Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/63142)

The rewrite relation of the following TRS is considered.

0(0(0(1(0(2(0(2(1(2(0(2(2(x1))))))))))))) → 0(0(1(0(1(1(0(2(1(0(0(0(1(0(1(1(0(x1))))))))))))))))) (1)
0(0(0(1(1(2(1(2(1(1(0(0(0(x1))))))))))))) → 1(0(0(2(2(2(2(1(1(2(0(2(0(0(2(1(0(x1))))))))))))))))) (2)
0(1(0(1(0(0(1(0(0(2(1(2(0(x1))))))))))))) → 0(1(0(2(0(0(2(1(0(0(0(0(0(1(0(0(0(x1))))))))))))))))) (3)
0(1(2(0(2(0(1(1(1(1(0(0(2(x1))))))))))))) → 0(0(0(0(0(2(0(2(2(0(2(2(2(0(0(0(0(x1))))))))))))))))) (4)
0(1(2(1(1(0(0(2(2(1(0(2(2(x1))))))))))))) → 1(0(0(2(1(0(0(2(0(0(0(2(0(2(2(2(2(x1))))))))))))))))) (5)
0(1(2(2(0(0(2(0(0(0(2(0(2(x1))))))))))))) → 2(1(0(0(0(2(1(1(0(2(0(1(0(2(1(0(2(x1))))))))))))))))) (6)
0(2(0(1(0(1(1(0(1(2(0(0(1(x1))))))))))))) → 0(1(1(0(0(0(2(1(1(1(0(2(0(0(2(0(1(x1))))))))))))))))) (7)
1(0(0(1(0(2(2(0(0(1(2(0(0(x1))))))))))))) → 0(0(0(0(0(1(0(1(0(1(1(0(1(0(0(2(0(x1))))))))))))))))) (8)
1(0(1(1(1(2(2(2(2(1(0(0(0(x1))))))))))))) → 2(1(0(0(1(0(1(0(2(2(1(1(0(0(2(2(2(x1))))))))))))))))) (9)
1(1(0(0(1(0(0(0(0(1(1(1(2(x1))))))))))))) → 1(1(0(2(1(0(0(2(1(0(1(0(0(2(0(1(2(x1))))))))))))))))) (10)
1(1(2(0(1(0(2(1(2(0(1(0(2(x1))))))))))))) → 1(1(2(1(0(1(0(2(1(1(1(0(1(0(2(0(2(x1))))))))))))))))) (11)
1(1(2(2(1(1(2(1(0(0(1(0(2(x1))))))))))))) → 0(0(2(0(2(0(0(0(2(0(0(2(0(0(2(2(2(x1))))))))))))))))) (12)
2(0(0(0(1(1(2(1(0(2(2(0(0(x1))))))))))))) → 1(1(0(2(0(1(0(2(2(1(1(1(0(2(2(0(0(x1))))))))))))))))) (13)
2(0(0(1(1(2(2(1(0(2(2(2(2(x1))))))))))))) → 1(0(2(2(1(0(1(2(1(0(1(0(0(2(0(2(0(x1))))))))))))))))) (14)
2(0(0(2(1(2(1(1(0(1(0(0(2(x1))))))))))))) → 2(1(1(1(1(0(2(0(1(0(1(0(2(1(0(0(2(x1))))))))))))))))) (15)
2(1(1(2(2(0(2(1(0(0(0(1(0(x1))))))))))))) → 1(1(2(0(0(2(0(0(1(0(0(2(0(0(0(1(0(x1))))))))))))))))) (16)
2(1(2(1(1(2(1(0(0(1(0(1(0(x1))))))))))))) → 2(2(1(1(1(0(0(0(0(1(0(0(2(2(0(1(0(x1))))))))))))))))) (17)
2(1(2(2(0(2(1(0(2(0(2(1(0(x1))))))))))))) → 1(0(1(2(0(0(2(0(0(2(1(0(2(1(0(0(0(x1))))))))))))))))) (18)
2(2(0(1(1(1(1(0(1(0(1(2(0(x1))))))))))))) → 0(1(1(0(1(1(0(0(2(2(0(1(1(0(1(0(0(x1))))))))))))))))) (19)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
2(2(0(2(1(2(0(2(0(1(0(0(0(x1))))))))))))) → 0(1(1(0(1(0(0(0(1(2(0(1(1(0(1(0(0(x1))))))))))))))))) (20)
0(0(0(1(1(2(1(2(1(1(0(0(0(x1))))))))))))) → 0(1(2(0(0(2(0(2(1(1(2(2(2(2(0(0(1(x1))))))))))))))))) (21)
0(2(1(2(0(0(1(0(0(1(0(1(0(x1))))))))))))) → 0(0(0(1(0(0(0(0(0(1(2(0(0(2(0(1(0(x1))))))))))))))))) (22)
2(0(0(1(1(1(1(0(2(0(2(1(0(x1))))))))))))) → 0(0(0(0(2(2(2(0(2(2(0(2(0(0(0(0(0(x1))))))))))))))))) (23)
2(2(0(1(2(2(0(0(1(1(2(1(0(x1))))))))))))) → 2(2(2(2(0(2(0(0(0(2(0(0(1(2(0(0(1(x1))))))))))))))))) (24)
2(0(2(0(0(0(2(0(0(2(2(1(0(x1))))))))))))) → 2(0(1(2(0(1(0(2(0(1(1(2(0(0(0(1(2(x1))))))))))))))))) (25)
1(0(0(2(1(0(1(1(0(1(0(2(0(x1))))))))))))) → 1(0(2(0(0(2(0(1(1(1(2(0(0(0(1(1(0(x1))))))))))))))))) (26)
0(0(2(1(0(0(2(2(0(1(0(0(1(x1))))))))))))) → 0(2(0(0(1(0(1(1(0(1(0(1(0(0(0(0(0(x1))))))))))))))))) (27)
0(0(0(1(2(2(2(2(1(1(1(0(1(x1))))))))))))) → 2(2(2(0(0(1(1(2(2(0(1(0(1(0(0(1(2(x1))))))))))))))))) (28)
2(1(1(1(0(0(0(0(1(0(0(1(1(x1))))))))))))) → 2(1(0(2(0(0(1(0(1(2(0(0(1(2(0(1(1(x1))))))))))))))))) (29)
2(0(1(0(2(1(2(0(1(0(2(1(1(x1))))))))))))) → 2(0(2(0(1(0(1(1(1(2(0(1(0(1(2(1(1(x1))))))))))))))))) (30)
2(0(1(0(0(1(2(1(1(2(2(1(1(x1))))))))))))) → 2(2(2(0(0(2(0(0(2(0(0(0(2(0(2(0(0(x1))))))))))))))))) (31)
0(0(2(2(0(1(2(1(1(0(0(0(2(x1))))))))))))) → 0(0(2(2(0(1(1(1(2(2(0(1(0(2(0(1(1(x1))))))))))))))))) (32)
2(2(2(2(0(1(2(2(1(1(0(0(2(x1))))))))))))) → 0(2(0(2(0(0(1(0(1(2(1(0(1(2(2(0(1(x1))))))))))))))))) (33)
2(0(0(1(0(1(1(2(1(2(0(0(2(x1))))))))))))) → 2(0(0(1(2(0(1(0(1(0(2(0(1(1(1(1(2(x1))))))))))))))))) (34)
0(1(0(0(0(1(2(0(2(2(1(1(2(x1))))))))))))) → 0(1(0(0(0(2(0(0(1(0(0(2(0(0(2(1(1(x1))))))))))))))))) (35)
