YES(O(1),O(n^1)) 3.29/1.25 YES(O(1),O(n^1)) 3.29/1.25 3.29/1.25 We are left with following problem, upon which TcT provides the 3.29/1.25 certificate YES(O(1),O(n^1)). 3.29/1.25 3.29/1.25 Strict Trs: 3.29/1.25 { del(.(x, .(y, z))) -> f(=(x, y), x, y, z) 3.29/1.25 , f(true(), x, y, z) -> del(.(y, z)) 3.29/1.25 , f(false(), x, y, z) -> .(x, del(.(y, z))) 3.29/1.25 , =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 3.29/1.25 , =(.(x, y), nil()) -> false() 3.29/1.25 , =(nil(), .(y, z)) -> false() 3.29/1.25 , =(nil(), nil()) -> true() } 3.29/1.25 Obligation: 3.29/1.25 runtime complexity 3.29/1.25 Answer: 3.29/1.25 YES(O(1),O(n^1)) 3.29/1.25 3.29/1.25 We add the following weak dependency pairs: 3.29/1.25 3.29/1.25 Strict DPs: 3.29/1.25 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) 3.29/1.25 , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 3.29/1.25 , =^#(.(x, y), nil()) -> c_5() 3.29/1.25 , =^#(nil(), .(y, z)) -> c_6() 3.29/1.25 , =^#(nil(), nil()) -> c_7() } 3.29/1.25 3.29/1.25 and mark the set of starting terms. 3.29/1.25 3.29/1.25 We are left with following problem, upon which TcT provides the 3.29/1.25 certificate YES(O(1),O(n^1)). 3.29/1.25 3.29/1.25 Strict DPs: 3.29/1.25 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) 3.29/1.25 , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 3.29/1.25 , =^#(.(x, y), nil()) -> c_5() 3.29/1.25 , =^#(nil(), .(y, z)) -> c_6() 3.29/1.25 , =^#(nil(), nil()) -> c_7() } 3.29/1.25 Strict Trs: 3.29/1.25 { del(.(x, .(y, z))) -> f(=(x, y), x, y, z) 3.29/1.25 , f(true(), x, y, z) -> del(.(y, z)) 3.29/1.25 , f(false(), x, y, z) -> .(x, del(.(y, z))) 3.29/1.25 , =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 3.29/1.25 , =(.(x, y), nil()) -> false() 3.29/1.25 , =(nil(), .(y, z)) -> false() 3.29/1.25 , =(nil(), nil()) -> true() } 3.29/1.25 Obligation: 3.29/1.25 runtime complexity 3.29/1.25 Answer: 3.29/1.25 YES(O(1),O(n^1)) 3.29/1.25 3.29/1.25 We replace rewrite rules by usable rules: 3.29/1.25 3.29/1.25 Strict Usable Rules: 3.29/1.25 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 3.29/1.25 , =(.(x, y), nil()) -> false() 3.29/1.25 , =(nil(), .(y, z)) -> false() 3.29/1.25 , =(nil(), nil()) -> true() } 3.29/1.25 3.29/1.25 We are left with following problem, upon which TcT provides the 3.29/1.25 certificate YES(O(1),O(n^1)). 3.29/1.25 3.29/1.25 Strict DPs: 3.29/1.25 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) 3.29/1.25 , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 3.29/1.25 , =^#(.(x, y), nil()) -> c_5() 3.29/1.25 , =^#(nil(), .(y, z)) -> c_6() 3.29/1.25 , =^#(nil(), nil()) -> c_7() } 3.29/1.25 Strict Trs: 3.29/1.25 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 3.29/1.25 , =(.(x, y), nil()) -> false() 3.29/1.25 , =(nil(), .(y, z)) -> false() 3.29/1.25 , =(nil(), nil()) -> true() } 3.29/1.25 Obligation: 3.29/1.25 runtime complexity 3.29/1.25 Answer: 3.29/1.25 YES(O(1),O(n^1)) 3.29/1.25 3.29/1.25 The weightgap principle applies (using the following constant 3.29/1.25 growth matrix-interpretation) 3.29/1.25 3.29/1.25 The following argument positions are usable: 3.29/1.25 Uargs(c_1) = {1}, Uargs(f^#) = {1}, Uargs(c_2) = {1}, 3.29/1.25 Uargs(c_3) = {2} 3.29/1.25 3.29/1.25 TcT has computed the following constructor-restricted matrix 3.29/1.25 interpretation. 