The rewrite relation of the following TRS is considered.
| 2(5(3(0(x1)))) | → | 1(0(0(1(3(0(4(5(1(2(x1)))))))))) | (1) |
| 1(3(5(4(3(x1))))) | → | 2(1(4(1(4(0(3(0(1(1(x1)))))))))) | (2) |
| 5(1(3(5(0(x1))))) | → | 5(1(4(3(0(4(4(5(2(1(x1)))))))))) | (3) |
| 5(4(4(2(5(x1))))) | → | 4(3(1(1(1(1(5(3(3(5(x1)))))))))) | (4) |
| 2(2(5(0(5(4(x1)))))) | → | 2(1(4(1(3(3(2(2(5(4(x1)))))))))) | (5) |
| 3(0(5(5(4(3(x1)))))) | → | 3(3(0(3(2(3(5(5(1(0(x1)))))))))) | (6) |
| 3(5(4(3(5(2(x1)))))) | → | 2(0(5(2(0(5(2(2(3(2(x1)))))))))) | (7) |
| 4(4(2(5(5(0(x1)))))) | → | 4(4(0(0(3(3(3(2(2(3(x1)))))))))) | (8) |
| 4(5(3(5(5(0(x1)))))) | → | 4(2(2(3(0(2(4(1(1(5(x1)))))))))) | (9) |
| 5(4(5(1(1(2(x1)))))) | → | 5(4(0(3(3(3(3(2(5(5(x1)))))))))) | (10) |
| 5(5(5(5(5(3(x1)))))) | → | 5(5(0(1(4(0(0(5(0(1(x1)))))))))) | (11) |
| 3(5(0(0(5(4(3(x1))))))) | → | 0(1(2(1(1(5(5(2(1(0(x1)))))))))) | (12) |
| 3(5(4(2(5(2(3(x1))))))) | → | 4(0(4(0(0(2(2(3(4(4(x1)))))))))) | (13) |
| 3(5(4(5(1(4(0(x1))))))) | → | 1(1(1(0(0(3(3(1(2(5(x1)))))))))) | (14) |
{2(☐), 5(☐), 3(☐), 0(☐), 1(☐), 4(☐)}
We obtain the transformed TRS| 5(1(3(5(0(x1))))) | → | 5(1(4(3(0(4(4(5(2(1(x1)))))))))) | (3) |
| 2(2(5(0(5(4(x1)))))) | → | 2(1(4(1(3(3(2(2(5(4(x1)))))))))) | (5) |
| 3(0(5(5(4(3(x1)))))) | → | 3(3(0(3(2(3(5(5(1(0(x1)))))))))) | (6) |
| 4(4(2(5(5(0(x1)))))) | → | 4(4(0(0(3(3(3(2(2(3(x1)))))))))) | (8) |
| 4(5(3(5(5(0(x1)))))) | → | 4(2(2(3(0(2(4(1(1(5(x1)))))))))) | (9) |
| 5(4(5(1(1(2(x1)))))) | → | 5(4(0(3(3(3(3(2(5(5(x1)))))))))) | (10) |
| 5(5(5(5(5(3(x1)))))) | → | 5(5(0(1(4(0(0(5(0(1(x1)))))))))) | (11) |
| 2(2(5(3(0(x1))))) | → | 2(1(0(0(1(3(0(4(5(1(2(x1))))))))))) | (15) |
| 5(2(5(3(0(x1))))) | → | 5(1(0(0(1(3(0(4(5(1(2(x1))))))))))) | (16) |
| 3(2(5(3(0(x1))))) | → | 3(1(0(0(1(3(0(4(5(1(2(x1))))))))))) | (17) |
| 0(2(5(3(0(x1))))) | → | 0(1(0(0(1(3(0(4(5(1(2(x1))))))))))) | (18) |
| 1(2(5(3(0(x1))))) | → | 1(1(0(0(1(3(0(4(5(1(2(x1))))))))))) | (19) |
| 4(2(5(3(0(x1))))) | → | 4(1(0(0(1(3(0(4(5(1(2(x1))))))))))) | (20) |
