YES(O(1),O(n^10)) 238.20/86.04 YES(O(1),O(n^10)) 238.20/86.04 238.20/86.04 We are left with following problem, upon which TcT provides the 238.20/86.04 certificate YES(O(1),O(n^10)). 238.20/86.04 238.20/86.04 Strict Trs: 238.20/86.04 { f_0(x) -> a() 238.20/86.04 , f_1(x) -> g_1(x, x) 238.20/86.04 , g_1(s(x), y) -> b(f_0(y), g_1(x, y)) 238.20/86.04 , f_2(x) -> g_2(x, x) 238.20/86.04 , g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 238.20/86.04 , f_3(x) -> g_3(x, x) 238.20/86.04 , g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 238.20/86.04 , f_4(x) -> g_4(x, x) 238.20/86.04 , g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 238.20/86.04 , f_5(x) -> g_5(x, x) 238.20/86.04 , g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 238.20/86.04 , f_6(x) -> g_6(x, x) 238.20/86.04 , g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 238.20/86.04 , f_7(x) -> g_7(x, x) 238.20/86.04 , g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 238.20/86.04 , f_8(x) -> g_8(x, x) 238.20/86.04 , g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 238.20/86.04 , f_9(x) -> g_9(x, x) 238.20/86.04 , g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 238.20/86.04 , f_10(x) -> g_10(x, x) 238.20/86.04 , g_10(s(x), y) -> b(f_9(y), g_10(x, y)) } 238.20/86.04 Obligation: 238.20/86.04 runtime complexity 238.20/86.04 Answer: 238.20/86.04 YES(O(1),O(n^10)) 238.20/86.04 238.20/86.04 We add the following weak dependency pairs: 238.20/86.04 238.20/86.04 Strict DPs: 238.20/86.04 { f_0^#(x) -> c_1() 238.20/86.04 , f_1^#(x) -> c_2(g_1^#(x, x)) 238.20/86.04 , g_1^#(s(x), y) -> c_3(f_0^#(y), g_1^#(x, y)) 238.20/86.04 , f_2^#(x) -> c_4(g_2^#(x, x)) 238.20/86.04 , g_2^#(s(x), y) -> c_5(f_1^#(y), g_2^#(x, y)) 238.20/86.04 , f_3^#(x) -> c_6(g_3^#(x, x)) 238.20/86.04 , g_3^#(s(x), y) -> c_7(f_2^#(y), g_3^#(x, y)) 238.20/86.04 , f_4^#(x) -> c_8(g_4^#(x, x)) 238.20/86.04 , g_4^#(s(x), y) -> c_9(f_3^#(y), g_4^#(x, y)) 238.20/86.04 , f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.04 , g_5^#(s(x), y) -> c_11(f_4^#(y), g_5^#(x, y)) 238.20/86.04 , f_6^#(x) -> c_12(g_6^#(x, x)) 238.20/86.04 , g_6^#(s(x), y) -> c_13(f_5^#(y), g_6^#(x, y)) 238.20/86.04 , f_7^#(x) -> c_14(g_7^#(x, x)) 238.20/86.04 , g_7^#(s(x), y) -> c_15(f_6^#(y), g_7^#(x, y)) 238.20/86.04 , f_8^#(x) -> c_16(g_8^#(x, x)) 238.20/86.04 , g_8^#(s(x), y) -> c_17(f_7^#(y), g_8^#(x, y)) 238.20/86.04 , f_9^#(x) -> c_18(g_9^#(x, x)) 238.20/86.04 , g_9^#(s(x), y) -> c_19(f_8^#(y), g_9^#(x, y)) 238.20/86.04 , f_10^#(x) -> c_20(g_10^#(x, x)) 238.20/86.04 , g_10^#(s(x), y) -> c_21(f_9^#(y), g_10^#(x, y)) } 238.20/86.04 238.20/86.04 and mark the set of starting terms. 238.20/86.04 238.20/86.04 We are left with following problem, upon which TcT provides the 238.20/86.04 certificate YES(O(1),O(n^10)). 238.20/86.04 238.20/86.04 Strict DPs: 238.20/86.04 { f_0^#(x) -> c_1() 238.20/86.04 , f_1^#(x) -> c_2(g_1^#(x, x)) 238.20/86.04 , g_1^#(s(x), y) -> c_3(f_0^#(y), g_1^#(x, y)) 238.20/86.04 , f_2^#(x) -> c_4(g_2^#(x, x)) 238.20/86.04 , g_2^#(s(x), y) -> c_5(f_1^#(y), g_2^#(x, y)) 238.20/86.04 , f_3^#(x) -> c_6(g_3^#(x, x)) 238.20/86.04 , g_3^#(s(x), y) -> c_7(f_2^#(y), g_3^#(x, y)) 238.20/86.04 , f_4^#(x) -> c_8(g_4^#(x, x)) 238.20/86.04 , g_4^#(s(x), y) -> c_9(f_3^#(y), g_4^#(x, y)) 238.20/86.04 , f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.04 , g_5^#(s(x), y) -> c_11(f_4^#(y), g_5^#(x, y)) 238.20/86.04 , f_6^#(x) -> c_12(g_6^#(x, x)) 238.20/86.04 , g_6^#(s(x), y) -> c_13(f_5^#(y), g_6^#(x, y)) 238.20/86.04 , f_7^#(x) -> c_14(g_7^#(x, x)) 238.20/86.04 , g_7^#(s(x), y) -> c_15(f_6^#(y), g_7^#(x, y)) 238.20/86.04 , f_8^#(x) -> c_16(g_8^#(x, x)) 238.20/86.04 , g_8^#(s(x), y) -> c_17(f_7^#(y), g_8^#(x, y)) 238.20/86.04 , f_9^#(x) -> c_18(g_9^#(x, x)) 238.20/86.04 , g_9^#(s(x), y) -> c_19(f_8^#(y), g_9^#(x, y)) 238.20/86.04 , f_10^#(x) -> c_20(g_10^#(x, x)) 238.20/86.04 , g_10^#(s(x), y) -> c_21(f_9^#(y), g_10^#(x, y)) } 238.20/86.04 Strict Trs: 238.20/86.04 { f_0(x) -> a() 238.20/86.04 , f_1(x) -> g_1(x, x) 238.20/86.04 , g_1(s(x), y) -> b(f_0(y), g_1(x, y)) 238.20/86.04 , f_2(x) -> g_2(x, x) 238.20/86.04 , g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 238.20/86.04 , f_3(x) -> g_3(x, x) 238.20/86.04 , g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 238.20/86.04 , f_4(x) -> g_4(x, x) 238.20/86.04 , g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 238.20/86.04 , f_5(x) -> g_5(x, x) 238.20/86.04 , g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 238.20/86.04 , f_6(x) -> g_6(x, x) 238.20/86.04 , g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 238.20/86.04 , f_7(x) -> g_7(x, x) 238.20/86.04 , g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 238.20/86.04 , f_8(x) -> g_8(x, x) 238.20/86.04 , g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 238.20/86.04 , f_9(x) -> g_9(x, x) 238.20/86.04 , g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 238.20/86.04 , f_10(x) -> g_10(x, x) 238.20/86.04 , g_10(s(x), y) -> b(f_9(y), g_10(x, y)) } 238.20/86.04 Obligation: 238.20/86.04 runtime complexity 238.20/86.04 Answer: 238.20/86.04 YES(O(1),O(n^10)) 238.20/86.04 238.20/86.04 No rule is usable, rules are removed from the input problem. 238.20/86.04 238.20/86.04 We are left with following problem, upon which TcT provides the 238.20/86.04 certificate YES(O(1),O(n^10)). 238.20/86.04 238.20/86.04 Strict DPs: 238.20/86.04 { f_0^#(x) -> c_1() 238.20/86.04 , f_1^#(x) -> c_2(g_1^#(x, x)) 238.20/86.04 , g_1^#(s(x), y) -> c_3(f_0^#(y), g_1^#(x, y)) 238.20/86.04 , f_2^#(x) -> c_4(g_2^#(x, x)) 238.20/86.04 , g_2^#(s(x), y) -> c_5(f_1^#(y), g_2^#(x, y)) 238.20/86.04 , f_3^#(x) -> c_6(g_3^#(x, x)) 238.20/86.04 , g_3^#(s(x), y) -> c_7(f_2^#(y), g_3^#(x, y)) 238.20/86.04 , f_4^#(x) -> c_8(g_4^#(x, x)) 238.20/86.04 , g_4^#(s(x), y) -> c_9(f_3^#(y), g_4^#(x, y)) 238.20/86.04 , f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.04 , g_5^#(s(x), y) -> c_11(f_4^#(y), g_5^#(x, y)) 238.20/86.04 , f_6^#(x) -> c_12(g_6^#(x, x)) 238.20/86.04 , g_6^#(s(x), y) -> c_13(f_5^#(y), g_6^#(x, y)) 238.20/86.04 , f_7^#(x) -> c_14(g_7^#(x, x)) 238.20/86.04 , g_7^#(s(x), y) -> c_15(f_6^#(y), g_7^#(x, y)) 238.20/86.04 , f_8^#(x) -> c_16(g_8^#(x, x)) 238.20/86.04 , g_8^#(s(x), y) -> c_17(f_7^#(y), g_8^#(x, y)) 238.20/86.04 , f_9^#(x) -> c_18(g_9^#(x, x)) 238.20/86.04 , g_9^#(s(x), y) -> c_19(f_8^#(y), g_9^#(x, y)) 238.20/86.04 , f_10^#(x) -> c_20(g_10^#(x, x)) 238.20/86.04 , g_10^#(s(x), y) -> c_21(f_9^#(y), g_10^#(x, y)) } 238.20/86.04 Obligation: 238.20/86.04 runtime complexity 238.20/86.04 Answer: 238.20/86.04 YES(O(1),O(n^10)) 238.20/86.04 238.20/86.04 The weightgap principle applies (using the following constant 238.20/86.04 growth matrix-interpretation) 238.20/86.04 238.20/86.04 The following argument positions are usable: 238.20/86.04 Uargs(c_2) = {1}, Uargs(c_3) = {1, 2}, Uargs(c_4) = {1}, 238.20/86.04 Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}, Uargs(c_7) = {1, 2}, 238.20/86.04 Uargs(c_8) = {1}, Uargs(c_9) = {1, 2}, Uargs(c_10) = {1}, 238.20/86.04 Uargs(c_11) = {1, 2}, Uargs(c_12) = {1}, Uargs(c_13) = {1, 2}, 238.20/86.04 Uargs(c_14) = {1}, Uargs(c_15) = {1, 2}, Uargs(c_16) = {1}, 238.20/86.04 Uargs(c_17) = {1, 2}, Uargs(c_18) = {1}, Uargs(c_19) = {1, 2}, 238.20/86.04 Uargs(c_20) = {1}, Uargs(c_21) = {1, 2} 238.20/86.04 238.20/86.04 TcT has computed the following constructor-restricted matrix 238.20/86.04 interpretation. 238.20/86.04 238.20/86.04 [s](x1) = [1 2] x1 + [1] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [f_0^#](x1) = [0 0] x1 + [1] 238.20/86.04 [1 1] [1] 238.20/86.04 238.20/86.04 [c_1] = [0] 238.20/86.04 [1] 238.20/86.04 238.20/86.04 [f_1^#](x1) = [0 0] x1 + [1] 238.20/86.04 [2 2] [1] 238.20/86.04 238.20/86.04 [c_2](x1) = [1 0] x1 + [1] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_1^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 238.20/86.04 [2 2] [1 1] [2] 238.20/86.04 238.20/86.04 [c_3](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 238.20/86.04 [0 1] [0 1] [0] 238.20/86.04 238.20/86.04 [f_2^#](x1) = [0 0] x1 + [2] 238.20/86.04 [2 2] [2] 238.20/86.04 238.20/86.04 [c_4](x1) = [1 0] x1 + [0] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_2^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1] 238.20/86.04 [1 2] [1 0] [1] 238.20/86.04 238.20/86.04 [c_5](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 238.20/86.04 [0 1] [0 1] [1] 238.20/86.04 238.20/86.04 [f_3^#](x1) = [0 0] x1 + [1] 238.20/86.04 [1 2] [1] 238.20/86.04 238.20/86.04 [c_6](x1) = [1 0] x1 + [1] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_3^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 238.20/86.04 [2 2] [1 1] [2] 238.20/86.04 238.20/86.04 [c_7](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 238.20/86.04 [0 1] [0 1] [1] 238.20/86.04 238.20/86.04 [f_4^#](x1) = [0 0] x1 + [1] 238.20/86.04 [2 1] [1] 238.20/86.04 238.20/86.04 [c_8](x1) = [1 0] x1 + [2] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_4^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 238.20/86.04 [2 2] [1 1] [2] 238.20/86.04 238.20/86.04 [c_9](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 238.20/86.04 [0 1] [0 1] [1] 238.20/86.04 238.20/86.04 [f_5^#](x1) = [0 0] x1 + [1] 238.20/86.04 [2 1] [1] 238.20/86.04 238.20/86.04 [c_10](x1) = [1 0] x1 + [1] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_5^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 238.20/86.04 [2 2] [1 1] [2] 238.20/86.04 238.20/86.04 [c_11](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 238.20/86.04 [0 1] [0 1] [1] 238.20/86.04 238.20/86.04 [f_6^#](x1) = [0 0] x1 + [1] 238.20/86.04 [2 1] [1] 238.20/86.04 238.20/86.04 [c_12](x1) = [1 0] x1 + [2] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_6^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 238.20/86.04 [1 1] [1 1] [2] 238.20/86.04 238.20/86.04 [c_13](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 238.20/86.04 [0 1] [0 1] [1] 238.20/86.04 238.20/86.04 [f_7^#](x1) = [0 0] x1 + [1] 238.20/86.04 [2 2] [1] 238.20/86.04 238.20/86.04 [c_14](x1) = [1 0] x1 + [2] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_7^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 238.20/86.04 [2 2] [1 1] [2] 238.20/86.04 238.20/86.04 [c_15](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 238.20/86.04 [0 1] [0 1] [1] 238.20/86.04 238.20/86.04 [f_8^#](x1) = [0 0] x1 + [1] 238.20/86.04 [2 1] [1] 238.20/86.04 238.20/86.04 [c_16](x1) = [1 0] x1 + [0] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_8^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 238.20/86.04 [2 2] [1 1] [2] 238.20/86.04 238.20/86.04 [c_17](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 238.20/86.04 [0 1] [0 1] [0] 238.20/86.04 238.20/86.04 [f_9^#](x1) = [1] 238.20/86.04 [1] 238.20/86.04 238.20/86.04 [c_18](x1) = [1 0] x1 + [2] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_9^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 238.20/86.04 [2 2] [1 1] [2] 238.20/86.04 238.20/86.04 [c_19](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 238.20/86.04 [0 1] [0 1] [1] 238.20/86.04 238.20/86.04 [f_10^#](x1) = [2 2] x1 + [2] 238.20/86.04 [2 2] [2] 238.20/86.04 238.20/86.04 [c_20](x1) = [1 0] x1 + [1] 238.20/86.04 [0 1] [1] 238.20/86.04 238.20/86.04 [g_10^#](x1, x2) = [2 2] x1 + [0 0] x2 + [2] 238.20/86.04 [2 2] [2 2] [2] 238.20/86.04 238.20/86.04 [c_21](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 238.20/86.04 [0 1] [0 1] [1] 238.20/86.04 238.20/86.04 The order satisfies the following ordering constraints: 238.20/86.04 238.20/86.04 [f_0^#(x)] = [0 0] x + [1] 238.20/86.04 [1 1] [1] 238.20/86.04 > [0] 238.20/86.04 [1] 238.20/86.04 = [c_1()] 238.20/86.04 238.20/86.04 [f_1^#(x)] = [0 0] x + [1] 238.20/86.04 [2 2] [1] 238.20/86.04 ? [0 0] x + [3] 238.20/86.04 [3 3] [3] 238.20/86.04 = [c_2(g_1^#(x, x))] 238.20/86.04 238.20/86.04 [g_1^#(s(x), y)] = [0 0] x + [0 0] y + [2] 238.20/86.04 [2 6] [1 1] [6] 238.20/86.04 ? [0 0] x + [0 0] y + [3] 238.20/86.04 [2 2] [2 2] [3] 238.20/86.04 = [c_3(f_0^#(y), g_1^#(x, y))] 238.20/86.04 238.20/86.04 [f_2^#(x)] = [0 0] x + [2] 238.20/86.04 [2 2] [2] 238.20/86.04 > [0 0] x + [1] 238.20/86.04 [2 2] [2] 238.20/86.04 = [c_4(g_2^#(x, x))] 238.20/86.04 238.20/86.04 [g_2^#(s(x), y)] = [0 0] x + [0 0] y + [1] 238.20/86.04 [1 4] [1 0] [4] 238.20/86.04 ? [0 0] x + [0 0] y + [3] 238.20/86.04 [1 2] [3 2] [3] 238.20/86.04 = [c_5(f_1^#(y), g_2^#(x, y))] 238.20/86.04 238.20/86.04 [f_3^#(x)] = [0 0] x + [1] 238.20/86.04 [1 2] [1] 238.20/86.04 ? [0 0] x + [3] 238.20/86.04 [3 3] [3] 238.20/86.04 = [c_6(g_3^#(x, x))] 238.20/86.04 238.20/86.04 [g_3^#(s(x), y)] = [0 0] x + [0 0] y + [2] 238.20/86.04 [2 6] [1 1] [6] 238.20/86.04 ? [0 0] x + [0 0] y + [6] 238.20/86.04 [2 2] [3 3] [5] 238.20/86.04 = [c_7(f_2^#(y), g_3^#(x, y))] 238.20/86.04 238.20/86.04 [f_4^#(x)] = [0 0] x + [1] 238.20/86.04 [2 1] [1] 238.20/86.04 ? [0 0] x + [4] 238.20/86.04 [3 3] [3] 238.20/86.04 = [c_8(g_4^#(x, x))] 238.20/86.04 238.20/86.04 [g_4^#(s(x), y)] = [0 0] x + [0 0] y + [2] 238.20/86.04 [2 6] [1 1] [6] 238.20/86.04 ? [0 0] x + [0 0] y + [3] 238.20/86.04 [2 2] [2 3] [4] 238.20/86.04 = [c_9(f_3^#(y), g_4^#(x, y))] 238.20/86.04 238.20/86.04 [f_5^#(x)] = [0 0] x + [1] 238.20/86.04 [2 1] [1] 238.20/86.04 ? [0 0] x + [3] 238.20/86.04 [3 3] [3] 238.20/86.04 = [c_10(g_5^#(x, x))] 238.20/86.04 238.20/86.04 [g_5^#(s(x), y)] = [0 0] x + [0 0] y + [2] 238.20/86.04 [2 6] [1 1] [6] 238.20/86.04 ? [0 0] x + [0 0] y + [3] 238.20/86.04 [2 2] [3 2] [4] 238.20/86.04 = [c_11(f_4^#(y), g_5^#(x, y))] 238.20/86.04 238.20/86.04 [f_6^#(x)] = [0 0] x + [1] 238.20/86.04 [2 1] [1] 238.20/86.04 ? [0 0] x + [4] 238.20/86.04 [2 2] [3] 238.20/86.04 = [c_12(g_6^#(x, x))] 238.20/86.04 238.20/86.04 [g_6^#(s(x), y)] = [0 0] x + [0 0] y + [2] 238.20/86.04 [1 3] [1 1] [4] 238.20/86.04 ? [0 0] x + [0 0] y + [3] 238.20/86.04 [1 1] [3 2] [4] 238.20/86.04 = [c_13(f_5^#(y), g_6^#(x, y))] 238.20/86.04 238.20/86.04 [f_7^#(x)] = [0 0] x + [1] 238.20/86.04 [2 2] [1] 238.20/86.04 ? [0 0] x + [4] 238.20/86.04 [3 3] [3] 238.20/86.04 = [c_14(g_7^#(x, x))] 238.20/86.04 238.20/86.04 [g_7^#(s(x), y)] = [0 0] x + [0 0] y + [2] 238.20/86.04 [2 6] [1 1] [6] 238.20/86.04 ? [0 0] x + [0 0] y + [4] 238.20/86.04 [2 2] [3 2] [4] 238.20/86.04 = [c_15(f_6^#(y), g_7^#(x, y))] 238.20/86.04 238.20/86.04 [f_8^#(x)] = [0 0] x + [1] 238.20/86.04 [2 1] [1] 238.20/86.04 ? [0 0] x + [2] 238.20/86.04 [3 3] [3] 238.20/86.04 = [c_16(g_8^#(x, x))] 238.20/86.04 238.20/86.04 [g_8^#(s(x), y)] = [0 0] x + [0 0] y + [2] 238.20/86.04 [2 6] [1 1] [6] 238.20/86.04 ? [0 0] x + [0 0] y + [4] 238.20/86.04 [2 2] [3 3] [3] 238.20/86.04 = [c_17(f_7^#(y), g_8^#(x, y))] 238.20/86.04 238.20/86.04 [f_9^#(x)] = [1] 238.20/86.04 [1] 238.20/86.04 ? [0 0] x + [4] 238.20/86.04 [3 3] [3] 238.20/86.04 = [c_18(g_9^#(x, x))] 238.20/86.04 238.20/86.04 [g_9^#(s(x), y)] = [0 0] x + [0 0] y + [2] 238.20/86.04 [2 6] [1 1] [6] 238.20/86.04 ? [0 0] x + [0 0] y + [4] 238.20/86.04 [2 2] [3 2] [4] 238.20/86.04 = [c_19(f_8^#(y), g_9^#(x, y))] 238.20/86.04 238.20/86.04 [f_10^#(x)] = [2 2] x + [2] 238.20/86.04 [2 2] [2] 238.20/86.04 ? [2 2] x + [3] 238.20/86.04 [4 4] [3] 238.20/86.04 = [c_20(g_10^#(x, x))] 238.20/86.04 238.20/86.04 [g_10^#(s(x), y)] = [2 6] x + [0 0] y + [6] 238.20/86.04 [2 6] [2 2] [6] 238.20/86.04 > [2 2] x + [0 0] y + [3] 238.20/86.04 [2 2] [2 2] [4] 238.20/86.04 = [c_21(f_9^#(y), g_10^#(x, y))] 238.20/86.04 238.20/86.04 238.20/86.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 238.20/86.04 238.20/86.04 We are left with following problem, upon which TcT provides the 238.20/86.04 certificate YES(O(1),O(n^10)). 238.20/86.04 238.20/86.04 Strict DPs: 238.20/86.04 { f_1^#(x) -> c_2(g_1^#(x, x)) 238.20/86.04 , g_1^#(s(x), y) -> c_3(f_0^#(y), g_1^#(x, y)) 238.20/86.04 , g_2^#(s(x), y) -> c_5(f_1^#(y), g_2^#(x, y)) 238.20/86.04 , f_3^#(x) -> c_6(g_3^#(x, x)) 238.20/86.04 , g_3^#(s(x), y) -> c_7(f_2^#(y), g_3^#(x, y)) 238.20/86.04 , f_4^#(x) -> c_8(g_4^#(x, x)) 238.20/86.04 , g_4^#(s(x), y) -> c_9(f_3^#(y), g_4^#(x, y)) 238.20/86.04 , f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.04 , g_5^#(s(x), y) -> c_11(f_4^#(y), g_5^#(x, y)) 238.20/86.04 , f_6^#(x) -> c_12(g_6^#(x, x)) 238.20/86.04 , g_6^#(s(x), y) -> c_13(f_5^#(y), g_6^#(x, y)) 238.20/86.04 , f_7^#(x) -> c_14(g_7^#(x, x)) 238.20/86.04 , g_7^#(s(x), y) -> c_15(f_6^#(y), g_7^#(x, y)) 238.20/86.04 , f_8^#(x) -> c_16(g_8^#(x, x)) 238.20/86.04 , g_8^#(s(x), y) -> c_17(f_7^#(y), g_8^#(x, y)) 238.20/86.04 , f_9^#(x) -> c_18(g_9^#(x, x)) 238.20/86.04 , g_9^#(s(x), y) -> c_19(f_8^#(y), g_9^#(x, y)) 238.20/86.04 , f_10^#(x) -> c_20(g_10^#(x, x)) } 238.20/86.04 Weak DPs: 238.20/86.04 { f_0^#(x) -> c_1() 238.20/86.04 , f_2^#(x) -> c_4(g_2^#(x, x)) 238.20/86.04 , g_10^#(s(x), y) -> c_21(f_9^#(y), g_10^#(x, y)) } 238.20/86.04 Obligation: 238.20/86.04 runtime complexity 238.20/86.04 Answer: 238.20/86.04 YES(O(1),O(n^10)) 238.20/86.04 238.20/86.04 We estimate the number of application of {18} by applications of 238.20/86.04 Pre({18}) = {}. Here rules are labeled as follows: 238.20/86.04 238.20/86.04 DPs: 238.20/86.04 { 1: f_1^#(x) -> c_2(g_1^#(x, x)) 238.20/86.04 , 2: g_1^#(s(x), y) -> c_3(f_0^#(y), g_1^#(x, y)) 238.20/86.04 , 3: g_2^#(s(x), y) -> c_5(f_1^#(y), g_2^#(x, y)) 238.20/86.04 , 4: f_3^#(x) -> c_6(g_3^#(x, x)) 238.20/86.04 , 5: g_3^#(s(x), y) -> c_7(f_2^#(y), g_3^#(x, y)) 238.20/86.04 , 6: f_4^#(x) -> c_8(g_4^#(x, x)) 238.20/86.04 , 7: g_4^#(s(x), y) -> c_9(f_3^#(y), g_4^#(x, y)) 238.20/86.04 , 8: f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.04 , 9: g_5^#(s(x), y) -> c_11(f_4^#(y), g_5^#(x, y)) 238.20/86.04 , 10: f_6^#(x) -> c_12(g_6^#(x, x)) 238.20/86.04 , 11: g_6^#(s(x), y) -> c_13(f_5^#(y), g_6^#(x, y)) 238.20/86.04 , 12: f_7^#(x) -> c_14(g_7^#(x, x)) 238.20/86.04 , 13: g_7^#(s(x), y) -> c_15(f_6^#(y), g_7^#(x, y)) 238.20/86.04 , 14: f_8^#(x) -> c_16(g_8^#(x, x)) 238.20/86.04 , 15: g_8^#(s(x), y) -> c_17(f_7^#(y), g_8^#(x, y)) 238.20/86.04 , 16: f_9^#(x) -> c_18(g_9^#(x, x)) 238.20/86.04 , 17: g_9^#(s(x), y) -> c_19(f_8^#(y), g_9^#(x, y)) 238.20/86.04 , 18: f_10^#(x) -> c_20(g_10^#(x, x)) 238.20/86.04 , 19: f_0^#(x) -> c_1() 238.20/86.04 , 20: f_2^#(x) -> c_4(g_2^#(x, x)) 238.20/86.04 , 21: g_10^#(s(x), y) -> c_21(f_9^#(y), g_10^#(x, y)) } 238.20/86.04 238.20/86.04 We are left with following problem, upon which TcT provides the 238.20/86.04 certificate YES(O(1),O(n^10)). 238.20/86.04 238.20/86.04 Strict DPs: 238.20/86.04 { f_1^#(x) -> c_2(g_1^#(x, x)) 238.20/86.04 , g_1^#(s(x), y) -> c_3(f_0^#(y), g_1^#(x, y)) 238.20/86.04 , g_2^#(s(x), y) -> c_5(f_1^#(y), g_2^#(x, y)) 238.20/86.04 , f_3^#(x) -> c_6(g_3^#(x, x)) 238.20/86.04 , g_3^#(s(x), y) -> c_7(f_2^#(y), g_3^#(x, y)) 238.20/86.04 , f_4^#(x) -> c_8(g_4^#(x, x)) 238.20/86.04 , g_4^#(s(x), y) -> c_9(f_3^#(y), g_4^#(x, y)) 238.20/86.04 , f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.04 , g_5^#(s(x), y) -> c_11(f_4^#(y), g_5^#(x, y)) 238.20/86.04 , f_6^#(x) -> c_12(g_6^#(x, x)) 238.20/86.04 , g_6^#(s(x), y) -> c_13(f_5^#(y), g_6^#(x, y)) 238.20/86.04 , f_7^#(x) -> c_14(g_7^#(x, x)) 238.20/86.04 , g_7^#(s(x), y) -> c_15(f_6^#(y), g_7^#(x, y)) 238.20/86.04 , f_8^#(x) -> c_16(g_8^#(x, x)) 238.20/86.04 , g_8^#(s(x), y) -> c_17(f_7^#(y), g_8^#(x, y)) 238.20/86.04 , f_9^#(x) -> c_18(g_9^#(x, x)) 238.20/86.04 , g_9^#(s(x), y) -> c_19(f_8^#(y), g_9^#(x, y)) } 238.20/86.04 Weak DPs: 238.20/86.04 { f_0^#(x) -> c_1() 238.20/86.04 , f_2^#(x) -> c_4(g_2^#(x, x)) 238.20/86.04 , f_10^#(x) -> c_20(g_10^#(x, x)) 238.20/86.04 , g_10^#(s(x), y) -> c_21(f_9^#(y), g_10^#(x, y)) } 238.20/86.04 Obligation: 238.20/86.04 runtime complexity 238.20/86.04 Answer: 238.20/86.04 YES(O(1),O(n^10)) 238.20/86.04 238.20/86.04 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.04 closed under successors. The DPs are removed. 238.20/86.04 238.20/86.04 { f_0^#(x) -> c_1() } 238.20/86.04 238.20/86.04 We are left with following problem, upon which TcT provides the 238.20/86.04 certificate YES(O(1),O(n^10)). 238.20/86.04 238.20/86.04 Strict DPs: 238.20/86.04 { f_1^#(x) -> c_2(g_1^#(x, x)) 238.20/86.04 , g_1^#(s(x), y) -> c_3(f_0^#(y), g_1^#(x, y)) 238.20/86.04 , g_2^#(s(x), y) -> c_5(f_1^#(y), g_2^#(x, y)) 238.20/86.04 , f_3^#(x) -> c_6(g_3^#(x, x)) 238.20/86.04 , g_3^#(s(x), y) -> c_7(f_2^#(y), g_3^#(x, y)) 238.20/86.04 , f_4^#(x) -> c_8(g_4^#(x, x)) 238.20/86.04 , g_4^#(s(x), y) -> c_9(f_3^#(y), g_4^#(x, y)) 238.20/86.04 , f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.04 , g_5^#(s(x), y) -> c_11(f_4^#(y), g_5^#(x, y)) 238.20/86.04 , f_6^#(x) -> c_12(g_6^#(x, x)) 238.20/86.04 , g_6^#(s(x), y) -> c_13(f_5^#(y), g_6^#(x, y)) 238.20/86.04 , f_7^#(x) -> c_14(g_7^#(x, x)) 238.20/86.04 , g_7^#(s(x), y) -> c_15(f_6^#(y), g_7^#(x, y)) 238.20/86.04 , f_8^#(x) -> c_16(g_8^#(x, x)) 238.20/86.04 , g_8^#(s(x), y) -> c_17(f_7^#(y), g_8^#(x, y)) 238.20/86.04 , f_9^#(x) -> c_18(g_9^#(x, x)) 238.20/86.04 , g_9^#(s(x), y) -> c_19(f_8^#(y), g_9^#(x, y)) } 238.20/86.04 Weak DPs: 238.20/86.04 { f_2^#(x) -> c_4(g_2^#(x, x)) 238.20/86.04 , f_10^#(x) -> c_20(g_10^#(x, x)) 238.20/86.04 , g_10^#(s(x), y) -> c_21(f_9^#(y), g_10^#(x, y)) } 238.20/86.04 Obligation: 238.20/86.04 runtime complexity 238.20/86.04 Answer: 238.20/86.04 YES(O(1),O(n^10)) 238.20/86.04 238.20/86.04 Due to missing edges in the dependency-graph, the right-hand sides 238.20/86.04 of following rules could be simplified: 238.20/86.04 238.20/86.04 { g_1^#(s(x), y) -> c_3(f_0^#(y), g_1^#(x, y)) } 238.20/86.04 238.20/86.04 We are left with following problem, upon which TcT provides the 238.20/86.04 certificate YES(O(1),O(n^10)). 