YES(O(1),O(n^1)) 8.92/2.91 YES(O(1),O(n^1)) 8.92/2.91 8.92/2.91 We are left with following problem, upon which TcT provides the 8.92/2.91 certificate YES(O(1),O(n^1)). 8.92/2.91 8.92/2.91 Strict Trs: 8.92/2.91 { f(x, c(y)) -> f(x, s(f(y, y))) 8.92/2.91 , f(s(x), y) -> f(x, s(c(y))) } 8.92/2.91 Obligation: 8.92/2.91 innermost runtime complexity 8.92/2.91 Answer: 8.92/2.91 YES(O(1),O(n^1)) 8.92/2.91 8.92/2.91 We add the following dependency tuples: 8.92/2.91 8.92/2.91 Strict DPs: 8.92/2.91 { f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y)) 8.92/2.91 , f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) } 8.92/2.91 8.92/2.91 and mark the set of starting terms. 8.92/2.91 8.92/2.91 We are left with following problem, upon which TcT provides the 8.92/2.91 certificate YES(O(1),O(n^1)). 8.92/2.91 8.92/2.91 Strict DPs: 8.92/2.91 { f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y)) 8.92/2.91 , f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) } 8.92/2.91 Weak Trs: 8.92/2.91 { f(x, c(y)) -> f(x, s(f(y, y))) 8.92/2.91 , f(s(x), y) -> f(x, s(c(y))) } 8.92/2.91 Obligation: 8.92/2.91 innermost runtime complexity 8.92/2.91 Answer: 8.92/2.91 YES(O(1),O(n^1)) 8.92/2.91 8.92/2.91 We use the processor 'matrix interpretation of dimension 2' to 8.92/2.91 orient following rules strictly. 8.92/2.91 8.92/2.91 DPs: 8.92/2.91 { 2: f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) } 8.92/2.91 8.92/2.91 Sub-proof: 8.92/2.91 ---------- 8.92/2.91 The following argument positions are usable: 8.92/2.91 Uargs(c_1) = {1, 2}, Uargs(c_2) = {1} 8.92/2.91 8.92/2.91 TcT has computed the following constructor-based matrix 8.92/2.91 interpretation satisfying not(EDA) and not(IDA(1)). 8.92/2.91 8.92/2.91 [f](x1, x2) = [0 0] x1 + [0] 8.92/2.91 [0 1] [0] 8.92/2.91 8.92/2.91 [c](x1) = [1 4] x1 + [4] 8.92/2.91 [0 0] [0] 8.92/2.91 8.92/2.91 [s](x1) = [0 0] x1 + [0] 8.92/2.91 [0 1] [2] 8.92/2.91 8.92/2.91 [f^#](x1, x2) = [0 1] x1 + [2 0] x2 + [0] 8.92/2.91 [0 0] [0 1] [0] 8.92/2.91 8.92/2.91 [c_1](x1, x2) = [1 4] x1 + [1 3] x2 + [0] 8.92/2.91 [0 0] [0 0] [0] 8.92/2.91 8.92/2.91 [c_2](x1) = [1 0] x1 + [1] 8.92/2.91 [0 0] [0] 8.92/2.91 8.92/2.91 The order satisfies the following ordering constraints: 8.92/2.91 8.92/2.91 [f(x, c(y))] = [0 0] x + [0] 8.92/2.91 [0 1] [0] 8.92/2.91 >= [0 0] x + [0] 8.92/2.91 [0 1] [0] 8.92/2.91 = [f(x, s(f(y, y)))] 8.92/2.91 8.92/2.91 [f(s(x), y)] = [0 0] x + [0] 8.92/2.91 [0 1] [2] 8.92/2.91 >= [0 0] x + [0] 8.92/2.91 [0 1] [0] 8.92/2.91 = [f(x, s(c(y)))] 8.92/2.91 8.92/2.91 [f^#(x, c(y))] = [0 1] x + [2 8] y + [8] 8.92/2.91 [0 0] [0 0] [0] 8.92/2.91 >= [0 1] x + [2 8] y + [8] 8.92/2.91 [0 0] [0 0] [0] 8.92/2.91 = [c_1(f^#(x, s(f(y, y))), f^#(y, y))] 8.92/2.91 8.92/2.91 [f^#(s(x), y)] = [0 1] x + [2 0] y + [2] 8.92/2.91 [0 0] [0 1] [0] 8.92/2.91 > [0 1] x + [1] 8.92/2.91 [0 0] [0] 8.92/2.91 = [c_2(f^#(x, s(c(y))))] 8.92/2.91 8.92/2.91 8.92/2.91 The strictly oriented rules are moved into the weak component. 8.92/2.91 8.92/2.91 We are left with following problem, upon which TcT provides the 8.92/2.91 certificate YES(O(1),O(n^1)). 8.92/2.91 8.92/2.91 Strict DPs: { f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y)) } 8.92/2.91 Weak DPs: { f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) } 8.92/2.91 Weak Trs: 8.92/2.91 { f(x, c(y)) -> f(x, s(f(y, y))) 8.92/2.91 , f(s(x), y) -> f(x, s(c(y))) } 8.92/2.91 Obligation: 8.92/2.91 innermost runtime complexity 8.92/2.91 Answer: 8.92/2.91 YES(O(1),O(n^1)) 8.92/2.91 8.92/2.91 The following weak DPs constitute a sub-graph of the DG that is 8.92/2.91 closed under successors. The DPs are removed. 