| 2(1(3(5(4(3(x1)))))) | → | 2(2(1(4(1(4(0(3(0(1(1(x1))))))))))) | (21) |
| 5(1(3(5(4(3(x1)))))) | → | 5(2(1(4(1(4(0(3(0(1(1(x1))))))))))) | (22) |
| 3(1(3(5(4(3(x1)))))) | → | 3(2(1(4(1(4(0(3(0(1(1(x1))))))))))) | (23) |
| 0(1(3(5(4(3(x1)))))) | → | 0(2(1(4(1(4(0(3(0(1(1(x1))))))))))) | (24) |
| 1(1(3(5(4(3(x1)))))) | → | 1(2(1(4(1(4(0(3(0(1(1(x1))))))))))) | (25) |
| 4(1(3(5(4(3(x1)))))) | → | 4(2(1(4(1(4(0(3(0(1(1(x1))))))))))) | (26) |
| 2(5(4(4(2(5(x1)))))) | → | 2(4(3(1(1(1(1(5(3(3(5(x1))))))))))) | (27) |
| 5(5(4(4(2(5(x1)))))) | → | 5(4(3(1(1(1(1(5(3(3(5(x1))))))))))) | (28) |
| 3(5(4(4(2(5(x1)))))) | → | 3(4(3(1(1(1(1(5(3(3(5(x1))))))))))) | (29) |
| 0(5(4(4(2(5(x1)))))) | → | 0(4(3(1(1(1(1(5(3(3(5(x1))))))))))) | (30) |
| 1(5(4(4(2(5(x1)))))) | → | 1(4(3(1(1(1(1(5(3(3(5(x1))))))))))) | (31) |
| 4(5(4(4(2(5(x1)))))) | → | 4(4(3(1(1(1(1(5(3(3(5(x1))))))))))) | (32) |
| 2(3(5(4(3(5(2(x1))))))) | → | 2(2(0(5(2(0(5(2(2(3(2(x1))))))))))) | (33) |
| 5(3(5(4(3(5(2(x1))))))) | → | 5(2(0(5(2(0(5(2(2(3(2(x1))))))))))) | (34) |
| 3(3(5(4(3(5(2(x1))))))) | → | 3(2(0(5(2(0(5(2(2(3(2(x1))))))))))) | (35) |
| 0(3(5(4(3(5(2(x1))))))) | → | 0(2(0(5(2(0(5(2(2(3(2(x1))))))))))) | (36) |
| 1(3(5(4(3(5(2(x1))))))) | → | 1(2(0(5(2(0(5(2(2(3(2(x1))))))))))) | (37) |
| 4(3(5(4(3(5(2(x1))))))) | → | 4(2(0(5(2(0(5(2(2(3(2(x1))))))))))) | (38) |
| 2(3(5(0(0(5(4(3(x1)))))))) | → | 2(0(1(2(1(1(5(5(2(1(0(x1))))))))))) | (39) |
| 5(3(5(0(0(5(4(3(x1)))))))) | → | 5(0(1(2(1(1(5(5(2(1(0(x1))))))))))) | (40) |
| 3(3(5(0(0(5(4(3(x1)))))))) | → | 3(0(1(2(1(1(5(5(2(1(0(x1))))))))))) | (41) |
| 0(3(5(0(0(5(4(3(x1)))))))) | → | 0(0(1(2(1(1(5(5(2(1(0(x1))))))))))) | (42) |
| 1(3(5(0(0(5(4(3(x1)))))))) | → | 1(0(1(2(1(1(5(5(2(1(0(x1))))))))))) | (43) |
| 4(3(5(0(0(5(4(3(x1)))))))) | → | 4(0(1(2(1(1(5(5(2(1(0(x1))))))))))) | (44) |
| 2(3(5(4(2(5(2(3(x1)))))))) | → | 2(4(0(4(0(0(2(2(3(4(4(x1))))))))))) | (45) |
| 5(3(5(4(2(5(2(3(x1)))))))) | → | 5(4(0(4(0(0(2(2(3(4(4(x1))))))))))) | (46) |