238.20/86.04 238.20/86.04 Strict DPs: 238.20/86.04 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.04 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.04 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.04 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.04 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.04 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.04 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.04 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.04 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.04 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.04 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.04 , f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.04 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.04 , f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.04 , g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) 238.20/86.04 , f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 , g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) } 238.20/86.05 Weak DPs: 238.20/86.05 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.05 , f_10^#(x) -> c_19(g_10^#(x, x)) 238.20/86.05 , g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) } 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(n^10)) 238.20/86.05 238.20/86.05 Consider the dependency graph 238.20/86.05 238.20/86.05 1: f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.05 -->_1 g_1^#(s(x), y) -> c_2(g_1^#(x, y)) :2 238.20/86.05 238.20/86.05 2: g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.05 -->_1 g_1^#(s(x), y) -> c_2(g_1^#(x, y)) :2 238.20/86.05 238.20/86.05 3: g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.05 -->_2 g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) :3 238.20/86.05 -->_1 f_1^#(x) -> c_1(g_1^#(x, x)) :1 238.20/86.05 238.20/86.05 4: f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.05 -->_1 g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) :5 238.20/86.05 238.20/86.05 5: g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.05 -->_1 f_2^#(x) -> c_18(g_2^#(x, x)) :18 238.20/86.05 -->_2 g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) :5 238.20/86.05 238.20/86.05 6: f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.05 -->_1 g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) :7 238.20/86.05 238.20/86.05 7: g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.05 -->_2 g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) :7 238.20/86.05 -->_1 f_3^#(x) -> c_4(g_3^#(x, x)) :4 238.20/86.05 238.20/86.05 8: f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.05 -->_1 g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) :9 238.20/86.05 238.20/86.05 9: g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.05 -->_2 g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) :9 238.20/86.05 -->_1 f_4^#(x) -> c_6(g_4^#(x, x)) :6 238.20/86.05 238.20/86.05 10: f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.05 -->_1 g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) :11 238.20/86.05 238.20/86.05 11: g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.05 -->_2 g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) :11 238.20/86.05 -->_1 f_5^#(x) -> c_8(g_5^#(x, x)) :8 238.20/86.05 238.20/86.05 12: f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.05 -->_1 g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) :13 238.20/86.05 238.20/86.05 13: g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.05 -->_2 g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) :13 238.20/86.05 -->_1 f_6^#(x) -> c_10(g_6^#(x, x)) :10 238.20/86.05 238.20/86.05 14: f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.05 -->_1 g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) :15 238.20/86.05 238.20/86.05 15: g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) 238.20/86.05 -->_2 g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) :15 238.20/86.05 -->_1 f_7^#(x) -> c_12(g_7^#(x, x)) :12 238.20/86.05 238.20/86.05 16: f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 -->_1 g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) :17 238.20/86.05 238.20/86.05 17: g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) 238.20/86.05 -->_2 g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) :17 238.20/86.05 -->_1 f_8^#(x) -> c_14(g_8^#(x, x)) :14 238.20/86.05 238.20/86.05 18: f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.05 -->_1 g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) :3 238.20/86.05 238.20/86.05 19: f_10^#(x) -> c_19(g_10^#(x, x)) 238.20/86.05 -->_1 g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) :20 238.20/86.05 238.20/86.05 20: g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) 238.20/86.05 -->_2 g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) :20 238.20/86.05 -->_1 f_9^#(x) -> c_16(g_9^#(x, x)) :16 238.20/86.05 238.20/86.05 238.20/86.05 Following roots of the dependency graph are removed, as the 238.20/86.05 considered set of starting terms is closed under reduction with 238.20/86.05 respect to these rules (modulo compound contexts). 238.20/86.05 238.20/86.05 { f_10^#(x) -> c_19(g_10^#(x, x)) } 238.20/86.05 238.20/86.05 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(n^10)). 238.20/86.05 238.20/86.05 Strict DPs: 238.20/86.05 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.05 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.05 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.05 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.05 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.05 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.05 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.05 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.05 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.05 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.05 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.05 , f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.05 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.05 , f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.05 , g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) 238.20/86.05 , f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 , g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) } 238.20/86.05 Weak DPs: 238.20/86.05 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.05 , g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) } 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(n^10)) 238.20/86.05 238.20/86.05 We decompose the input problem according to the dependency graph 238.20/86.05 into the upper component 238.20/86.05 238.20/86.05 { g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) } 238.20/86.05 238.20/86.05 and lower component 238.20/86.05 238.20/86.05 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.05 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.05 , f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.05 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.05 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.05 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.05 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.05 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.05 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.05 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.05 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.05 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.05 , f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.05 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.05 , f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.05 , g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) 238.20/86.05 , f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 , g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) } 238.20/86.05 238.20/86.05 Further, following extension rules are added to the lower 238.20/86.05 component. 238.20/86.05 238.20/86.05 { g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 238.20/86.05 TcT solves the upper component with certificate YES(O(1),O(n^1)). 238.20/86.05 238.20/86.05 Sub-proof: 238.20/86.05 ---------- 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(n^1)). 238.20/86.05 238.20/86.05 Strict DPs: { g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) } 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(n^1)) 238.20/86.05 238.20/86.05 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.05 orient following rules strictly. 238.20/86.05 238.20/86.05 DPs: 238.20/86.05 { 1: g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) } 238.20/86.05 238.20/86.05 Sub-proof: 238.20/86.05 ---------- 238.20/86.05 The following argument positions are usable: 238.20/86.05 Uargs(c_20) = {2} 238.20/86.05 238.20/86.05 TcT has computed the following constructor-based matrix 238.20/86.05 interpretation satisfying not(EDA). 238.20/86.05 238.20/86.05 [s](x1) = [1] x1 + [4] 238.20/86.05 238.20/86.05 [f_9^#](x1) = [7] x1 + [7] 238.20/86.05 238.20/86.05 [g_10^#](x1, x2) = [2] x1 + [0] 238.20/86.05 238.20/86.05 [c_20](x1, x2) = [1] x2 + [5] 238.20/86.05 238.20/86.05 The order satisfies the following ordering constraints: 238.20/86.05 238.20/86.05 [g_10^#(s(x), y)] = [2] x + [8] 238.20/86.05 > [2] x + [5] 238.20/86.05 = [c_20(f_9^#(y), g_10^#(x, y))] 238.20/86.05 238.20/86.05 238.20/86.05 The strictly oriented rules are moved into the weak component. 238.20/86.05 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(1)). 238.20/86.05 238.20/86.05 Weak DPs: { g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) } 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(1)) 238.20/86.05 238.20/86.05 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.05 closed under successors. The DPs are removed. 238.20/86.05 238.20/86.05 { g_10^#(s(x), y) -> c_20(f_9^#(y), g_10^#(x, y)) } 238.20/86.05 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(1)). 238.20/86.05 238.20/86.05 Rules: Empty 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(1)) 238.20/86.05 238.20/86.05 Empty rules are trivially bounded 238.20/86.05 238.20/86.05 We return to the main proof. 238.20/86.05 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(n^9)). 238.20/86.05 238.20/86.05 Strict DPs: 238.20/86.05 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.05 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.05 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.05 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.05 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.05 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.05 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.05 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.05 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.05 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.05 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.05 , f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.05 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.05 , f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.05 , g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) 238.20/86.05 , f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 , g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) } 238.20/86.05 Weak DPs: 238.20/86.05 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.05 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(n^9)) 238.20/86.05 238.20/86.05 We decompose the input problem according to the dependency graph 238.20/86.05 into the upper component 238.20/86.05 238.20/86.05 { f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 , g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) 238.20/86.05 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 238.20/86.05 and lower component 238.20/86.05 238.20/86.05 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.05 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.05 , f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.05 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.05 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.05 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.05 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.05 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.05 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.05 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.05 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.05 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.05 , f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.05 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.05 , f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.05 , g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) } 238.20/86.05 238.20/86.05 Further, following extension rules are added to the lower 238.20/86.05 component. 238.20/86.05 238.20/86.05 { f_9^#(x) -> g_9^#(x, x) 238.20/86.05 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.05 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.05 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 238.20/86.05 TcT solves the upper component with certificate YES(O(1),O(n^1)). 238.20/86.05 238.20/86.05 Sub-proof: 238.20/86.05 ---------- 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(n^1)). 238.20/86.05 238.20/86.05 Strict DPs: 238.20/86.05 { f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 , g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) } 238.20/86.05 Weak DPs: 238.20/86.05 { g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(n^1)) 238.20/86.05 238.20/86.05 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.05 orient following rules strictly. 238.20/86.05 238.20/86.05 DPs: 238.20/86.05 { 1: f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 , 2: g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) } 238.20/86.05 238.20/86.05 Sub-proof: 238.20/86.05 ---------- 238.20/86.05 The following argument positions are usable: 238.20/86.05 Uargs(c_16) = {1}, Uargs(c_17) = {2} 238.20/86.05 238.20/86.05 TcT has computed the following constructor-based matrix 238.20/86.05 interpretation satisfying not(EDA). 238.20/86.05 238.20/86.05 [s](x1) = [1] x1 + [4] 238.20/86.05 238.20/86.05 [f_8^#](x1) = [7] x1 + [7] 238.20/86.05 238.20/86.05 [f_9^#](x1) = [7] x1 + [1] 238.20/86.05 238.20/86.05 [g_9^#](x1, x2) = [2] x1 + [0] 238.20/86.05 238.20/86.05 [g_10^#](x1, x2) = [7] x2 + [1] 238.20/86.05 238.20/86.05 [c_16](x1) = [2] x1 + [0] 238.20/86.05 238.20/86.05 [c_17](x1, x2) = [1] x2 + [1] 238.20/86.05 238.20/86.05 The order satisfies the following ordering constraints: 238.20/86.05 238.20/86.05 [f_9^#(x)] = [7] x + [1] 238.20/86.05 > [4] x + [0] 238.20/86.05 = [c_16(g_9^#(x, x))] 238.20/86.05 238.20/86.05 [g_9^#(s(x), y)] = [2] x + [8] 238.20/86.05 > [2] x + [1] 238.20/86.05 = [c_17(f_8^#(y), g_9^#(x, y))] 238.20/86.05 238.20/86.05 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.05 >= [7] y + [1] 238.20/86.05 = [f_9^#(y)] 238.20/86.05 238.20/86.05 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.05 >= [7] y + [1] 238.20/86.05 = [g_10^#(x, y)] 238.20/86.05 238.20/86.05 238.20/86.05 The strictly oriented rules are moved into the weak component. 238.20/86.05 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(1)). 238.20/86.05 238.20/86.05 Weak DPs: 238.20/86.05 { f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 , g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) 238.20/86.05 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(1)) 238.20/86.05 238.20/86.05 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.05 closed under successors. The DPs are removed. 238.20/86.05 238.20/86.05 { f_9^#(x) -> c_16(g_9^#(x, x)) 238.20/86.05 , g_9^#(s(x), y) -> c_17(f_8^#(y), g_9^#(x, y)) 238.20/86.05 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(1)). 238.20/86.05 238.20/86.05 Rules: Empty 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(1)) 238.20/86.05 238.20/86.05 Empty rules are trivially bounded 238.20/86.05 238.20/86.05 We return to the main proof. 238.20/86.05 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(n^8)). 238.20/86.05 238.20/86.05 Strict DPs: 238.20/86.05 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.05 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.05 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.05 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.05 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.05 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.05 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.05 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.05 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.05 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.05 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.05 , f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.05 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.05 , f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.05 , g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) } 238.20/86.05 Weak DPs: 238.20/86.05 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.05 , f_9^#(x) -> g_9^#(x, x) 238.20/86.05 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.05 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.05 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(n^8)) 238.20/86.05 238.20/86.05 We decompose the input problem according to the dependency graph 238.20/86.05 into the upper component 238.20/86.05 238.20/86.05 { f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.05 , g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) 238.20/86.05 , f_9^#(x) -> g_9^#(x, x) 238.20/86.05 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.05 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.05 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 238.20/86.05 and lower component 238.20/86.05 238.20/86.05 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.05 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.05 , f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.05 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.05 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.05 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.05 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.05 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.05 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.05 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.05 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.05 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.05 , f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.05 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) } 238.20/86.05 238.20/86.05 Further, following extension rules are added to the lower 238.20/86.05 component. 238.20/86.05 238.20/86.05 { f_8^#(x) -> g_8^#(x, x) 238.20/86.05 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.05 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.05 , f_9^#(x) -> g_9^#(x, x) 238.20/86.05 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.05 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.05 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 238.20/86.05 TcT solves the upper component with certificate YES(O(1),O(n^1)). 238.20/86.05 238.20/86.05 Sub-proof: 238.20/86.05 ---------- 238.20/86.05 We are left with following problem, upon which TcT provides the 238.20/86.05 certificate YES(O(1),O(n^1)). 238.20/86.05 238.20/86.05 Strict DPs: 238.20/86.05 { f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.05 , g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) } 238.20/86.05 Weak DPs: 238.20/86.05 { f_9^#(x) -> g_9^#(x, x) 238.20/86.05 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.05 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.05 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.05 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.05 Obligation: 238.20/86.05 runtime complexity 238.20/86.05 Answer: 238.20/86.05 YES(O(1),O(n^1)) 238.20/86.05 238.20/86.05 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.05 orient following rules strictly. 