0(1(0(1(0(0(1(2(1(1(2(1(2(x1))))))))))))) → 0(1(0(2(2(0(0(1(0(0(0(0(1(1(1(2(2(x1))))))))))))))))) (36)
0(1(2(0(2(0(1(2(0(2(2(1(2(x1))))))))))))) → 0(0(0(1(2(0(1(2(0(0(2(0(0(2(1(0(1(x1))))))))))))))))) (37)
0(2(1(0(1(0(1(1(1(1(0(2(2(x1))))))))))))) → 0(0(1(0(1(1(0(2(2(0(0(1(1(0(1(1(0(x1))))))))))))))))) (38)

1.1 Closure Under Flat Contexts

Using the flat contexts

{2(☐), 0(☐), 1(☐)}

We obtain the transformed TRS
0(0(0(1(1(2(1(2(1(1(0(0(0(x1))))))))))))) → 0(1(2(0(0(2(0(2(1(1(2(2(2(2(0(0(1(x1))))))))))))))))) (21)
0(2(1(2(0(0(1(0(0(1(0(1(0(x1))))))))))))) → 0(0(0(1(0(0(0(0(0(1(2(0(0(2(0(1(0(x1))))))))))))))))) (22)
2(2(0(1(2(2(0(0(1(1(2(1(0(x1))))))))))))) → 2(2(2(2(0(2(0(0(0(2(0(0(1(2(0(0(1(x1))))))))))))))))) (24)
2(0(2(0(0(0(2(0(0(2(2(1(0(x1))))))))))))) → 2(0(1(2(0(1(0(2(0(1(1(2(0(0(0(1(2(x1))))))))))))))))) (25)
1(0(0(2(1(0(1(1(0(1(0(2(0(x1))))))))))))) → 1(0(2(0(0(2(0(1(1(1(2(0(0(0(1(1(0(x1))))))))))))))))) (26)
0(0(2(1(0(0(2(2(0(1(0(0(1(x1))))))))))))) → 0(2(0(0(1(0(1(1(0(1(0(1(0(0(0(0(0(x1))))))))))))))))) (27)
2(1(1(1(0(0(0(0(1(0(0(1(1(x1))))))))))))) → 2(1(0(2(0(0(1(0(1(2(0(0(1(2(0(1(1(x1))))))))))))))))) (29)
2(0(1(0(2(1(2(0(1(0(2(1(1(x1))))))))))))) → 2(0(2(0(1(0(1(1(1(2(0(1(0(1(2(1(1(x1))))))))))))))))) (30)
2(0(1(0(0(1(2(1(1(2(2(1(1(x1))))))))))))) → 2(2(2(0(0(2(0(0(2(0(0(0(2(0(2(0(0(x1))))))))))))))))) (31)
0(0(2(2(0(1(2(1(1(0(0(0(2(x1))))))))))))) → 0(0(2(2(0(1(1(1(2(2(0(1(0(2(0(1(1(x1))))))))))))))))) (32)
2(0(0(1(0(1(1(2(1(2(0(0(2(x1))))))))))))) → 2(0(0(1(2(0(1(0(1(0(2(0(1(1(1(1(2(x1))))))))))))))))) (34)
0(1(0(0(0(1(2(0(2(2(1(1(2(x1))))))))))))) → 0(1(0(0(0(2(0(0(1(0(0(2(0(0(2(1(1(x1))))))))))))))))) (35)
0(1(0(1(0(0(1(2(1(1(2(1(2(x1))))))))))))) → 0(1(0(2(2(0(0(1(0(0(0(0(1(1(1(2(2(x1))))))))))))))))) (36)
0(1(2(0(2(0(1(2(0(2(2(1(2(x1))))))))))))) → 0(0(0(1(2(0(1(2(0(0(2(0(0(2(1(0(1(x1))))))))))))))))) (37)
0(2(1(0(1(0(1(1(1(1(0(2(2(x1))))))))))))) → 0(0(1(0(1(1(0(2(2(0(0(1(1(0(1(1(0(x1))))))))))))))))) (38)
2(2(2(0(2(1(2(0(2(0(1(0(0(0(x1)))))))))))))) → 2(0(1(1(0(1(0(0(0(1(2(0(1(1(0(1(0(0(x1)))))))))))))))))) (39)
0(2(2(0(2(1(2(0(2(0(1(0(0(0(x1)))))))))))))) → 0(0(1(1(0(1(0(0(0(1(2(0(1(1(0(1(0(0(x1)))))))))))))))))) (40)
1(2(2(0(2(1(2(0(2(0(1(0(0(0(x1)))))))))))))) → 1(0(1(1(0(1(0(0(0(1(2(0(1(1(0(1(0(0(x1)))))))))))))))))) (41)
2(2(0(0(1(1(1(1(0(2(0(2(1(0(x1)))))))))))))) → 2(0(0(0(0(2(2(2(0(2(2(0(2(0(0(0(0(0(x1)))))))))))))))))) (42)
0(2(0(0(1(1(1(1(0(2(0(2(1(0(x1)))))))))))))) → 0(0(0(0(0(2(2(2(0(2(2(0(2(0(0(0(0(0(x1)))))))))))))))))) (43)
1(2(0(0(1(1(1(1(0(2(0(2(1(0(x1)))))))))))))) → 1(0(0(0(0(2(2(2(0(2(2(0(2(0(0(0(0(0(x1)))))))))))))))))) (44)
2(0(0(0(1(2(2(2(2(1(1(1(0(1(x1)))))))))))))) → 2(2(2(2(0(0(1(1(2(2(0(1(0(1(0(0(1(2(x1)))))))))))))))))) (45)
0(0(0(0(1(2(2(2(2(1(1(1(0(1(x1)))))))))))))) → 0(2(2(2(0(0(1(1(2(2(0(1(0(1(0(0(1(2(x1)))))))))))))))))) (46)
1(0(0(0(1(2(2(2(2(1(1(1(0(1(x1)))))))))))))) → 1(2(2(2(0(0(1(1(2(2(0(1(0(1(0(0(1(2(x1)))))))))))))))))) (47)
2(2(2(2(2(0(1(2(2(1(1(0(0(2(x1)))))))))))))) → 2(0(2(0(2(0(0(1(0(1(2(1(0(1(2(2(0(1(x1)))))))))))))))))) (48)
0(2(2(2(2(0(1(2(2(1(1(0(0(2(x1)))))))))))))) → 0(0(2(0(2(0(0(1(0(1(2(1(0(1(2(2(0(1(x1)))))))))))))))))) (49)
1(2(2(2(2(0(1(2(2(1(1(0(0(2(x1)))))))))))))) → 1(0(2(0(2(0(0(1(0(1(2(1(0(1(2(2(0(1(x1)))))))))))))))))) (50)

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS
00(00(01(11(12(21(12(21(11(10(00(00(00(x1))))))))))))) → 01(12(20(00(02(20(02(21(11(12(22(22(22(20(00(01(10(x1))))))))))))))))) (51)
00(00(01(11(12(21(12(21(11(10(00(00(01(x1))))))))))))) → 01(12(20(00(02(20(02(21(11(12(22(22(22(20(00(01(11(x1))))))))))))))))) (52)
00(00(01(11(12(21(12(21(11(10(00(00(02(x1))))))))))))) → 01(12(20(00(02(20(02(21(11(12(22(22(22(20(00(01(12(x1))))))))))))))))) (53)
02(21(12(20(00(01(10(00(01(10(01(10(00(x1))))))))))))) → 00(00(01(10(00(00(00(00(01(12(20(00(02(20(01(10(00(x1))))))))))))))))) (54)