| 3(3(5(4(2(5(2(3(x1)))))))) | → | 3(4(0(4(0(0(2(2(3(4(4(x1))))))))))) | (47) |
| 0(3(5(4(2(5(2(3(x1)))))))) | → | 0(4(0(4(0(0(2(2(3(4(4(x1))))))))))) | (48) |
| 1(3(5(4(2(5(2(3(x1)))))))) | → | 1(4(0(4(0(0(2(2(3(4(4(x1))))))))))) | (49) |
| 4(3(5(4(2(5(2(3(x1)))))))) | → | 4(4(0(4(0(0(2(2(3(4(4(x1))))))))))) | (50) |
| 2(3(5(4(5(1(4(0(x1)))))))) | → | 2(1(1(1(0(0(3(3(1(2(5(x1))))))))))) | (51) |
| 5(3(5(4(5(1(4(0(x1)))))))) | → | 5(1(1(1(0(0(3(3(1(2(5(x1))))))))))) | (52) |
| 3(3(5(4(5(1(4(0(x1)))))))) | → | 3(1(1(1(0(0(3(3(1(2(5(x1))))))))))) | (53) |
| 0(3(5(4(5(1(4(0(x1)))))))) | → | 0(1(1(1(0(0(3(3(1(2(5(x1))))))))))) | (54) |
| 1(3(5(4(5(1(4(0(x1)))))))) | → | 1(1(1(1(0(0(3(3(1(2(5(x1))))))))))) | (55) |
| 4(3(5(4(5(1(4(0(x1)))))))) | → | 4(1(1(1(0(0(3(3(1(2(5(x1))))))))))) | (56) |
Root-labeling is applied.
We obtain the labeled TRSThere are 294 ruless (increase limit for explicit display).
| [51(x1)] | = | 1 · x1 |
| [13(x1)] | = | 1 · x1 + 5 |
| [35(x1)] | = | 1 · x1 + 69 |
| [50(x1)] | = | 1 · x1 + 2 |
| [05(x1)] | = | 1 · x1 + 36 |
| [14(x1)] | = | 1 · x1 |
| [43(x1)] | = | 1 · x1 |
| [30(x1)] | = | 1 · x1 |
| [04(x1)] | = | 1 · x1 |
| [44(x1)] | = | 1 · x1 |
| [45(x1)] | = | 1 · x1 + 66 |
| [52(x1)] | = | 1 · x1 |
| [21(x1)] | = | 1 · x1 + 1 |
| [15(x1)] | = | 1 · x1 |
| [01(x1)] | = | 1 · x1 |
| [11(x1)] | = | 1 · x1 + 8 |
| [03(x1)] | = | 1 · x1 |
| [00(x1)] | = | 1 · x1 |
| [10(x1)] | = | 1 · x1 |
| [02(x1)] | = | 1 · x1 |
| [12(x1)] | = | 1 · x1 + 7 |
| [22(x1)] | = | 1 · x1 |
| [25(x1)] | = | 1 · x1 + 40 |
| [54(x1)] | = | 1 · x1 + 38 |
| [41(x1)] | = | 1 · x1 + 32 |
| [33(x1)] | = | 1 · x1 |
| [32(x1)] | = | 1 · x1 |
| [40(x1)] | = | 1 · x1 |
| [42(x1)] | = | 1 · x1 + 94 |
| [55(x1)] | = | 1 · x1 + 2 |
| [23(x1)] | = | 1 · x1 + 4 |
| [31(x1)] | = | 1 · x1 |
| [34(x1)] | = | 1 · x1 |
| [53(x1)] | = | 1 · x1 + 43 |
| [24(x1)] | = | 1 · x1 |
| [20(x1)] | = | 1 · x1 + 3 |
There are 271 ruless (increase limit for explicit display).