238.20/86.05 238.20/86.05 DPs: 238.20/86.05 { 2: g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) 238.20/86.05 , 3: f_9^#(x) -> g_9^#(x, x) } 238.20/86.05 238.20/86.05 Sub-proof: 238.20/86.05 ---------- 238.20/86.05 The following argument positions are usable: 238.20/86.05 Uargs(c_14) = {1}, Uargs(c_15) = {2} 238.20/86.05 238.20/86.05 TcT has computed the following constructor-based matrix 238.20/86.05 interpretation satisfying not(EDA). 238.20/86.05 238.20/86.05 [s](x1) = [1] x1 + [4] 238.20/86.05 238.20/86.05 [f_7^#](x1) = [7] x1 + [7] 238.20/86.05 238.20/86.05 [f_8^#](x1) = [5] x1 + [0] 238.20/86.05 238.20/86.05 [g_8^#](x1, x2) = [2] x1 + [0] 238.20/86.05 238.20/86.05 [f_9^#](x1) = [7] x1 + [1] 238.20/86.05 238.20/86.05 [g_9^#](x1, x2) = [5] x2 + [0] 238.20/86.05 238.20/86.05 [g_10^#](x1, x2) = [7] x2 + [1] 238.20/86.05 238.20/86.05 [c_14](x1) = [2] x1 + [0] 238.20/86.05 238.20/86.05 [c_15](x1, x2) = [1] x2 + [1] 238.20/86.05 238.20/86.05 The order satisfies the following ordering constraints: 238.20/86.05 238.20/86.05 [f_8^#(x)] = [5] x + [0] 238.20/86.05 >= [4] x + [0] 238.20/86.05 = [c_14(g_8^#(x, x))] 238.20/86.05 238.20/86.05 [g_8^#(s(x), y)] = [2] x + [8] 238.20/86.05 > [2] x + [1] 238.20/86.05 = [c_15(f_7^#(y), g_8^#(x, y))] 238.20/86.05 238.20/86.05 [f_9^#(x)] = [7] x + [1] 238.20/86.05 > [5] x + [0] 238.20/86.05 = [g_9^#(x, x)] 238.20/86.05 238.20/86.05 [g_9^#(s(x), y)] = [5] y + [0] 238.20/86.06 >= [5] y + [0] 238.20/86.06 = [f_8^#(y)] 238.20/86.06 238.20/86.06 [g_9^#(s(x), y)] = [5] y + [0] 238.20/86.06 >= [5] y + [0] 238.20/86.06 = [g_9^#(x, y)] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.06 >= [7] y + [1] 238.20/86.06 = [f_9^#(y)] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.06 >= [7] y + [1] 238.20/86.06 = [g_10^#(x, y)] 238.20/86.06 238.20/86.06 238.20/86.06 The strictly oriented rules are moved into the weak component. 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Strict DPs: { f_8^#(x) -> c_14(g_8^#(x, x)) } 238.20/86.06 Weak DPs: 238.20/86.06 { g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.06 closed under successors. The DPs are removed. 238.20/86.06 238.20/86.06 { g_8^#(s(x), y) -> c_15(f_7^#(y), g_8^#(x, y)) } 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Strict DPs: { f_8^#(x) -> c_14(g_8^#(x, x)) } 238.20/86.06 Weak DPs: 238.20/86.06 { f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 Due to missing edges in the dependency-graph, the right-hand sides 238.20/86.06 of following rules could be simplified: 238.20/86.06 238.20/86.06 { f_8^#(x) -> c_14(g_8^#(x, x)) } 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Strict DPs: { f_8^#(x) -> c_1() } 238.20/86.06 Weak DPs: 238.20/86.06 { f_9^#(x) -> c_2(g_9^#(x, x)) 238.20/86.06 , g_9^#(s(x), y) -> c_3(f_8^#(y)) 238.20/86.06 , g_9^#(s(x), y) -> c_4(g_9^#(x, y)) 238.20/86.06 , g_10^#(s(x), y) -> c_5(f_9^#(y)) 238.20/86.06 , g_10^#(s(x), y) -> c_6(g_10^#(x, y)) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.06 orient following rules strictly. 238.20/86.06 238.20/86.06 DPs: 238.20/86.06 { 1: f_8^#(x) -> c_1() 238.20/86.06 , 3: g_9^#(s(x), y) -> c_3(f_8^#(y)) } 238.20/86.06 238.20/86.06 Sub-proof: 238.20/86.06 ---------- 238.20/86.06 The following argument positions are usable: 238.20/86.06 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, 238.20/86.06 Uargs(c_5) = {1}, Uargs(c_6) = {1} 238.20/86.06 238.20/86.06 TcT has computed the following constructor-restricted matrix 238.20/86.06 interpretation. Note that the diagonal of the component-wise maxima 238.20/86.06 of interpretation-entries (of constructors) contains no more than 0 238.20/86.06 non-zero entries. 238.20/86.06 238.20/86.06 [s](x1) = [0] 238.20/86.06 238.20/86.06 [f_7^#](x1) = [0] 238.20/86.06 238.20/86.06 [f_8^#](x1) = [1] 238.20/86.06 238.20/86.06 [g_8^#](x1, x2) = [0] 238.20/86.06 238.20/86.06 [f_9^#](x1) = [4] 238.20/86.06 238.20/86.06 [g_9^#](x1, x2) = [4] 238.20/86.06 238.20/86.06 [g_10^#](x1, x2) = [4] x2 + [4] 238.20/86.06 238.20/86.06 [c_14](x1) = [0] 238.20/86.06 238.20/86.06 [c_15](x1, x2) = [0] 238.20/86.06 238.20/86.06 [c] = [0] 238.20/86.06 238.20/86.06 [c_1] = [0] 238.20/86.06 238.20/86.06 [c_2](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [c_3](x1) = [1] x1 + [1] 238.20/86.06 238.20/86.06 [c_4](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [c_5](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [c_6](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 The order satisfies the following ordering constraints: 238.20/86.06 238.20/86.06 [f_8^#(x)] = [1] 238.20/86.06 > [0] 238.20/86.06 = [c_1()] 238.20/86.06 238.20/86.06 [f_9^#(x)] = [4] 238.20/86.06 >= [4] 238.20/86.06 = [c_2(g_9^#(x, x))] 238.20/86.06 238.20/86.06 [g_9^#(s(x), y)] = [4] 238.20/86.06 > [2] 238.20/86.06 = [c_3(f_8^#(y))] 238.20/86.06 238.20/86.06 [g_9^#(s(x), y)] = [4] 238.20/86.06 >= [4] 238.20/86.06 = [c_4(g_9^#(x, y))] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.06 >= [4] 238.20/86.06 = [c_5(f_9^#(y))] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.06 >= [4] y + [4] 238.20/86.06 = [c_6(g_10^#(x, y))] 238.20/86.06 238.20/86.06 238.20/86.06 The strictly oriented rules are moved into the weak component. 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Weak DPs: 238.20/86.06 { f_8^#(x) -> c_1() 238.20/86.06 , f_9^#(x) -> c_2(g_9^#(x, x)) 238.20/86.06 , g_9^#(s(x), y) -> c_3(f_8^#(y)) 238.20/86.06 , g_9^#(s(x), y) -> c_4(g_9^#(x, y)) 238.20/86.06 , g_10^#(s(x), y) -> c_5(f_9^#(y)) 238.20/86.06 , g_10^#(s(x), y) -> c_6(g_10^#(x, y)) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.06 closed under successors. The DPs are removed. 238.20/86.06 238.20/86.06 { f_8^#(x) -> c_1() 238.20/86.06 , f_9^#(x) -> c_2(g_9^#(x, x)) 238.20/86.06 , g_9^#(s(x), y) -> c_3(f_8^#(y)) 238.20/86.06 , g_9^#(s(x), y) -> c_4(g_9^#(x, y)) 238.20/86.06 , g_10^#(s(x), y) -> c_5(f_9^#(y)) 238.20/86.06 , g_10^#(s(x), y) -> c_6(g_10^#(x, y)) } 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Rules: Empty 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 Empty rules are trivially bounded 238.20/86.06 238.20/86.06 We return to the main proof. 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(n^7)). 238.20/86.06 238.20/86.06 Strict DPs: 238.20/86.06 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.06 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.06 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.06 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.06 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.06 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.06 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.06 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.06 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.06 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.06 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.06 , f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.06 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) } 238.20/86.06 Weak DPs: 238.20/86.06 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(n^7)) 238.20/86.06 238.20/86.06 We decompose the input problem according to the dependency graph 238.20/86.06 into the upper component 238.20/86.06 238.20/86.06 { f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.06 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 238.20/86.06 and lower component 238.20/86.06 238.20/86.06 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.06 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.06 , f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.06 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.06 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.06 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.06 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.06 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.06 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.06 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.06 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.06 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) } 238.20/86.06 238.20/86.06 Further, following extension rules are added to the lower 238.20/86.06 component. 238.20/86.06 238.20/86.06 { f_7^#(x) -> g_7^#(x, x) 238.20/86.06 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.06 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 238.20/86.06 TcT solves the upper component with certificate YES(O(1),O(n^1)). 238.20/86.06 238.20/86.06 Sub-proof: 238.20/86.06 ---------- 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(n^1)). 238.20/86.06 238.20/86.06 Strict DPs: 238.20/86.06 { f_7^#(x) -> c_12(g_7^#(x, x)) 238.20/86.06 , g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) } 238.20/86.06 Weak DPs: 238.20/86.06 { f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(n^1)) 238.20/86.06 238.20/86.06 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.06 orient following rules strictly. 238.20/86.06 238.20/86.06 DPs: 238.20/86.06 { 2: g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.06 , 6: f_9^#(x) -> g_9^#(x, x) } 238.20/86.06 238.20/86.06 Sub-proof: 238.20/86.06 ---------- 238.20/86.06 The following argument positions are usable: 238.20/86.06 Uargs(c_12) = {1}, Uargs(c_13) = {2} 238.20/86.06 238.20/86.06 TcT has computed the following constructor-based matrix 238.20/86.06 interpretation satisfying not(EDA). 238.20/86.06 238.20/86.06 [s](x1) = [1] x1 + [4] 238.20/86.06 238.20/86.06 [f_6^#](x1) = [7] x1 + [7] 238.20/86.06 238.20/86.06 [f_7^#](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [g_7^#](x1, x2) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [f_8^#](x1) = [5] x1 + [0] 238.20/86.06 238.20/86.06 [g_8^#](x1, x2) = [1] x2 + [0] 238.20/86.06 238.20/86.06 [f_9^#](x1) = [7] x1 + [1] 238.20/86.06 238.20/86.06 [g_9^#](x1, x2) = [5] x2 + [0] 238.20/86.06 238.20/86.06 [g_10^#](x1, x2) = [7] x2 + [1] 238.20/86.06 238.20/86.06 [c_12](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [c_13](x1, x2) = [1] x2 + [1] 238.20/86.06 238.20/86.06 The order satisfies the following ordering constraints: 238.20/86.06 238.20/86.06 [f_7^#(x)] = [1] x + [0] 238.20/86.06 >= [1] x + [0] 238.20/86.06 = [c_12(g_7^#(x, x))] 238.20/86.06 238.20/86.06 [g_7^#(s(x), y)] = [1] x + [4] 238.20/86.06 > [1] x + [1] 238.20/86.06 = [c_13(f_6^#(y), g_7^#(x, y))] 238.20/86.06 238.20/86.06 [f_8^#(x)] = [5] x + [0] 238.20/86.06 >= [1] x + [0] 238.20/86.06 = [g_8^#(x, x)] 238.20/86.06 238.20/86.06 [g_8^#(s(x), y)] = [1] y + [0] 238.20/86.06 >= [1] y + [0] 238.20/86.06 = [f_7^#(y)] 238.20/86.06 238.20/86.06 [g_8^#(s(x), y)] = [1] y + [0] 238.20/86.06 >= [1] y + [0] 238.20/86.06 = [g_8^#(x, y)] 238.20/86.06 238.20/86.06 [f_9^#(x)] = [7] x + [1] 238.20/86.06 > [5] x + [0] 238.20/86.06 = [g_9^#(x, x)] 238.20/86.06 238.20/86.06 [g_9^#(s(x), y)] = [5] y + [0] 238.20/86.06 >= [5] y + [0] 238.20/86.06 = [f_8^#(y)] 238.20/86.06 238.20/86.06 [g_9^#(s(x), y)] = [5] y + [0] 238.20/86.06 >= [5] y + [0] 238.20/86.06 = [g_9^#(x, y)] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.06 >= [7] y + [1] 238.20/86.06 = [f_9^#(y)] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.06 >= [7] y + [1] 238.20/86.06 = [g_10^#(x, y)] 238.20/86.06 238.20/86.06 238.20/86.06 The strictly oriented rules are moved into the weak component. 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Strict DPs: { f_7^#(x) -> c_12(g_7^#(x, x)) } 238.20/86.06 Weak DPs: 238.20/86.06 { g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.06 closed under successors. The DPs are removed. 238.20/86.06 238.20/86.06 { g_7^#(s(x), y) -> c_13(f_6^#(y), g_7^#(x, y)) } 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Strict DPs: { f_7^#(x) -> c_12(g_7^#(x, x)) } 238.20/86.06 Weak DPs: 238.20/86.06 { f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 Due to missing edges in the dependency-graph, the right-hand sides 238.20/86.06 of following rules could be simplified: 238.20/86.06 238.20/86.06 { f_7^#(x) -> c_12(g_7^#(x, x)) } 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Strict DPs: { f_7^#(x) -> c_1() } 238.20/86.06 Weak DPs: 238.20/86.06 { f_8^#(x) -> c_2(g_8^#(x, x)) 238.20/86.06 , g_8^#(s(x), y) -> c_3(f_7^#(y)) 238.20/86.06 , g_8^#(s(x), y) -> c_4(g_8^#(x, y)) 238.20/86.06 , f_9^#(x) -> c_5(g_9^#(x, x)) 238.20/86.06 , g_9^#(s(x), y) -> c_6(f_8^#(y)) 238.20/86.06 , g_9^#(s(x), y) -> c_7(g_9^#(x, y)) 238.20/86.06 , g_10^#(s(x), y) -> c_8(f_9^#(y)) 238.20/86.06 , g_10^#(s(x), y) -> c_9(g_10^#(x, y)) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.06 orient following rules strictly. 238.20/86.06 238.20/86.06 DPs: 238.20/86.06 { 1: f_7^#(x) -> c_1() 238.20/86.06 , 6: g_9^#(s(x), y) -> c_6(f_8^#(y)) } 238.20/86.06 238.20/86.06 Sub-proof: 238.20/86.06 ---------- 238.20/86.06 The following argument positions are usable: 238.20/86.06 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, 238.20/86.06 Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, 238.20/86.06 Uargs(c_8) = {1}, Uargs(c_9) = {1} 238.20/86.06 238.20/86.06 TcT has computed the following constructor-restricted matrix 238.20/86.06 interpretation. Note that the diagonal of the component-wise maxima 238.20/86.06 of interpretation-entries (of constructors) contains no more than 0 238.20/86.06 non-zero entries. 238.20/86.06 238.20/86.06 [s](x1) = [0] 238.20/86.06 238.20/86.06 [f_6^#](x1) = [0] 238.20/86.06 238.20/86.06 [f_7^#](x1) = [1] 238.20/86.06 238.20/86.06 [g_7^#](x1, x2) = [0] 238.20/86.06 238.20/86.06 [f_8^#](x1) = [3] 238.20/86.06 238.20/86.06 [g_8^#](x1, x2) = [2] 238.20/86.06 238.20/86.06 [f_9^#](x1) = [4] 238.20/86.06 238.20/86.06 [g_9^#](x1, x2) = [4] 238.20/86.06 238.20/86.06 [g_10^#](x1, x2) = [4] x2 + [4] 238.20/86.06 238.20/86.06 [c_12](x1) = [0] 238.20/86.06 238.20/86.06 [c_13](x1, x2) = [0] 238.20/86.06 238.20/86.06 [c] = [0] 238.20/86.06 238.20/86.06 [c_1] = [0] 238.20/86.06 238.20/86.06 [c_2](x1) = [1] x1 + [1] 238.20/86.06 238.20/86.06 [c_3](x1) = [1] x1 + [1] 238.20/86.06 238.20/86.06 [c_4](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [c_5](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [c_6](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [c_7](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [c_8](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 [c_9](x1) = [1] x1 + [0] 238.20/86.06 238.20/86.06 The order satisfies the following ordering constraints: 238.20/86.06 238.20/86.06 [f_7^#(x)] = [1] 238.20/86.06 > [0] 238.20/86.06 = [c_1()] 238.20/86.06 238.20/86.06 [f_8^#(x)] = [3] 238.20/86.06 >= [3] 238.20/86.06 = [c_2(g_8^#(x, x))] 238.20/86.06 238.20/86.06 [g_8^#(s(x), y)] = [2] 238.20/86.06 >= [2] 238.20/86.06 = [c_3(f_7^#(y))] 238.20/86.06 238.20/86.06 [g_8^#(s(x), y)] = [2] 238.20/86.06 >= [2] 238.20/86.06 = [c_4(g_8^#(x, y))] 238.20/86.06 238.20/86.06 [f_9^#(x)] = [4] 238.20/86.06 >= [4] 238.20/86.06 = [c_5(g_9^#(x, x))] 238.20/86.06 238.20/86.06 [g_9^#(s(x), y)] = [4] 238.20/86.06 > [3] 238.20/86.06 = [c_6(f_8^#(y))] 238.20/86.06 238.20/86.06 [g_9^#(s(x), y)] = [4] 238.20/86.06 >= [4] 238.20/86.06 = [c_7(g_9^#(x, y))] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.06 >= [4] 238.20/86.06 = [c_8(f_9^#(y))] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.06 >= [4] y + [4] 238.20/86.06 = [c_9(g_10^#(x, y))] 238.20/86.06 238.20/86.06 238.20/86.06 We return to the main proof. Consider the set of all dependency 238.20/86.06 pairs 238.20/86.06 238.20/86.06 : 238.20/86.06 { 1: f_7^#(x) -> c_1() 238.20/86.06 , 2: f_8^#(x) -> c_2(g_8^#(x, x)) 238.20/86.06 , 3: g_8^#(s(x), y) -> c_3(f_7^#(y)) 238.20/86.06 , 4: g_8^#(s(x), y) -> c_4(g_8^#(x, y)) 238.20/86.06 , 5: f_9^#(x) -> c_5(g_9^#(x, x)) 238.20/86.06 , 6: g_9^#(s(x), y) -> c_6(f_8^#(y)) 238.20/86.06 , 7: g_9^#(s(x), y) -> c_7(g_9^#(x, y)) 238.20/86.06 , 8: g_10^#(s(x), y) -> c_8(f_9^#(y)) 238.20/86.06 , 9: g_10^#(s(x), y) -> c_9(g_10^#(x, y)) } 238.20/86.06 238.20/86.06 Processor 'matrix interpretation of dimension 1' induces the 238.20/86.06 complexity certificate YES(?,O(1)) on application of dependency 238.20/86.06 pairs {1,6}. These cover all (indirect) predecessors of dependency 238.20/86.06 pairs {1,2,6}, their number of application is equally bounded. The 238.20/86.06 dependency pairs are shifted into the weak component. 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Weak DPs: 238.20/86.06 { f_7^#(x) -> c_1() 238.20/86.06 , f_8^#(x) -> c_2(g_8^#(x, x)) 238.20/86.06 , g_8^#(s(x), y) -> c_3(f_7^#(y)) 238.20/86.06 , g_8^#(s(x), y) -> c_4(g_8^#(x, y)) 238.20/86.06 , f_9^#(x) -> c_5(g_9^#(x, x)) 238.20/86.06 , g_9^#(s(x), y) -> c_6(f_8^#(y)) 238.20/86.06 , g_9^#(s(x), y) -> c_7(g_9^#(x, y)) 238.20/86.06 , g_10^#(s(x), y) -> c_8(f_9^#(y)) 238.20/86.06 , g_10^#(s(x), y) -> c_9(g_10^#(x, y)) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.06 closed under successors. The DPs are removed. 238.20/86.06 238.20/86.06 { f_7^#(x) -> c_1() 238.20/86.06 , f_8^#(x) -> c_2(g_8^#(x, x)) 238.20/86.06 , g_8^#(s(x), y) -> c_3(f_7^#(y)) 238.20/86.06 , g_8^#(s(x), y) -> c_4(g_8^#(x, y)) 238.20/86.06 , f_9^#(x) -> c_5(g_9^#(x, x)) 238.20/86.06 , g_9^#(s(x), y) -> c_6(f_8^#(y)) 238.20/86.06 , g_9^#(s(x), y) -> c_7(g_9^#(x, y)) 238.20/86.06 , g_10^#(s(x), y) -> c_8(f_9^#(y)) 238.20/86.06 , g_10^#(s(x), y) -> c_9(g_10^#(x, y)) } 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Rules: Empty 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 Empty rules are trivially bounded 238.20/86.06 238.20/86.06 We return to the main proof. 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(n^6)). 238.20/86.06 238.20/86.06 Strict DPs: 238.20/86.06 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.06 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.06 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.06 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.06 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.06 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.06 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.06 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.06 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.06 , f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.06 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) } 238.20/86.06 Weak DPs: 238.20/86.06 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.06 , f_7^#(x) -> g_7^#(x, x) 238.20/86.06 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.06 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(n^6)) 238.20/86.06 238.20/86.06 We decompose the input problem according to the dependency graph 238.20/86.06 into the upper component 238.20/86.06 238.20/86.06 { f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.06 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.06 , f_7^#(x) -> g_7^#(x, x) 238.20/86.06 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.06 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 238.20/86.06 and lower component 238.20/86.06 238.20/86.06 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.06 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.06 , f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.06 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.06 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.06 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.06 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.06 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.06 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.06 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) } 238.20/86.06 238.20/86.06 Further, following extension rules are added to the lower 238.20/86.06 component. 238.20/86.06 238.20/86.06 { f_6^#(x) -> g_6^#(x, x) 238.20/86.06 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.06 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.06 , f_7^#(x) -> g_7^#(x, x) 238.20/86.06 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.06 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 238.20/86.06 TcT solves the upper component with certificate YES(O(1),O(n^1)). 238.20/86.06 238.20/86.06 Sub-proof: 238.20/86.06 ---------- 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(n^1)). 238.20/86.06 238.20/86.06 Strict DPs: 238.20/86.06 { f_6^#(x) -> c_10(g_6^#(x, x)) 238.20/86.06 , g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) } 238.20/86.06 Weak DPs: 238.20/86.06 { f_7^#(x) -> g_7^#(x, x) 238.20/86.06 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.06 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(n^1)) 238.20/86.06 238.20/86.06 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.06 orient following rules strictly. 238.20/86.06 238.20/86.06 DPs: 238.20/86.06 { 2: g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.06 , 9: f_9^#(x) -> g_9^#(x, x) } 238.20/86.06 238.20/86.06 Sub-proof: 238.20/86.06 ---------- 238.20/86.06 The following argument positions are usable: 238.20/86.06 Uargs(c_10) = {1}, Uargs(c_11) = {2} 238.20/86.06 238.20/86.06 TcT has computed the following constructor-based matrix 238.20/86.06 interpretation satisfying not(EDA). 