8.92/2.91 8.92/2.91 { f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) } 8.92/2.91 8.92/2.91 We are left with following problem, upon which TcT provides the 8.92/2.91 certificate YES(O(1),O(n^1)). 8.92/2.91 8.92/2.91 Strict DPs: { f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y)) } 8.92/2.91 Weak Trs: 8.92/2.91 { f(x, c(y)) -> f(x, s(f(y, y))) 8.92/2.91 , f(s(x), y) -> f(x, s(c(y))) } 8.92/2.91 Obligation: 8.92/2.91 innermost runtime complexity 8.92/2.91 Answer: 8.92/2.91 YES(O(1),O(n^1)) 8.92/2.91 8.92/2.91 Due to missing edges in the dependency-graph, the right-hand sides 8.92/2.91 of following rules could be simplified: 8.92/2.91 8.92/2.91 { f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y)) } 8.92/2.91 8.92/2.91 We are left with following problem, upon which TcT provides the 8.92/2.91 certificate YES(O(1),O(n^1)). 8.92/2.91 8.92/2.91 Strict DPs: { f^#(x, c(y)) -> c_1(f^#(y, y)) } 8.92/2.91 Weak Trs: 8.92/2.91 { f(x, c(y)) -> f(x, s(f(y, y))) 8.92/2.91 , f(s(x), y) -> f(x, s(c(y))) } 8.92/2.91 Obligation: 8.92/2.91 innermost runtime complexity 8.92/2.91 Answer: 8.92/2.91 YES(O(1),O(n^1)) 8.92/2.91 8.92/2.91 No rule is usable, rules are removed from the input problem. 8.92/2.91 8.92/2.91 We are left with following problem, upon which TcT provides the 8.92/2.91 certificate YES(O(1),O(n^1)). 8.92/2.91 8.92/2.91 Strict DPs: { f^#(x, c(y)) -> c_1(f^#(y, y)) } 8.92/2.91 Obligation: 8.92/2.91 innermost runtime complexity 8.92/2.91 Answer: 8.92/2.91 YES(O(1),O(n^1)) 8.92/2.91 8.92/2.91 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 8.92/2.91 to orient following rules strictly. 8.92/2.91 8.92/2.91 DPs: 8.92/2.91 { 1: f^#(x, c(y)) -> c_1(f^#(y, y)) } 8.92/2.91 8.92/2.91 Sub-proof: 8.92/2.91 ---------- 8.92/2.91 The input was oriented with the instance of 'Small Polynomial Path 8.92/2.91 Order (PS,1-bounded)' as induced by the safe mapping 8.92/2.91 8.92/2.91 safe(f) = {}, safe(c) = {1}, safe(s) = {1}, safe(f^#) = {1}, 8.92/2.91 safe(c_1) = {}, safe(c_2) = {}, safe(c) = {}, safe(c_1) = {} 8.92/2.91 8.92/2.91 and precedence 8.92/2.91 8.92/2.91 empty . 8.92/2.91 8.92/2.91 Following symbols are considered recursive: 8.92/2.91 8.92/2.91 {f^#} 8.92/2.91 8.92/2.91 The recursion depth is 1. 8.92/2.91 8.92/2.91 Further, following argument filtering is employed: 8.92/2.91 8.92/2.91 pi(f) = [], pi(c) = [1], pi(s) = [], pi(f^#) = [1, 2], 8.92/2.91 pi(c_1) = [], pi(c_2) = [], pi(c) = [], pi(c_1) = [1] 8.92/2.91 8.92/2.91 Usable defined function symbols are a subset of: 8.92/2.91 8.92/2.91 {f^#} 8.92/2.91 8.92/2.91 For your convenience, here are the satisfied ordering constraints: 8.92/2.91 8.92/2.91 pi(f^#(x, c(y))) = f^#(c(; y); x) 8.92/2.91 > c_1(f^#(y; y);) 8.92/2.91 = pi(c_1(f^#(y, y))) 8.92/2.91 8.92/2.91 8.92/2.91 The strictly oriented rules are moved into the weak component. 8.92/2.91 8.92/2.91 We are left with following problem, upon which TcT provides the 8.92/2.91 certificate YES(O(1),O(1)). 8.92/2.91 8.92/2.91 Weak DPs: { f^#(x, c(y)) -> c_1(f^#(y, y)) } 8.92/2.91 Obligation: 8.92/2.91 innermost runtime complexity 8.92/2.91 Answer: 8.92/2.91 YES(O(1),O(1)) 8.92/2.91 8.92/2.91 The following weak DPs constitute a sub-graph of the DG that is 8.92/2.91 closed under successors. The DPs are removed. 8.92/2.91 8.92/2.91 { f^#(x, c(y)) -> c_1(f^#(y, y)) } 8.92/2.91 8.92/2.91 We are left with following problem, upon which TcT provides the 8.92/2.91 certificate YES(O(1),O(1)). 8.92/2.91 8.92/2.91 Rules: Empty 8.92/2.91 Obligation: 8.92/2.91 innermost runtime complexity 8.92/2.91 Answer: 8.92/2.91 YES(O(1),O(1)) 8.92/2.91 8.92/2.91 Empty rules are trivially bounded 8.92/2.91 8.92/2.91 Hurray, we answered YES(O(1),O(n^1)) 8.92/2.92 EOF