Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/88208)

The rewrite relation of the following TRS is considered.

0(0(0(1(0(2(2(3(3(2(3(3(0(x1))))))))))))) → 3(3(1(1(3(3(3(1(3(1(3(2(1(0(3(3(1(x1))))))))))))))))) (1)
0(0(2(2(1(0(2(3(1(3(1(1(0(x1))))))))))))) → 1(0(1(0(2(1(3(3(3(3(2(3(2(3(1(1(0(x1))))))))))))))))) (2)
0(1(1(0(2(1(0(1(3(0(1(3(3(x1))))))))))))) → 0(3(3(3(0(2(3(3(0(3(3(2(1(3(3(1(3(x1))))))))))))))))) (3)
0(1(1(1(3(3(0(3(2(3(3(1(2(x1))))))))))))) → 3(3(1(1(0(2(1(2(3(0(3(3(3(2(3(0(2(x1))))))))))))))))) (4)
0(1(1(3(0(1(1(1(1(3(1(3(1(x1))))))))))))) → 2(3(1(1(0(3(3(1(3(2(1(3(3(1(2(1(3(x1))))))))))))))))) (5)
0(1(2(3(2(2(2(1(2(3(0(0(0(x1))))))))))))) → 1(1(3(3(0(3(3(1(2(3(1(3(3(0(0(1(1(x1))))))))))))))))) (6)
0(2(3(0(0(0(0(2(3(1(0(3(1(x1))))))))))))) → 1(1(2(2(1(2(3(0(2(1(3(1(2(3(3(2(1(x1))))))))))))))))) (7)
0(2(3(3(0(0(2(2(3(0(3(1(3(x1))))))))))))) → 3(1(3(3(2(1(0(0(1(0(3(3(3(1(3(2(1(x1))))))))))))))))) (8)
0(3(3(0(3(0(3(0(3(3(3(0(0(x1))))))))))))) → 1(1(0(3(3(1(1(2(3(2(3(3(3(0(2(1(1(x1))))))))))))))))) (9)
1(0(1(3(2(2(2(1(2(1(3(3(2(x1))))))))))))) → 3(3(1(0(0(2(3(3(3(2(3(1(3(1(3(1(1(x1))))))))))))))))) (10)
1(1(0(3(2(0(1(0(3(3(2(0(2(x1))))))))))))) → 1(3(0(0(3(2(1(1(2(3(3(2(1(1(1(1(2(x1))))))))))))))))) (11)
1(2(0(1(0(3(1(3(0(2(1(2(3(x1))))))))))))) → 1(3(1(3(1(0(1(0(0(3(3(3(1(3(2(3(3(x1))))))))))))))))) (12)
1(2(1(0(1(1(2(3(2(1(1(0(0(x1))))))))))))) → 3(1(3(3(3(3(3(1(1(2(3(2(1(0(1(0(1(x1))))))))))))))))) (13)
1(2(2(1(1(2(0(3(3(3(0(3(2(x1))))))))))))) → 1(3(1(3(3(3(3(1(3(2(1(3(3(2(2(3(2(x1))))))))))))))))) (14)
1(2(2(3(3(3(3(3(0(0(3(2(2(x1))))))))))))) → 1(3(3(3(3(1(1(2(3(1(1(3(1(0(0(1(0(x1))))))))))))))))) (15)
2(0(1(1(2(1(0(3(2(1(3(1(3(x1))))))))))))) → 0(3(3(0(2(2(1(3(1(2(3(1(3(1(3(3(3(x1))))))))))))))))) (16)
2(0(2(0(2(1(3(0(0(0(2(3(0(x1))))))))))))) → 2(3(1(2(3(3(2(2(0(3(1(1(3(3(3(2(0(x1))))))))))))))))) (17)
2(0(2(2(1(3(1(3(0(1(3(2(1(x1))))))))))))) → 2(0(1(3(3(1(3(3(2(3(0(2(1(1(0(2(1(x1))))))))))))))))) (18)
2(1(3(0(2(3(2(2(1(2(3(2(3(x1))))))))))))) → 2(1(1(1(2(1(2(1(3(1(3(1(1(0(3(3(3(x1))))))))))))))))) (19)