3.29/1.25 3.29/1.25 [.](x1, x2) = [0] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [=](x1, x2) = [1] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [true] = [0] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [false] = [0] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [nil] = [0] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [u] = [0] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [v] = [0] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [and](x1, x2) = [0] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [del^#](x1) = [0] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [c_1](x1) = [1 0] x1 + [2] 3.29/1.25 [0 1] [2] 3.29/1.25 3.29/1.25 [f^#](x1, x2, x3, x4) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [0 3.29/1.25 0] x4 + [0] 3.29/1.25 [0 0] [1 1] [1 0] [2 3.29/1.25 1] [0] 3.29/1.25 3.29/1.25 [c_2](x1) = [1 0] x1 + [2] 3.29/1.25 [0 1] [2] 3.29/1.25 3.29/1.25 [c_3](x1, x2) = [0 0] x1 + [1 0] x2 + [2] 3.29/1.25 [1 1] [0 1] [2] 3.29/1.25 3.29/1.25 [=^#](x1, x2) = [0] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [c_4](x1, x2) = [2] 3.29/1.25 [2] 3.29/1.25 3.29/1.25 [c_5] = [1] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [c_6] = [1] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 [c_7] = [1] 3.29/1.25 [0] 3.29/1.25 3.29/1.25 The order satisfies the following ordering constraints: 3.29/1.25 3.29/1.25 [=(.(x, y), .(u(), v()))] = [1] 3.29/1.25 [0] 3.29/1.25 > [0] 3.29/1.25 [0] 3.29/1.25 = [and(=(x, u()), =(y, v()))] 3.29/1.25 3.29/1.25 [=(.(x, y), nil())] = [1] 3.29/1.25 [0] 3.29/1.25 > [0] 3.29/1.25 [0] 3.29/1.25 = [false()] 3.29/1.25 3.29/1.25 [=(nil(), .(y, z))] = [1] 3.29/1.25 [0] 3.29/1.25 > [0] 3.29/1.25 [0] 3.29/1.25 = [false()] 3.29/1.25 3.29/1.25 [=(nil(), nil())] = [1] 3.29/1.25 [0] 3.29/1.25 > [0] 3.29/1.25 [0] 3.29/1.25 = [true()] 3.29/1.25 3.29/1.25 [del^#(.(x, .(y, z)))] = [0] 3.29/1.25 [0] 3.29/1.25 ? [0 0] x + [0 0] y + [0 0] z + [4] 3.29/1.25 [1 1] [1 0] [2 1] [2] 3.29/1.25 = [c_1(f^#(=(x, y), x, y, z))] 3.29/1.25 3.29/1.25 [f^#(true(), x, y, z)] = [0 0] x + [0 0] y + [0 0] z + [0] 3.29/1.25 [1 1] [1 0] [2 1] [0] 3.29/1.25 ? [2] 3.29/1.25 [2] 3.29/1.25 = [c_2(del^#(.(y, z)))] 3.29/1.25 3.29/1.25 [f^#(false(), x, y, z)] = [0 0] x + [0 0] y + [0 0] z + [0] 3.29/1.25 [1 1] [1 0] [2 1] [0] 3.29/1.25 ? [0 0] x + [2] 3.29/1.25 [1 1] [2] 3.29/1.25 = [c_3(x, del^#(.(y, z)))] 3.29/1.25 3.29/1.25 [=^#(.(x, y), .(u(), v()))] = [0] 3.29/1.25 [0] 3.29/1.25 ? [2] 3.29/1.25 [2] 3.29/1.25 = [c_4(=^#(x, u()), =^#(y, v()))] 3.29/1.25 3.29/1.25 [=^#(.(x, y), nil())] = [0] 3.29/1.25 [0] 3.29/1.25 ? [1] 3.29/1.25 [0] 3.29/1.25 = [c_5()] 3.29/1.25 3.29/1.25 [=^#(nil(), .(y, z))] = [0] 3.29/1.25 [0] 3.29/1.25 ? [1] 3.29/1.25 [0] 3.29/1.25 = [c_6()] 3.29/1.25 3.29/1.25 [=^#(nil(), nil())] = [0] 3.29/1.25 [0] 3.29/1.25 ? [1] 3.29/1.25 [0] 3.29/1.25 = [c_7()] 3.29/1.25 3.29/1.25 3.29/1.25 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 3.29/1.25 3.29/1.25 We are left with following problem, upon which TcT provides the 3.29/1.25 certificate YES(O(1),O(n^1)). 3.29/1.25 3.29/1.25 Strict DPs: 3.29/1.25 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) 3.29/1.25 , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 3.29/1.25 , =^#(.(x, y), nil()) -> c_5() 3.29/1.25 , =^#(nil(), .(y, z)) -> c_6() 3.29/1.25 , =^#(nil(), nil()) -> c_7() } 3.29/1.25 Weak Trs: 3.29/1.25 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 3.29/1.25 , =(.