YES(O(1),O(n^1)) 4.41/1.51 YES(O(1),O(n^1)) 4.41/1.51 4.41/1.51 We are left with following problem, upon which TcT provides the 4.41/1.51 certificate YES(O(1),O(n^1)). 4.41/1.51 4.41/1.51 Strict Trs: 4.41/1.51 { del(.(x, .(y, z))) -> f(=(x, y), x, y, z) 4.41/1.51 , f(true(), x, y, z) -> del(.(y, z)) 4.41/1.51 , f(false(), x, y, z) -> .(x, del(.(y, z))) 4.41/1.51 , =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 4.41/1.51 , =(.(x, y), nil()) -> false() 4.41/1.51 , =(nil(), .(y, z)) -> false() 4.41/1.51 , =(nil(), nil()) -> true() } 4.41/1.51 Obligation: 4.41/1.51 innermost runtime complexity 4.41/1.51 Answer: 4.41/1.51 YES(O(1),O(n^1)) 4.41/1.51 4.41/1.51 We add the following weak dependency pairs: 4.41/1.51 4.41/1.51 Strict DPs: 4.41/1.51 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) 4.41/1.51 , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 4.41/1.51 , =^#(.(x, y), nil()) -> c_5() 4.41/1.51 , =^#(nil(), .(y, z)) -> c_6() 4.41/1.51 , =^#(nil(), nil()) -> c_7() } 4.41/1.51 4.41/1.51 and mark the set of starting terms. 4.41/1.51 4.41/1.51 We are left with following problem, upon which TcT provides the 4.41/1.51 certificate YES(O(1),O(n^1)). 4.41/1.51 4.41/1.51 Strict DPs: 4.41/1.51 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) 4.41/1.51 , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 4.41/1.51 , =^#(.(x, y), nil()) -> c_5() 4.41/1.51 , =^#(nil(), .(y, z)) -> c_6() 4.41/1.51 , =^#(nil(), nil()) -> c_7() } 4.41/1.51 Strict Trs: 4.41/1.51 { del(.(x, .(y, z))) -> f(=(x, y), x, y, z) 4.41/1.51 , f(true(), x, y, z) -> del(.(y, z)) 4.41/1.51 , f(false(), x, y, z) -> .(x, del(.(y, z))) 4.41/1.51 , =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 4.41/1.51 , =(.(x, y), nil()) -> false() 4.41/1.51 , =(nil(), .(y, z)) -> false() 4.41/1.51 , =(nil(), nil()) -> true() } 4.41/1.51 Obligation: 4.41/1.51 innermost runtime complexity 4.41/1.51 Answer: 4.41/1.51 YES(O(1),O(n^1)) 4.41/1.51 4.41/1.51 We replace rewrite rules by usable rules: 4.41/1.51 4.41/1.51 Strict Usable Rules: 4.41/1.51 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 4.41/1.51 , =(.(x, y), nil()) -> false() 4.41/1.51 , =(nil(), .(y, z)) -> false() 4.41/1.51 , =(nil(), nil()) -> true() } 4.41/1.51 4.41/1.51 We are left with following problem, upon which TcT provides the 4.41/1.51 certificate YES(O(1),O(n^1)). 4.41/1.51 4.41/1.51 Strict DPs: 4.41/1.51 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) 4.41/1.51 , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 4.41/1.51 , =^#(.(x, y), nil()) -> c_5() 4.41/1.51 , =^#(nil(), .(y, z)) -> c_6() 4.41/1.51 , =^#(nil(), nil()) -> c_7() } 4.41/1.51 Strict Trs: 4.41/1.51 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 4.41/1.51 , =(.(x, y), nil()) -> false() 4.41/1.51 , =(nil(), .(y, z)) -> false() 4.41/1.51 , =(nil(), nil()) -> true() } 4.41/1.51 Obligation: 4.41/1.51 innermost runtime complexity 4.41/1.51 Answer: 4.41/1.51 YES(O(1),O(n^1)) 4.41/1.51 4.41/1.51 The weightgap principle applies (using the following constant 4.41/1.51 growth matrix-interpretation) 4.41/1.51 4.41/1.51 The following argument positions are usable: 4.41/1.51 Uargs(c_1) = {1}, Uargs(f^#) = {1}, Uargs(c_2) = {1}, 4.41/1.51 Uargs(c_3) = {1} 4.41/1.51 4.41/1.51 TcT has computed the following constructor-restricted matrix 4.41/1.51 interpretation. 