2(3(1(0(3(3(3(0(0(3(0(0(3(x1))))))))))))) → 1(1(2(2(3(3(3(3(1(3(3(1(2(3(3(3(1(x1))))))))))))))))) (20)
3(0(1(2(2(0(3(1(0(1(2(0(1(x1))))))))))))) → 3(3(2(2(3(1(0(2(3(3(1(2(1(0(1(1(1(x1))))))))))))))))) (21)
3(0(2(2(1(0(0(3(1(2(1(1(1(x1))))))))))))) → 3(2(3(1(3(1(2(3(1(0(0(2(3(1(2(3(1(x1))))))))))))))))) (22)
3(0(2(2(1(3(3(1(1(0(1(0(2(x1))))))))))))) → 3(2(1(1(1(3(3(1(3(1(3(1(1(2(1(3(3(x1))))))))))))))))) (23)
3(0(3(2(1(3(2(3(2(3(2(3(2(x1))))))))))))) → 3(2(3(3(0(0(3(1(1(2(3(1(1(1(3(1(2(x1))))))))))))))))) (24)
3(0(3(3(1(2(3(0(0(2(2(0(1(x1))))))))))))) → 3(3(1(2(0(1(1(0(3(3(3(0(2(3(3(1(0(x1))))))))))))))))) (25)
3(1(1(3(3(2(0(3(3(1(1(0(1(x1))))))))))))) → 3(1(1(3(3(3(1(0(2(3(1(1(3(1(3(3(3(x1))))))))))))))))) (26)
3(2(0(3(3(1(3(2(0(2(0(0(1(x1))))))))))))) → 3(3(3(3(1(0(2(3(2(1(3(1(3(3(2(1(3(x1))))))))))))))))) (27)
3(2(1(2(0(3(2(3(2(3(0(2(3(x1))))))))))))) → 3(3(3(1(3(2(1(0(0(2(0(0(3(0(3(3(3(x1))))))))))))))))) (28)
3(2(2(2(0(0(2(3(2(3(1(1(1(x1))))))))))))) → 3(2(3(3(3(1(1(3(2(1(2(2(3(2(2(3(1(x1))))))))))))))))) (29)
3(3(2(0(0(0(0(3(1(1(0(0(2(x1))))))))))))) → 3(1(2(3(3(2(1(1(1(1(2(1(2(1(0(3(1(x1))))))))))))))))) (30)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{0(☐), 1(☐), 2(☐), 3(☐)}

We obtain the transformed TRS
0(1(1(0(2(1(0(1(3(0(1(3(3(x1))))))))))))) → 0(3(3(3(0(2(3(3(0(3(3(2(1(3(3(1(3(x1))))))))))))))))) (3)
1(1(0(3(2(0(1(0(3(3(2(0(2(x1))))))))))))) → 1(3(0(0(3(2(1(1(2(3(3(2(1(1(1(1(2(x1))))))))))))))))) (11)
1(2(0(1(0(3(1(3(0(2(1(2(3(x1))))))))))))) → 1(3(1(3(1(0(1(0(0(3(3(3(1(3(2(3(3(x1))))))))))))))))) (12)
1(2(2(1(1(2(0(3(3(3(0(3(2(x1))))))))))))) → 1(3(1(3(3(3(3(1(3(2(1(3(3(2(2(3(2(x1))))))))))))))))) (14)
1(2(2(3(3(3(3(3(0(0(3(2(2(x1))))))))))))) → 1(3(3(3(3(1(1(2(3(1(1(3(1(0(0(1(0(x1))))))))))))))))) (15)
2(0(2(0(2(1(3(0(0(0(2(3(0(x1))))))))))))) → 2(3(1(2(3(3(2(2(0(3(1(1(3(3(3(2(0(x1))))))))))))))))) (17)
2(0(2(2(1(3(1(3(0(1(3(2(1(x1))))))))))))) → 2(0(1(3(3(1(3(3(2(3(0(2(1(1(0(2(1(x1))))))))))))))))) (18)
2(1(3(0(2(3(2(2(1(2(3(2(3(x1))))))))))))) → 2(1(1(1(2(1(2(1(3(1(3(1(1(0(3(3(3(x1))))))))))))))))) (19)
3(0(1(2(2(0(3(1(0(1(2(0(1(x1))))))))))))) → 3(3(2(2(3(1(0(2(3(3(1(2(1(0(1(1(1(x1))))))))))))))))) (21)
