YES(O(1),O(n^2)) 176.45/75.61 YES(O(1),O(n^2)) 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(n^2)). 176.45/75.61 176.45/75.61 Strict Trs: 176.45/75.61 { fac(s(x)) -> *(fac(p(s(x))), s(x)) 176.45/75.61 , p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(n^2)) 176.45/75.61 176.45/75.61 We add the following dependency tuples: 176.45/75.61 176.45/75.61 Strict DPs: 176.45/75.61 { fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) 176.45/75.61 , p^#(s(s(x))) -> c_2(p^#(s(x))) 176.45/75.61 , p^#(s(0())) -> c_3() } 176.45/75.61 176.45/75.61 and mark the set of starting terms. 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(n^2)). 176.45/75.61 176.45/75.61 Strict DPs: 176.45/75.61 { fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) 176.45/75.61 , p^#(s(s(x))) -> c_2(p^#(s(x))) 176.45/75.61 , p^#(s(0())) -> c_3() } 176.45/75.61 Weak Trs: 176.45/75.61 { fac(s(x)) -> *(fac(p(s(x))), s(x)) 176.45/75.61 , p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(n^2)) 176.45/75.61 176.45/75.61 We estimate the number of application of {3} by applications of 176.45/75.61 Pre({3}) = {1,2}. Here rules are labeled as follows: 176.45/75.61 176.45/75.61 DPs: 176.45/75.61 { 1: fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) 176.45/75.61 , 2: p^#(s(s(x))) -> c_2(p^#(s(x))) 176.45/75.61 , 3: p^#(s(0())) -> c_3() } 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(n^2)). 176.45/75.61 176.45/75.61 Strict DPs: 176.45/75.61 { fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) 176.45/75.61 , p^#(s(s(x))) -> c_2(p^#(s(x))) } 176.45/75.61 Weak DPs: { p^#(s(0())) -> c_3() } 176.45/75.61 Weak Trs: 176.45/75.61 { fac(s(x)) -> *(fac(p(s(x))), s(x)) 176.45/75.61 , p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(n^2)) 176.45/75.61 176.45/75.61 The following weak DPs constitute a sub-graph of the DG that is 176.45/75.61 closed under successors. The DPs are removed. 176.45/75.61 176.45/75.61 { p^#(s(0())) -> c_3() } 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(n^2)). 176.45/75.61 176.45/75.61 Strict DPs: 176.45/75.61 { fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) 176.45/75.61 , p^#(s(s(x))) -> c_2(p^#(s(x))) } 176.45/75.61 Weak Trs: 176.45/75.61 { fac(s(x)) -> *(fac(p(s(x))), s(x)) 176.45/75.61 , p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(n^2)) 176.45/75.61 176.45/75.61 We replace rewrite rules by usable rules: 176.45/75.61 176.45/75.61 Weak Usable Rules: 176.45/75.61 { p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(n^2)). 176.45/75.61 176.45/75.61 Strict DPs: 176.45/75.61 { fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) 176.45/75.61 , p^#(s(s(x))) -> c_2(p^#(s(x))) } 176.45/75.61 Weak Trs: 176.45/75.61 { p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(n^2)) 176.45/75.61 176.45/75.61 We decompose the input problem according to the dependency graph 176.45/75.61 into the upper component 176.45/75.61 176.45/75.61 { fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) } 176.45/75.61 176.45/75.61 and lower component 176.45/75.61 176.45/75.61 { p^#(s(s(x))) -> c_2(p^#(s(x))) } 176.45/75.61 176.45/75.61 Further, following extension rules are added to the lower 176.45/75.61 component. 176.45/75.61 176.45/75.61 { fac^#(s(x)) -> fac^#(p(s(x))) 176.45/75.61 , fac^#(s(x)) -> p^#(s(x)) } 176.45/75.61 176.45/75.61 TcT solves the upper component with certificate YES(O(1),O(n^1)). 