| [22(x1)] | = | 1 · x1 + 2 |
| [25(x1)] | = | 1 · x1 |
| [50(x1)] | = | 1 · x1 |
| [05(x1)] | = | 1 · x1 + 1 |
| [54(x1)] | = | 1 · x1 + 4 |
| [45(x1)] | = | 1 · x1 |
| [21(x1)] | = | 1 · x1 |
| [14(x1)] | = | 1 · x1 |
| [41(x1)] | = | 1 · x1 + 1 |
| [13(x1)] | = | 1 · x1 |
| [33(x1)] | = | 1 · x1 |
| [32(x1)] | = | 1 · x1 |
| [43(x1)] | = | 1 · x1 |
| [40(x1)] | = | 1 · x1 |
| [44(x1)] | = | 1 · x1 |
| [42(x1)] | = | 1 · x1 |
| [51(x1)] | = | 1 · x1 |
| [11(x1)] | = | 1 · x1 |
| [12(x1)] | = | 1 · x1 + 2 |
| [23(x1)] | = | 1 · x1 |
| [03(x1)] | = | 1 · x1 + 1 |
| [55(x1)] | = | 1 · x1 |
| [53(x1)] | = | 1 · x1 |
| [30(x1)] | = | 1 · x1 |
| [10(x1)] | = | 1 · x1 |
| [00(x1)] | = | 1 · x1 |
| [01(x1)] | = | 1 · x1 |
| [04(x1)] | = | 1 · x1 |
| [31(x1)] | = | 1 · x1 |
| [15(x1)] | = | 1 · x1 |
| [35(x1)] | = | 1 · x1 |
| [52(x1)] | = | 1 · x1 |
| [20(x1)] | = | 1 · x1 |
| [24(x1)] | = | 1 · x1 |
| 54(45(51(11(12(23(x1)))))) | → | 54(40(03(33(33(33(32(25(55(53(x1)))))))))) | (89) |
| 22(25(53(30(05(x1))))) | → | 21(10(00(01(13(30(04(45(51(12(25(x1))))))))))) | (99) |
| 22(25(53(30(03(x1))))) | → | 21(10(00(01(13(30(04(45(51(12(23(x1))))))))))) | (101) |
| 12(25(53(30(05(x1))))) | → | 11(10(00(01(13(30(04(45(51(12(25(x1))))))))))) | (123) |
| 12(25(53(30(03(x1))))) | → | 11(10(00(01(13(30(04(45(51(12(23(x1))))))))))) | (125) |
| [22(x1)] | = | 1 · x1 |
| [25(x1)] | = | 1 · x1 |
| [50(x1)] | = | 1 · x1 + 1 |
| [05(x1)] | = | 1 · x1 |
| [54(x1)] | = | 1 · x1 |
| [45(x1)] | = | 1 · x1 |
| [21(x1)] | = | 1 · x1 |
| [14(x1)] | = | 1 · x1 |
| [41(x1)] | = | 1 · x1 |
| [13(x1)] | = | 1 · x1 |
| [33(x1)] | = | 1 · x1 |
| [32(x1)] | = | 1 · x1 |
| [43(x1)] | = | 1 · x1 |
| [40(x1)] | = | 1 · x1 |
| [44(x1)] | = | 1 · x1 |
| [42(x1)] | = | 1 · x1 |
| [55(x1)] | = | 1 · x1 |
| [31(x1)] | = | 1 · x1 |
| [11(x1)] | = | 1 · x1 |
| [15(x1)] | = | 1 · x1 |
| [53(x1)] | = | 1 · x1 |
| [35(x1)] | = | 1 · x1 |
| [51(x1)] | = | 1 · x1 |
| [52(x1)] | = | 1 · x1 |
| [20(x1)] | = | 1 · x1 |
| [23(x1)] | = | 1 · x1 |
| [24(x1)] | = | 1 · x1 |
| 22(25(50(05(54(45(x1)))))) | → | 21(14(41(13(33(32(22(25(54(45(x1)))))))))) | (63) |
| 22(25(50(05(54(41(x1)))))) | → | 21(14(41(13(33(32(22(25(54(41(x1)))))))))) | (64) |
| 22(25(50(05(54(43(x1)))))) | → | 21(14(41(13(33(32(22(25(54(43(x1)))))))))) | (65) |