3(0(2(2(1(0(0(3(1(2(1(1(1(x1))))))))))))) → 3(2(3(1(3(1(2(3(1(0(0(2(3(1(2(3(1(x1))))))))))))))))) (22)
3(0(2(2(1(3(3(1(1(0(1(0(2(x1))))))))))))) → 3(2(1(1(1(3(3(1(3(1(3(1(1(2(1(3(3(x1))))))))))))))))) (23)
3(0(3(2(1(3(2(3(2(3(2(3(2(x1))))))))))))) → 3(2(3(3(0(0(3(1(1(2(3(1(1(1(3(1(2(x1))))))))))))))))) (24)
3(0(3(3(1(2(3(0(0(2(2(0(1(x1))))))))))))) → 3(3(1(2(0(1(1(0(3(3(3(0(2(3(3(1(0(x1))))))))))))))))) (25)
3(1(1(3(3(2(0(3(3(1(1(0(1(x1))))))))))))) → 3(1(1(3(3(3(1(0(2(3(1(1(3(1(3(3(3(x1))))))))))))))))) (26)
3(2(0(3(3(1(3(2(0(2(0(0(1(x1))))))))))))) → 3(3(3(3(1(0(2(3(2(1(3(1(3(3(2(1(3(x1))))))))))))))))) (27)
3(2(1(2(0(3(2(3(2(3(0(2(3(x1))))))))))))) → 3(3(3(1(3(2(1(0(0(2(0(0(3(0(3(3(3(x1))))))))))))))))) (28)
3(2(2(2(0(0(2(3(2(3(1(1(1(x1))))))))))))) → 3(2(3(3(3(1(1(3(2(1(2(2(3(2(2(3(1(x1))))))))))))))))) (29)
3(3(2(0(0(0(0(3(1(1(0(0(2(x1))))))))))))) → 3(1(2(3(3(2(1(1(1(1(2(1(2(1(0(3(1(x1))))))))))))))))) (30)
0(0(0(0(1(0(2(2(3(3(2(3(3(0(x1)))))))))))))) → 0(3(3(1(1(3(3(3(1(3(1(3(2(1(0(3(3(1(x1)))))))))))))))))) (31)
1(0(0(0(1(0(2(2(3(3(2(3(3(0(x1)))))))))))))) → 1(3(3(1(1(3(3(3(1(3(1(3(2(1(0(3(3(1(x1)))))))))))))))))) (32)
2(0(0(0(1(0(2(2(3(3(2(3(3(0(x1)))))))))))))) → 2(3(3(1(1(3(3(3(1(3(1(3(2(1(0(3(3(1(x1)))))))))))))))))) (33)
3(0(0(0(1(0(2(2(3(3(2(3(3(0(x1)))))))))))))) → 3(3(3(1(1(3(3(3(1(3(1(3(2(1(0(3(3(1(x1)))))))))))))))))) (34)
0(0(0(2(2(1(0(2(3(1(3(1(1(0(x1)))))))))))))) → 0(1(0(1(0(2(1(3(3(3(3(2(3(2(3(1(1(0(x1)))))))))))))))))) (35)
1(0(0(2(2(1(0(2(3(1(3(1(1(0(x1)))))))))))))) → 1(1(0(1(0(2(1(3(3(3(3(2(3(2(3(1(1(0(x1)))))))))))))))))) (36)
2(0(0(2(2(1(0(2(3(1(3(1(1(0(x1)))))))))))))) → 2(1(0(1(0(2(1(3(3(3(3(2(3(2(3(1(1(0(x1)))))))))))))))))) (37)
3(0(0(2(2(1(0(2(3(1(3(1(1(0(x1)))))))))))))) → 3(1(0(1(0(2(1(3(3(3(3(2(3(2(3(1(1(0(x1)))))))))))))))))) (38)
0(0(1(1(1(3(3(0(3(2(3(3(1(2(x1)))))))))))))) → 0(3(3(1(1(0(2(1(2(3(0(3(3(3(2(3(0(2(x1)))))))))))))))))) (39)
1(0(1(1(1(3(3(0(3(2(3(3(1(2(x1)))))))))))))) → 1(3(3(1(1(0(2(1(2(3(0(3(3(3(2(3(0(2(x1)))))))))))))))))) (40)
