00/0.67 , prime^#(s(0())) -> c_2() 0.00/0.67 , prime1^#(x, 0()) -> c_4() 0.00/0.67 , prime1^#(x, s(0())) -> c_5() 0.00/0.67 , divp^#(x, y) -> c_7() } 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(n^1)) 0.00/0.67 0.00/0.67 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.67 closed under successors. The DPs are removed. 0.00/0.67 0.00/0.67 { prime^#(0()) -> c_1() 0.00/0.67 , prime^#(s(0())) -> c_2() 0.00/0.67 , prime1^#(x, 0()) -> c_4() 0.00/0.67 , prime1^#(x, s(0())) -> c_5() 0.00/0.67 , divp^#(x, y) -> c_7() } 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(n^1)). 0.00/0.67 0.00/0.67 Strict DPs: 0.00/0.67 { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) 0.00/0.67 , prime1^#(x, s(s(y))) -> 0.00/0.67 c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(n^1)) 0.00/0.67 0.00/0.67 Due to missing edges in the dependency-graph, the right-hand sides 0.00/0.67 of following rules could be simplified: 0.00/0.67 0.00/0.67 { prime1^#(x, s(s(y))) -> 0.00/0.67 c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) } 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(n^1)). 0.00/0.67 0.00/0.67 Strict DPs: 0.00/0.67 { prime^#(s(s(x))) -> c_1(prime1^#(s(s(x)), s(x))) 0.00/0.67 , prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) } 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(n^1)) 0.00/0.67 0.00/0.67 Consider the dependency graph 0.00/0.67 0.00/0.67 1: prime^#(s(s(x))) -> c_1(prime1^#(s(s(x)), s(x))) 0.00/0.67 -->_1 prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) :2 0.00/0.67 0.00/0.67 2: prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) 0.00/0.67 -->_1 prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) :2 0.00/0.67 0.00/0.67 0.00/0.67 Following roots of the dependency graph are removed, as the 0.00/0.67 considered set of starting terms is closed under reduction with 0.00/0.67 respect to these rules (modulo compound contexts). 0.00/0.67 0.00/0.67 { prime^#(s(s(x))) -> c_1(prime1^#(s(s(x)), s(x))) } 0.00/0.67 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(n^1)). 0.00/0.67 0.00/0.67 Strict DPs: { prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) } 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(n^1)) 0.00/0.67 0.00/0.67 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 0.00/0.67 to orient following rules strictly. 0.00/0.67 0.00/0.67 DPs: 0.00/0.67 { 1: prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) } 0.00/0.67 0.00/0.67 Sub-proof: 0.00/0.67 ---------- 0.00/0.67 The input was oriented with the instance of 'Small Polynomial Path 0.00/0.67 Order (PS,1-bounded)' as induced by the safe mapping 0.00/0.67 0.00/0.67 safe(s) = {1}, safe(prime1^#) = {1}, safe(c_2) = {} 0.00/0.67 0.00/0.67 and precedence 0.00/0.67 0.00/0.67 empty . 0.00/0.67 0.00/0.67 Following symbols are considered recursive: 0.00/0.67 0.00/0.67 {prime1^#} 0.00/0.67 0.00/0.67 The recursion depth is 1. 0.00/0.67 0.00/0.67 Further, following argument filtering is employed: 0.00/0.67 0.00/0.67 pi(s) = [1], pi(prime1^#) = [1, 2], pi(c_2) = [1] 0.00/0.67 0.00/0.67 Usable defined function symbols are a subset of: 0.00/0.67 0.00/0.67 {prime1^#} 0.00/0.67 0.00/0.67 For your convenience, here are the satisfied ordering constraints: 0.00/0.67 0.00/0.67 pi(prime1^#(x, s(s(y)))) = prime1^#(s(; s(; y)); x) 0.00/0.67 > c_2(prime1^#(s(; y); x);) 0.00/0.67 = pi(c_2(prime1^#(x, s(y)))) 0.00/0.67 0.00/0.67 0.00/0.67 The strictly oriented rules are moved into the weak component. 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(1)). 0.00/0.67 0.00/0.67 Weak DPs: { prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) } 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(1)) 0.00/0.67 0.00/0.67 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.67 closed under successors. The DPs are removed. 0.00/0.67 0.00/0.67 { prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) } 0.00/0.67 0.00/0.67 We are left with following problem, upon which TcT provides the 0.00/0.67 certificate YES(O(1),O(1)). 0.00/0.67 0.00/0.67 Rules: Empty 0.00/0.67 Obligation: 0.00/0.67 innermost runtime complexity 0.00/0.67 Answer: 0.00/0.67 YES(O(1),O(1)) 0.00/0.67 0.00/0.67 Empty rules are trivially bounded 0.00/0.67 0.00/0.67 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.67 EOF