02(21(12(20(00(01(10(00(01(10(01(10(01(x1))))))))))))) → 00(00(01(10(00(00(00(00(01(12(20(00(02(20(01(10(01(x1))))))))))))))))) (55)
02(21(12(20(00(01(10(00(01(10(01(10(02(x1))))))))))))) → 00(00(01(10(00(00(00(00(01(12(20(00(02(20(01(10(02(x1))))))))))))))))) (56)
22(20(01(12(22(20(00(01(11(12(21(10(00(x1))))))))))))) → 22(22(22(20(02(20(00(00(02(20(00(01(12(20(00(01(10(x1))))))))))))))))) (57)
22(20(01(12(22(20(00(01(11(12(21(10(01(x1))))))))))))) → 22(22(22(20(02(20(00(00(02(20(00(01(12(20(00(01(11(x1))))))))))))))))) (58)
22(20(01(12(22(20(00(01(11(12(21(10(02(x1))))))))))))) → 22(22(22(20(02(20(00(00(02(20(00(01(12(20(00(01(12(x1))))))))))))))))) (59)
20(02(20(00(00(02(20(00(02(22(21(10(00(x1))))))))))))) → 20(01(12(20(01(10(02(20(01(11(12(20(00(00(01(12(20(x1))))))))))))))))) (60)
20(02(20(00(00(02(20(00(02(22(21(10(01(x1))))))))))))) → 20(01(12(20(01(10(02(20(01(11(12(20(00(00(01(12(21(x1))))))))))))))))) (61)
20(02(20(00(00(02(20(00(02(22(21(10(02(x1))))))))))))) → 20(01(12(20(01(10(02(20(01(11(12(20(00(00(01(12(22(x1))))))))))))))))) (62)
10(00(02(21(10(01(11(10(01(10(02(20(00(x1))))))))))))) → 10(02(20(00(02(20(01(11(11(12(20(00(00(01(11(10(00(x1))))))))))))))))) (63)
10(00(02(21(10(01(11(10(01(10(02(20(01(x1))))))))))))) → 10(02(20(00(02(20(01(11(11(12(20(00(00(01(11(10(01(x1))))))))))))))))) (64)
10(00(02(21(10(01(11(10(01(10(02(20(02(x1))))))))))))) → 10(02(20(00(02(20(01(11(11(12(20(00(00(01(11(10(02(x1))))))))))))))))) (65)
00(02(21(10(00(02(22(20(01(10(00(01(10(x1))))))))))))) → 02(20(00(01(10(01(11(10(01(10(01(10(00(00(00(00(00(x1))))))))))))))))) (66)
00(02(21(10(00(02(22(20(01(10(00(01(11(x1))))))))))))) → 02(20(00(01(10(01(11(10(01(10(01(10(00(00(00(00(01(x1))))))))))))))))) (67)
00(02(21(10(00(02(22(20(01(10(00(01(12(x1))))))))))))) → 02(20(00(01(10(01(11(10(01(10(01(10(00(00(00(00(02(x1))))))))))))))))) (68)
21(11(11(10(00(00(00(01(10(00(01(11(10(x1))))))))))))) → 21(10(02(20(00(01(10(01(12(20(00(01(12(20(01(11(10(x1))))))))))))))))) (69)
21(11(11(10(00(00(00(01(10(00(01(11(11(x1))))))))))))) → 21(10(02(20(00(01(10(01(12(20(00(01(12(20(01(11(11(x1))))))))))))))))) (70)
21(11(11(10(00(00(00(01(10(00(01(11(12(x1))))))))))))) → 21(10(02(20(00(01(10(01(12(20(00(01(12(20(01(11(12(x1))))))))))))))))) (71)
20(01(10(02(21(12(20(01(10(02(21(11(10(x1))))))))))))) → 20(02(20(01(10(01(11(11(12(20(01(10(01(12(21(11(10(x1))))))))))))))))) (72)
20(01(10(02(21(12(20(01(10(02(21(11(11(x1))))))))))))) → 20(02(20(01(10(01(11(11(12(20(01(10(01(12(21(11(11(x1))))))))))))))))) (73)
20(01(10(02(21(12(20(01(10(02(21(11(12(x1))))))))))))) → 20(02(20(01(10(01(11(11(12(20(01(10(01(12(21(11(12(x1))))))))))))))))) (74)
20(01(10(00(01(12(21(11(12(22(21(11(10(x1))))))))))))) → 22(22(20(00(02(20(00(02(20(00(00(02(20(02(20(00(00(x1))))))))))))))))) (75)
20(01(10(00(01(12(21(11(12(22(21(11(11(x1))))))))))))) → 22(22(20(00(02(20(00(02(20(00(00(02(20(02(20(00(01(x1))))))))))))))))) (76)
20(01(10(00(01(12(21(11(12(22(21(11(12(x1))))))))))))) → 22(22(20(00(02(20(00(02(20(00(00(02(20(02(20(00(02(x1))))))))))))))))) (77)
00(02(22(20(01(12(21(11(10(00(00(02(20(x1))))))))))))) → 00(02(22(20(01(11(11(12(22(20(01(10(02(20(01(11(10(x1))))))))))))))))) (78)
00(02(22(20(01(12(21(11(10(00(00(02(21(x1))))))))))))) → 00(02(22(20(01(11(11(12(22(20(01(10(02(20(01(11(11(x1))))))))))))))))) (79)
00(02(22(20(01(12(21(11(10(00(00(02(22(x1))))))))))))) → 00(02(22(20(01(11(11(12(22(20(01(10(02(20(01(11(12(x1))))))))))))))))) (80)
20(00(01(10(01(11(12(21(12(20(00(02(20(x1))))))))))))) → 20(00(01(12(20(01(10(01(10(02(20(01(11(11(11(12(20(x1))))))))))))))))) (81)
20(00(01(10(01(11(12(21(12(20(00(02(21(x1))))))))))))) → 20(00(01(12(20(01(10(01(10(02(20(01(11(11(11(12(21(x1))))))))))))))))) (82)