2(0(1(1(1(3(3(0(3(2(3(3(1(2(x1)))))))))))))) → 2(3(3(1(1(0(2(1(2(3(0(3(3(3(2(3(0(2(x1)))))))))))))))))) (41)
3(0(1(1(1(3(3(0(3(2(3(3(1(2(x1)))))))))))))) → 3(3(3(1(1(0(2(1(2(3(0(3(3(3(2(3(0(2(x1)))))))))))))))))) (42)
0(0(1(1(3(0(1(1(1(1(3(1(3(1(x1)))))))))))))) → 0(2(3(1(1(0(3(3(1(3(2(1(3(3(1(2(1(3(x1)))))))))))))))))) (43)
1(0(1(1(3(0(1(1(1(1(3(1(3(1(x1)))))))))))))) → 1(2(3(1(1(0(3(3(1(3(2(1(3(3(1(2(1(3(x1)))))))))))))))))) (44)
2(0(1(1(3(0(1(1(1(1(3(1(3(1(x1)))))))))))))) → 2(2(3(1(1(0(3(3(1(3(2(1(3(3(1(2(1(3(x1)))))))))))))))))) (45)
3(0(1(1(3(0(1(1(1(1(3(1(3(1(x1)))))))))))))) → 3(2(3(1(1(0(3(3(1(3(2(1(3(3(1(2(1(3(x1)))))))))))))))))) (46)
0(0(1(2(3(2(2(2(1(2(3(0(0(0(x1)))))))))))))) → 0(1(1(3(3(0(3(3(1(2(3(1(3(3(0(0(1(1(x1)))))))))))))))))) (47)
1(0(1(2(3(2(2(2(1(2(3(0(0(0(x1)))))))))))))) → 1(1(1(3(3(0(3(3(1(2(3(1(3(3(0(0(1(1(x1)))))))))))))))))) (48)
2(0(1(2(3(2(2(2(1(2(3(0(0(0(x1)))))))))))))) → 2(1(1(3(3(0(3(3(1(2(3(1(3(3(0(0(1(1(x1)))))))))))))))))) (49)
3(0(1(2(3(2(2(2(1(2(3(0(0(0(x1)))))))))))))) → 3(1(1(3(3(0(3(3(1(2(3(1(3(3(0(0(1(1(x1)))))))))))))))))) (50)
0(0(2(3(0(0(0(0(2(3(1(0(3(1(x1)))))))))))))) → 0(1(1(2(2(1(2(3(0(2(1(3(1(2(3(3(2(1(x1)))))))))))))))))) (51)
1(0(2(3(0(0(0(0(2(3(1(0(3(1(x1)))))))))))))) → 1(1(1(2(2(1(2(3(0(2(1(3(1(2(3(3(2(1(x1)))))))))))))))))) (52)
2(0(2(3(0(0(0(0(2(3(1(0(3(1(x1)))))))))))))) → 2(1(1(2(2(1(2(3(0(2(1(3(1(2(3(3(2(1(x1)))))))))))))))))) (53)
3(0(2(3(0(0(0(0(2(3(1(0(3(1(x1)))))))))))))) → 3(1(1(2(2(1(2(3(0(2(1(3(1(2(3(3(2(1(x1)))))))))))))))))) (54)
0(0(2(3(3(0(0(2(2(3(0(3(1(3(x1)))))))))))))) → 0(3(1(3(3(2(1(0(0(1(0(3(3(3(1(3(2(1(x1)))))))))))))))))) (55)
1(0(2(3(3(0(0(2(2(3(0(3(1(3(x1)))))))))))))) → 1(3(1(3(3(2(1(0(0(1(0(3(3(3(1(3(2(1(x1)))))))))))))))))) (56)
2(0(2(3(3(0(0(2(2(3(0(3(1(3(x1)))))))))))))) → 2(3(1(3(3(2(1(0(0(1(0(3(3(3(1(3(2(1(x1)))))))))))))))))) (57)
3(0(2(3(3(0(0(2(2(3(0(3(1(3(x1)))))))))))))) → 3(3(1(3(3(2(1(0(0(1(0(3(3(3(1(3(2(1(x1)))))))))))))))))) (58)
0(0(3(3(0(3(0(3(0(3(3(3(0(0(x1)))))))))))))) → 0(1(1(0(3(3(1(1(2(3(2(3(3(3(0(2(1(1(x1)))))))))))))))))) (59)
1(0(3(3(0(3(0(3(0(3(3(3(0(0(x1)))))))))))))) → 1(1(1(0(3(3(1(1(2(3(2(3(3(3(0(2(1(1(x1)))))))))))))))))) (60)