238.20/86.06 238.20/86.06 [s](x1) = [1] x1 + [4] 238.20/86.06 238.20/86.06 [f_5^#](x1) = [7] x1 + [7] 238.20/86.06 238.20/86.06 [f_6^#](x1) = [4] x1 + [0] 238.20/86.06 238.20/86.06 [g_6^#](x1, x2) = [2] x1 + [0] 238.20/86.06 238.20/86.06 [f_7^#](x1) = [4] x1 + [0] 238.20/86.06 238.20/86.06 [g_7^#](x1, x2) = [4] x2 + [0] 238.20/86.06 238.20/86.06 [f_8^#](x1) = [7] x1 + [0] 238.20/86.06 238.20/86.06 [g_8^#](x1, x2) = [4] x2 + [0] 238.20/86.06 238.20/86.06 [f_9^#](x1) = [7] x1 + [1] 238.20/86.06 238.20/86.06 [g_9^#](x1, x2) = [7] x2 + [0] 238.20/86.06 238.20/86.06 [g_10^#](x1, x2) = [7] x2 + [1] 238.20/86.06 238.20/86.06 [c_10](x1) = [2] x1 + [0] 238.20/86.06 238.20/86.06 [c_11](x1, x2) = [1] x2 + [5] 238.20/86.06 238.20/86.06 The order satisfies the following ordering constraints: 238.20/86.06 238.20/86.06 [f_6^#(x)] = [4] x + [0] 238.20/86.06 >= [4] x + [0] 238.20/86.06 = [c_10(g_6^#(x, x))] 238.20/86.06 238.20/86.06 [g_6^#(s(x), y)] = [2] x + [8] 238.20/86.06 > [2] x + [5] 238.20/86.06 = [c_11(f_5^#(y), g_6^#(x, y))] 238.20/86.06 238.20/86.06 [f_7^#(x)] = [4] x + [0] 238.20/86.06 >= [4] x + [0] 238.20/86.06 = [g_7^#(x, x)] 238.20/86.06 238.20/86.06 [g_7^#(s(x), y)] = [4] y + [0] 238.20/86.06 >= [4] y + [0] 238.20/86.06 = [f_6^#(y)] 238.20/86.06 238.20/86.06 [g_7^#(s(x), y)] = [4] y + [0] 238.20/86.06 >= [4] y + [0] 238.20/86.06 = [g_7^#(x, y)] 238.20/86.06 238.20/86.06 [f_8^#(x)] = [7] x + [0] 238.20/86.06 >= [4] x + [0] 238.20/86.06 = [g_8^#(x, x)] 238.20/86.06 238.20/86.06 [g_8^#(s(x), y)] = [4] y + [0] 238.20/86.06 >= [4] y + [0] 238.20/86.06 = [f_7^#(y)] 238.20/86.06 238.20/86.06 [g_8^#(s(x), y)] = [4] y + [0] 238.20/86.06 >= [4] y + [0] 238.20/86.06 = [g_8^#(x, y)] 238.20/86.06 238.20/86.06 [f_9^#(x)] = [7] x + [1] 238.20/86.06 > [7] x + [0] 238.20/86.06 = [g_9^#(x, x)] 238.20/86.06 238.20/86.06 [g_9^#(s(x), y)] = [7] y + [0] 238.20/86.06 >= [7] y + [0] 238.20/86.06 = [f_8^#(y)] 238.20/86.06 238.20/86.06 [g_9^#(s(x), y)] = [7] y + [0] 238.20/86.06 >= [7] y + [0] 238.20/86.06 = [g_9^#(x, y)] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.06 >= [7] y + [1] 238.20/86.06 = [f_9^#(y)] 238.20/86.06 238.20/86.06 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.06 >= [7] y + [1] 238.20/86.06 = [g_10^#(x, y)] 238.20/86.06 238.20/86.06 238.20/86.06 The strictly oriented rules are moved into the weak component. 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Strict DPs: { f_6^#(x) -> c_10(g_6^#(x, x)) } 238.20/86.06 Weak DPs: 238.20/86.06 { g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) 238.20/86.06 , f_7^#(x) -> g_7^#(x, x) 238.20/86.06 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.06 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.06 closed under successors. The DPs are removed. 238.20/86.06 238.20/86.06 { g_6^#(s(x), y) -> c_11(f_5^#(y), g_6^#(x, y)) } 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Strict DPs: { f_6^#(x) -> c_10(g_6^#(x, x)) } 238.20/86.06 Weak DPs: 238.20/86.06 { f_7^#(x) -> g_7^#(x, x) 238.20/86.06 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.06 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.06 , f_8^#(x) -> g_8^#(x, x) 238.20/86.06 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.06 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.06 , f_9^#(x) -> g_9^#(x, x) 238.20/86.06 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.06 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.06 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.06 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 Due to missing edges in the dependency-graph, the right-hand sides 238.20/86.06 of following rules could be simplified: 238.20/86.06 238.20/86.06 { f_6^#(x) -> c_10(g_6^#(x, x)) } 238.20/86.06 238.20/86.06 We are left with following problem, upon which TcT provides the 238.20/86.06 certificate YES(O(1),O(1)). 238.20/86.06 238.20/86.06 Strict DPs: { f_6^#(x) -> c_1() } 238.20/86.06 Weak DPs: 238.20/86.06 { f_7^#(x) -> c_2(g_7^#(x, x)) 238.20/86.06 , g_7^#(s(x), y) -> c_3(f_6^#(y)) 238.20/86.06 , g_7^#(s(x), y) -> c_4(g_7^#(x, y)) 238.20/86.06 , f_8^#(x) -> c_5(g_8^#(x, x)) 238.20/86.06 , g_8^#(s(x), y) -> c_6(f_7^#(y)) 238.20/86.06 , g_8^#(s(x), y) -> c_7(g_8^#(x, y)) 238.20/86.06 , f_9^#(x) -> c_8(g_9^#(x, x)) 238.20/86.06 , g_9^#(s(x), y) -> c_9(f_8^#(y)) 238.20/86.06 , g_9^#(s(x), y) -> c_10(g_9^#(x, y)) 238.20/86.06 , g_10^#(s(x), y) -> c_11(f_9^#(y)) 238.20/86.06 , g_10^#(s(x), y) -> c_12(g_10^#(x, y)) } 238.20/86.06 Obligation: 238.20/86.06 runtime complexity 238.20/86.06 Answer: 238.20/86.06 YES(O(1),O(1)) 238.20/86.06 238.20/86.06 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.06 orient following rules strictly. 238.20/86.06 238.20/86.06 DPs: 238.20/86.06 { 3: g_7^#(s(x), y) -> c_3(f_6^#(y)) 238.20/86.06 , 6: g_8^#(s(x), y) -> c_6(f_7^#(y)) } 238.20/86.06 238.20/86.06 Sub-proof: 238.20/86.06 ---------- 238.20/86.06 The following argument positions are usable: 238.20/86.06 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, 238.20/86.06 Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, 238.20/86.06 Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(c_10) = {1}, 238.20/86.06 Uargs(c_11) = {1}, Uargs(c_12) = {1} 238.20/86.06 238.20/86.06 TcT has computed the following constructor-restricted matrix 238.20/86.06 interpretation. Note that the diagonal of the component-wise maxima 238.20/86.06 of interpretation-entries (of constructors) contains no more than 0 238.20/86.06 non-zero entries. 238.20/86.06 238.20/86.06 [s](x1) = [0] 238.20/86.06 238.20/86.06 [f_5^#](x1) = [0] 238.20/86.06 238.20/86.06 [f_6^#](x1) = [0] 238.20/86.06 238.20/86.06 [g_6^#](x1, x2) = [0] 238.20/86.06 238.20/86.06 [f_7^#](x1) = [3] 238.20/86.06 238.20/86.06 [g_7^#](x1, x2) = [2] 238.20/86.06 238.20/86.06 [f_8^#](x1) = [4] 238.20/86.06 238.20/86.06 [g_8^#](x1, x2) = [4] 238.20/86.06 238.20/86.06 [f_9^#](x1) = [4] 238.20/86.06 238.20/86.07 [g_9^#](x1, x2) = [4] 238.20/86.07 238.20/86.07 [g_10^#](x1, x2) = [4] x2 + [4] 238.20/86.07 238.20/86.07 [c_10](x1) = [0] 238.20/86.07 238.20/86.07 [c_11](x1, x2) = [0] 238.20/86.07 238.20/86.07 [c] = [0] 238.20/86.07 238.20/86.07 [c_1] = [0] 238.20/86.07 238.20/86.07 [c_2](x1) = [1] x1 + [1] 238.20/86.07 238.20/86.07 [c_3](x1) = [2] x1 + [1] 238.20/86.07 238.20/86.07 [c_4](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_5](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_6](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_7](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_8](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_9](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_10](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_11](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_12](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 The order satisfies the following ordering constraints: 238.20/86.07 238.20/86.07 [f_6^#(x)] = [0] 238.20/86.07 >= [0] 238.20/86.07 = [c_1()] 238.20/86.07 238.20/86.07 [f_7^#(x)] = [3] 238.20/86.07 >= [3] 238.20/86.07 = [c_2(g_7^#(x, x))] 238.20/86.07 238.20/86.07 [g_7^#(s(x), y)] = [2] 238.20/86.07 > [1] 238.20/86.07 = [c_3(f_6^#(y))] 238.20/86.07 238.20/86.07 [g_7^#(s(x), y)] = [2] 238.20/86.07 >= [2] 238.20/86.07 = [c_4(g_7^#(x, y))] 238.20/86.07 238.20/86.07 [f_8^#(x)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_5(g_8^#(x, x))] 238.20/86.07 238.20/86.07 [g_8^#(s(x), y)] = [4] 238.20/86.07 > [3] 238.20/86.07 = [c_6(f_7^#(y))] 238.20/86.07 238.20/86.07 [g_8^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_7(g_8^#(x, y))] 238.20/86.07 238.20/86.07 [f_9^#(x)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_8(g_9^#(x, x))] 238.20/86.07 238.20/86.07 [g_9^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_9(f_8^#(y))] 238.20/86.07 238.20/86.07 [g_9^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_10(g_9^#(x, y))] 238.20/86.07 238.20/86.07 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_11(f_9^#(y))] 238.20/86.07 238.20/86.07 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.07 >= [4] y + [4] 238.20/86.07 = [c_12(g_10^#(x, y))] 238.20/86.07 238.20/86.07 238.20/86.07 We return to the main proof. Consider the set of all dependency 238.20/86.07 pairs 238.20/86.07 238.20/86.07 : 238.20/86.07 { 1: f_6^#(x) -> c_1() 238.20/86.07 , 2: f_7^#(x) -> c_2(g_7^#(x, x)) 238.20/86.07 , 3: g_7^#(s(x), y) -> c_3(f_6^#(y)) 238.20/86.07 , 4: g_7^#(s(x), y) -> c_4(g_7^#(x, y)) 238.20/86.07 , 5: f_8^#(x) -> c_5(g_8^#(x, x)) 238.20/86.07 , 6: g_8^#(s(x), y) -> c_6(f_7^#(y)) 238.20/86.07 , 7: g_8^#(s(x), y) -> c_7(g_8^#(x, y)) 238.20/86.07 , 8: f_9^#(x) -> c_8(g_9^#(x, x)) 238.20/86.07 , 9: g_9^#(s(x), y) -> c_9(f_8^#(y)) 238.20/86.07 , 10: g_9^#(s(x), y) -> c_10(g_9^#(x, y)) 238.20/86.07 , 11: g_10^#(s(x), y) -> c_11(f_9^#(y)) 238.20/86.07 , 12: g_10^#(s(x), y) -> c_12(g_10^#(x, y)) } 238.20/86.07 238.20/86.07 Processor 'matrix interpretation of dimension 1' induces the 238.20/86.07 complexity certificate YES(?,O(1)) on application of dependency 238.20/86.07 pairs {3,6}. These cover all (indirect) predecessors of dependency 238.20/86.07 pairs {1,2,3,6}, their number of application is equally bounded. 238.20/86.07 The dependency pairs are shifted into the weak component. 238.20/86.07 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(1)). 238.20/86.07 238.20/86.07 Weak DPs: 238.20/86.07 { f_6^#(x) -> c_1() 238.20/86.07 , f_7^#(x) -> c_2(g_7^#(x, x)) 238.20/86.07 , g_7^#(s(x), y) -> c_3(f_6^#(y)) 238.20/86.07 , g_7^#(s(x), y) -> c_4(g_7^#(x, y)) 238.20/86.07 , f_8^#(x) -> c_5(g_8^#(x, x)) 238.20/86.07 , g_8^#(s(x), y) -> c_6(f_7^#(y)) 238.20/86.07 , g_8^#(s(x), y) -> c_7(g_8^#(x, y)) 238.20/86.07 , f_9^#(x) -> c_8(g_9^#(x, x)) 238.20/86.07 , g_9^#(s(x), y) -> c_9(f_8^#(y)) 238.20/86.07 , g_9^#(s(x), y) -> c_10(g_9^#(x, y)) 238.20/86.07 , g_10^#(s(x), y) -> c_11(f_9^#(y)) 238.20/86.07 , g_10^#(s(x), y) -> c_12(g_10^#(x, y)) } 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(1)) 238.20/86.07 238.20/86.07 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.07 closed under successors. The DPs are removed. 238.20/86.07 238.20/86.07 { f_6^#(x) -> c_1() 238.20/86.07 , f_7^#(x) -> c_2(g_7^#(x, x)) 238.20/86.07 , g_7^#(s(x), y) -> c_3(f_6^#(y)) 238.20/86.07 , g_7^#(s(x), y) -> c_4(g_7^#(x, y)) 238.20/86.07 , f_8^#(x) -> c_5(g_8^#(x, x)) 238.20/86.07 , g_8^#(s(x), y) -> c_6(f_7^#(y)) 238.20/86.07 , g_8^#(s(x), y) -> c_7(g_8^#(x, y)) 238.20/86.07 , f_9^#(x) -> c_8(g_9^#(x, x)) 238.20/86.07 , g_9^#(s(x), y) -> c_9(f_8^#(y)) 238.20/86.07 , g_9^#(s(x), y) -> c_10(g_9^#(x, y)) 238.20/86.07 , g_10^#(s(x), y) -> c_11(f_9^#(y)) 238.20/86.07 , g_10^#(s(x), y) -> c_12(g_10^#(x, y)) } 238.20/86.07 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(1)). 238.20/86.07 238.20/86.07 Rules: Empty 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(1)) 238.20/86.07 238.20/86.07 Empty rules are trivially bounded 238.20/86.07 238.20/86.07 We return to the main proof. 238.20/86.07 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(n^5)). 238.20/86.07 238.20/86.07 Strict DPs: 238.20/86.07 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.07 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.07 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.07 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.07 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.07 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.07 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.07 , f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.07 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) } 238.20/86.07 Weak DPs: 238.20/86.07 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.07 , f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(n^5)) 238.20/86.07 238.20/86.07 We decompose the input problem according to the dependency graph 238.20/86.07 into the upper component 238.20/86.07 238.20/86.07 { f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.07 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.07 , f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 238.20/86.07 and lower component 238.20/86.07 238.20/86.07 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.07 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.07 , f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.07 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.07 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.07 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.07 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.07 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) } 238.20/86.07 238.20/86.07 Further, following extension rules are added to the lower 238.20/86.07 component. 238.20/86.07 238.20/86.07 { f_5^#(x) -> g_5^#(x, x) 238.20/86.07 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.07 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.07 , f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 238.20/86.07 TcT solves the upper component with certificate YES(O(1),O(n^1)). 238.20/86.07 238.20/86.07 Sub-proof: 238.20/86.07 ---------- 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(n^1)). 238.20/86.07 238.20/86.07 Strict DPs: 238.20/86.07 { f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.07 , g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) } 238.20/86.07 Weak DPs: 238.20/86.07 { f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(n^1)) 238.20/86.07 238.20/86.07 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.07 orient following rules strictly. 238.20/86.07 238.20/86.07 DPs: 238.20/86.07 { 2: g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.07 , 12: f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , 16: g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 238.20/86.07 Sub-proof: 238.20/86.07 ---------- 238.20/86.07 The following argument positions are usable: 238.20/86.07 Uargs(c_8) = {1}, Uargs(c_9) = {2} 238.20/86.07 238.20/86.07 TcT has computed the following constructor-based matrix 238.20/86.07 interpretation satisfying not(EDA). 238.20/86.07 238.20/86.07 [s](x1) = [1] x1 + [1] 238.20/86.07 238.20/86.07 [f_4^#](x1) = [7] x1 + [7] 238.20/86.07 238.20/86.07 [f_5^#](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [g_5^#](x1, x2) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [f_6^#](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [g_6^#](x1, x2) = [1] x2 + [0] 238.20/86.07 238.20/86.07 [f_7^#](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [g_7^#](x1, x2) = [1] x2 + [0] 238.20/86.07 238.20/86.07 [f_8^#](x1) = [5] x1 + [0] 238.20/86.07 238.20/86.07 [g_8^#](x1, x2) = [1] x2 + [0] 238.20/86.07 238.20/86.07 [f_9^#](x1) = [7] x1 + [1] 238.20/86.07 238.20/86.07 [g_9^#](x1, x2) = [5] x2 + [0] 238.20/86.07 238.20/86.07 [g_10^#](x1, x2) = [1] x1 + [7] x2 + [0] 238.20/86.07 238.20/86.07 [c_8](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_9](x1, x2) = [1] x2 + [0] 238.20/86.07 238.20/86.07 The order satisfies the following ordering constraints: 238.20/86.07 238.20/86.07 [f_5^#(x)] = [1] x + [0] 238.20/86.07 >= [1] x + [0] 238.20/86.07 = [c_8(g_5^#(x, x))] 238.20/86.07 238.20/86.07 [g_5^#(s(x), y)] = [1] x + [1] 238.20/86.07 > [1] x + [0] 238.20/86.07 = [c_9(f_4^#(y), g_5^#(x, y))] 238.20/86.07 238.20/86.07 [f_6^#(x)] = [1] x + [0] 238.20/86.07 >= [1] x + [0] 238.20/86.07 = [g_6^#(x, x)] 238.20/86.07 238.20/86.07 [g_6^#(s(x), y)] = [1] y + [0] 238.20/86.07 >= [1] y + [0] 238.20/86.07 = [f_5^#(y)] 238.20/86.07 238.20/86.07 [g_6^#(s(x), y)] = [1] y + [0] 238.20/86.07 >= [1] y + [0] 238.20/86.07 = [g_6^#(x, y)] 238.20/86.07 238.20/86.07 [f_7^#(x)] = [1] x + [0] 238.20/86.07 >= [1] x + [0] 238.20/86.07 = [g_7^#(x, x)] 238.20/86.07 238.20/86.07 [g_7^#(s(x), y)] = [1] y + [0] 238.20/86.07 >= [1] y + [0] 238.20/86.07 = [f_6^#(y)] 238.20/86.07 238.20/86.07 [g_7^#(s(x), y)] = [1] y + [0] 238.20/86.07 >= [1] y + [0] 238.20/86.07 = [g_7^#(x, y)] 238.20/86.07 238.20/86.07 [f_8^#(x)] = [5] x + [0] 238.20/86.07 >= [1] x + [0] 238.20/86.07 = [g_8^#(x, x)] 238.20/86.07 238.20/86.07 [g_8^#(s(x), y)] = [1] y + [0] 238.20/86.07 >= [1] y + [0] 238.20/86.07 = [f_7^#(y)] 238.20/86.07 238.20/86.07 [g_8^#(s(x), y)] = [1] y + [0] 238.20/86.07 >= [1] y + [0] 238.20/86.07 = [g_8^#(x, y)] 238.20/86.07 238.20/86.07 [f_9^#(x)] = [7] x + [1] 238.20/86.07 > [5] x + [0] 238.20/86.07 = [g_9^#(x, x)] 238.20/86.07 238.20/86.07 [g_9^#(s(x), y)] = [5] y + [0] 238.20/86.07 >= [5] y + [0] 238.20/86.07 = [f_8^#(y)] 238.20/86.07 238.20/86.07 [g_9^#(s(x), y)] = [5] y + [0] 238.20/86.07 >= [5] y + [0] 238.20/86.07 = [g_9^#(x, y)] 238.20/86.07 238.20/86.07 [g_10^#(s(x), y)] = [1] x + [7] y + [1] 238.20/86.07 >= [7] y + [1] 238.20/86.07 = [f_9^#(y)] 238.20/86.07 238.20/86.07 [g_10^#(s(x), y)] = [1] x + [7] y + [1] 238.20/86.07 > [1] x + [7] y + [0] 238.20/86.07 = [g_10^#(x, y)] 238.20/86.07 238.20/86.07 238.20/86.07 We return to the main proof. Consider the set of all dependency 238.20/86.07 pairs 238.20/86.07 238.20/86.07 : 238.20/86.07 { 1: f_5^#(x) -> c_8(g_5^#(x, x)) 238.20/86.07 , 2: g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.07 , 3: f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , 4: g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , 5: g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , 6: f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , 7: g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , 8: g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , 9: f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , 10: g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , 11: g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , 12: f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , 13: g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , 14: g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , 15: g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , 16: g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 238.20/86.07 Processor 'matrix interpretation of dimension 1' induces the 238.20/86.07 complexity certificate YES(?,O(n^1)) on application of dependency 238.20/86.07 pairs {2,12,16}. These cover all (indirect) predecessors of 238.20/86.07 dependency pairs {2,12,15,16}, their number of application is 238.20/86.07 equally bounded. The dependency pairs are shifted into the weak 238.20/86.07 component. 238.20/86.07 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(1)). 238.20/86.07 238.20/86.07 Strict DPs: { f_5^#(x) -> c_8(g_5^#(x, x)) } 238.20/86.07 Weak DPs: 238.20/86.07 { g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) 238.20/86.07 , f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(1)) 238.20/86.07 238.20/86.07 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.07 closed under successors. The DPs are removed. 238.20/86.07 238.20/86.07 { g_5^#(s(x), y) -> c_9(f_4^#(y), g_5^#(x, y)) } 238.20/86.07 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(1)). 238.20/86.07 238.20/86.07 Strict DPs: { f_5^#(x) -> c_8(g_5^#(x, x)) } 238.20/86.07 Weak DPs: 238.20/86.07 { f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(1)) 238.20/86.07 238.20/86.07 Due to missing edges in the dependency-graph, the right-hand sides 238.20/86.07 of following rules could be simplified: 238.20/86.07 238.20/86.07 { f_5^#(x) -> c_8(g_5^#(x, x)) } 238.20/86.07 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(1)). 238.20/86.07 238.20/86.07 Strict DPs: { f_5^#(x) -> c_1() } 238.20/86.07 Weak DPs: 238.20/86.07 { f_6^#(x) -> c_2(g_6^#(x, x)) 238.20/86.07 , g_6^#(s(x), y) -> c_3(f_5^#(y)) 238.20/86.07 , g_6^#(s(x), y) -> c_4(g_6^#(x, y)) 238.20/86.07 , f_7^#(x) -> c_5(g_7^#(x, x)) 238.20/86.07 , g_7^#(s(x), y) -> c_6(f_6^#(y)) 238.20/86.07 , g_7^#(s(x), y) -> c_7(g_7^#(x, y)) 238.20/86.07 , f_8^#(x) -> c_8(g_8^#(x, x)) 238.20/86.07 , g_8^#(s(x), y) -> c_9(f_7^#(y)) 238.20/86.07 , g_8^#(s(x), y) -> c_10(g_8^#(x, y)) 238.20/86.07 , f_9^#(x) -> c_11(g_9^#(x, x)) 238.20/86.07 , g_9^#(s(x), y) -> c_12(f_8^#(y)) 238.20/86.07 , g_9^#(s(x), y) -> c_13(g_9^#(x, y)) 238.20/86.07 , g_10^#(s(x), y) -> c_14(f_9^#(y)) 238.20/86.07 , g_10^#(s(x), y) -> c_15(g_10^#(x, y)) } 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(1)) 238.20/86.07 238.20/86.07 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.07 orient following rules strictly. 238.20/86.07 238.20/86.07 DPs: 238.20/86.07 { 1: f_5^#(x) -> c_1() } 238.20/86.07 238.20/86.07 Sub-proof: 238.20/86.07 ---------- 238.20/86.07 The following argument positions are usable: 238.20/86.07 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, 238.20/86.07 Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, 238.20/86.07 Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(c_10) = {1}, 238.20/86.07 Uargs(c_11) = {1}, Uargs(c_12) = {1}, Uargs(c_13) = {1}, 238.20/86.07 Uargs(c_14) = {1}, Uargs(c_15) = {1} 238.20/86.07 238.20/86.07 TcT has computed the following constructor-restricted matrix 238.20/86.07 interpretation. Note that the diagonal of the component-wise maxima 238.20/86.07 of interpretation-entries (of constructors) contains no more than 0 238.20/86.07 non-zero entries. 