4.41/1.51 4.41/1.51 [.](x1, x2) = [0] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [=](x1, x2) = [1] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [true] = [0] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [false] = [0] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [nil] = [0] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [u] = [0] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [v] = [0] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [and](x1, x2) = [0] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [del^#](x1) = [0] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [c_1](x1) = [1 0] x1 + [2] 4.41/1.51 [0 1] [2] 4.41/1.51 4.41/1.51 [f^#](x1, x2, x3, x4) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [0 4.41/1.51 0] x4 + [0] 4.41/1.51 [0 0] [1 0] [1 1] [2 4.41/1.51 2] [0] 4.41/1.51 4.41/1.51 [c_2](x1) = [1 0] x1 + [2] 4.41/1.51 [0 1] [2] 4.41/1.51 4.41/1.51 [c_3](x1) = [1 0] x1 + [2] 4.41/1.51 [0 1] [2] 4.41/1.51 4.41/1.51 [=^#](x1, x2) = [0] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [c_4](x1, x2) = [2] 4.41/1.51 [2] 4.41/1.51 4.41/1.51 [c_5] = [1] 4.41/1.51 [1] 4.41/1.51 4.41/1.51 [c_6] = [1] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 [c_7] = [1] 4.41/1.51 [0] 4.41/1.51 4.41/1.51 The order satisfies the following ordering constraints: 4.41/1.51 4.41/1.51 [=(.(x, y), .(u(), v()))] = [1] 4.41/1.51 [0] 4.41/1.51 > [0] 4.41/1.51 [0] 4.41/1.51 = [and(=(x, u()), =(y, v()))] 4.41/1.51 4.41/1.51 [=(.(x, y), nil())] = [1] 4.41/1.51 [0] 4.41/1.51 > [0] 4.41/1.51 [0] 4.41/1.51 = [false()] 4.41/1.51 4.41/1.51 [=(nil(), .(y, z))] = [1] 4.41/1.51 [0] 4.41/1.51 > [0] 4.41/1.51 [0] 4.41/1.51 = [false()] 4.41/1.51 4.41/1.51 [=(nil(), nil())] = [1] 4.41/1.51 [0] 4.41/1.51 > [0] 4.41/1.51 [0] 4.41/1.51 = [true()] 4.41/1.51 4.41/1.51 [del^#(.(x, .(y, z)))] = [0] 4.41/1.51 [0] 4.41/1.51 ? [0 0] x + [0 0] y + [0 0] z + [4] 4.41/1.51 [1 0] [1 1] [2 2] [2] 4.41/1.51 = [c_1(f^#(=(x, y), x, y, z))] 4.41/1.51 4.41/1.51 [f^#(true(), x, y, z)] = [0 0] x + [0 0] y + [0 0] z + [0] 4.41/1.51 [1 0] [1 1] [2 2] [0] 4.41/1.51 ? [2] 4.41/1.51 [2] 4.41/1.51 = [c_2(del^#(.(y, z)))] 4.41/1.51 4.41/1.51 [f^#(false(), x, y, z)] = [0 0] x + [0 0] y + [0 0] z + [0] 4.41/1.51 [1 0] [1 1] [2 2] [0] 4.41/1.51 ? [2] 4.41/1.51 [2] 4.41/1.51 = [c_3(del^#(.(y, z)))] 4.41/1.51 4.41/1.51 [=^#(.(x, y), .(u(), v()))] = [0] 4.41/1.51 [0] 4.41/1.51 ? [2] 4.41/1.51 [2] 4.41/1.51 = [c_4(=^#(x, u()), =^#(y, v()))] 4.41/1.51 4.41/1.51 [=^#(.(x, y), nil())] = [0] 4.41/1.51 [0] 4.41/1.51 ? [1] 4.41/1.51 [1] 4.41/1.51 = [c_5()] 4.41/1.51 4.41/1.51 [=^#(nil(), .(y, z))] = [0] 4.41/1.51 [0] 4.41/1.51 ? [1] 4.41/1.51 [0] 4.41/1.51 = [c_6()] 4.41/1.51 4.41/1.51 [=^#(nil(), nil())] = [0] 4.41/1.51 [0] 4.41/1.51 ? [1] 4.41/1.51 [0] 4.41/1.51 = [c_7()] 4.41/1.51 4.41/1.51 4.41/1.51 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 4.41/1.51 4.41/1.51 We are left with following problem, upon which TcT provides the 4.41/1.51 certificate YES(O(1),O(n^1)). 4.41/1.51 4.41/1.51 Strict DPs: 4.41/1.51 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) 4.41/1.51 , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 4.41/1.51 , =^#(.(x, y), nil()) -> c_5() 4.41/1.51 , =^#(nil(), .(y, z)) -> c_6() 4.41/1.51 , =^#(nil(), nil()) -> c_7() } 4.41/1.51 Weak Trs: 4.41/1.51 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 4.41/1.51 , =(.