2(0(3(3(0(3(0(3(0(3(3(3(0(0(x1)))))))))))))) → 2(1(1(0(3(3(1(1(2(3(2(3(3(3(0(2(1(1(x1)))))))))))))))))) (61)
3(0(3(3(0(3(0(3(0(3(3(3(0(0(x1)))))))))))))) → 3(1(1(0(3(3(1(1(2(3(2(3(3(3(0(2(1(1(x1)))))))))))))))))) (62)
0(1(0(1(3(2(2(2(1(2(1(3(3(2(x1)))))))))))))) → 0(3(3(1(0(0(2(3(3(3(2(3(1(3(1(3(1(1(x1)))))))))))))))))) (63)
1(1(0(1(3(2(2(2(1(2(1(3(3(2(x1)))))))))))))) → 1(3(3(1(0(0(2(3(3(3(2(3(1(3(1(3(1(1(x1)))))))))))))))))) (64)
2(1(0(1(3(2(2(2(1(2(1(3(3(2(x1)))))))))))))) → 2(3(3(1(0(0(2(3(3(3(2(3(1(3(1(3(1(1(x1)))))))))))))))))) (65)
3(1(0(1(3(2(2(2(1(2(1(3(3(2(x1)))))))))))))) → 3(3(3(1(0(0(2(3(3(3(2(3(1(3(1(3(1(1(x1)))))))))))))))))) (66)
0(1(2(1(0(1(1(2(3(2(1(1(0(0(x1)))))))))))))) → 0(3(1(3(3(3(3(3(1(1(2(3(2(1(0(1(0(1(x1)))))))))))))))))) (67)
1(1(2(1(0(1(1(2(3(2(1(1(0(0(x1)))))))))))))) → 1(3(1(3(3(3(3(3(1(1(2(3(2(1(0(1(0(1(x1)))))))))))))))))) (68)
2(1(2(1(0(1(1(2(3(2(1(1(0(0(x1)))))))))))))) → 2(3(1(3(3(3(3(3(1(1(2(3(2(1(0(1(0(1(x1)))))))))))))))))) (69)
3(1(2(1(0(1(1(2(3(2(1(1(0(0(x1)))))))))))))) → 3(3(1(3(3(3(3(3(1(1(2(3(2(1(0(1(0(1(x1)))))))))))))))))) (70)
0(2(0(1(1(2(1(0(3(2(1(3(1(3(x1)))))))))))))) → 0(0(3(3(0(2(2(1(3(1(2(3(1(3(1(3(3(3(x1)))))))))))))))))) (71)
1(2(0(1(1(2(1(0(3(2(1(3(1(3(x1)))))))))))))) → 1(0(3(3(0(2(2(1(3(1(2(3(1(3(1(3(3(3(x1)))))))))))))))))) (72)
2(2(0(1(1(2(1(0(3(2(1(3(1(3(x1)))))))))))))) → 2(0(3(3(0(2(2(1(3(1(2(3(1(3(1(3(3(3(x1)))))))))))))))))) (73)
3(2(0(1(1(2(1(0(3(2(1(3(1(3(x1)))))))))))))) → 3(0(3(3(0(2(2(1(3(1(2(3(1(3(1(3(3(3(x1)))))))))))))))))) (74)
0(2(3(1(0(3(3(3(0(0(3(0(0(3(x1)))))))))))))) → 0(1(1(2(2(3(3(3(3(1(3(3(1(2(3(3(3(1(x1)))))))))))))))))) (75)
1(2(3(1(0(3(3(3(0(0(3(0(0(3(x1)))))))))))))) → 1(1(1(2(2(3(3(3(3(1(3(3(1(2(3(3(3(1(x1)))))))))))))))))) (76)
2(2(3(1(0(3(3(3(0(0(3(0(0(3(x1)))))))))))))) → 2(1(1(2(2(3(3(3(3(1(3(3(1(2(3(3(3(1(x1)))))))))))))))))) (77)
3(2(3(1(0(3(3(3(0(0(3(0(0(3(x1)))))))))))))) → 3(1(1(2(2(3(3(3(3(1(3(3(1(2(3(3(3(1(x1)))))))))))))))))) (78)

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 264 ruless (increase limit for explicit display).