20(00(01(10(01(11(12(21(12(20(00(02(22(x1))))))))))))) → 20(00(01(12(20(01(10(01(10(02(20(01(11(11(11(12(22(x1))))))))))))))))) (83)
01(10(00(00(01(12(20(02(22(21(11(12(20(x1))))))))))))) → 01(10(00(00(02(20(00(01(10(00(02(20(00(02(21(11(10(x1))))))))))))))))) (84)
01(10(00(00(01(12(20(02(22(21(11(12(21(x1))))))))))))) → 01(10(00(00(02(20(00(01(10(00(02(20(00(02(21(11(11(x1))))))))))))))))) (85)
01(10(00(00(01(12(20(02(22(21(11(12(22(x1))))))))))))) → 01(10(00(00(02(20(00(01(10(00(02(20(00(02(21(11(12(x1))))))))))))))))) (86)
01(10(01(10(00(01(12(21(11(12(21(12(20(x1))))))))))))) → 01(10(02(22(20(00(01(10(00(00(00(01(11(11(12(22(20(x1))))))))))))))))) (87)
01(10(01(10(00(01(12(21(11(12(21(12(21(x1))))))))))))) → 01(10(02(22(20(00(01(10(00(00(00(01(11(11(12(22(21(x1))))))))))))))))) (88)
01(10(01(10(00(01(12(21(11(12(21(12(22(x1))))))))))))) → 01(10(02(22(20(00(01(10(00(00(00(01(11(11(12(22(22(x1))))))))))))))))) (89)
01(12(20(02(20(01(12(20(02(22(21(12(20(x1))))))))))))) → 00(00(01(12(20(01(12(20(00(02(20(00(02(21(10(01(10(x1))))))))))))))))) (90)
01(12(20(02(20(01(12(20(02(22(21(12(21(x1))))))))))))) → 00(00(01(12(20(01(12(20(00(02(20(00(02(21(10(01(11(x1))))))))))))))))) (91)
01(12(20(02(20(01(12(20(02(22(21(12(22(x1))))))))))))) → 00(00(01(12(20(01(12(20(00(02(20(00(02(21(10(01(12(x1))))))))))))))))) (92)
02(21(10(01(10(01(11(11(11(10(02(22(20(x1))))))))))))) → 00(01(10(01(11(10(02(22(20(00(01(11(10(01(11(10(00(x1))))))))))))))))) (93)
02(21(10(01(10(01(11(11(11(10(02(22(21(x1))))))))))))) → 00(01(10(01(11(10(02(22(20(00(01(11(10(01(11(10(01(x1))))))))))))))))) (94)
02(21(10(01(10(01(11(11(11(10(02(22(22(x1))))))))))))) → 00(01(10(01(11(10(02(22(20(00(01(11(10(01(11(10(02(x1))))))))))))))))) (95)
22(22(20(02(21(12(20(02(20(01(10(00(00(00(x1)))))))))))))) → 20(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(00(x1)))))))))))))))))) (96)
22(22(20(02(21(12(20(02(20(01(10(00(00(01(x1)))))))))))))) → 20(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(01(x1)))))))))))))))))) (97)
22(22(20(02(21(12(20(02(20(01(10(00(00(02(x1)))))))))))))) → 20(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(02(x1)))))))))))))))))) (98)
02(22(20(02(21(12(20(02(20(01(10(00(00(00(x1)))))))))))))) → 00(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(00(x1)))))))))))))))))) (99)
02(22(20(02(21(12(20(02(20(01(10(00(00(01(x1)))))))))))))) → 00(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(01(x1)))))))))))))))))) (100)
02(22(20(02(21(12(20(02(20(01(10(00(00(02(x1)))))))))))))) → 00(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(02(x1)))))))))))))))))) (101)
12(22(20(02(21(12(20(02(20(01(10(00(00(00(x1)))))))))))))) → 10(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(00(x1)))))))))))))))))) (102)
12(22(20(02(21(12(20(02(20(01(10(00(00(01(x1)))))))))))))) → 10(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(01(x1)))))))))))))))))) (103)
12(22(20(02(21(12(20(02(20(01(10(00(00(02(x1)))))))))))))) → 10(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(02(x1)))))))))))))))))) (104)
22(20(00(01(11(11(11(10(02(20(02(21(10(00(x1)))))))))))))) → 20(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(00(x1)))))))))))))))))) (105)
22(20(00(01(11(11(11(10(02(20(02(21(10(01(x1)))))))))))))) → 20(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(01(x1)))))))))))))))))) (106)
22(20(00(01(11(11(11(10(02(20(02(21(10(02(x1)))))))))))))) → 20(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(02(x1)))))))))))))))))) (107)
02(20(00(01(11(11(11(10(02(20(02(21(10(00(x1)))))))))))))) → 00(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(00(x1)))))))))))))))))) (108)
02(20(00(01(11(11(11(10(02(20(02(21(10(01(x1)))))))))))))) → 00(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(01(x1)))))))))))))))))) (109)