(x, y), nil()) -> false() 4.41/1.51 , =(nil(), .(y, z)) -> false() 4.41/1.51 , =(nil(), nil()) -> true() } 4.41/1.51 Obligation: 4.41/1.51 innermost runtime complexity 4.41/1.51 Answer: 4.41/1.51 YES(O(1),O(n^1)) 4.41/1.51 4.41/1.51 We estimate the number of application of {4,5,6,7} by applications 4.41/1.51 of Pre({4,5,6,7}) = {}. Here rules are labeled as follows: 4.41/1.51 4.41/1.51 DPs: 4.41/1.51 { 1: del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , 2: f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , 3: f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) 4.41/1.51 , 4: =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 4.41/1.51 , 5: =^#(.(x, y), nil()) -> c_5() 4.41/1.51 , 6: =^#(nil(), .(y, z)) -> c_6() 4.41/1.51 , 7: =^#(nil(), nil()) -> c_7() } 4.41/1.51 4.41/1.51 We are left with following problem, upon which TcT provides the 4.41/1.51 certificate YES(O(1),O(n^1)). 4.41/1.51 4.41/1.51 Strict DPs: 4.41/1.51 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) } 4.41/1.51 Weak DPs: 4.41/1.51 { =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 4.41/1.51 , =^#(.(x, y), nil()) -> c_5() 4.41/1.51 , =^#(nil(), .(y, z)) -> c_6() 4.41/1.51 , =^#(nil(), nil()) -> c_7() } 4.41/1.51 Weak Trs: 4.41/1.51 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 4.41/1.51 , =(.(x, y), nil()) -> false() 4.41/1.51 , =(nil(), .(y, z)) -> false() 4.41/1.51 , =(nil(), nil()) -> true() } 4.41/1.51 Obligation: 4.41/1.51 innermost runtime complexity 4.41/1.51 Answer: 4.41/1.51 YES(O(1),O(n^1)) 4.41/1.51 4.41/1.51 The following weak DPs constitute a sub-graph of the DG that is 4.41/1.51 closed under successors. The DPs are removed. 4.41/1.51 4.41/1.51 { =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 4.41/1.51 , =^#(.(x, y), nil()) -> c_5() 4.41/1.51 , =^#(nil(), .(y, z)) -> c_6() 4.41/1.51 , =^#(nil(), nil()) -> c_7() } 4.41/1.51 4.41/1.51 We are left with following problem, upon which TcT provides the 4.41/1.51 certificate YES(O(1),O(n^1)). 4.41/1.51 4.41/1.51 Strict DPs: 4.41/1.51 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) } 4.41/1.51 Weak Trs: 4.41/1.51 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 4.41/1.51 , =(.(x, y), nil()) -> false() 4.41/1.51 , =(nil(), .(y, z)) -> false() 4.41/1.51 , =(nil(), nil()) -> true() } 4.41/1.51 Obligation: 4.41/1.51 innermost runtime complexity 4.41/1.51 Answer: 4.41/1.51 YES(O(1),O(n^1)) 4.41/1.51 4.41/1.51 We use the processor 'matrix interpretation of dimension 1' to 4.41/1.51 orient following rules strictly. 4.41/1.51 4.41/1.51 DPs: 4.41/1.51 { 1: del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) } 4.41/1.51 4.41/1.51 Sub-proof: 4.41/1.51 ---------- 4.41/1.51 The following argument positions are usable: 4.41/1.51 Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1} 4.41/1.51 4.41/1.51 TcT has computed the following constructor-based matrix 4.41/1.51 interpretation satisfying not(EDA). 4.41/1.51 4.41/1.51 [.](x1, x2) = [1] x2 + [1] 4.41/1.51 4.41/1.51 [=](x1, x2) = [0] 4.41/1.51 4.41/1.51 [true] = [0] 4.41/1.51 4.41/1.51 [false] = [0] 4.41/1.51 4.41/1.51 [nil] = [0] 4.41/1.51 4.41/1.51 [u] = [0] 4.41/1.51 4.41/1.51 [v] = [0] 4.41/1.51 4.41/1.51 [and](x1, x2) = [3] 4.41/1.51 4.41/1.51 [del^#](x1) = [1] x1 + [0] 4.41/1.51 4.41/1.51 [c_1](x1) = [1] x1 + [0] 4.41/1.51 4.41/1.51 [f^#](x1, x2, x3, x4) = [1] x4 + [1] 4.41/1.51 4.41/1.51 [c_2](x1) = [1] x1 + [0] 4.41/1.51 4.41/1.51 [c_3](x1) = [1] x1 + [0] 4.41/1.51 4.41/1.51 The order satisfies the following ordering constraints: 4.41/1.51 4.41/1.51 [=(.