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[01(x1)] = 1 · x1 + 36
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 8
[02(x1)] = 1 · x1 + 7
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1 + 20
[33(x1)] = 1 · x1
[03(x1)] = 1 · x1
[23(x1)] = 1 · x1 + 2
[32(x1)] = 1 · x1 + 34
[31(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 43
[00(x1)] = 1 · x1 + 44
[12(x1)] = 1 · x1 + 22
[22(x1)] = 1 · x1 + 55
all of the following rules can be deleted.

There are 232 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[32(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 2
[03(x1)] = 1 · x1
[23(x1)] = 1 · x1
[30(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 1
[22(x1)] = 1 · x1 + 2
[11(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
10(01(11(13(30(01(11(11(11(13(31(13(31(10(x1)))))))))))))) → 12(23(31(11(10(03(33(31(13(32(21(13(33(31(12(21(13(30(x1)))))))))))))))))) (203)
10(01(11(13(30(01(11(11(11(13(31(13(31(12(x1)))))))))))))) → 12(23(31(11(10(03(33(31(13(32(21(13(33(31(12(21(13(32(x1)))))))))))))))))) (205)
30(01(11(13(30(01(11(11(11(13(31(13(31(10(x1)))))))))))))) → 32(23(31(11(10(03(33(31(13(32(21(13(33(31(12(21(13(30(x1)))))))))))))))))) (211)
30(01(11(13(30(01(11(11(11(13(31(13(31(12(x1)))))))))))))) → 32(23(31(11(10(03(33(31(13(32(21(13(33(31(12(21(13(32(x1)))))))))))))))))) (213)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1
[03(x1)] = 1 · x1
[23(x1)] = 1 · x1 + 4
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 7
[00(x1)] = 1 · x1 + 2
[22(x1)] = 1 · x1 + 7
[11(x1)] = 1 · x1 + 5
[01(x1)] = 1 · x1
all of the following rules can be deleted.
32(21(12(20(03(32(23(32(23(30(02(23(30(x1))))))))))))) → 33(33(31(13(32(21(10(00(02(20(00(03(30(03(33(33(30(x1))))))))))))))))) (139)
32(21(12(20(03(32(23(32(23(30(02(23(31(x1))))))))))))) → 33(33(31(13(32(21(10(00(02(20(00(03(30(03(33(33(31(x1))))))))))))))))) (140)
32(21(12(20(03(32(23(32(23(30(02(23(32(x1))))))))))))) → 33(33(31(13(32(21(10(00(02(20(00(03(30(03(33(33(32(x1))))))))))))))))) (141)
32(21(12(20(03(32(23(32(23(30(02(23(33(x1))))))))))))) → 33(33(31(13(32(21(10(00(02(20(00(03(30(03(33(33(33(x1))))))))))))))))) (142)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1 + 1
[11(x1)] = 1 · x1
[01(x1)] = 1 · x1
[33(x1)] = 1 · x1
[32(x1)] = 1 · x1
[03(x1)] = 1 · x1
[30(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 3
all of the following rules can be deleted.
00(00(02(22(21(10(02(23(31(13(31(11(10(00(x1)))))))))))))) → 01(10(01(10(02(21(13(33(33(33(32(23(32(23(31(11(10(00(x1)))))))))))))))))) (167)
00(00(02(22(21(10(02(23(31(13(31(11(10(01(x1)))))))))))))) → 01(10(01(10(02(21(13(33(33(33(32(23(32(23(31(11(10(01(x1)))))))))))))))))) (168)
00(00(02(22(21(10(02(23(31(13(31(11(10(02(x1)))))))))))))) → 01(10(01(10(02(21(13(33(33(33(32(23(32(23(31(11(10(02(x1)))))))))))))))))) (169)
00(00(02(22(21(10(02(23(31(13(31(11(10(03(x1)))))))))))))) → 01(10(01(10(02(21(13(33(33(33(32(23(32(23(31(11(10(03(x1)))))))))))))))))) (170)
10(00(02(22(21(10(02(23(31(13(31(11(10(00(x1)))))))))))))) → 11(10(01(10(02(21(13(33(33(33(32(23(32(23(31(11(10(00(x1)))))))))))))))))) (171)
10(00(02(22(21(10(02(23(31(13(31(11(10(01(x1)))))))))))))) → 11(10(01(10(02(21(13(33(33(33(32(23(32(23(31(11(10(01(x1)))))))))))))))))) (172)
10(00(02(22(21(10(02(23(31(13(31(11(10(02(x1)))))))))))))) → 11(10(01(10(02(21(13(33(33(33(32(23(32(23(31(11(10(02(x1)))))))))))))))))) (173)