02(20(00(01(11(11(11(10(02(20(02(21(10(02(x1)))))))))))))) → 00(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(02(x1)))))))))))))))))) (110)
12(20(00(01(11(11(11(10(02(20(02(21(10(00(x1)))))))))))))) → 10(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(00(x1)))))))))))))))))) (111)
12(20(00(01(11(11(11(10(02(20(02(21(10(01(x1)))))))))))))) → 10(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(01(x1)))))))))))))))))) (112)
12(20(00(01(11(11(11(10(02(20(02(21(10(02(x1)))))))))))))) → 10(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(02(x1)))))))))))))))))) (113)
20(00(00(01(12(22(22(22(21(11(11(10(01(10(x1)))))))))))))) → 22(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(20(x1)))))))))))))))))) (114)
20(00(00(01(12(22(22(22(21(11(11(10(01(11(x1)))))))))))))) → 22(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(21(x1)))))))))))))))))) (115)
20(00(00(01(12(22(22(22(21(11(11(10(01(12(x1)))))))))))))) → 22(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(22(x1)))))))))))))))))) (116)
00(00(00(01(12(22(22(22(21(11(11(10(01(10(x1)))))))))))))) → 02(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(20(x1)))))))))))))))))) (117)
00(00(00(01(12(22(22(22(21(11(11(10(01(11(x1)))))))))))))) → 02(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(21(x1)))))))))))))))))) (118)
00(00(00(01(12(22(22(22(21(11(11(10(01(12(x1)))))))))))))) → 02(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(22(x1)))))))))))))))))) (119)
10(00(00(01(12(22(22(22(21(11(11(10(01(10(x1)))))))))))))) → 12(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(20(x1)))))))))))))))))) (120)
10(00(00(01(12(22(22(22(21(11(11(10(01(11(x1)))))))))))))) → 12(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(21(x1)))))))))))))))))) (121)
10(00(00(01(12(22(22(22(21(11(11(10(01(12(x1)))))))))))))) → 12(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(22(x1)))))))))))))))))) (122)
22(22(22(22(20(01(12(22(21(11(10(00(02(20(x1)))))))))))))) → 20(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(10(x1)))))))))))))))))) (123)
22(22(22(22(20(01(12(22(21(11(10(00(02(21(x1)))))))))))))) → 20(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(11(x1)))))))))))))))))) (124)
22(22(22(22(20(01(12(22(21(11(10(00(02(22(x1)))))))))))))) → 20(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(12(x1)))))))))))))))))) (125)
02(22(22(22(20(01(12(22(21(11(10(00(02(20(x1)))))))))))))) → 00(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(10(x1)))))))))))))))))) (126)
02(22(22(22(20(01(12(22(21(11(10(00(02(21(x1)))))))))))))) → 00(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(11(x1)))))))))))))))))) (127)
02(22(22(22(20(01(12(22(21(11(10(00(02(22(x1)))))))))))))) → 00(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(12(x1)))))))))))))))))) (128)
12(22(22(22(20(01(12(22(21(11(10(00(02(20(x1)))))))))))))) → 10(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(10(x1)))))))))))))))))) (129)
12(22(22(22(20(01(12(22(21(11(10(00(02(21(x1)))))))))))))) → 10(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(11(x1)))))))))))))))))) (130)
12(22(22(22(20(01(12(22(21(11(10(00(02(22(x1)))))))))))))) → 10(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(12(x1)))))))))))))))))) (131)

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 1
[10(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
00(00(01(11(12(21(12(21(11(10(00(00(00(x1))))))))))))) → 01(12(20(00(02(20(02(21(11(12(22(22(22(20(00(01(10(x1))))))))))))))))) (51)
00(00(01(11(12(21(12(21(11(10(00(00(01(x1))))))))))))) → 01(12(20(00(02(20(02(21(11(12(22(22(22(20(00(01(11(x1))))))))))))))))) (52)
00(00(01(11(12(21(12(21(11(10(00(00(02(x1))))))))))))) → 01(12(20(00(02(20(02(21(11(12(22(22(22(20(00(01(12(x1))))))))))))))))) (53)
02(21(12(20(00(01(10(00(01(10(01(10(00(x1))))))))))))) → 00(00(01(10(00(00(00(00(01(12(20(00(02(20(01(10(00(x1))))))))))))))))) (54)