238.20/86.07 238.20/86.07 [s](x1) = [0] 238.20/86.07 238.20/86.07 [f_4^#](x1) = [0] 238.20/86.07 238.20/86.07 [f_5^#](x1) = [4] 238.20/86.07 238.20/86.07 [g_5^#](x1, x2) = [0] 238.20/86.07 238.20/86.07 [f_6^#](x1) = [4] 238.20/86.07 238.20/86.07 [g_6^#](x1, x2) = [4] 238.20/86.07 238.20/86.07 [f_7^#](x1) = [4] 238.20/86.07 238.20/86.07 [g_7^#](x1, x2) = [4] 238.20/86.07 238.20/86.07 [f_8^#](x1) = [4] 238.20/86.07 238.20/86.07 [g_8^#](x1, x2) = [4] 238.20/86.07 238.20/86.07 [f_9^#](x1) = [4] 238.20/86.07 238.20/86.07 [g_9^#](x1, x2) = [4] 238.20/86.07 238.20/86.07 [g_10^#](x1, x2) = [7] x2 + [4] 238.20/86.07 238.20/86.07 [c_8](x1) = [0] 238.20/86.07 238.20/86.07 [c_9](x1, x2) = [0] 238.20/86.07 238.20/86.07 [c] = [0] 238.20/86.07 238.20/86.07 [c_1] = [3] 238.20/86.07 238.20/86.07 [c_2](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_3](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_4](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_5](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_6](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_7](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_8](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_9](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_10](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_11](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_12](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_13](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_14](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_15](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 The order satisfies the following ordering constraints: 238.20/86.07 238.20/86.07 [f_5^#(x)] = [4] 238.20/86.07 > [3] 238.20/86.07 = [c_1()] 238.20/86.07 238.20/86.07 [f_6^#(x)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_2(g_6^#(x, x))] 238.20/86.07 238.20/86.07 [g_6^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_3(f_5^#(y))] 238.20/86.07 238.20/86.07 [g_6^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_4(g_6^#(x, y))] 238.20/86.07 238.20/86.07 [f_7^#(x)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_5(g_7^#(x, x))] 238.20/86.07 238.20/86.07 [g_7^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_6(f_6^#(y))] 238.20/86.07 238.20/86.07 [g_7^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_7(g_7^#(x, y))] 238.20/86.07 238.20/86.07 [f_8^#(x)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_8(g_8^#(x, x))] 238.20/86.07 238.20/86.07 [g_8^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_9(f_7^#(y))] 238.20/86.07 238.20/86.07 [g_8^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_10(g_8^#(x, y))] 238.20/86.07 238.20/86.07 [f_9^#(x)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_11(g_9^#(x, x))] 238.20/86.07 238.20/86.07 [g_9^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_12(f_8^#(y))] 238.20/86.07 238.20/86.07 [g_9^#(s(x), y)] = [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_13(g_9^#(x, y))] 238.20/86.07 238.20/86.07 [g_10^#(s(x), y)] = [7] y + [4] 238.20/86.07 >= [4] 238.20/86.07 = [c_14(f_9^#(y))] 238.20/86.07 238.20/86.07 [g_10^#(s(x), y)] = [7] y + [4] 238.20/86.07 >= [7] y + [4] 238.20/86.07 = [c_15(g_10^#(x, y))] 238.20/86.07 238.20/86.07 238.20/86.07 The strictly oriented rules are moved into the weak component. 238.20/86.07 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(1)). 238.20/86.07 238.20/86.07 Weak DPs: 238.20/86.07 { f_5^#(x) -> c_1() 238.20/86.07 , f_6^#(x) -> c_2(g_6^#(x, x)) 238.20/86.07 , g_6^#(s(x), y) -> c_3(f_5^#(y)) 238.20/86.07 , g_6^#(s(x), y) -> c_4(g_6^#(x, y)) 238.20/86.07 , f_7^#(x) -> c_5(g_7^#(x, x)) 238.20/86.07 , g_7^#(s(x), y) -> c_6(f_6^#(y)) 238.20/86.07 , g_7^#(s(x), y) -> c_7(g_7^#(x, y)) 238.20/86.07 , f_8^#(x) -> c_8(g_8^#(x, x)) 238.20/86.07 , g_8^#(s(x), y) -> c_9(f_7^#(y)) 238.20/86.07 , g_8^#(s(x), y) -> c_10(g_8^#(x, y)) 238.20/86.07 , f_9^#(x) -> c_11(g_9^#(x, x)) 238.20/86.07 , g_9^#(s(x), y) -> c_12(f_8^#(y)) 238.20/86.07 , g_9^#(s(x), y) -> c_13(g_9^#(x, y)) 238.20/86.07 , g_10^#(s(x), y) -> c_14(f_9^#(y)) 238.20/86.07 , g_10^#(s(x), y) -> c_15(g_10^#(x, y)) } 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(1)) 238.20/86.07 238.20/86.07 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.07 closed under successors. The DPs are removed. 238.20/86.07 238.20/86.07 { f_5^#(x) -> c_1() 238.20/86.07 , f_6^#(x) -> c_2(g_6^#(x, x)) 238.20/86.07 , g_6^#(s(x), y) -> c_3(f_5^#(y)) 238.20/86.07 , g_6^#(s(x), y) -> c_4(g_6^#(x, y)) 238.20/86.07 , f_7^#(x) -> c_5(g_7^#(x, x)) 238.20/86.07 , g_7^#(s(x), y) -> c_6(f_6^#(y)) 238.20/86.07 , g_7^#(s(x), y) -> c_7(g_7^#(x, y)) 238.20/86.07 , f_8^#(x) -> c_8(g_8^#(x, x)) 238.20/86.07 , g_8^#(s(x), y) -> c_9(f_7^#(y)) 238.20/86.07 , g_8^#(s(x), y) -> c_10(g_8^#(x, y)) 238.20/86.07 , f_9^#(x) -> c_11(g_9^#(x, x)) 238.20/86.07 , g_9^#(s(x), y) -> c_12(f_8^#(y)) 238.20/86.07 , g_9^#(s(x), y) -> c_13(g_9^#(x, y)) 238.20/86.07 , g_10^#(s(x), y) -> c_14(f_9^#(y)) 238.20/86.07 , g_10^#(s(x), y) -> c_15(g_10^#(x, y)) } 238.20/86.07 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(1)). 238.20/86.07 238.20/86.07 Rules: Empty 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(1)) 238.20/86.07 238.20/86.07 Empty rules are trivially bounded 238.20/86.07 238.20/86.07 We return to the main proof. 238.20/86.07 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(n^4)). 238.20/86.07 238.20/86.07 Strict DPs: 238.20/86.07 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.07 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.07 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.07 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.07 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.07 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.07 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) } 238.20/86.07 Weak DPs: 238.20/86.07 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.07 , f_5^#(x) -> g_5^#(x, x) 238.20/86.07 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.07 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.07 , f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(n^4)) 238.20/86.07 238.20/86.07 We decompose the input problem according to the dependency graph 238.20/86.07 into the upper component 238.20/86.07 238.20/86.07 { f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.07 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.07 , f_5^#(x) -> g_5^#(x, x) 238.20/86.07 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.07 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.07 , f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 238.20/86.07 and lower component 238.20/86.07 238.20/86.07 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.07 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.07 , f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.07 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.07 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.07 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) } 238.20/86.07 238.20/86.07 Further, following extension rules are added to the lower 238.20/86.07 component. 238.20/86.07 238.20/86.07 { f_4^#(x) -> g_4^#(x, x) 238.20/86.07 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.07 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.07 , f_5^#(x) -> g_5^#(x, x) 238.20/86.07 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.07 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.07 , f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 238.20/86.07 TcT solves the upper component with certificate YES(O(1),O(n^1)). 238.20/86.07 238.20/86.07 Sub-proof: 238.20/86.07 ---------- 238.20/86.07 We are left with following problem, upon which TcT provides the 238.20/86.07 certificate YES(O(1),O(n^1)). 238.20/86.07 238.20/86.07 Strict DPs: 238.20/86.07 { f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.07 , g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) } 238.20/86.07 Weak DPs: 238.20/86.07 { f_5^#(x) -> g_5^#(x, x) 238.20/86.07 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.07 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.07 , f_6^#(x) -> g_6^#(x, x) 238.20/86.07 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.07 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.07 , f_7^#(x) -> g_7^#(x, x) 238.20/86.07 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.07 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.07 , f_8^#(x) -> g_8^#(x, x) 238.20/86.07 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.07 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.07 , f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.07 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.07 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.07 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 Obligation: 238.20/86.07 runtime complexity 238.20/86.07 Answer: 238.20/86.07 YES(O(1),O(n^1)) 238.20/86.07 238.20/86.07 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.07 orient following rules strictly. 238.20/86.07 238.20/86.07 DPs: 238.20/86.07 { 2: g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.07 , 15: f_9^#(x) -> g_9^#(x, x) 238.20/86.07 , 19: g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.07 238.20/86.07 Sub-proof: 238.20/86.07 ---------- 238.20/86.07 The following argument positions are usable: 238.20/86.07 Uargs(c_6) = {1}, Uargs(c_7) = {2} 238.20/86.07 238.20/86.07 TcT has computed the following constructor-based matrix 238.20/86.07 interpretation satisfying not(EDA). 238.20/86.07 238.20/86.07 [s](x1) = [1] x1 + [1] 238.20/86.07 238.20/86.07 [f_3^#](x1) = [7] x1 + [7] 238.20/86.07 238.20/86.07 [f_4^#](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [g_4^#](x1, x2) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [f_5^#](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [g_5^#](x1, x2) = [1] x2 + [0] 238.20/86.07 238.20/86.07 [f_6^#](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [g_6^#](x1, x2) = [1] x2 + [0] 238.20/86.07 238.20/86.07 [f_7^#](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [g_7^#](x1, x2) = [1] x2 + [0] 238.20/86.07 238.20/86.07 [f_8^#](x1) = [7] x1 + [0] 238.20/86.07 238.20/86.07 [g_8^#](x1, x2) = [1] x2 + [0] 238.20/86.07 238.20/86.07 [f_9^#](x1) = [7] x1 + [1] 238.20/86.07 238.20/86.07 [g_9^#](x1, x2) = [7] x2 + [0] 238.20/86.07 238.20/86.07 [g_10^#](x1, x2) = [1] x1 + [7] x2 + [0] 238.20/86.07 238.20/86.07 [c_6](x1) = [1] x1 + [0] 238.20/86.07 238.20/86.07 [c_7](x1, x2) = [1] x2 + [0] 238.20/86.07 238.20/86.07 The order satisfies the following ordering constraints: 238.20/86.07 238.20/86.07 [f_4^#(x)] = [1] x + [0] 238.20/86.07 >= [1] x + [0] 238.20/86.07 = [c_6(g_4^#(x, x))] 238.20/86.07 238.20/86.07 [g_4^#(s(x), y)] = [1] x + [1] 238.20/86.07 > [1] x + [0] 238.20/86.07 = [c_7(f_3^#(y), g_4^#(x, y))] 238.20/86.07 238.20/86.07 [f_5^#(x)] = [1] x + [0] 238.20/86.07 >= [1] x + [0] 238.20/86.08 = [g_5^#(x, x)] 238.20/86.08 238.20/86.08 [g_5^#(s(x), y)] = [1] y + [0] 238.20/86.08 >= [1] y + [0] 238.20/86.08 = [f_4^#(y)] 238.20/86.08 238.20/86.08 [g_5^#(s(x), y)] = [1] y + [0] 238.20/86.08 >= [1] y + [0] 238.20/86.08 = [g_5^#(x, y)] 238.20/86.08 238.20/86.08 [f_6^#(x)] = [1] x + [0] 238.20/86.08 >= [1] x + [0] 238.20/86.08 = [g_6^#(x, x)] 238.20/86.08 238.20/86.08 [g_6^#(s(x), y)] = [1] y + [0] 238.20/86.08 >= [1] y + [0] 238.20/86.08 = [f_5^#(y)] 238.20/86.08 238.20/86.08 [g_6^#(s(x), y)] = [1] y + [0] 238.20/86.08 >= [1] y + [0] 238.20/86.08 = [g_6^#(x, y)] 238.20/86.08 238.20/86.08 [f_7^#(x)] = [1] x + [0] 238.20/86.08 >= [1] x + [0] 238.20/86.08 = [g_7^#(x, x)] 238.20/86.08 238.20/86.08 [g_7^#(s(x), y)] = [1] y + [0] 238.20/86.08 >= [1] y + [0] 238.20/86.08 = [f_6^#(y)] 238.20/86.08 238.20/86.08 [g_7^#(s(x), y)] = [1] y + [0] 238.20/86.08 >= [1] y + [0] 238.20/86.08 = [g_7^#(x, y)] 238.20/86.08 238.20/86.08 [f_8^#(x)] = [7] x + [0] 238.20/86.08 >= [1] x + [0] 238.20/86.08 = [g_8^#(x, x)] 238.20/86.08 238.20/86.08 [g_8^#(s(x), y)] = [1] y + [0] 238.20/86.08 >= [1] y + [0] 238.20/86.08 = [f_7^#(y)] 238.20/86.08 238.20/86.08 [g_8^#(s(x), y)] = [1] y + [0] 238.20/86.08 >= [1] y + [0] 238.20/86.08 = [g_8^#(x, y)] 238.20/86.08 238.20/86.08 [f_9^#(x)] = [7] x + [1] 238.20/86.08 > [7] x + [0] 238.20/86.08 = [g_9^#(x, x)] 238.20/86.08 238.20/86.08 [g_9^#(s(x), y)] = [7] y + [0] 238.20/86.08 >= [7] y + [0] 238.20/86.08 = [f_8^#(y)] 238.20/86.08 238.20/86.08 [g_9^#(s(x), y)] = [7] y + [0] 238.20/86.08 >= [7] y + [0] 238.20/86.08 = [g_9^#(x, y)] 238.20/86.08 238.20/86.08 [g_10^#(s(x), y)] = [1] x + [7] y + [1] 238.20/86.08 >= [7] y + [1] 238.20/86.08 = [f_9^#(y)] 238.20/86.08 238.20/86.08 [g_10^#(s(x), y)] = [1] x + [7] y + [1] 238.20/86.08 > [1] x + [7] y + [0] 238.20/86.08 = [g_10^#(x, y)] 238.20/86.08 238.20/86.08 238.20/86.08 We return to the main proof. Consider the set of all dependency 238.20/86.08 pairs 238.20/86.08 238.20/86.08 : 238.20/86.08 { 1: f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.08 , 2: g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.08 , 3: f_5^#(x) -> g_5^#(x, x) 238.20/86.08 , 4: g_5^#(s(x), y) -> f_4^#(y) 238.20/86.08 , 5: g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.08 , 6: f_6^#(x) -> g_6^#(x, x) 238.20/86.08 , 7: g_6^#(s(x), y) -> f_5^#(y) 238.20/86.08 , 8: g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.08 , 9: f_7^#(x) -> g_7^#(x, x) 238.20/86.08 , 10: g_7^#(s(x), y) -> f_6^#(y) 238.20/86.08 , 11: g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.08 , 12: f_8^#(x) -> g_8^#(x, x) 238.20/86.08 , 13: g_8^#(s(x), y) -> f_7^#(y) 238.20/86.08 , 14: g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.08 , 15: f_9^#(x) -> g_9^#(x, x) 238.20/86.08 , 16: g_9^#(s(x), y) -> f_8^#(y) 238.20/86.08 , 17: g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.08 , 18: g_10^#(s(x), y) -> f_9^#(y) 238.20/86.08 , 19: g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.08 238.20/86.08 Processor 'matrix interpretation of dimension 1' induces the 238.20/86.08 complexity certificate YES(?,O(n^1)) on application of dependency 238.20/86.08 pairs {2,15,19}. These cover all (indirect) predecessors of 238.20/86.08 dependency pairs {2,15,18,19}, their number of application is 238.20/86.08 equally bounded. The dependency pairs are shifted into the weak 238.20/86.08 component. 238.20/86.08 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(1)). 238.20/86.08 238.20/86.08 Strict DPs: { f_4^#(x) -> c_6(g_4^#(x, x)) } 238.20/86.08 Weak DPs: 238.20/86.08 { g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) 238.20/86.08 , f_5^#(x) -> g_5^#(x, x) 238.20/86.08 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.08 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.08 , f_6^#(x) -> g_6^#(x, x) 238.20/86.08 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.08 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.08 , f_7^#(x) -> g_7^#(x, x) 238.20/86.08 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.08 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.08 , f_8^#(x) -> g_8^#(x, x) 238.20/86.08 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.08 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.08 , f_9^#(x) -> g_9^#(x, x) 238.20/86.08 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.08 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.08 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.08 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(1)) 238.20/86.08 238.20/86.08 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.08 closed under successors. The DPs are removed. 238.20/86.08 238.20/86.08 { g_4^#(s(x), y) -> c_7(f_3^#(y), g_4^#(x, y)) } 238.20/86.08 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(1)). 238.20/86.08 238.20/86.08 Strict DPs: { f_4^#(x) -> c_6(g_4^#(x, x)) } 238.20/86.08 Weak DPs: 238.20/86.08 { f_5^#(x) -> g_5^#(x, x) 238.20/86.08 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.08 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.08 , f_6^#(x) -> g_6^#(x, x) 238.20/86.08 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.08 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.08 , f_7^#(x) -> g_7^#(x, x) 238.20/86.08 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.08 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.08 , f_8^#(x) -> g_8^#(x, x) 238.20/86.08 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.08 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.08 , f_9^#(x) -> g_9^#(x, x) 238.20/86.08 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.08 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.08 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.08 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(1)) 238.20/86.08 238.20/86.08 Due to missing edges in the dependency-graph, the right-hand sides 238.20/86.08 of following rules could be simplified: 238.20/86.08 238.20/86.08 { f_4^#(x) -> c_6(g_4^#(x, x)) } 238.20/86.08 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(1)). 238.20/86.08 238.20/86.08 Strict DPs: { f_4^#(x) -> c_1() } 238.20/86.08 Weak DPs: 238.20/86.08 { f_5^#(x) -> c_2(g_5^#(x, x)) 238.20/86.08 , g_5^#(s(x), y) -> c_3(f_4^#(y)) 238.20/86.08 , g_5^#(s(x), y) -> c_4(g_5^#(x, y)) 238.20/86.08 , f_6^#(x) -> c_5(g_6^#(x, x)) 238.20/86.08 , g_6^#(s(x), y) -> c_6(f_5^#(y)) 238.20/86.08 , g_6^#(s(x), y) -> c_7(g_6^#(x, y)) 238.20/86.08 , f_7^#(x) -> c_8(g_7^#(x, x)) 238.20/86.08 , g_7^#(s(x), y) -> c_9(f_6^#(y)) 238.20/86.08 , g_7^#(s(x), y) -> c_10(g_7^#(x, y)) 238.20/86.08 , f_8^#(x) -> c_11(g_8^#(x, x)) 238.20/86.08 , g_8^#(s(x), y) -> c_12(f_7^#(y)) 238.20/86.08 , g_8^#(s(x), y) -> c_13(g_8^#(x, y)) 238.20/86.08 , f_9^#(x) -> c_14(g_9^#(x, x)) 238.20/86.08 , g_9^#(s(x), y) -> c_15(f_8^#(y)) 238.20/86.08 , g_9^#(s(x), y) -> c_16(g_9^#(x, y)) 238.20/86.08 , g_10^#(s(x), y) -> c_17(f_9^#(y)) 238.20/86.08 , g_10^#(s(x), y) -> c_18(g_10^#(x, y)) } 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(1)) 238.20/86.08 238.20/86.08 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.08 orient following rules strictly. 238.20/86.08 238.20/86.08 DPs: 238.20/86.08 { 1: f_4^#(x) -> c_1() } 238.20/86.08 238.20/86.08 Sub-proof: 238.20/86.08 ---------- 238.20/86.08 The following argument positions are usable: 238.20/86.08 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, 238.20/86.08 Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, 238.20/86.08 Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(c_10) = {1}, 238.20/86.08 Uargs(c_11) = {1}, Uargs(c_12) = {1}, Uargs(c_13) = {1}, 238.20/86.08 Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}, 238.20/86.08 Uargs(c_17) = {1}, Uargs(c_18) = {1} 238.20/86.08 238.20/86.08 TcT has computed the following constructor-restricted matrix 238.20/86.08 interpretation. Note that the diagonal of the component-wise maxima 238.20/86.08 of interpretation-entries (of constructors) contains no more than 0 238.20/86.08 non-zero entries. 