(x, y), nil()) -> false() 3.29/1.25 , =(nil(), .(y, z)) -> false() 3.29/1.25 , =(nil(), nil()) -> true() } 3.29/1.25 Obligation: 3.29/1.25 runtime complexity 3.29/1.25 Answer: 3.29/1.25 YES(O(1),O(n^1)) 3.29/1.25 3.29/1.25 We estimate the number of application of {4,5,6,7} by applications 3.29/1.25 of Pre({4,5,6,7}) = {3}. Here rules are labeled as follows: 3.29/1.25 3.29/1.25 DPs: 3.29/1.25 { 1: del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , 2: f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , 3: f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) 3.29/1.25 , 4: =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 3.29/1.25 , 5: =^#(.(x, y), nil()) -> c_5() 3.29/1.25 , 6: =^#(nil(), .(y, z)) -> c_6() 3.29/1.25 , 7: =^#(nil(), nil()) -> c_7() } 3.29/1.25 3.29/1.25 We are left with following problem, upon which TcT provides the 3.29/1.25 certificate YES(O(1),O(n^1)). 3.29/1.25 3.29/1.25 Strict DPs: 3.29/1.25 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) } 3.29/1.25 Weak DPs: 3.29/1.25 { =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 3.29/1.25 , =^#(.(x, y), nil()) -> c_5() 3.29/1.25 , =^#(nil(), .(y, z)) -> c_6() 3.29/1.25 , =^#(nil(), nil()) -> c_7() } 3.29/1.25 Weak Trs: 3.29/1.25 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 3.29/1.25 , =(.(x, y), nil()) -> false() 3.29/1.25 , =(nil(), .(y, z)) -> false() 3.29/1.25 , =(nil(), nil()) -> true() } 3.29/1.25 Obligation: 3.29/1.25 runtime complexity 3.29/1.25 Answer: 3.29/1.25 YES(O(1),O(n^1)) 3.29/1.25 3.29/1.25 The following weak DPs constitute a sub-graph of the DG that is 3.29/1.25 closed under successors. The DPs are removed. 3.29/1.25 3.29/1.25 { =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) 3.29/1.25 , =^#(.(x, y), nil()) -> c_5() 3.29/1.25 , =^#(nil(), .(y, z)) -> c_6() 3.29/1.25 , =^#(nil(), nil()) -> c_7() } 3.29/1.25 3.29/1.25 We are left with following problem, upon which TcT provides the 3.29/1.25 certificate YES(O(1),O(n^1)). 3.29/1.25 3.29/1.25 Strict DPs: 3.29/1.25 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) } 3.29/1.25 Weak Trs: 3.29/1.25 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 3.29/1.25 , =(.(x, y), nil()) -> false() 3.29/1.25 , =(nil(), .(y, z)) -> false() 3.29/1.25 , =(nil(), nil()) -> true() } 3.29/1.25 Obligation: 3.29/1.25 runtime complexity 3.29/1.25 Answer: 3.29/1.25 YES(O(1),O(n^1)) 3.29/1.25 3.29/1.25 We use the processor 'matrix interpretation of dimension 1' to 3.29/1.25 orient following rules strictly. 3.29/1.25 3.29/1.25 DPs: 3.29/1.25 { 2: f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , 3: f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) } 3.29/1.25 3.29/1.25 Sub-proof: 3.29/1.25 ---------- 3.29/1.25 The following argument positions are usable: 3.29/1.25 Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {2} 3.29/1.25 3.29/1.25 TcT has computed the following constructor-based matrix 3.29/1.25 interpretation satisfying not(EDA). 3.29/1.25 3.29/1.25 [.](x1, x2) = [1] x2 + [1] 3.29/1.25 3.29/1.25 [=](x1, x2) = [0] 3.29/1.25 3.29/1.25 [true] = [0] 3.29/1.25 3.29/1.25 [false] = [0] 3.29/1.25 3.29/1.25 [nil] = [0] 3.29/1.25 3.29/1.25 [u] = [0] 3.29/1.25 3.29/1.25 [v] = [0] 3.29/1.25 3.29/1.25 [and](x1, x2) = [3] 3.29/1.25 3.29/1.25 [del^#](x1) = [1] x1 + [2] 3.29/1.25 3.29/1.25 [c_1](x1) = [1] x1 + [0] 3.29/1.25 3.29/1.25 [f^#](x1, x2, x3, x4) = [1] x4 + [4] 3.29/1.25 3.29/1.25 [c_2](x1) = [1] x1 + [0] 3.29/1.25 3.29/1.25 [c_3](x1, x2) = [1] x2 + [0] 3.29/1.25 3.29/1.25 The order satisfies the following ordering constraints: 3.29/1.25 3.29/1.25 [=(.