02(21(12(20(00(01(10(00(01(10(01(10(01(x1))))))))))))) → 00(00(01(10(00(00(00(00(01(12(20(00(02(20(01(10(01(x1))))))))))))))))) (55)
02(21(12(20(00(01(10(00(01(10(01(10(02(x1))))))))))))) → 00(00(01(10(00(00(00(00(01(12(20(00(02(20(01(10(02(x1))))))))))))))))) (56)
22(20(01(12(22(20(00(01(11(12(21(10(00(x1))))))))))))) → 22(22(22(20(02(20(00(00(02(20(00(01(12(20(00(01(10(x1))))))))))))))))) (57)
22(20(01(12(22(20(00(01(11(12(21(10(01(x1))))))))))))) → 22(22(22(20(02(20(00(00(02(20(00(01(12(20(00(01(11(x1))))))))))))))))) (58)
22(20(01(12(22(20(00(01(11(12(21(10(02(x1))))))))))))) → 22(22(22(20(02(20(00(00(02(20(00(01(12(20(00(01(12(x1))))))))))))))))) (59)
20(02(20(00(00(02(20(00(02(22(21(10(00(x1))))))))))))) → 20(01(12(20(01(10(02(20(01(11(12(20(00(00(01(12(20(x1))))))))))))))))) (60)
20(02(20(00(00(02(20(00(02(22(21(10(02(x1))))))))))))) → 20(01(12(20(01(10(02(20(01(11(12(20(00(00(01(12(22(x1))))))))))))))))) (62)
10(00(02(21(10(01(11(10(01(10(02(20(00(x1))))))))))))) → 10(02(20(00(02(20(01(11(11(12(20(00(00(01(11(10(00(x1))))))))))))))))) (63)
10(00(02(21(10(01(11(10(01(10(02(20(01(x1))))))))))))) → 10(02(20(00(02(20(01(11(11(12(20(00(00(01(11(10(01(x1))))))))))))))))) (64)
10(00(02(21(10(01(11(10(01(10(02(20(02(x1))))))))))))) → 10(02(20(00(02(20(01(11(11(12(20(00(00(01(11(10(02(x1))))))))))))))))) (65)
00(02(21(10(00(02(22(20(01(10(00(01(10(x1))))))))))))) → 02(20(00(01(10(01(11(10(01(10(01(10(00(00(00(00(00(x1))))))))))))))))) (66)
00(02(21(10(00(02(22(20(01(10(00(01(11(x1))))))))))))) → 02(20(00(01(10(01(11(10(01(10(01(10(00(00(00(00(01(x1))))))))))))))))) (67)
00(02(21(10(00(02(22(20(01(10(00(01(12(x1))))))))))))) → 02(20(00(01(10(01(11(10(01(10(01(10(00(00(00(00(02(x1))))))))))))))))) (68)
20(01(10(02(21(12(20(01(10(02(21(11(10(x1))))))))))))) → 20(02(20(01(10(01(11(11(12(20(01(10(01(12(21(11(10(x1))))))))))))))))) (72)
20(01(10(02(21(12(20(01(10(02(21(11(11(x1))))))))))))) → 20(02(20(01(10(01(11(11(12(20(01(10(01(12(21(11(11(x1))))))))))))))))) (73)
20(01(10(02(21(12(20(01(10(02(21(11(12(x1))))))))))))) → 20(02(20(01(10(01(11(11(12(20(01(10(01(12(21(11(12(x1))))))))))))))))) (74)
20(01(10(00(01(12(21(11(12(22(21(11(10(x1))))))))))))) → 22(22(20(00(02(20(00(02(20(00(00(02(20(02(20(00(00(x1))))))))))))))))) (75)
20(01(10(00(01(12(21(11(12(22(21(11(11(x1))))))))))))) → 22(22(20(00(02(20(00(02(20(00(00(02(20(02(20(00(01(x1))))))))))))))))) (76)
20(01(10(00(01(12(21(11(12(22(21(11(12(x1))))))))))))) → 22(22(20(00(02(20(00(02(20(00(00(02(20(02(20(00(02(x1))))))))))))))))) (77)
00(02(22(20(01(12(21(11(10(00(00(02(20(x1))))))))))))) → 00(02(22(20(01(11(11(12(22(20(01(10(02(20(01(11(10(x1))))))))))))))))) (78)
00(02(22(20(01(12(21(11(10(00(00(02(21(x1))))))))))))) → 00(02(22(20(01(11(11(12(22(20(01(10(02(20(01(11(11(x1))))))))))))))))) (79)
00(02(22(20(01(12(21(11(10(00(00(02(22(x1))))))))))))) → 00(02(22(20(01(11(11(12(22(20(01(10(02(20(01(11(12(x1))))))))))))))))) (80)
20(00(01(10(01(11(12(21(12(20(00(02(20(x1))))))))))))) → 20(00(01(12(20(01(10(01(10(02(20(01(11(11(11(12(20(x1))))))))))))))))) (81)
20(00(01(10(01(11(12(21(12(20(00(02(21(x1))))))))))))) → 20(00(01(12(20(01(10(01(10(02(20(01(11(11(11(12(21(x1))))))))))))))))) (82)
20(00(01(10(01(11(12(21(12(20(00(02(22(x1))))))))))))) → 20(00(01(12(20(01(10(01(10(02(20(01(11(11(11(12(22(x1))))))))))))))))) (83)
01(10(00(00(01(12(20(02(22(21(11(12(21(x1))))))))))))) → 01(10(00(00(02(20(00(01(10(00(02(20(00(02(21(11(11(x1))))))))))))))))) (85)
01(10(01(10(00(01(12(21(11(12(21(12(20(x1))))))))))))) → 01(10(02(22(20(00(01(10(00(00(00(01(11(11(12(22(20(x1))))))))))))))))) (87)
01(10(01(10(00(01(12(21(11(12(21(12(21(x1))))))))))))) → 01(10(02(22(20(00(01(10(00(00(00(01(11(11(12(22(21(x1))))))))))))))))) (88)