| 22(25(50(05(54(40(x1)))))) | → | 21(14(41(13(33(32(22(25(54(40(x1)))))))))) | (66) |
| 22(25(50(05(54(44(x1)))))) | → | 21(14(41(13(33(32(22(25(54(44(x1)))))))))) | (67) |
| 22(25(50(05(54(42(x1)))))) | → | 21(14(41(13(33(32(22(25(54(42(x1)))))))))) | (68) |
| [55(x1)] | = | 1 · x1 |
| [54(x1)] | = | 1 · x1 |
| [44(x1)] | = | 1 · x1 + 1 |
| [42(x1)] | = | 1 · x1 |
| [25(x1)] | = | 1 · x1 |
| [43(x1)] | = | 1 · x1 |
| [31(x1)] | = | 1 · x1 |
| [11(x1)] | = | 1 · x1 |
| [15(x1)] | = | 1 · x1 |
| [53(x1)] | = | 1 · x1 |
| [33(x1)] | = | 1 · x1 |
| [35(x1)] | = | 1 · x1 |
| [51(x1)] | = | 1 · x1 |
| [50(x1)] | = | 1 · x1 |
| [52(x1)] | = | 1 · x1 |
| [20(x1)] | = | 1 · x1 |
| [05(x1)] | = | 1 · x1 |
| [22(x1)] | = | 1 · x1 |
| [23(x1)] | = | 1 · x1 |
| [32(x1)] | = | 1 · x1 |
| [21(x1)] | = | 1 · x1 |
| [24(x1)] | = | 1 · x1 |
| 55(54(44(42(25(55(x1)))))) | → | 54(43(31(11(11(11(15(53(33(35(55(x1))))))))))) | (177) |
| 55(54(44(42(25(51(x1)))))) | → | 54(43(31(11(11(11(15(53(33(35(51(x1))))))))))) | (178) |
| 55(54(44(42(25(53(x1)))))) | → | 54(43(31(11(11(11(15(53(33(35(53(x1))))))))))) | (179) |
| 55(54(44(42(25(50(x1)))))) | → | 54(43(31(11(11(11(15(53(33(35(50(x1))))))))))) | (180) |
| 55(54(44(42(25(54(x1)))))) | → | 54(43(31(11(11(11(15(53(33(35(54(x1))))))))))) | (181) |
| 55(54(44(42(25(52(x1)))))) | → | 54(43(31(11(11(11(15(53(33(35(52(x1))))))))))) | (182) |
| [43(x1)] | = | 1 · x1 |
| [35(x1)] | = | 1 · x1 |
| [54(x1)] | = | 1 · x1 + 1 |
| [52(x1)] | = | 1 · x1 |
| [25(x1)] | = | 1 · x1 |
| [42(x1)] | = | 1 · x1 |
| [20(x1)] | = | 1 · x1 |
| [05(x1)] | = | 1 · x1 |
| [22(x1)] | = | 1 · x1 |
| [23(x1)] | = | 1 · x1 |
| [32(x1)] | = | 1 · x1 |
| [21(x1)] | = | 1 · x1 |
| [24(x1)] | = | 1 · x1 |
| 43(35(54(43(35(52(25(x1))))))) | → | 42(20(05(52(20(05(52(22(23(32(25(x1))))))))))) | (237) |
| 43(35(54(43(35(52(21(x1))))))) | → | 42(20(05(52(20(05(52(22(23(32(21(x1))))))))))) | (238) |
| 43(35(54(43(35(52(23(x1))))))) | → | 42(20(05(52(20(05(52(22(23(32(23(x1))))))))))) | (239) |
| 43(35(54(43(35(52(20(x1))))))) | → | 42(20(05(52(20(05(52(22(23(32(20(x1))))))))))) | (240) |
| 43(35(54(43(35(52(24(x1))))))) | → | 42(20(05(52(20(05(52(22(23(32(24(x1))))))))))) | (241) |
| 43(35(54(43(35(52(22(x1))))))) | → | 42(20(05(52(20(05(52(22(23(32(22(x1))))))))))) | (242) |
There are no rules in the TRS. Hence, it is terminating.