176.45/75.61 176.45/75.61 Sub-proof: 176.45/75.61 ---------- 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(n^1)). 176.45/75.61 176.45/75.61 Strict DPs: { fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) } 176.45/75.61 Weak Trs: 176.45/75.61 { p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(n^1)) 176.45/75.61 176.45/75.61 We use the processor 'matrix interpretation of dimension 2' to 176.45/75.61 orient following rules strictly. 176.45/75.61 176.45/75.61 DPs: 176.45/75.61 { 1: fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) } 176.45/75.61 176.45/75.61 Sub-proof: 176.45/75.61 ---------- 176.45/75.61 The following argument positions are usable: 176.45/75.61 Uargs(c_1) = {1} 176.45/75.61 176.45/75.61 TcT has computed the following constructor-based matrix 176.45/75.61 interpretation satisfying not(EDA) and not(IDA(1)). 176.45/75.61 176.45/75.61 [s](x1) = [1 0] x1 + [4] 176.45/75.61 [1 0] [0] 176.45/75.61 176.45/75.61 [p](x1) = [0 1] x1 + [0] 176.45/75.61 [1 0] [0] 176.45/75.61 176.45/75.61 [0] = [0] 176.45/75.61 [0] 176.45/75.61 176.45/75.61 [fac^#](x1) = [2 0] x1 + [0] 176.45/75.61 [0 0] [4] 176.45/75.61 176.45/75.61 [c_1](x1, x2) = [1 0] x1 + [5] 176.45/75.61 [0 0] [3] 176.45/75.61 176.45/75.61 [p^#](x1) = [0] 176.45/75.61 [0] 176.45/75.61 176.45/75.61 The order satisfies the following ordering constraints: 176.45/75.61 176.45/75.61 [p(s(s(x)))] = [1 0] x + [4] 176.45/75.61 [1 0] [8] 176.45/75.61 >= [1 0] x + [4] 176.45/75.61 [1 0] [0] 176.45/75.61 = [s(p(s(x)))] 176.45/75.61 176.45/75.61 [p(s(0()))] = [0] 176.45/75.61 [4] 176.45/75.61 >= [0] 176.45/75.61 [0] 176.45/75.61 = [0()] 176.45/75.61 176.45/75.61 [fac^#(s(x))] = [2 0] x + [8] 176.45/75.61 [0 0] [4] 176.45/75.61 > [2 0] x + [5] 176.45/75.61 [0 0] [3] 176.45/75.61 = [c_1(fac^#(p(s(x))), p^#(s(x)))] 176.45/75.61 176.45/75.61 176.45/75.61 The strictly oriented rules are moved into the weak component. 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(1)). 176.45/75.61 176.45/75.61 Weak DPs: { fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) } 176.45/75.61 Weak Trs: 176.45/75.61 { p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(1)) 176.45/75.61 176.45/75.61 The following weak DPs constitute a sub-graph of the DG that is 176.45/75.61 closed under successors. The DPs are removed. 176.45/75.61 176.45/75.61 { fac^#(s(x)) -> c_1(fac^#(p(s(x))), p^#(s(x))) } 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(1)). 176.45/75.61 176.45/75.61 Weak Trs: 176.45/75.61 { p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(1)) 176.45/75.61 176.45/75.61 No rule is usable, rules are removed from the input problem. 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(1)). 176.45/75.61 176.45/75.61 Rules: Empty 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(1)) 176.45/75.61 176.45/75.61 Empty rules are trivially bounded 176.45/75.61 176.45/75.61 We return to the main proof. 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(n^1)). 176.45/75.61 176.45/75.61 Strict DPs: { p^#(s(s(x))) -> c_2(p^#(s(x))) } 176.45/75.61 Weak DPs: 176.45/75.61 { fac^#(s(x)) -> fac^#(p(s(x))) 176.45/75.61 , fac^#(s(x)) -> p^#(s(x)) } 176.45/75.61 Weak Trs: 176.45/75.61 { p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(n^1)) 176.45/75.61 176.45/75.61 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 176.45/75.61 to orient following rules strictly. 