10(00(02(22(21(10(02(23(31(13(31(11(10(03(x1)))))))))))))) → 11(10(01(10(02(21(13(33(33(33(32(23(32(23(31(11(10(03(x1)))))))))))))))))) (174)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 1
[11(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[30(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[22(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
10(01(11(11(13(33(30(03(32(23(33(31(12(20(x1)))))))))))))) → 13(33(31(11(10(02(21(12(23(30(03(33(33(32(23(30(02(20(x1)))))))))))))))))) (187)
10(01(11(11(13(33(30(03(32(23(33(31(12(21(x1)))))))))))))) → 13(33(31(11(10(02(21(12(23(30(03(33(33(32(23(30(02(21(x1)))))))))))))))))) (188)
10(01(11(11(13(33(30(03(32(23(33(31(12(22(x1)))))))))))))) → 13(33(31(11(10(02(21(12(23(30(03(33(33(32(23(30(02(22(x1)))))))))))))))))) (189)
10(01(11(11(13(33(30(03(32(23(33(31(12(23(x1)))))))))))))) → 13(33(31(11(10(02(21(12(23(30(03(33(33(32(23(30(02(23(x1)))))))))))))))))) (190)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1
[23(x1)] = 1 · x1
[30(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1
[03(x1)] = 1 · x1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
00(02(23(30(00(00(00(02(23(31(10(03(31(10(x1)))))))))))))) → 01(11(12(22(21(12(23(30(02(21(13(31(12(23(33(32(21(10(x1)))))))))))))))))) (231)
00(02(23(30(00(00(00(02(23(31(10(03(31(11(x1)))))))))))))) → 01(11(12(22(21(12(23(30(02(21(13(31(12(23(33(32(21(11(x1)))))))))))))))))) (232)
00(02(23(30(00(00(00(02(23(31(10(03(31(12(x1)))))))))))))) → 01(11(12(22(21(12(23(30(02(21(13(31(12(23(33(32(21(12(x1)))))))))))))))))) (233)
00(02(23(30(00(00(00(02(23(31(10(03(31(13(x1)))))))))))))) → 01(11(12(22(21(12(23(30(02(21(13(31(12(23(33(32(21(13(x1)))))))))))))))))) (234)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[23(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1
[31(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
all of the following rules can be deleted.
10(02(23(30(00(00(00(02(23(31(10(03(31(10(x1)))))))))))))) → 11(11(12(22(21(12(23(30(02(21(13(31(12(23(33(32(21(10(x1)))))))))))))))))) (235)
10(02(23(30(00(00(00(02(23(31(10(03(31(11(x1)))))))))))))) → 11(11(12(22(21(12(23(30(02(21(13(31(12(23(33(32(21(11(x1)))))))))))))))))) (236)
10(02(23(30(00(00(00(02(23(31(10(03(31(12(x1)))))))))))))) → 11(11(12(22(21(12(23(30(02(21(13(31(12(23(33(32(21(12(x1)))))))))))))))))) (237)
10(02(23(30(00(00(00(02(23(31(10(03(31(13(x1)))))))))))))) → 11(11(12(22(21(12(23(30(02(21(13(31(12(23(33(32(21(13(x1)))))))))))))))))) (238)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 · x1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1
[33(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1
all of the following rules can be deleted.
02(20(01(11(12(21(10(03(32(21(13(31(13(30(x1)))))))))))))) → 00(03(33(30(02(22(21(13(31(12(23(31(13(31(13(33(33(30(x1)))))))))))))))))) (311)
02(20(01(11(12(21(10(03(32(21(13(31(13(31(x1)))))))))))))) → 00(03(33(30(02(22(21(13(31(12(23(31(13(31(13(33(33(31(x1)))))))))))))))))) (312)
02(20(01(11(12(21(10(03(32(21(13(31(13(32(x1)))))))))))))) → 00(03(33(30(02(22(21(13(31(12(23(31(13(31(13(33(33(32(x1)))))))))))))))))) (313)
02(20(01(11(12(21(10(03(32(21(13(31(13(33(x1)))))))))))))) → 00(03(33(30(02(22(21(13(31(12(23(31(13(31(13(33(33(33(x1)))))))))))))))))) (314)

1.1.1.1.1.1.1.1.1.1.1 R is empty

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