238.20/86.08 238.20/86.08 [s](x1) = [0] 238.20/86.08 238.20/86.08 [f_3^#](x1) = [0] 238.20/86.08 238.20/86.08 [f_4^#](x1) = [1] 238.20/86.08 238.20/86.08 [g_4^#](x1, x2) = [0] 238.20/86.08 238.20/86.08 [f_5^#](x1) = [1] 238.20/86.08 238.20/86.08 [g_5^#](x1, x2) = [1] 238.20/86.08 238.20/86.08 [f_6^#](x1) = [2] 238.20/86.08 238.20/86.08 [g_6^#](x1, x2) = [1] 238.20/86.08 238.20/86.08 [f_7^#](x1) = [2] 238.20/86.08 238.20/86.08 [g_7^#](x1, x2) = [2] 238.20/86.08 238.20/86.08 [f_8^#](x1) = [4] 238.20/86.08 238.20/86.08 [g_8^#](x1, x2) = [2] 238.20/86.08 238.20/86.08 [f_9^#](x1) = [4] 238.20/86.08 238.20/86.08 [g_9^#](x1, x2) = [4] 238.20/86.08 238.20/86.08 [g_10^#](x1, x2) = [4] x2 + [4] 238.20/86.08 238.20/86.08 [c_6](x1) = [0] 238.20/86.08 238.20/86.08 [c_7](x1, x2) = [0] 238.20/86.08 238.20/86.08 [c] = [0] 238.20/86.08 238.20/86.08 [c_1] = [0] 238.20/86.08 238.20/86.08 [c_2](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_3](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_4](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_5](x1) = [2] x1 + [0] 238.20/86.08 238.20/86.08 [c_6](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_7](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_8](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_9](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_10](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_11](x1) = [2] x1 + [0] 238.20/86.08 238.20/86.08 [c_12](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_13](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_14](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_15](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_16](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_17](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_18](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 The order satisfies the following ordering constraints: 238.20/86.08 238.20/86.08 [f_4^#(x)] = [1] 238.20/86.08 > [0] 238.20/86.08 = [c_1()] 238.20/86.08 238.20/86.08 [f_5^#(x)] = [1] 238.20/86.08 >= [1] 238.20/86.08 = [c_2(g_5^#(x, x))] 238.20/86.08 238.20/86.08 [g_5^#(s(x), y)] = [1] 238.20/86.08 >= [1] 238.20/86.08 = [c_3(f_4^#(y))] 238.20/86.08 238.20/86.08 [g_5^#(s(x), y)] = [1] 238.20/86.08 >= [1] 238.20/86.08 = [c_4(g_5^#(x, y))] 238.20/86.08 238.20/86.08 [f_6^#(x)] = [2] 238.20/86.08 >= [2] 238.20/86.08 = [c_5(g_6^#(x, x))] 238.20/86.08 238.20/86.08 [g_6^#(s(x), y)] = [1] 238.20/86.08 >= [1] 238.20/86.08 = [c_6(f_5^#(y))] 238.20/86.08 238.20/86.08 [g_6^#(s(x), y)] = [1] 238.20/86.08 >= [1] 238.20/86.08 = [c_7(g_6^#(x, y))] 238.20/86.08 238.20/86.08 [f_7^#(x)] = [2] 238.20/86.08 >= [2] 238.20/86.08 = [c_8(g_7^#(x, x))] 238.20/86.08 238.20/86.08 [g_7^#(s(x), y)] = [2] 238.20/86.08 >= [2] 238.20/86.08 = [c_9(f_6^#(y))] 238.20/86.08 238.20/86.08 [g_7^#(s(x), y)] = [2] 238.20/86.08 >= [2] 238.20/86.08 = [c_10(g_7^#(x, y))] 238.20/86.08 238.20/86.08 [f_8^#(x)] = [4] 238.20/86.08 >= [4] 238.20/86.08 = [c_11(g_8^#(x, x))] 238.20/86.08 238.20/86.08 [g_8^#(s(x), y)] = [2] 238.20/86.08 >= [2] 238.20/86.08 = [c_12(f_7^#(y))] 238.20/86.08 238.20/86.08 [g_8^#(s(x), y)] = [2] 238.20/86.08 >= [2] 238.20/86.08 = [c_13(g_8^#(x, y))] 238.20/86.08 238.20/86.08 [f_9^#(x)] = [4] 238.20/86.08 >= [4] 238.20/86.08 = [c_14(g_9^#(x, x))] 238.20/86.08 238.20/86.08 [g_9^#(s(x), y)] = [4] 238.20/86.08 >= [4] 238.20/86.08 = [c_15(f_8^#(y))] 238.20/86.08 238.20/86.08 [g_9^#(s(x), y)] = [4] 238.20/86.08 >= [4] 238.20/86.08 = [c_16(g_9^#(x, y))] 238.20/86.08 238.20/86.08 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.08 >= [4] 238.20/86.08 = [c_17(f_9^#(y))] 238.20/86.08 238.20/86.08 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.08 >= [4] y + [4] 238.20/86.08 = [c_18(g_10^#(x, y))] 238.20/86.08 238.20/86.08 238.20/86.08 The strictly oriented rules are moved into the weak component. 238.20/86.08 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(1)). 238.20/86.08 238.20/86.08 Weak DPs: 238.20/86.08 { f_4^#(x) -> c_1() 238.20/86.08 , f_5^#(x) -> c_2(g_5^#(x, x)) 238.20/86.08 , g_5^#(s(x), y) -> c_3(f_4^#(y)) 238.20/86.08 , g_5^#(s(x), y) -> c_4(g_5^#(x, y)) 238.20/86.08 , f_6^#(x) -> c_5(g_6^#(x, x)) 238.20/86.08 , g_6^#(s(x), y) -> c_6(f_5^#(y)) 238.20/86.08 , g_6^#(s(x), y) -> c_7(g_6^#(x, y)) 238.20/86.08 , f_7^#(x) -> c_8(g_7^#(x, x)) 238.20/86.08 , g_7^#(s(x), y) -> c_9(f_6^#(y)) 238.20/86.08 , g_7^#(s(x), y) -> c_10(g_7^#(x, y)) 238.20/86.08 , f_8^#(x) -> c_11(g_8^#(x, x)) 238.20/86.08 , g_8^#(s(x), y) -> c_12(f_7^#(y)) 238.20/86.08 , g_8^#(s(x), y) -> c_13(g_8^#(x, y)) 238.20/86.08 , f_9^#(x) -> c_14(g_9^#(x, x)) 238.20/86.08 , g_9^#(s(x), y) -> c_15(f_8^#(y)) 238.20/86.08 , g_9^#(s(x), y) -> c_16(g_9^#(x, y)) 238.20/86.08 , g_10^#(s(x), y) -> c_17(f_9^#(y)) 238.20/86.08 , g_10^#(s(x), y) -> c_18(g_10^#(x, y)) } 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(1)) 238.20/86.08 238.20/86.08 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.08 closed under successors. The DPs are removed. 238.20/86.08 238.20/86.08 { f_4^#(x) -> c_1() 238.20/86.08 , f_5^#(x) -> c_2(g_5^#(x, x)) 238.20/86.08 , g_5^#(s(x), y) -> c_3(f_4^#(y)) 238.20/86.08 , g_5^#(s(x), y) -> c_4(g_5^#(x, y)) 238.20/86.08 , f_6^#(x) -> c_5(g_6^#(x, x)) 238.20/86.08 , g_6^#(s(x), y) -> c_6(f_5^#(y)) 238.20/86.08 , g_6^#(s(x), y) -> c_7(g_6^#(x, y)) 238.20/86.08 , f_7^#(x) -> c_8(g_7^#(x, x)) 238.20/86.08 , g_7^#(s(x), y) -> c_9(f_6^#(y)) 238.20/86.08 , g_7^#(s(x), y) -> c_10(g_7^#(x, y)) 238.20/86.08 , f_8^#(x) -> c_11(g_8^#(x, x)) 238.20/86.08 , g_8^#(s(x), y) -> c_12(f_7^#(y)) 238.20/86.08 , g_8^#(s(x), y) -> c_13(g_8^#(x, y)) 238.20/86.08 , f_9^#(x) -> c_14(g_9^#(x, x)) 238.20/86.08 , g_9^#(s(x), y) -> c_15(f_8^#(y)) 238.20/86.08 , g_9^#(s(x), y) -> c_16(g_9^#(x, y)) 238.20/86.08 , g_10^#(s(x), y) -> c_17(f_9^#(y)) 238.20/86.08 , g_10^#(s(x), y) -> c_18(g_10^#(x, y)) } 238.20/86.08 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(1)). 238.20/86.08 238.20/86.08 Rules: Empty 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(1)) 238.20/86.08 238.20/86.08 Empty rules are trivially bounded 238.20/86.08 238.20/86.08 We return to the main proof. 238.20/86.08 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(n^3)). 238.20/86.08 238.20/86.08 Strict DPs: 238.20/86.08 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.08 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.08 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) 238.20/86.08 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.08 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) } 238.20/86.08 Weak DPs: 238.20/86.08 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.08 , f_4^#(x) -> g_4^#(x, x) 238.20/86.08 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.08 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.08 , f_5^#(x) -> g_5^#(x, x) 238.20/86.08 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.08 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.08 , f_6^#(x) -> g_6^#(x, x) 238.20/86.08 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.08 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.08 , f_7^#(x) -> g_7^#(x, x) 238.20/86.08 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.08 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.08 , f_8^#(x) -> g_8^#(x, x) 238.20/86.08 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.08 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.08 , f_9^#(x) -> g_9^#(x, x) 238.20/86.08 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.08 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.08 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.08 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(n^3)) 238.20/86.08 238.20/86.08 We decompose the input problem according to the dependency graph 238.20/86.08 into the upper component 238.20/86.08 238.20/86.08 { f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.08 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.08 , f_4^#(x) -> g_4^#(x, x) 238.20/86.08 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.08 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.08 , f_5^#(x) -> g_5^#(x, x) 238.20/86.08 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.08 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.08 , f_6^#(x) -> g_6^#(x, x) 238.20/86.08 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.08 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.08 , f_7^#(x) -> g_7^#(x, x) 238.20/86.08 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.08 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.08 , f_8^#(x) -> g_8^#(x, x) 238.20/86.08 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.08 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.08 , f_9^#(x) -> g_9^#(x, x) 238.20/86.08 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.08 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.08 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.08 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.08 238.20/86.08 and lower component 238.20/86.08 238.20/86.08 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.08 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.08 , f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.08 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) } 238.20/86.08 238.20/86.08 Further, following extension rules are added to the lower 238.20/86.08 component. 238.20/86.08 238.20/86.08 { f_3^#(x) -> g_3^#(x, x) 238.20/86.08 , g_3^#(s(x), y) -> f_2^#(y) 238.20/86.08 , g_3^#(s(x), y) -> g_3^#(x, y) 238.20/86.08 , f_4^#(x) -> g_4^#(x, x) 238.20/86.08 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.08 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.08 , f_5^#(x) -> g_5^#(x, x) 238.20/86.08 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.08 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.08 , f_6^#(x) -> g_6^#(x, x) 238.20/86.08 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.08 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.08 , f_7^#(x) -> g_7^#(x, x) 238.20/86.08 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.08 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.08 , f_8^#(x) -> g_8^#(x, x) 238.20/86.08 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.08 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.08 , f_9^#(x) -> g_9^#(x, x) 238.20/86.08 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.08 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.08 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.08 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.08 238.20/86.08 TcT solves the upper component with certificate YES(O(1),O(n^1)). 238.20/86.08 238.20/86.08 Sub-proof: 238.20/86.08 ---------- 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(n^1)). 238.20/86.08 238.20/86.08 Strict DPs: 238.20/86.08 { f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.08 , g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) } 238.20/86.08 Weak DPs: 238.20/86.08 { f_4^#(x) -> g_4^#(x, x) 238.20/86.08 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.08 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.08 , f_5^#(x) -> g_5^#(x, x) 238.20/86.08 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.08 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.08 , f_6^#(x) -> g_6^#(x, x) 238.20/86.08 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.08 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.08 , f_7^#(x) -> g_7^#(x, x) 238.20/86.08 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.08 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.08 , f_8^#(x) -> g_8^#(x, x) 238.20/86.08 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.08 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.08 , f_9^#(x) -> g_9^#(x, x) 238.20/86.08 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.08 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.08 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.08 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(n^1)) 238.20/86.08 238.20/86.08 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.08 orient following rules strictly. 238.20/86.08 238.20/86.08 DPs: 238.20/86.08 { 2: g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.08 , 18: f_9^#(x) -> g_9^#(x, x) } 238.20/86.08 238.20/86.08 Sub-proof: 238.20/86.08 ---------- 238.20/86.08 The following argument positions are usable: 238.20/86.08 Uargs(c_4) = {1}, Uargs(c_5) = {2} 238.20/86.08 238.20/86.08 TcT has computed the following constructor-based matrix 238.20/86.08 interpretation satisfying not(EDA). 238.20/86.08 238.20/86.08 [s](x1) = [1] x1 + [4] 238.20/86.08 238.20/86.08 [f_2^#](x1) = [7] x1 + [7] 238.20/86.08 238.20/86.08 [f_3^#](x1) = [4] x1 + [0] 238.20/86.08 238.20/86.08 [g_3^#](x1, x2) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [f_4^#](x1) = [4] x1 + [0] 238.20/86.08 238.20/86.08 [g_4^#](x1, x2) = [4] x2 + [0] 238.20/86.08 238.20/86.08 [f_5^#](x1) = [4] x1 + [0] 238.20/86.08 238.20/86.08 [g_5^#](x1, x2) = [4] x2 + [0] 238.20/86.08 238.20/86.08 [f_6^#](x1) = [4] x1 + [0] 238.20/86.08 238.20/86.08 [g_6^#](x1, x2) = [4] x2 + [0] 238.20/86.08 238.20/86.08 [f_7^#](x1) = [4] x1 + [0] 238.20/86.08 238.20/86.08 [g_7^#](x1, x2) = [4] x2 + [0] 238.20/86.08 238.20/86.08 [f_8^#](x1) = [7] x1 + [0] 238.20/86.08 238.20/86.08 [g_8^#](x1, x2) = [4] x2 + [0] 238.20/86.08 238.20/86.08 [f_9^#](x1) = [7] x1 + [1] 238.20/86.08 238.20/86.08 [g_9^#](x1, x2) = [7] x2 + [0] 238.20/86.08 238.20/86.08 [g_10^#](x1, x2) = [7] x2 + [1] 238.20/86.08 238.20/86.08 [c_4](x1) = [1] x1 + [0] 238.20/86.08 238.20/86.08 [c_5](x1, x2) = [1] x2 + [1] 238.20/86.08 238.20/86.08 The order satisfies the following ordering constraints: 238.20/86.08 238.20/86.08 [f_3^#(x)] = [4] x + [0] 238.20/86.08 >= [1] x + [0] 238.20/86.08 = [c_4(g_3^#(x, x))] 238.20/86.08 238.20/86.08 [g_3^#(s(x), y)] = [1] x + [4] 238.20/86.08 > [1] x + [1] 238.20/86.08 = [c_5(f_2^#(y), g_3^#(x, y))] 238.20/86.08 238.20/86.08 [f_4^#(x)] = [4] x + [0] 238.20/86.08 >= [4] x + [0] 238.20/86.08 = [g_4^#(x, x)] 238.20/86.08 238.20/86.08 [g_4^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [f_3^#(y)] 238.20/86.08 238.20/86.08 [g_4^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [g_4^#(x, y)] 238.20/86.08 238.20/86.08 [f_5^#(x)] = [4] x + [0] 238.20/86.08 >= [4] x + [0] 238.20/86.08 = [g_5^#(x, x)] 238.20/86.08 238.20/86.08 [g_5^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [f_4^#(y)] 238.20/86.08 238.20/86.08 [g_5^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [g_5^#(x, y)] 238.20/86.08 238.20/86.08 [f_6^#(x)] = [4] x + [0] 238.20/86.08 >= [4] x + [0] 238.20/86.08 = [g_6^#(x, x)] 238.20/86.08 238.20/86.08 [g_6^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [f_5^#(y)] 238.20/86.08 238.20/86.08 [g_6^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [g_6^#(x, y)] 238.20/86.08 238.20/86.08 [f_7^#(x)] = [4] x + [0] 238.20/86.08 >= [4] x + [0] 238.20/86.08 = [g_7^#(x, x)] 238.20/86.08 238.20/86.08 [g_7^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [f_6^#(y)] 238.20/86.08 238.20/86.08 [g_7^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [g_7^#(x, y)] 238.20/86.08 238.20/86.08 [f_8^#(x)] = [7] x + [0] 238.20/86.08 >= [4] x + [0] 238.20/86.08 = [g_8^#(x, x)] 238.20/86.08 238.20/86.08 [g_8^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [f_7^#(y)] 238.20/86.08 238.20/86.08 [g_8^#(s(x), y)] = [4] y + [0] 238.20/86.08 >= [4] y + [0] 238.20/86.08 = [g_8^#(x, y)] 238.20/86.08 238.20/86.08 [f_9^#(x)] = [7] x + [1] 238.20/86.08 > [7] x + [0] 238.20/86.08 = [g_9^#(x, x)] 238.20/86.08 238.20/86.08 [g_9^#(s(x), y)] = [7] y + [0] 238.20/86.08 >= [7] y + [0] 238.20/86.08 = [f_8^#(y)] 238.20/86.08 238.20/86.08 [g_9^#(s(x), y)] = [7] y + [0] 238.20/86.08 >= [7] y + [0] 238.20/86.08 = [g_9^#(x, y)] 238.20/86.08 238.20/86.08 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.08 >= [7] y + [1] 238.20/86.08 = [f_9^#(y)] 238.20/86.08 238.20/86.08 [g_10^#(s(x), y)] = [7] y + [1] 238.20/86.08 >= [7] y + [1] 238.20/86.08 = [g_10^#(x, y)] 238.20/86.08 238.20/86.08 238.20/86.08 The strictly oriented rules are moved into the weak component. 238.20/86.08 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(1)). 238.20/86.08 238.20/86.08 Strict DPs: { f_3^#(x) -> c_4(g_3^#(x, x)) } 238.20/86.08 Weak DPs: 238.20/86.08 { g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) 238.20/86.08 , f_4^#(x) -> g_4^#(x, x) 238.20/86.08 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.08 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.08 , f_5^#(x) -> g_5^#(x, x) 238.20/86.08 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.08 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.08 , f_6^#(x) -> g_6^#(x, x) 238.20/86.08 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.08 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.08 , f_7^#(x) -> g_7^#(x, x) 238.20/86.08 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.08 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.08 , f_8^#(x) -> g_8^#(x, x) 238.20/86.08 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.08 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.08 , f_9^#(x) -> g_9^#(x, x) 238.20/86.08 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.08 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.08 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.08 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(1)) 238.20/86.08 238.20/86.08 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.08 closed under successors. The DPs are removed. 238.20/86.08 238.20/86.08 { g_3^#(s(x), y) -> c_5(f_2^#(y), g_3^#(x, y)) } 238.20/86.08 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(1)). 238.20/86.08 238.20/86.08 Strict DPs: { f_3^#(x) -> c_4(g_3^#(x, x)) } 238.20/86.08 Weak DPs: 238.20/86.08 { f_4^#(x) -> g_4^#(x, x) 238.20/86.08 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.08 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.08 , f_5^#(x) -> g_5^#(x, x) 238.20/86.08 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.08 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.08 , f_6^#(x) -> g_6^#(x, x) 238.20/86.08 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.08 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.08 , f_7^#(x) -> g_7^#(x, x) 238.20/86.08 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.08 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.08 , f_8^#(x) -> g_8^#(x, x) 238.20/86.08 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.08 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.08 , f_9^#(x) -> g_9^#(x, x) 238.20/86.08 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.08 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.08 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.08 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(1)) 238.20/86.08 238.20/86.08 Due to missing edges in the dependency-graph, the right-hand sides 238.20/86.08 of following rules could be simplified: 238.20/86.08 238.20/86.08 { f_3^#(x) -> c_4(g_3^#(x, x)) } 238.20/86.08 238.20/86.08 We are left with following problem, upon which TcT provides the 238.20/86.08 certificate YES(O(1),O(1)). 238.20/86.08 238.20/86.08 Strict DPs: { f_3^#(x) -> c_1() } 238.20/86.08 Weak DPs: 238.20/86.08 { f_4^#(x) -> c_2(g_4^#(x, x)) 238.20/86.08 , g_4^#(s(x), y) -> c_3(f_3^#(y)) 238.20/86.08 , g_4^#(s(x), y) -> c_4(g_4^#(x, y)) 238.20/86.08 , f_5^#(x) -> c_5(g_5^#(x, x)) 238.20/86.08 , g_5^#(s(x), y) -> c_6(f_4^#(y)) 238.20/86.08 , g_5^#(s(x), y) -> c_7(g_5^#(x, y)) 238.20/86.08 , f_6^#(x) -> c_8(g_6^#(x, x)) 238.20/86.08 , g_6^#(s(x), y) -> c_9(f_5^#(y)) 238.20/86.08 , g_6^#(s(x), y) -> c_10(g_6^#(x, y)) 238.20/86.08 , f_7^#(x) -> c_11(g_7^#(x, x)) 238.20/86.08 , g_7^#(s(x), y) -> c_12(f_6^#(y)) 238.20/86.08 , g_7^#(s(x), y) -> c_13(g_7^#(x, y)) 238.20/86.08 , f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.08 , g_8^#(s(x), y) -> c_15(f_7^#(y)) 238.20/86.08 , g_8^#(s(x), y) -> c_16(g_8^#(x, y)) 238.20/86.08 , f_9^#(x) -> c_17(g_9^#(x, x)) 238.20/86.08 , g_9^#(s(x), y) -> c_18(f_8^#(y)) 238.20/86.08 , g_9^#(s(x), y) -> c_19(g_9^#(x, y)) 238.20/86.08 , g_10^#(s(x), y) -> c_20(f_9^#(y)) 238.20/86.08 , g_10^#(s(x), y) -> c_21(g_10^#(x, y)) } 238.20/86.08 Obligation: 238.20/86.08 runtime complexity 238.20/86.08 Answer: 238.20/86.08 YES(O(1),O(1)) 238.20/86.08 238.20/86.08 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.08 orient following rules strictly. 238.20/86.08 238.20/86.08 DPs: 238.20/86.08 { 1: f_3^#(x) -> c_1() } 238.20/86.08 238.20/86.08 Sub-proof: 238.20/86.08 ---------- 238.20/86.08 The following argument positions are usable: 238.20/86.08 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, 238.20/86.08 Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, 238.20/86.08 Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(c_10) = {1}, 238.20/86.08 Uargs(c_11) = {1}, Uargs(c_12) = {1}, Uargs(c_13) = {1}, 238.20/86.08 Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}, 238.20/86.08 Uargs(c_17) = {1}, Uargs(c_18) = {1}, Uargs(c_19) = {1}, 238.20/86.08 Uargs(c_20) = {1}, Uargs(c_21) = {1} 238.20/86.08 238.20/86.08 TcT has computed the following constructor-restricted matrix 238.20/86.08 interpretation. Note that the diagonal of the component-wise maxima 238.20/86.08 of interpretation-entries (of constructors) contains no more than 0 238.20/86.08 non-zero entries. 