(x, y), .(u(), v()))] = [0] 4.41/1.51 ? [3] 4.41/1.51 = [and(=(x, u()), =(y, v()))] 4.41/1.51 4.41/1.51 [=(.(x, y), nil())] = [0] 4.41/1.51 >= [0] 4.41/1.51 = [false()] 4.41/1.51 4.41/1.51 [=(nil(), .(y, z))] = [0] 4.41/1.51 >= [0] 4.41/1.51 = [false()] 4.41/1.51 4.41/1.51 [=(nil(), nil())] = [0] 4.41/1.51 >= [0] 4.41/1.51 = [true()] 4.41/1.51 4.41/1.51 [del^#(.(x, .(y, z)))] = [1] z + [2] 4.41/1.51 > [1] z + [1] 4.41/1.51 = [c_1(f^#(=(x, y), x, y, z))] 4.41/1.51 4.41/1.51 [f^#(true(), x, y, z)] = [1] z + [1] 4.41/1.51 >= [1] z + [1] 4.41/1.51 = [c_2(del^#(.(y, z)))] 4.41/1.51 4.41/1.51 [f^#(false(), x, y, z)] = [1] z + [1] 4.41/1.51 >= [1] z + [1] 4.41/1.51 = [c_3(del^#(.(y, z)))] 4.41/1.51 4.41/1.51 4.41/1.51 We return to the main proof. Consider the set of all dependency 4.41/1.51 pairs 4.41/1.51 4.41/1.51 : 4.41/1.51 { 1: del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , 2: f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , 3: f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) } 4.41/1.51 4.41/1.51 Processor 'matrix interpretation of dimension 1' induces the 4.41/1.51 complexity certificate YES(?,O(n^1)) on application of dependency 4.41/1.51 pairs {1}. These cover all (indirect) predecessors of dependency 4.41/1.51 pairs {1,2,3}, their number of application is equally bounded. The 4.41/1.51 dependency pairs are shifted into the weak component. 4.41/1.51 4.41/1.51 We are left with following problem, upon which TcT provides the 4.41/1.51 certificate YES(O(1),O(1)). 4.41/1.51 4.41/1.51 Weak DPs: 4.41/1.51 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) } 4.41/1.51 Weak Trs: 4.41/1.51 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 4.41/1.51 , =(.(x, y), nil()) -> false() 4.41/1.51 , =(nil(), .(y, z)) -> false() 4.41/1.51 , =(nil(), nil()) -> true() } 4.41/1.51 Obligation: 4.41/1.51 innermost runtime complexity 4.41/1.51 Answer: 4.41/1.51 YES(O(1),O(1)) 4.41/1.51 4.41/1.51 The following weak DPs constitute a sub-graph of the DG that is 4.41/1.51 closed under successors. The DPs are removed. 4.41/1.51 4.41/1.51 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 4.41/1.51 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 4.41/1.51 , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) } 4.41/1.51 4.41/1.51 We are left with following problem, upon which TcT provides the 4.41/1.51 certificate YES(O(1),O(1)). 4.41/1.51 4.41/1.51 Weak Trs: 4.41/1.51 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 4.41/1.51 , =(.(x, y), nil()) -> false() 4.41/1.51 , =(nil(), .(y, z)) -> false() 4.41/1.51 , =(nil(), nil()) -> true() } 4.41/1.51 Obligation: 4.41/1.51 innermost runtime complexity 4.41/1.51 Answer: 4.41/1.51 YES(O(1),O(1)) 4.41/1.51 4.41/1.51 No rule is usable, rules are removed from the input problem. 4.41/1.51 4.41/1.51 We are left with following problem, upon which TcT provides the 4.41/1.51 certificate YES(O(1),O(1)). 4.41/1.51 4.41/1.51 Rules: Empty 4.41/1.51 Obligation: 4.41/1.51 innermost runtime complexity 4.41/1.51 Answer: 4.41/1.51 YES(O(1),O(1)) 4.41/1.51 4.41/1.51 Empty rules are trivially bounded 4.41/1.51 4.41/1.51 Hurray, we answered YES(O(1),O(n^1)) 4.41/1.52 EOF