176.45/75.61 176.45/75.61 DPs: 176.45/75.61 { 1: p^#(s(s(x))) -> c_2(p^#(s(x))) 176.45/75.61 , 3: fac^#(s(x)) -> p^#(s(x)) } 176.45/75.61 Trs: { p(s(0())) -> 0() } 176.45/75.61 176.45/75.61 Sub-proof: 176.45/75.61 ---------- 176.45/75.61 The input was oriented with the instance of 'Small Polynomial Path 176.45/75.61 Order (PS,1-bounded)' as induced by the safe mapping 176.45/75.61 176.45/75.61 safe(s) = {1}, safe(p) = {1}, safe(0) = {}, safe(fac^#) = {}, 176.45/75.61 safe(p^#) = {}, safe(c_2) = {} 176.45/75.61 176.45/75.61 and precedence 176.45/75.61 176.45/75.61 fac^# > p, fac^# > p^#, p ~ p^# . 176.45/75.61 176.45/75.61 Following symbols are considered recursive: 176.45/75.61 176.45/75.61 {p, p^#} 176.45/75.61 176.45/75.61 The recursion depth is 1. 176.45/75.61 176.45/75.61 Further, following argument filtering is employed: 176.45/75.61 176.45/75.61 pi(s) = [1], pi(p) = 1, pi(0) = [], pi(fac^#) = [1], pi(p^#) = [1], 176.45/75.61 pi(c_2) = [1] 176.45/75.61 176.45/75.61 Usable defined function symbols are a subset of: 176.45/75.61 176.45/75.61 {p, fac^#, p^#} 176.45/75.61 176.45/75.61 For your convenience, here are the satisfied ordering constraints: 176.45/75.61 176.45/75.61 pi(fac^#(s(x))) = fac^#(s(; x);) 176.45/75.61 >= fac^#(s(; x);) 176.45/75.61 = pi(fac^#(p(s(x)))) 176.45/75.61 176.45/75.61 pi(fac^#(s(x))) = fac^#(s(; x);) 176.45/75.61 > p^#(s(; x);) 176.45/75.61 = pi(p^#(s(x))) 176.45/75.61 176.45/75.61 pi(p^#(s(s(x)))) = p^#(s(; s(; x));) 176.45/75.61 > c_2(p^#(s(; x););) 176.45/75.61 = pi(c_2(p^#(s(x)))) 176.45/75.61 176.45/75.61 pi(p(s(s(x)))) = s(; s(; x)) 176.45/75.61 >= s(; s(; x)) 176.45/75.61 = pi(s(p(s(x)))) 176.45/75.61 176.45/75.61 pi(p(s(0()))) = s(; 0()) 176.45/75.61 > 0() 176.45/75.61 = pi(0()) 176.45/75.61 176.45/75.61 176.45/75.61 The strictly oriented rules are moved into the weak component. 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(1)). 176.45/75.61 176.45/75.61 Weak DPs: 176.45/75.61 { fac^#(s(x)) -> fac^#(p(s(x))) 176.45/75.61 , fac^#(s(x)) -> p^#(s(x)) 176.45/75.61 , p^#(s(s(x))) -> c_2(p^#(s(x))) } 176.45/75.61 Weak Trs: 176.45/75.61 { p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(1)) 176.45/75.61 176.45/75.61 The following weak DPs constitute a sub-graph of the DG that is 176.45/75.61 closed under successors. The DPs are removed. 176.45/75.61 176.45/75.61 { fac^#(s(x)) -> fac^#(p(s(x))) 176.45/75.61 , fac^#(s(x)) -> p^#(s(x)) 176.45/75.61 , p^#(s(s(x))) -> c_2(p^#(s(x))) } 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(1)). 176.45/75.61 176.45/75.61 Weak Trs: 176.45/75.61 { p(s(s(x))) -> s(p(s(x))) 176.45/75.61 , p(s(0())) -> 0() } 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(1)) 176.45/75.61 176.45/75.61 No rule is usable, rules are removed from the input problem. 176.45/75.61 176.45/75.61 We are left with following problem, upon which TcT provides the 176.45/75.61 certificate YES(O(1),O(1)). 176.45/75.61 176.45/75.61 Rules: Empty 176.45/75.61 Obligation: 176.45/75.61 innermost runtime complexity 176.45/75.61 Answer: 176.45/75.61 YES(O(1),O(1)) 176.45/75.61 176.45/75.61 Empty rules are trivially bounded 176.45/75.61 176.45/75.61 Hurray, we answered YES(O(1),O(n^2)) 176.45/75.67 EOF