238.20/86.08 238.20/86.08 [s](x1) = [0] 238.20/86.08 238.20/86.08 [f_2^#](x1) = [0] 238.20/86.08 238.20/86.08 [f_3^#](x1) = [2] 238.20/86.08 238.20/86.08 [g_3^#](x1, x2) = [0] 238.20/86.08 238.20/86.08 [f_4^#](x1) = [2] 238.20/86.08 238.20/86.08 [g_4^#](x1, x2) = [2] 238.20/86.08 238.20/86.08 [f_5^#](x1) = [2] 238.20/86.08 238.20/86.08 [g_5^#](x1, x2) = [2] 238.20/86.08 238.20/86.08 [f_6^#](x1) = [2] 238.20/86.08 238.20/86.08 [g_6^#](x1, x2) = [2] 238.20/86.08 238.20/86.08 [f_7^#](x1) = [2] 238.20/86.08 238.20/86.08 [g_7^#](x1, x2) = [2] 238.20/86.08 238.20/86.09 [f_8^#](x1) = [4] 238.20/86.09 238.20/86.09 [g_8^#](x1, x2) = [4] 238.20/86.09 238.20/86.09 [f_9^#](x1) = [4] 238.20/86.09 238.20/86.09 [g_9^#](x1, x2) = [4] 238.20/86.09 238.20/86.09 [g_10^#](x1, x2) = [4] x2 + [4] 238.20/86.09 238.20/86.09 [c_4](x1) = [0] 238.20/86.09 238.20/86.09 [c_5](x1, x2) = [0] 238.20/86.09 238.20/86.09 [c] = [0] 238.20/86.09 238.20/86.09 [c_1] = [0] 238.20/86.09 238.20/86.09 [c_2](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_3](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_4](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_5](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_6](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_7](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_8](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_9](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_10](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_11](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_12](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_13](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_14](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_15](x1) = [2] x1 + [0] 238.20/86.09 238.20/86.09 [c_16](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_17](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_18](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_19](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_20](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_21](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 The order satisfies the following ordering constraints: 238.20/86.09 238.20/86.09 [f_3^#(x)] = [2] 238.20/86.09 > [0] 238.20/86.09 = [c_1()] 238.20/86.09 238.20/86.09 [f_4^#(x)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_2(g_4^#(x, x))] 238.20/86.09 238.20/86.09 [g_4^#(s(x), y)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_3(f_3^#(y))] 238.20/86.09 238.20/86.09 [g_4^#(s(x), y)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_4(g_4^#(x, y))] 238.20/86.09 238.20/86.09 [f_5^#(x)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_5(g_5^#(x, x))] 238.20/86.09 238.20/86.09 [g_5^#(s(x), y)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_6(f_4^#(y))] 238.20/86.09 238.20/86.09 [g_5^#(s(x), y)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_7(g_5^#(x, y))] 238.20/86.09 238.20/86.09 [f_6^#(x)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_8(g_6^#(x, x))] 238.20/86.09 238.20/86.09 [g_6^#(s(x), y)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_9(f_5^#(y))] 238.20/86.09 238.20/86.09 [g_6^#(s(x), y)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_10(g_6^#(x, y))] 238.20/86.09 238.20/86.09 [f_7^#(x)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_11(g_7^#(x, x))] 238.20/86.09 238.20/86.09 [g_7^#(s(x), y)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_12(f_6^#(y))] 238.20/86.09 238.20/86.09 [g_7^#(s(x), y)] = [2] 238.20/86.09 >= [2] 238.20/86.09 = [c_13(g_7^#(x, y))] 238.20/86.09 238.20/86.09 [f_8^#(x)] = [4] 238.20/86.09 >= [4] 238.20/86.09 = [c_14(g_8^#(x, x))] 238.20/86.09 238.20/86.09 [g_8^#(s(x), y)] = [4] 238.20/86.09 >= [4] 238.20/86.09 = [c_15(f_7^#(y))] 238.20/86.09 238.20/86.09 [g_8^#(s(x), y)] = [4] 238.20/86.09 >= [4] 238.20/86.09 = [c_16(g_8^#(x, y))] 238.20/86.09 238.20/86.09 [f_9^#(x)] = [4] 238.20/86.09 >= [4] 238.20/86.09 = [c_17(g_9^#(x, x))] 238.20/86.09 238.20/86.09 [g_9^#(s(x), y)] = [4] 238.20/86.09 >= [4] 238.20/86.09 = [c_18(f_8^#(y))] 238.20/86.09 238.20/86.09 [g_9^#(s(x), y)] = [4] 238.20/86.09 >= [4] 238.20/86.09 = [c_19(g_9^#(x, y))] 238.20/86.09 238.20/86.09 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.09 >= [4] 238.20/86.09 = [c_20(f_9^#(y))] 238.20/86.09 238.20/86.09 [g_10^#(s(x), y)] = [4] y + [4] 238.20/86.09 >= [4] y + [4] 238.20/86.09 = [c_21(g_10^#(x, y))] 238.20/86.09 238.20/86.09 238.20/86.09 The strictly oriented rules are moved into the weak component. 238.20/86.09 238.20/86.09 We are left with following problem, upon which TcT provides the 238.20/86.09 certificate YES(O(1),O(1)). 238.20/86.09 238.20/86.09 Weak DPs: 238.20/86.09 { f_3^#(x) -> c_1() 238.20/86.09 , f_4^#(x) -> c_2(g_4^#(x, x)) 238.20/86.09 , g_4^#(s(x), y) -> c_3(f_3^#(y)) 238.20/86.09 , g_4^#(s(x), y) -> c_4(g_4^#(x, y)) 238.20/86.09 , f_5^#(x) -> c_5(g_5^#(x, x)) 238.20/86.09 , g_5^#(s(x), y) -> c_6(f_4^#(y)) 238.20/86.09 , g_5^#(s(x), y) -> c_7(g_5^#(x, y)) 238.20/86.09 , f_6^#(x) -> c_8(g_6^#(x, x)) 238.20/86.09 , g_6^#(s(x), y) -> c_9(f_5^#(y)) 238.20/86.09 , g_6^#(s(x), y) -> c_10(g_6^#(x, y)) 238.20/86.09 , f_7^#(x) -> c_11(g_7^#(x, x)) 238.20/86.09 , g_7^#(s(x), y) -> c_12(f_6^#(y)) 238.20/86.09 , g_7^#(s(x), y) -> c_13(g_7^#(x, y)) 238.20/86.09 , f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.09 , g_8^#(s(x), y) -> c_15(f_7^#(y)) 238.20/86.09 , g_8^#(s(x), y) -> c_16(g_8^#(x, y)) 238.20/86.09 , f_9^#(x) -> c_17(g_9^#(x, x)) 238.20/86.09 , g_9^#(s(x), y) -> c_18(f_8^#(y)) 238.20/86.09 , g_9^#(s(x), y) -> c_19(g_9^#(x, y)) 238.20/86.09 , g_10^#(s(x), y) -> c_20(f_9^#(y)) 238.20/86.09 , g_10^#(s(x), y) -> c_21(g_10^#(x, y)) } 238.20/86.09 Obligation: 238.20/86.09 runtime complexity 238.20/86.09 Answer: 238.20/86.09 YES(O(1),O(1)) 238.20/86.09 238.20/86.09 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.09 closed under successors. The DPs are removed. 238.20/86.09 238.20/86.09 { f_3^#(x) -> c_1() 238.20/86.09 , f_4^#(x) -> c_2(g_4^#(x, x)) 238.20/86.09 , g_4^#(s(x), y) -> c_3(f_3^#(y)) 238.20/86.09 , g_4^#(s(x), y) -> c_4(g_4^#(x, y)) 238.20/86.09 , f_5^#(x) -> c_5(g_5^#(x, x)) 238.20/86.09 , g_5^#(s(x), y) -> c_6(f_4^#(y)) 238.20/86.09 , g_5^#(s(x), y) -> c_7(g_5^#(x, y)) 238.20/86.09 , f_6^#(x) -> c_8(g_6^#(x, x)) 238.20/86.09 , g_6^#(s(x), y) -> c_9(f_5^#(y)) 238.20/86.09 , g_6^#(s(x), y) -> c_10(g_6^#(x, y)) 238.20/86.09 , f_7^#(x) -> c_11(g_7^#(x, x)) 238.20/86.09 , g_7^#(s(x), y) -> c_12(f_6^#(y)) 238.20/86.09 , g_7^#(s(x), y) -> c_13(g_7^#(x, y)) 238.20/86.09 , f_8^#(x) -> c_14(g_8^#(x, x)) 238.20/86.09 , g_8^#(s(x), y) -> c_15(f_7^#(y)) 238.20/86.09 , g_8^#(s(x), y) -> c_16(g_8^#(x, y)) 238.20/86.09 , f_9^#(x) -> c_17(g_9^#(x, x)) 238.20/86.09 , g_9^#(s(x), y) -> c_18(f_8^#(y)) 238.20/86.09 , g_9^#(s(x), y) -> c_19(g_9^#(x, y)) 238.20/86.09 , g_10^#(s(x), y) -> c_20(f_9^#(y)) 238.20/86.09 , g_10^#(s(x), y) -> c_21(g_10^#(x, y)) } 238.20/86.09 238.20/86.09 We are left with following problem, upon which TcT provides the 238.20/86.09 certificate YES(O(1),O(1)). 238.20/86.09 238.20/86.09 Rules: Empty 238.20/86.09 Obligation: 238.20/86.09 runtime complexity 238.20/86.09 Answer: 238.20/86.09 YES(O(1),O(1)) 238.20/86.09 238.20/86.09 Empty rules are trivially bounded 238.20/86.09 238.20/86.09 We return to the main proof. 238.20/86.09 238.20/86.09 We are left with following problem, upon which TcT provides the 238.20/86.09 certificate YES(O(1),O(n^2)). 238.20/86.09 238.20/86.09 Strict DPs: 238.20/86.09 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.09 , g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.09 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) } 238.20/86.09 Weak DPs: 238.20/86.09 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.09 , f_3^#(x) -> g_3^#(x, x) 238.20/86.09 , g_3^#(s(x), y) -> f_2^#(y) 238.20/86.09 , g_3^#(s(x), y) -> g_3^#(x, y) 238.20/86.09 , f_4^#(x) -> g_4^#(x, x) 238.20/86.09 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.09 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.09 , f_5^#(x) -> g_5^#(x, x) 238.20/86.09 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.09 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.09 , f_6^#(x) -> g_6^#(x, x) 238.20/86.09 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.09 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.09 , f_7^#(x) -> g_7^#(x, x) 238.20/86.09 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.09 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.09 , f_8^#(x) -> g_8^#(x, x) 238.20/86.09 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.09 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.09 , f_9^#(x) -> g_9^#(x, x) 238.20/86.09 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.09 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.09 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.09 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.09 Obligation: 238.20/86.09 runtime complexity 238.20/86.09 Answer: 238.20/86.09 YES(O(1),O(n^2)) 238.20/86.09 238.20/86.09 We use the processor 'polynomial interpretation' to orient 238.20/86.09 following rules strictly. 238.20/86.09 238.20/86.09 DPs: 238.20/86.09 { 2: g_1^#(s(x), y) -> c_2(g_1^#(x, y)) } 238.20/86.09 238.20/86.09 Sub-proof: 238.20/86.09 ---------- 238.20/86.09 We consider the following typing: 238.20/86.09 238.20/86.09 s :: a -> a 238.20/86.09 f_1^# :: a -> c 238.20/86.09 g_1^# :: (a,a) -> b 238.20/86.09 f_2^# :: a -> e 238.20/86.09 g_2^# :: (a,a) -> d 238.20/86.09 f_3^# :: a -> e 238.20/86.09 g_3^# :: (a,a) -> e 238.20/86.09 f_4^# :: a -> e 238.20/86.09 g_4^# :: (a,a) -> e 238.20/86.09 f_5^# :: a -> e 238.20/86.09 g_5^# :: (a,a) -> e 238.20/86.09 f_6^# :: a -> e 238.20/86.09 g_6^# :: (a,a) -> e 238.20/86.09 f_7^# :: a -> e 238.20/86.09 g_7^# :: (a,a) -> e 238.20/86.09 f_8^# :: a -> e 238.20/86.09 g_8^# :: (a,a) -> e 238.20/86.09 f_9^# :: a -> e 238.20/86.09 g_9^# :: (a,a) -> e 238.20/86.09 g_10^# :: (a,a) -> e 238.20/86.09 c_1 :: b -> c 238.20/86.09 c_2 :: b -> b 238.20/86.09 c_3 :: (c,d) -> d 238.20/86.09 c_18 :: d -> e 238.20/86.09 238.20/86.09 The following argument positions are considered usable: 238.20/86.09 238.20/86.09 Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1, 2}, 238.20/86.09 Uargs(c_18) = {1} 238.20/86.09 238.20/86.09 TcT has computed the following constructor-restricted 238.20/86.09 typedpolynomial interpretation. 238.20/86.09 238.20/86.09 [s](x1) = 1 + x1 238.20/86.09 238.20/86.09 [f_1^#](x1) = x1 238.20/86.09 238.20/86.09 [g_1^#](x1, x2) = x1 238.20/86.09 238.20/86.09 [f_2^#](x1) = 3 + x1^2 238.20/86.09 238.20/86.09 [g_2^#](x1, x2) = x1*x2 238.20/86.09 238.20/86.09 [f_3^#](x1) = 3 + x1^2 238.20/86.09 238.20/86.09 [g_3^#](x1, x2) = 3 + x2^2 238.20/86.09 238.20/86.09 [f_4^#](x1) = 3 + x1^2 238.20/86.09 238.20/86.09 [g_4^#](x1, x2) = 3 + x2^2 238.20/86.09 238.20/86.09 [f_5^#](x1) = 3 + x1^2 238.20/86.09 238.20/86.09 [g_5^#](x1, x2) = 3 + x2^2 238.20/86.09 238.20/86.09 [f_6^#](x1) = 3 + x1^2 238.20/86.09 238.20/86.09 [g_6^#](x1, x2) = 3 + x2^2 238.20/86.09 238.20/86.09 [f_7^#](x1) = 3 + x1^2 238.20/86.09 238.20/86.09 [g_7^#](x1, x2) = 3 + x2^2 238.20/86.09 238.20/86.09 [f_8^#](x1) = 3 + 2*x1^2 238.20/86.09 238.20/86.09 [g_8^#](x1, x2) = 3 + 2*x2^2 238.20/86.09 238.20/86.09 [f_9^#](x1) = 3 + 3*x1 + 3*x1^2 238.20/86.09 238.20/86.09 [g_9^#](x1, x2) = 3 + 2*x2^2 238.20/86.09 238.20/86.09 [g_10^#](x1, x2) = 3 + 3*x1*x2 + 3*x2 + 3*x2^2 238.20/86.09 238.20/86.09 [c_1](x1) = x1 238.20/86.09 238.20/86.09 [c_2](x1) = x1 238.20/86.09 238.20/86.09 [c_3](x1, x2) = x1 + x2 238.20/86.09 238.20/86.09 [c_18](x1) = 3 + x1 238.20/86.09 238.20/86.09 238.20/86.09 This order satisfies the following ordering constraints. 238.20/86.09 238.20/86.09 [f_1^#(x)] = x 238.20/86.09 >= x 238.20/86.09 = [c_1(g_1^#(x, x))] 238.20/86.09 238.20/86.09 [g_1^#(s(x), y)] = 1 + x 238.20/86.09 > x 238.20/86.09 = [c_2(g_1^#(x, y))] 238.20/86.09 238.20/86.09 [f_2^#(x)] = 3 + x^2 238.20/86.09 >= 3 + x^2 238.20/86.09 = [c_18(g_2^#(x, x))] 238.20/86.09 238.20/86.09 [g_2^#(s(x), y)] = y + x*y 238.20/86.09 >= y + x*y 238.20/86.09 = [c_3(f_1^#(y), g_2^#(x, y))] 238.20/86.09 238.20/86.09 [f_3^#(x)] = 3 + x^2 238.20/86.09 >= 3 + x^2 238.20/86.09 = [g_3^#(x, x)] 238.20/86.09 238.20/86.09 [g_3^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [f_2^#(y)] 238.20/86.09 238.20/86.09 [g_3^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [g_3^#(x, y)] 238.20/86.09 238.20/86.09 [f_4^#(x)] = 3 + x^2 238.20/86.09 >= 3 + x^2 238.20/86.09 = [g_4^#(x, x)] 238.20/86.09 238.20/86.09 [g_4^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [f_3^#(y)] 238.20/86.09 238.20/86.09 [g_4^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [g_4^#(x, y)] 238.20/86.09 238.20/86.09 [f_5^#(x)] = 3 + x^2 238.20/86.09 >= 3 + x^2 238.20/86.09 = [g_5^#(x, x)] 238.20/86.09 238.20/86.09 [g_5^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [f_4^#(y)] 238.20/86.09 238.20/86.09 [g_5^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [g_5^#(x, y)] 238.20/86.09 238.20/86.09 [f_6^#(x)] = 3 + x^2 238.20/86.09 >= 3 + x^2 238.20/86.09 = [g_6^#(x, x)] 238.20/86.09 238.20/86.09 [g_6^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [f_5^#(y)] 238.20/86.09 238.20/86.09 [g_6^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [g_6^#(x, y)] 238.20/86.09 238.20/86.09 [f_7^#(x)] = 3 + x^2 238.20/86.09 >= 3 + x^2 238.20/86.09 = [g_7^#(x, x)] 238.20/86.09 238.20/86.09 [g_7^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [f_6^#(y)] 238.20/86.09 238.20/86.09 [g_7^#(s(x), y)] = 3 + y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [g_7^#(x, y)] 238.20/86.09 238.20/86.09 [f_8^#(x)] = 3 + 2*x^2 238.20/86.09 >= 3 + 2*x^2 238.20/86.09 = [g_8^#(x, x)] 238.20/86.09 238.20/86.09 [g_8^#(s(x), y)] = 3 + 2*y^2 238.20/86.09 >= 3 + y^2 238.20/86.09 = [f_7^#(y)] 238.20/86.09 238.20/86.09 [g_8^#(s(x), y)] = 3 + 2*y^2 238.20/86.09 >= 3 + 2*y^2 238.20/86.09 = [g_8^#(x, y)] 238.20/86.09 238.20/86.09 [f_9^#(x)] = 3 + 3*x + 3*x^2 238.20/86.09 >= 3 + 2*x^2 238.20/86.09 = [g_9^#(x, x)] 238.20/86.09 238.20/86.09 [g_9^#(s(x), y)] = 3 + 2*y^2 238.20/86.09 >= 3 + 2*y^2 238.20/86.09 = [f_8^#(y)] 238.20/86.09 238.20/86.09 [g_9^#(s(x), y)] = 3 + 2*y^2 238.20/86.09 >= 3 + 2*y^2 238.20/86.09 = [g_9^#(x, y)] 238.20/86.09 238.20/86.09 [g_10^#(s(x), y)] = 3 + 6*y + 3*x*y + 3*y^2 238.20/86.09 >= 3 + 3*y + 3*y^2 238.20/86.09 = [f_9^#(y)] 238.20/86.09 238.20/86.09 [g_10^#(s(x), y)] = 3 + 6*y + 3*x*y + 3*y^2 238.20/86.09 >= 3 + 3*x*y + 3*y + 3*y^2 238.20/86.09 = [g_10^#(x, y)] 238.20/86.09 238.20/86.09 238.20/86.09 The strictly oriented rules are moved into the weak component. 238.20/86.09 238.20/86.09 We are left with following problem, upon which TcT provides the 238.20/86.09 certificate YES(O(1),O(n^1)). 238.20/86.09 238.20/86.09 Strict DPs: 238.20/86.09 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.09 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) } 238.20/86.09 Weak DPs: 238.20/86.09 { g_1^#(s(x), y) -> c_2(g_1^#(x, y)) 238.20/86.09 , f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.09 , f_3^#(x) -> g_3^#(x, x) 238.20/86.09 , g_3^#(s(x), y) -> f_2^#(y) 238.20/86.09 , g_3^#(s(x), y) -> g_3^#(x, y) 238.20/86.09 , f_4^#(x) -> g_4^#(x, x) 238.20/86.09 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.09 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.09 , f_5^#(x) -> g_5^#(x, x) 238.20/86.09 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.09 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.09 , f_6^#(x) -> g_6^#(x, x) 238.20/86.09 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.09 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.09 , f_7^#(x) -> g_7^#(x, x) 238.20/86.09 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.09 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.09 , f_8^#(x) -> g_8^#(x, x) 238.20/86.09 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.09 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.09 , f_9^#(x) -> g_9^#(x, x) 238.20/86.09 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.09 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.09 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.09 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.09 Obligation: 238.20/86.09 runtime complexity 238.20/86.09 Answer: 238.20/86.09 YES(O(1),O(n^1)) 238.20/86.09 238.20/86.09 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.09 closed under successors. The DPs are removed. 238.20/86.09 238.20/86.09 { g_1^#(s(x), y) -> c_2(g_1^#(x, y)) } 238.20/86.09 238.20/86.09 We are left with following problem, upon which TcT provides the 238.20/86.09 certificate YES(O(1),O(n^1)). 238.20/86.09 238.20/86.09 Strict DPs: 238.20/86.09 { f_1^#(x) -> c_1(g_1^#(x, x)) 238.20/86.09 , g_2^#(s(x), y) -> c_3(f_1^#(y), g_2^#(x, y)) } 238.20/86.09 Weak DPs: 238.20/86.09 { f_2^#(x) -> c_18(g_2^#(x, x)) 238.20/86.09 , f_3^#(x) -> g_3^#(x, x) 238.20/86.09 , g_3^#(s(x), y) -> f_2^#(y) 238.20/86.09 , g_3^#(s(x), y) -> g_3^#(x, y) 238.20/86.09 , f_4^#(x) -> g_4^#(x, x) 238.20/86.09 , g_4^#(s(x), y) -> f_3^#(y) 238.20/86.09 , g_4^#(s(x), y) -> g_4^#(x, y) 238.20/86.09 , f_5^#(x) -> g_5^#(x, x) 238.20/86.09 , g_5^#(s(x), y) -> f_4^#(y) 238.20/86.09 , g_5^#(s(x), y) -> g_5^#(x, y) 238.20/86.09 , f_6^#(x) -> g_6^#(x, x) 238.20/86.09 , g_6^#(s(x), y) -> f_5^#(y) 238.20/86.09 , g_6^#(s(x), y) -> g_6^#(x, y) 238.20/86.09 , f_7^#(x) -> g_7^#(x, x) 238.20/86.09 , g_7^#(s(x), y) -> f_6^#(y) 238.20/86.09 , g_7^#(s(x), y) -> g_7^#(x, y) 238.20/86.09 , f_8^#(x) -> g_8^#(x, x) 238.20/86.09 , g_8^#(s(x), y) -> f_7^#(y) 238.20/86.09 , g_8^#(s(x), y) -> g_8^#(x, y) 238.20/86.09 , f_9^#(x) -> g_9^#(x, x) 238.20/86.09 , g_9^#(s(x), y) -> f_8^#(y) 238.20/86.09 , g_9^#(s(x), y) -> g_9^#(x, y) 238.20/86.09 , g_10^#(s(x), y) -> f_9^#(y) 238.20/86.09 , g_10^#(s(x), y) -> g_10^#(x, y) } 238.20/86.09 Obligation: 238.20/86.09 runtime complexity 238.20/86.09 Answer: 238.20/86.09 YES(O(1),O(n^1)) 238.20/86.09 238.20/86.09 Due to missing edges in the dependency-graph, the right-hand sides 238.20/86.09 of following rules could be simplified: 238.20/86.09 238.20/86.09 { f_1^#(x) -> c_1(g_1^#(x, x)) } 238.20/86.09 238.20/86.09 We are left with following problem, upon which TcT provides the 238.20/86.09 certificate YES(O(1),O(n^1)). 238.20/86.09 238.20/86.09 Strict DPs: 238.20/86.09 { f_1^#(x) -> c_1() 238.20/86.09 , g_2^#(s(x), y) -> c_2(f_1^#(y), g_2^#(x, y)) } 238.20/86.09 Weak DPs: 238.20/86.09 { f_2^#(x) -> c_3(g_2^#(x, x)) 238.20/86.09 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.09 , g_3^#(s(x), y) -> c_5(f_2^#(y)) 238.20/86.09 , g_3^#(s(x), y) -> c_6(g_3^#(x, y)) 238.20/86.09 , f_4^#(x) -> c_7(g_4^#(x, x)) 238.20/86.09 , g_4^#(s(x), y) -> c_8(f_3^#(y)) 238.20/86.09 , g_4^#(s(x), y) -> c_9(g_4^#(x, y)) 238.20/86.09 , f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.09 , g_5^#(s(x), y) -> c_11(f_4^#(y)) 238.20/86.09 , g_5^#(s(x), y) -> c_12(g_5^#(x, y)) 238.20/86.09 , f_6^#(x) -> c_13(g_6^#(x, x)) 238.20/86.09 , g_6^#(s(x), y) -> c_14(f_5^#(y)) 238.20/86.09 , g_6^#(s(x), y) -> c_15(g_6^#(x, y)) 238.20/86.09 , f_7^#(x) -> c_16(g_7^#(x, x)) 238.20/86.09 , g_7^#(s(x), y) -> c_17(f_6^#(y)) 238.20/86.09 , g_7^#(s(x), y) -> c_18(g_7^#(x, y)) 238.20/86.09 , f_8^#(x) -> c_19(g_8^#(x, x)) 238.20/86.09 , g_8^#(s(x), y) -> c_20(f_7^#(y)) 238.20/86.09 , g_8^#(s(x), y) -> c_21(g_8^#(x, y)) 238.20/86.09 , f_9^#(x) -> c_22(g_9^#(x, x)) 238.20/86.09 , g_9^#(s(x), y) -> c_23(f_8^#(y)) 238.20/86.09 , g_9^#(s(x), y) -> c_24(g_9^#(x, y)) 238.20/86.09 , g_10^#(s(x), y) -> c_25(f_9^#(y)) 238.20/86.09 , g_10^#(s(x), y) -> c_26(g_10^#(x, y)) } 238.20/86.09 Obligation: 238.20/86.09 runtime complexity 238.20/86.09 Answer: 238.20/86.09 YES(O(1),O(n^1)) 238.20/86.09 238.20/86.09 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.09 orient following rules strictly. 238.20/86.09 238.20/86.09 DPs: 238.20/86.09 { 1: f_1^#(x) -> c_1() } 238.20/86.09 238.20/86.09 Sub-proof: 238.20/86.09 ---------- 238.20/86.09 The following argument positions are usable: 238.20/86.09 Uargs(c_2) = {1, 2}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, 238.20/86.09 Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, 238.20/86.09 Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(c_10) = {1}, 238.20/86.09 Uargs(c_11) = {1}, Uargs(c_12) = {1}, Uargs(c_13) = {1}, 238.20/86.09 Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}, 238.20/86.09 Uargs(c_17) = {1}, Uargs(c_18) = {1}, Uargs(c_19) = {1}, 238.20/86.09 Uargs(c_20) = {1}, Uargs(c_21) = {1}, Uargs(c_22) = {1}, 238.20/86.09 Uargs(c_23) = {1}, Uargs(c_24) = {1}, Uargs(c_25) = {1}, 238.20/86.09 Uargs(c_26) = {1} 238.20/86.09 238.20/86.09 TcT has computed the following constructor-based matrix 238.20/86.09 interpretation satisfying not(EDA). 