01(10(01(10(00(01(12(21(11(12(21(12(22(x1))))))))))))) → 01(10(02(22(20(00(01(10(00(00(00(01(11(11(12(22(22(x1))))))))))))))))) (89)
01(12(20(02(20(01(12(20(02(22(21(12(21(x1))))))))))))) → 00(00(01(12(20(01(12(20(00(02(20(00(02(21(10(01(11(x1))))))))))))))))) (91)
02(21(10(01(10(01(11(11(11(10(02(22(20(x1))))))))))))) → 00(01(10(01(11(10(02(22(20(00(01(11(10(01(11(10(00(x1))))))))))))))))) (93)
02(21(10(01(10(01(11(11(11(10(02(22(21(x1))))))))))))) → 00(01(10(01(11(10(02(22(20(00(01(11(10(01(11(10(01(x1))))))))))))))))) (94)
02(21(10(01(10(01(11(11(11(10(02(22(22(x1))))))))))))) → 00(01(10(01(11(10(02(22(20(00(01(11(10(01(11(10(02(x1))))))))))))))))) (95)
22(22(20(02(21(12(20(02(20(01(10(00(00(00(x1)))))))))))))) → 20(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(00(x1)))))))))))))))))) (96)
22(22(20(02(21(12(20(02(20(01(10(00(00(01(x1)))))))))))))) → 20(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(01(x1)))))))))))))))))) (97)
22(22(20(02(21(12(20(02(20(01(10(00(00(02(x1)))))))))))))) → 20(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(02(x1)))))))))))))))))) (98)
02(22(20(02(21(12(20(02(20(01(10(00(00(00(x1)))))))))))))) → 00(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(00(x1)))))))))))))))))) (99)
02(22(20(02(21(12(20(02(20(01(10(00(00(01(x1)))))))))))))) → 00(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(01(x1)))))))))))))))))) (100)
02(22(20(02(21(12(20(02(20(01(10(00(00(02(x1)))))))))))))) → 00(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(02(x1)))))))))))))))))) (101)
12(22(20(02(21(12(20(02(20(01(10(00(00(00(x1)))))))))))))) → 10(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(00(x1)))))))))))))))))) (102)
12(22(20(02(21(12(20(02(20(01(10(00(00(01(x1)))))))))))))) → 10(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(01(x1)))))))))))))))))) (103)
12(22(20(02(21(12(20(02(20(01(10(00(00(02(x1)))))))))))))) → 10(01(11(10(01(10(00(00(01(12(20(01(11(10(01(10(00(02(x1)))))))))))))))))) (104)
22(20(00(01(11(11(11(10(02(20(02(21(10(00(x1)))))))))))))) → 20(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(00(x1)))))))))))))))))) (105)
22(20(00(01(11(11(11(10(02(20(02(21(10(01(x1)))))))))))))) → 20(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(01(x1)))))))))))))))))) (106)
22(20(00(01(11(11(11(10(02(20(02(21(10(02(x1)))))))))))))) → 20(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(02(x1)))))))))))))))))) (107)
02(20(00(01(11(11(11(10(02(20(02(21(10(00(x1)))))))))))))) → 00(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(00(x1)))))))))))))))))) (108)
02(20(00(01(11(11(11(10(02(20(02(21(10(01(x1)))))))))))))) → 00(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(01(x1)))))))))))))))))) (109)
02(20(00(01(11(11(11(10(02(20(02(21(10(02(x1)))))))))))))) → 00(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(02(x1)))))))))))))))))) (110)
12(20(00(01(11(11(11(10(02(20(02(21(10(00(x1)))))))))))))) → 10(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(00(x1)))))))))))))))))) (111)
12(20(00(01(11(11(11(10(02(20(02(21(10(01(x1)))))))))))))) → 10(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(01(x1)))))))))))))))))) (112)
12(20(00(01(11(11(11(10(02(20(02(21(10(02(x1)))))))))))))) → 10(00(00(00(02(22(22(20(02(22(20(02(20(00(00(00(00(02(x1)))))))))))))))))) (113)
20(00(00(01(12(22(22(22(21(11(11(10(01(10(x1)))))))))))))) → 22(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(20(x1)))))))))))))))))) (114)
20(00(00(01(12(22(22(22(21(11(11(10(01(12(x1)))))))))))))) → 22(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(22(x1)))))))))))))))))) (116)
00(00(00(01(12(22(22(22(21(11(11(10(01(10(x1)))))))))))))) → 02(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(20(x1)))))))))))))))))) (117)