(x, y), .(u(), v()))] = [0] 3.29/1.25 ? [3] 3.29/1.25 = [and(=(x, u()), =(y, v()))] 3.29/1.25 3.29/1.25 [=(.(x, y), nil())] = [0] 3.29/1.25 >= [0] 3.29/1.25 = [false()] 3.29/1.25 3.29/1.25 [=(nil(), .(y, z))] = [0] 3.29/1.25 >= [0] 3.29/1.25 = [false()] 3.29/1.25 3.29/1.25 [=(nil(), nil())] = [0] 3.29/1.25 >= [0] 3.29/1.25 = [true()] 3.29/1.25 3.29/1.25 [del^#(.(x, .(y, z)))] = [1] z + [4] 3.29/1.25 >= [1] z + [4] 3.29/1.25 = [c_1(f^#(=(x, y), x, y, z))] 3.29/1.25 3.29/1.25 [f^#(true(), x, y, z)] = [1] z + [4] 3.29/1.25 > [1] z + [3] 3.29/1.25 = [c_2(del^#(.(y, z)))] 3.29/1.25 3.29/1.25 [f^#(false(), x, y, z)] = [1] z + [4] 3.29/1.25 > [1] z + [3] 3.29/1.25 = [c_3(x, del^#(.(y, z)))] 3.29/1.25 3.29/1.25 3.29/1.25 We return to the main proof. Consider the set of all dependency 3.29/1.25 pairs 3.29/1.25 3.29/1.25 : 3.29/1.25 { 1: del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , 2: f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , 3: f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) } 3.29/1.25 3.29/1.25 Processor 'matrix interpretation of dimension 1' induces the 3.29/1.25 complexity certificate YES(?,O(n^1)) on application of dependency 3.29/1.25 pairs {2,3}. These cover all (indirect) predecessors of dependency 3.29/1.25 pairs {1,2,3}, their number of application is equally bounded. The 3.29/1.25 dependency pairs are shifted into the weak component. 3.29/1.25 3.29/1.25 We are left with following problem, upon which TcT provides the 3.29/1.25 certificate YES(O(1),O(1)). 3.29/1.25 3.29/1.25 Weak DPs: 3.29/1.25 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) } 3.29/1.25 Weak Trs: 3.29/1.25 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 3.29/1.25 , =(.(x, y), nil()) -> false() 3.29/1.25 , =(nil(), .(y, z)) -> false() 3.29/1.25 , =(nil(), nil()) -> true() } 3.29/1.25 Obligation: 3.29/1.25 runtime complexity 3.29/1.25 Answer: 3.29/1.25 YES(O(1),O(1)) 3.29/1.25 3.29/1.25 The following weak DPs constitute a sub-graph of the DG that is 3.29/1.25 closed under successors. The DPs are removed. 3.29/1.25 3.29/1.25 { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) 3.29/1.25 , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) 3.29/1.25 , f^#(false(), x, y, z) -> c_3(x, del^#(.(y, z))) } 3.29/1.25 3.29/1.25 We are left with following problem, upon which TcT provides the 3.29/1.25 certificate YES(O(1),O(1)). 3.29/1.25 3.29/1.25 Weak Trs: 3.29/1.25 { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) 3.29/1.25 , =(.(x, y), nil()) -> false() 3.29/1.25 , =(nil(), .(y, z)) -> false() 3.29/1.25 , =(nil(), nil()) -> true() } 3.29/1.25 Obligation: 3.29/1.25 runtime complexity 3.29/1.25 Answer: 3.29/1.25 YES(O(1),O(1)) 3.29/1.25 3.29/1.25 No rule is usable, rules are removed from the input problem. 3.29/1.25 3.29/1.25 We are left with following problem, upon which TcT provides the 3.29/1.25 certificate YES(O(1),O(1)). 3.29/1.25 3.29/1.25 Rules: Empty 3.29/1.25 Obligation: 3.29/1.25 runtime complexity 3.29/1.25 Answer: 3.29/1.25 YES(O(1),O(1)) 3.29/1.25 3.29/1.25 Empty rules are trivially bounded 3.29/1.25 3.29/1.25 Hurray, we answered YES(O(1),O(n^1)) 3.29/1.25 EOF