238.20/86.09 238.20/86.09 [s](x1) = [1] x1 + [4] 238.20/86.09 238.20/86.09 [f_1^#](x1) = [4] 238.20/86.09 238.20/86.09 [g_1^#](x1, x2) = [7] x1 + [7] x2 + [0] 238.20/86.09 238.20/86.09 [f_2^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_2^#](x1, x2) = [2] x1 + [0] 238.20/86.09 238.20/86.09 [f_3^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_3^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_4^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_4^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_5^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_5^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_6^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_6^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_7^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_7^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_8^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_8^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_9^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_9^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [g_10^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [c_1](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_2](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_3](x1, x2) = [7] x1 + [7] x2 + [0] 238.20/86.09 238.20/86.09 [c_18](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c] = [0] 238.20/86.09 238.20/86.09 [c_1] = [3] 238.20/86.09 238.20/86.09 [c_2](x1, x2) = [2] x1 + [1] x2 + [0] 238.20/86.09 238.20/86.09 [c_3](x1) = [2] x1 + [0] 238.20/86.09 238.20/86.09 [c_4](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_5](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_6](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_7](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_8](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_9](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_10](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_11](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_12](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_13](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_14](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_15](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_16](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_17](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_18](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_19](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_20](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_21](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_22](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_23](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_24](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_25](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_26](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 The order satisfies the following ordering constraints: 238.20/86.09 238.20/86.09 [f_1^#(x)] = [4] 238.20/86.09 > [3] 238.20/86.09 = [c_1()] 238.20/86.09 238.20/86.09 [f_2^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_3(g_2^#(x, x))] 238.20/86.09 238.20/86.09 [g_2^#(s(x), y)] = [2] x + [8] 238.20/86.09 >= [2] x + [8] 238.20/86.09 = [c_2(f_1^#(y), g_2^#(x, y))] 238.20/86.09 238.20/86.09 [f_3^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_4(g_3^#(x, x))] 238.20/86.09 238.20/86.09 [g_3^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_5(f_2^#(y))] 238.20/86.09 238.20/86.09 [g_3^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_6(g_3^#(x, y))] 238.20/86.09 238.20/86.09 [f_4^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_7(g_4^#(x, x))] 238.20/86.09 238.20/86.09 [g_4^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_8(f_3^#(y))] 238.20/86.09 238.20/86.09 [g_4^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_9(g_4^#(x, y))] 238.20/86.09 238.20/86.09 [f_5^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_10(g_5^#(x, x))] 238.20/86.09 238.20/86.09 [g_5^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_11(f_4^#(y))] 238.20/86.09 238.20/86.09 [g_5^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_12(g_5^#(x, y))] 238.20/86.09 238.20/86.09 [f_6^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_13(g_6^#(x, x))] 238.20/86.09 238.20/86.09 [g_6^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_14(f_5^#(y))] 238.20/86.09 238.20/86.09 [g_6^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_15(g_6^#(x, y))] 238.20/86.09 238.20/86.09 [f_7^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_16(g_7^#(x, x))] 238.20/86.09 238.20/86.09 [g_7^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_17(f_6^#(y))] 238.20/86.09 238.20/86.09 [g_7^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_18(g_7^#(x, y))] 238.20/86.09 238.20/86.09 [f_8^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_19(g_8^#(x, x))] 238.20/86.09 238.20/86.09 [g_8^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_20(f_7^#(y))] 238.20/86.09 238.20/86.09 [g_8^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_21(g_8^#(x, y))] 238.20/86.09 238.20/86.09 [f_9^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_22(g_9^#(x, x))] 238.20/86.09 238.20/86.09 [g_9^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_23(f_8^#(y))] 238.20/86.09 238.20/86.09 [g_9^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_24(g_9^#(x, y))] 238.20/86.09 238.20/86.09 [g_10^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_25(f_9^#(y))] 238.20/86.09 238.20/86.09 [g_10^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_26(g_10^#(x, y))] 238.20/86.09 238.20/86.09 238.20/86.09 The strictly oriented rules are moved into the weak component. 238.20/86.09 238.20/86.09 We are left with following problem, upon which TcT provides the 238.20/86.09 certificate YES(O(1),O(n^1)). 238.20/86.09 238.20/86.09 Strict DPs: { g_2^#(s(x), y) -> c_2(f_1^#(y), g_2^#(x, y)) } 238.20/86.09 Weak DPs: 238.20/86.09 { f_1^#(x) -> c_1() 238.20/86.09 , f_2^#(x) -> c_3(g_2^#(x, x)) 238.20/86.09 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.09 , g_3^#(s(x), y) -> c_5(f_2^#(y)) 238.20/86.09 , g_3^#(s(x), y) -> c_6(g_3^#(x, y)) 238.20/86.09 , f_4^#(x) -> c_7(g_4^#(x, x)) 238.20/86.09 , g_4^#(s(x), y) -> c_8(f_3^#(y)) 238.20/86.09 , g_4^#(s(x), y) -> c_9(g_4^#(x, y)) 238.20/86.09 , f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.09 , g_5^#(s(x), y) -> c_11(f_4^#(y)) 238.20/86.09 , g_5^#(s(x), y) -> c_12(g_5^#(x, y)) 238.20/86.09 , f_6^#(x) -> c_13(g_6^#(x, x)) 238.20/86.09 , g_6^#(s(x), y) -> c_14(f_5^#(y)) 238.20/86.09 , g_6^#(s(x), y) -> c_15(g_6^#(x, y)) 238.20/86.09 , f_7^#(x) -> c_16(g_7^#(x, x)) 238.20/86.09 , g_7^#(s(x), y) -> c_17(f_6^#(y)) 238.20/86.09 , g_7^#(s(x), y) -> c_18(g_7^#(x, y)) 238.20/86.09 , f_8^#(x) -> c_19(g_8^#(x, x)) 238.20/86.09 , g_8^#(s(x), y) -> c_20(f_7^#(y)) 238.20/86.09 , g_8^#(s(x), y) -> c_21(g_8^#(x, y)) 238.20/86.09 , f_9^#(x) -> c_22(g_9^#(x, x)) 238.20/86.09 , g_9^#(s(x), y) -> c_23(f_8^#(y)) 238.20/86.09 , g_9^#(s(x), y) -> c_24(g_9^#(x, y)) 238.20/86.09 , g_10^#(s(x), y) -> c_25(f_9^#(y)) 238.20/86.09 , g_10^#(s(x), y) -> c_26(g_10^#(x, y)) } 238.20/86.09 Obligation: 238.20/86.09 runtime complexity 238.20/86.09 Answer: 238.20/86.09 YES(O(1),O(n^1)) 238.20/86.09 238.20/86.09 The following weak DPs constitute a sub-graph of the DG that is 238.20/86.09 closed under successors. The DPs are removed. 238.20/86.09 238.20/86.09 { f_1^#(x) -> c_1() } 238.20/86.09 238.20/86.09 We are left with following problem, upon which TcT provides the 238.20/86.09 certificate YES(O(1),O(n^1)). 238.20/86.09 238.20/86.09 Strict DPs: { g_2^#(s(x), y) -> c_2(f_1^#(y), g_2^#(x, y)) } 238.20/86.09 Weak DPs: 238.20/86.09 { f_2^#(x) -> c_3(g_2^#(x, x)) 238.20/86.09 , f_3^#(x) -> c_4(g_3^#(x, x)) 238.20/86.09 , g_3^#(s(x), y) -> c_5(f_2^#(y)) 238.20/86.09 , g_3^#(s(x), y) -> c_6(g_3^#(x, y)) 238.20/86.09 , f_4^#(x) -> c_7(g_4^#(x, x)) 238.20/86.09 , g_4^#(s(x), y) -> c_8(f_3^#(y)) 238.20/86.09 , g_4^#(s(x), y) -> c_9(g_4^#(x, y)) 238.20/86.09 , f_5^#(x) -> c_10(g_5^#(x, x)) 238.20/86.09 , g_5^#(s(x), y) -> c_11(f_4^#(y)) 238.20/86.09 , g_5^#(s(x), y) -> c_12(g_5^#(x, y)) 238.20/86.09 , f_6^#(x) -> c_13(g_6^#(x, x)) 238.20/86.09 , g_6^#(s(x), y) -> c_14(f_5^#(y)) 238.20/86.09 , g_6^#(s(x), y) -> c_15(g_6^#(x, y)) 238.20/86.09 , f_7^#(x) -> c_16(g_7^#(x, x)) 238.20/86.09 , g_7^#(s(x), y) -> c_17(f_6^#(y)) 238.20/86.09 , g_7^#(s(x), y) -> c_18(g_7^#(x, y)) 238.20/86.09 , f_8^#(x) -> c_19(g_8^#(x, x)) 238.20/86.09 , g_8^#(s(x), y) -> c_20(f_7^#(y)) 238.20/86.09 , g_8^#(s(x), y) -> c_21(g_8^#(x, y)) 238.20/86.09 , f_9^#(x) -> c_22(g_9^#(x, x)) 238.20/86.09 , g_9^#(s(x), y) -> c_23(f_8^#(y)) 238.20/86.09 , g_9^#(s(x), y) -> c_24(g_9^#(x, y)) 238.20/86.09 , g_10^#(s(x), y) -> c_25(f_9^#(y)) 238.20/86.09 , g_10^#(s(x), y) -> c_26(g_10^#(x, y)) } 238.20/86.09 Obligation: 238.20/86.09 runtime complexity 238.20/86.09 Answer: 238.20/86.09 YES(O(1),O(n^1)) 238.20/86.09 238.20/86.09 Due to missing edges in the dependency-graph, the right-hand sides 238.20/86.09 of following rules could be simplified: 238.20/86.09 238.20/86.09 { g_2^#(s(x), y) -> c_2(f_1^#(y), g_2^#(x, y)) } 238.20/86.09 238.20/86.09 We are left with following problem, upon which TcT provides the 238.20/86.09 certificate YES(O(1),O(n^1)). 238.20/86.09 238.20/86.09 Strict DPs: { g_2^#(s(x), y) -> c_1(g_2^#(x, y)) } 238.20/86.09 Weak DPs: 238.20/86.09 { f_2^#(x) -> c_2(g_2^#(x, x)) 238.20/86.09 , f_3^#(x) -> c_3(g_3^#(x, x)) 238.20/86.09 , g_3^#(s(x), y) -> c_4(f_2^#(y)) 238.20/86.09 , g_3^#(s(x), y) -> c_5(g_3^#(x, y)) 238.20/86.09 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.20/86.09 , g_4^#(s(x), y) -> c_7(f_3^#(y)) 238.20/86.09 , g_4^#(s(x), y) -> c_8(g_4^#(x, y)) 238.20/86.09 , f_5^#(x) -> c_9(g_5^#(x, x)) 238.20/86.09 , g_5^#(s(x), y) -> c_10(f_4^#(y)) 238.20/86.09 , g_5^#(s(x), y) -> c_11(g_5^#(x, y)) 238.20/86.09 , f_6^#(x) -> c_12(g_6^#(x, x)) 238.20/86.09 , g_6^#(s(x), y) -> c_13(f_5^#(y)) 238.20/86.09 , g_6^#(s(x), y) -> c_14(g_6^#(x, y)) 238.20/86.09 , f_7^#(x) -> c_15(g_7^#(x, x)) 238.20/86.09 , g_7^#(s(x), y) -> c_16(f_6^#(y)) 238.20/86.09 , g_7^#(s(x), y) -> c_17(g_7^#(x, y)) 238.20/86.09 , f_8^#(x) -> c_18(g_8^#(x, x)) 238.20/86.09 , g_8^#(s(x), y) -> c_19(f_7^#(y)) 238.20/86.09 , g_8^#(s(x), y) -> c_20(g_8^#(x, y)) 238.20/86.09 , f_9^#(x) -> c_21(g_9^#(x, x)) 238.20/86.09 , g_9^#(s(x), y) -> c_22(f_8^#(y)) 238.20/86.09 , g_9^#(s(x), y) -> c_23(g_9^#(x, y)) 238.20/86.09 , g_10^#(s(x), y) -> c_24(f_9^#(y)) 238.20/86.09 , g_10^#(s(x), y) -> c_25(g_10^#(x, y)) } 238.20/86.09 Obligation: 238.20/86.09 runtime complexity 238.20/86.09 Answer: 238.20/86.09 YES(O(1),O(n^1)) 238.20/86.09 238.20/86.09 We use the processor 'matrix interpretation of dimension 1' to 238.20/86.09 orient following rules strictly. 238.20/86.09 238.20/86.09 DPs: 238.20/86.09 { 1: g_2^#(s(x), y) -> c_1(g_2^#(x, y)) } 238.20/86.09 238.20/86.09 Sub-proof: 238.20/86.09 ---------- 238.20/86.09 The following argument positions are usable: 238.20/86.09 Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, 238.20/86.09 Uargs(c_4) = {1}, Uargs(c_5) = {1}, Uargs(c_6) = {1}, 238.20/86.09 Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1}, 238.20/86.09 Uargs(c_10) = {1}, Uargs(c_11) = {1}, Uargs(c_12) = {1}, 238.20/86.09 Uargs(c_13) = {1}, Uargs(c_14) = {1}, Uargs(c_15) = {1}, 238.20/86.09 Uargs(c_16) = {1}, Uargs(c_17) = {1}, Uargs(c_18) = {1}, 238.20/86.09 Uargs(c_19) = {1}, Uargs(c_20) = {1}, Uargs(c_21) = {1}, 238.20/86.09 Uargs(c_22) = {1}, Uargs(c_23) = {1}, Uargs(c_24) = {1}, 238.20/86.09 Uargs(c_25) = {1} 238.20/86.09 238.20/86.09 TcT has computed the following constructor-based matrix 238.20/86.09 interpretation satisfying not(EDA). 238.20/86.09 238.20/86.09 [s](x1) = [1] x1 + [4] 238.20/86.09 238.20/86.09 [f_1^#](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [g_1^#](x1, x2) = [7] x1 + [7] x2 + [0] 238.20/86.09 238.20/86.09 [f_2^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_2^#](x1, x2) = [2] x1 + [0] 238.20/86.09 238.20/86.09 [f_3^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_3^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_4^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_4^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_5^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_5^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_6^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_6^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_7^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_7^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_8^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_8^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [f_9^#](x1) = [4] x1 + [0] 238.20/86.09 238.20/86.09 [g_9^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [g_10^#](x1, x2) = [4] x2 + [0] 238.20/86.09 238.20/86.09 [c_1](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_2](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_3](x1, x2) = [7] x1 + [7] x2 + [0] 238.20/86.09 238.20/86.09 [c_18](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c] = [0] 238.20/86.09 238.20/86.09 [c_1] = [0] 238.20/86.09 238.20/86.09 [c_2](x1, x2) = [7] x1 + [7] x2 + [0] 238.20/86.09 238.20/86.09 [c_3](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_4](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_5](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_6](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_7](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_8](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_9](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_10](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_11](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_12](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_13](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_14](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_15](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_16](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_17](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_18](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_19](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_20](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_21](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_22](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_23](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_24](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_25](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c_26](x1) = [7] x1 + [0] 238.20/86.09 238.20/86.09 [c] = [0] 238.20/86.09 238.20/86.09 [c_1](x1) = [1] x1 + [1] 238.20/86.09 238.20/86.09 [c_2](x1) = [2] x1 + [0] 238.20/86.09 238.20/86.09 [c_3](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_4](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_5](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_6](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_7](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_8](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_9](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_10](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_11](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_12](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_13](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_14](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_15](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_16](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_17](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_18](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_19](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_20](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_21](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_22](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_23](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_24](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 [c_25](x1) = [1] x1 + [0] 238.20/86.09 238.20/86.09 The order satisfies the following ordering constraints: 238.20/86.09 238.20/86.09 [f_2^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_2(g_2^#(x, x))] 238.20/86.09 238.20/86.09 [g_2^#(s(x), y)] = [2] x + [8] 238.20/86.09 > [2] x + [1] 238.20/86.09 = [c_1(g_2^#(x, y))] 238.20/86.09 238.20/86.09 [f_3^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_3(g_3^#(x, x))] 238.20/86.09 238.20/86.09 [g_3^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_4(f_2^#(y))] 238.20/86.09 238.20/86.09 [g_3^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_5(g_3^#(x, y))] 238.20/86.09 238.20/86.09 [f_4^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_6(g_4^#(x, x))] 238.20/86.09 238.20/86.09 [g_4^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_7(f_3^#(y))] 238.20/86.09 238.20/86.09 [g_4^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_8(g_4^#(x, y))] 238.20/86.09 238.20/86.09 [f_5^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_9(g_5^#(x, x))] 238.20/86.09 238.20/86.09 [g_5^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_10(f_4^#(y))] 238.20/86.09 238.20/86.09 [g_5^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_11(g_5^#(x, y))] 238.20/86.09 238.20/86.09 [f_6^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_12(g_6^#(x, x))] 238.20/86.09 238.20/86.09 [g_6^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_13(f_5^#(y))] 238.20/86.09 238.20/86.09 [g_6^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_14(g_6^#(x, y))] 238.20/86.09 238.20/86.09 [f_7^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_15(g_7^#(x, x))] 238.20/86.09 238.20/86.09 [g_7^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_16(f_6^#(y))] 238.20/86.09 238.20/86.09 [g_7^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_17(g_7^#(x, y))] 238.20/86.09 238.20/86.09 [f_8^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.20/86.09 = [c_18(g_8^#(x, x))] 238.20/86.09 238.20/86.09 [g_8^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_19(f_7^#(y))] 238.20/86.09 238.20/86.09 [g_8^#(s(x), y)] = [4] y + [0] 238.20/86.09 >= [4] y + [0] 238.20/86.09 = [c_20(g_8^#(x, y))] 238.20/86.09 238.20/86.09 [f_9^#(x)] = [4] x + [0] 238.20/86.09 >= [4] x + [0] 238.44/86.10 = [c_21(g_9^#(x, x))] 238.44/86.10 238.44/86.10 [g_9^#(s(x), y)] = [4] y + [0] 238.44/86.10 >= [4] y + [0] 238.44/86.10 = [c_22(f_8^#(y))] 238.44/86.10 238.44/86.10 [g_9^#(s(x), y)] = [4] y + [0] 238.44/86.10 >= [4] y + [0] 238.44/86.10 = [c_23(g_9^#(x, y))] 238.44/86.10 238.44/86.10 [g_10^#(s(x), y)] = [4] y + [0] 238.44/86.10 >= [4] y + [0] 238.44/86.10 = [c_24(f_9^#(y))] 238.44/86.10 238.44/86.10 [g_10^#(s(x), y)] = [4] y + [0] 238.44/86.10 >= [4] y + [0] 238.44/86.10 = [c_25(g_10^#(x, y))] 238.44/86.10 238.44/86.10 238.44/86.10 The strictly oriented rules are moved into the weak component. 238.44/86.10 238.44/86.10 We are left with following problem, upon which TcT provides the 238.44/86.10 certificate YES(O(1),O(1)). 238.44/86.10 238.44/86.10 Weak DPs: 238.44/86.10 { f_2^#(x) -> c_2(g_2^#(x, x)) 238.44/86.10 , g_2^#(s(x), y) -> c_1(g_2^#(x, y)) 238.44/86.10 , f_3^#(x) -> c_3(g_3^#(x, x)) 238.44/86.10 , g_3^#(s(x), y) -> c_4(f_2^#(y)) 238.44/86.10 , g_3^#(s(x), y) -> c_5(g_3^#(x, y)) 238.44/86.10 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.44/86.10 , g_4^#(s(x), y) -> c_7(f_3^#(y)) 238.44/86.10 , g_4^#(s(x), y) -> c_8(g_4^#(x, y)) 238.44/86.10 , f_5^#(x) -> c_9(g_5^#(x, x)) 238.44/86.10 , g_5^#(s(x), y) -> c_10(f_4^#(y)) 238.44/86.10 , g_5^#(s(x), y) -> c_11(g_5^#(x, y)) 238.44/86.10 , f_6^#(x) -> c_12(g_6^#(x, x)) 238.44/86.10 , g_6^#(s(x), y) -> c_13(f_5^#(y)) 238.44/86.10 , g_6^#(s(x), y) -> c_14(g_6^#(x, y)) 238.44/86.10 , f_7^#(x) -> c_15(g_7^#(x, x)) 238.44/86.10 , g_7^#(s(x), y) -> c_16(f_6^#(y)) 238.44/86.10 , g_7^#(s(x), y) -> c_17(g_7^#(x, y)) 238.44/86.10 , f_8^#(x) -> c_18(g_8^#(x, x)) 238.44/86.10 , g_8^#(s(x), y) -> c_19(f_7^#(y)) 238.44/86.10 , g_8^#(s(x), y) -> c_20(g_8^#(x, y)) 238.44/86.10 , f_9^#(x) -> c_21(g_9^#(x, x)) 238.44/86.10 , g_9^#(s(x), y) -> c_22(f_8^#(y)) 238.44/86.10 , g_9^#(s(x), y) -> c_23(g_9^#(x, y)) 238.44/86.10 , g_10^#(s(x), y) -> c_24(f_9^#(y)) 238.44/86.10 , g_10^#(s(x), y) -> c_25(g_10^#(x, y)) } 238.44/86.10 Obligation: 238.44/86.10 runtime complexity 238.44/86.10 Answer: 238.44/86.10 YES(O(1),O(1)) 238.44/86.10 238.44/86.10 The following weak DPs constitute a sub-graph of the DG that is 238.44/86.10 closed under successors. The DPs are removed. 238.44/86.10 238.44/86.10 { f_2^#(x) -> c_2(g_2^#(x, x)) 238.44/86.10 , g_2^#(s(x), y) -> c_1(g_2^#(x, y)) 238.44/86.10 , f_3^#(x) -> c_3(g_3^#(x, x)) 238.44/86.10 , g_3^#(s(x), y) -> c_4(f_2^#(y)) 238.44/86.10 , g_3^#(s(x), y) -> c_5(g_3^#(x, y)) 238.44/86.10 , f_4^#(x) -> c_6(g_4^#(x, x)) 238.44/86.10 , g_4^#(s(x), y) -> c_7(f_3^#(y)) 238.44/86.10 , g_4^#(s(x), y) -> c_8(g_4^#(x, y)) 238.44/86.10 , f_5^#(x) -> c_9(g_5^#(x, x)) 238.44/86.10 , g_5^#(s(x), y) -> c_10(f_4^#(y)) 238.44/86.10 , g_5^#(s(x), y) -> c_11(g_5^#(x, y)) 238.44/86.10 , f_6^#(x) -> c_12(g_6^#(x, x)) 238.44/86.10 , g_6^#(s(x), y) -> c_13(f_5^#(y)) 238.44/86.10 , g_6^#(s(x), y) -> c_14(g_6^#(x, y)) 238.44/86.10 , f_7^#(x) -> c_15(g_7^#(x, x)) 238.44/86.10 , g_7^#(s(x), y) -> c_16(f_6^#(y)) 238.44/86.10 , g_7^#(s(x), y) -> c_17(g_7^#(x, y)) 238.44/86.10 , f_8^#(x) -> c_18(g_8^#(x, x)) 238.44/86.10 , g_8^#(s(x), y) -> c_19(f_7^#(y)) 238.44/86.10 , g_8^#(s(x), y) -> c_20(g_8^#(x, y)) 238.44/86.10 , f_9^#(x) -> c_21(g_9^#(x, x)) 238.44/86.10 , g_9^#(s(x), y) -> c_22(f_8^#(y)) 238.44/86.10 , g_9^#(s(x), y) -> c_23(g_9^#(x, y)) 238.44/86.10 , g_10^#(s(x), y) -> c_24(f_9^#(y)) 238.44/86.10 , g_10^#(s(x), y) -> c_25(g_10^#(x, y)) } 238.44/86.10 238.44/86.10 We are left with following problem, upon which TcT provides the 238.44/86.10 certificate YES(O(1),O(1)). 238.44/86.10 238.44/86.10 Rules: Empty 238.44/86.10 Obligation: 238.44/86.10 runtime complexity 238.44/86.10 Answer: 238.44/86.10 YES(O(1),O(1)) 238.44/86.10 238.44/86.10 Empty rules are trivially bounded 238.44/86.10 238.44/86.10 Hurray, we answered YES(O(1),O(n^10)) 238.44/86.10 EOF