00(00(00(01(12(22(22(22(21(11(11(10(01(12(x1)))))))))))))) → 02(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(22(x1)))))))))))))))))) (119)
10(00(00(01(12(22(22(22(21(11(11(10(01(10(x1)))))))))))))) → 12(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(20(x1)))))))))))))))))) (120)
10(00(00(01(12(22(22(22(21(11(11(10(01(12(x1)))))))))))))) → 12(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(22(x1)))))))))))))))))) (122)
22(22(22(22(20(01(12(22(21(11(10(00(02(21(x1)))))))))))))) → 20(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(11(x1)))))))))))))))))) (124)
02(22(22(22(20(01(12(22(21(11(10(00(02(21(x1)))))))))))))) → 00(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(11(x1)))))))))))))))))) (127)
12(22(22(22(20(01(12(22(21(11(10(00(02(21(x1)))))))))))))) → 10(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(11(x1)))))))))))))))))) (130)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[20(x1)] = 1 · x1 + 3
[02(x1)] = 1 · x1 + 1
[00(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 6
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 5
all of the following rules can be deleted.
01(10(00(00(01(12(20(02(22(21(11(12(20(x1))))))))))))) → 01(10(00(00(02(20(00(01(10(00(02(20(00(02(21(11(10(x1))))))))))))))))) (84)
01(10(00(00(01(12(20(02(22(21(11(12(22(x1))))))))))))) → 01(10(00(00(02(20(00(01(10(00(02(20(00(02(21(11(12(x1))))))))))))))))) (86)
01(12(20(02(20(01(12(20(02(22(21(12(20(x1))))))))))))) → 00(00(01(12(20(01(12(20(00(02(20(00(02(21(10(01(10(x1))))))))))))))))) (90)
01(12(20(02(20(01(12(20(02(22(21(12(22(x1))))))))))))) → 00(00(01(12(20(01(12(20(00(02(20(00(02(21(10(01(12(x1))))))))))))))))) (92)
00(00(00(01(12(22(22(22(21(11(11(10(01(11(x1)))))))))))))) → 02(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(21(x1)))))))))))))))))) (118)
10(00(00(01(12(22(22(22(21(11(11(10(01(11(x1)))))))))))))) → 12(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(21(x1)))))))))))))))))) (121)
22(22(22(22(20(01(12(22(21(11(10(00(02(20(x1)))))))))))))) → 20(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(10(x1)))))))))))))))))) (123)
22(22(22(22(20(01(12(22(21(11(10(00(02(22(x1)))))))))))))) → 20(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(12(x1)))))))))))))))))) (125)
02(22(22(22(20(01(12(22(21(11(10(00(02(20(x1)))))))))))))) → 00(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(10(x1)))))))))))))))))) (126)
02(22(22(22(20(01(12(22(21(11(10(00(02(22(x1)))))))))))))) → 00(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(12(x1)))))))))))))))))) (128)
12(22(22(22(20(01(12(22(21(11(10(00(02(20(x1)))))))))))))) → 10(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(10(x1)))))))))))))))))) (129)
12(22(22(22(20(01(12(22(21(11(10(00(02(22(x1)))))))))))))) → 10(02(20(02(20(00(01(10(01(12(21(10(01(12(22(20(01(12(x1)))))))))))))))))) (131)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 2
[00(x1)] = 1 · x1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
20(02(20(00(00(02(20(00(02(22(21(10(01(x1))))))))))))) → 20(01(12(20(01(10(02(20(01(11(12(20(00(00(01(12(21(x1))))))))))))))))) (61)
20(00(00(01(12(22(22(22(21(11(11(10(01(11(x1)))))))))))))) → 22(22(22(20(00(01(11(12(22(20(01(10(01(10(00(01(12(21(x1)))))))))))))))))) (115)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1
[02(x1)] = 1 · x1
[20(x1)] = 1 · x1
[12(x1)] = 1 · x1
all of the following rules can be deleted.
21(11(11(10(00(00(00(01(10(00(01(11(10(x1))))))))))))) → 21(10(02(20(00(01(10(01(12(20(00(01(12(20(01(11(10(x1))))))))))))))))) (69)
21(11(11(10(00(00(00(01(10(00(01(11(11(x1))))))))))))) → 21(10(02(20(00(01(10(01(12(20(00(01(12(20(01(11(11(x1))))))))))))))))) (70)
21(11(11(10(00(00(00(01(10(00(01(11(12(x1))))))))))))) → 21(10(02(20(00(01(10(01(12(20(00(01(12(20(01(11(12(x1))))))))))))))))) (71)

1.1.1.1.1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.