Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/139025)

The rewrite relation of the following TRS is considered.

0(0(0(0(1(0(1(2(2(2(0(3(2(0(2(3(0(3(x1)))))))))))))))))) → 0(0(0(2(2(1(2(0(0(2(3(2(1(3(0(0(0(3(x1)))))))))))))))))) (1)
0(0(0(0(3(2(0(3(2(3(2(2(1(0(0(0(1(3(x1)))))))))))))))))) → 0(0(3(2(1(0(3(0(0(0(2(3(2(2(0(0(1(3(x1)))))))))))))))))) (2)
0(0(0(1(2(0(1(3(2(1(2(2(0(0(3(2(1(1(x1)))))))))))))))))) → 0(0(1(0(1(3(0(2(0(2(1(3(1(2(0(2(2(1(x1)))))))))))))))))) (3)
0(0(0(3(0(1(3(2(3(2(3(0(2(2(0(0(0(0(x1)))))))))))))))))) → 0(3(0(2(0(0(3(0(2(3(0(0(2(3(0(2(1(0(x1)))))))))))))))))) (4)
0(0(1(1(3(0(0(0(0(3(3(0(0(2(3(2(0(3(x1)))))))))))))))))) → 0(0(0(0(2(2(0(3(3(1(3(1(0(3(0(0(0(3(x1)))))))))))))))))) (5)
0(0(1(2(0(3(3(0(1(2(1(1(2(3(3(3(1(2(x1)))))))))))))))))) → 0(1(0(2(3(1(3(1(3(2(1(1(3(0(2(3(0(2(x1)))))))))))))))))) (6)
0(0(3(2(1(2(0(3(2(0(2(0(3(3(2(2(1(0(x1)))))))))))))))))) → 0(0(0(2(2(2(3(2(3(3(2(0(1(3(0(2(1(0(x1)))))))))))))))))) (7)
0(0(3(2(3(3(0(0(0(0(2(1(2(2(2(0(1(0(x1)))))))))))))))))) → 0(0(2(0(2(2(2(1(0(3(0(3(2(0(3(0(1(0(x1)))))))))))))))))) (8)
0(1(0(2(3(1(3(0(2(0(1(2(2(3(3(3(3(3(x1)))))))))))))))))) → 3(2(0(3(1(1(0(3(1(2(3(0(2(2(0(3(3(3(x1)))))))))))))))))) (9)
0(1(0(3(0(1(2(2(2(1(2(0(1(0(0(2(2(2(x1)))))))))))))))))) → 0(2(2(2(1(0(0(1(2(1(0(2(0(0(2(2(3(1(x1)))))))))))))))))) (10)
0(1(0(3(2(0(1(1(2(2(1(1(2(3(3(1(0(3(x1)))))))))))))))))) → 3(2(1(1(1(0(2(0(3(0(2(1(3(1(1(2(0(3(x1)))))))))))))))))) (11)
0(1(1(0(0(2(0(1(2(1(2(0(2(2(3(3(1(3(x1)))))))))))))))))) → 0(1(0(2(3(1(0(2(1(2(0(0(2(3(2(1(1(3(x1)))))))))))))))))) (12)
0(1(1(1(3(3(0(1(2(0(2(1(2(1(2(3(0(0(x1)))))))))))))))))) → 3(1(1(0(2(0(1(2(2(1(0(3(1(3(1(0(2(0(x1)))))))))))))))))) (13)
0(1(1(3(0(1(2(1(0(1(2(0(3(0(1(1(2(1(x1)))))))))))))))))) → 0(1(1(0(1(0(1(0(1(1(2(1(2(3(0(3(2(1(x1)))))))))))))))))) (14)
0(1(2(0(1(0(1(0(1(2(1(3(2(0(2(1(3(2(x1)))))))))))))))))) → 0(1(2(1(2(3(2(0(3(0(1(1(1(0(2(0(2(1(x1)))))))))))))))))) (15)
0(1(2(1(3(1(3(3(2(1(1(1(2(2(3(2(1(1(x1)))))))))))))))))) → 0(2(3(3(1(1(2(2(1(2(1(1(3(2(1(3(1(1(x1)))))))))))))))))) (16)
0(1(3(0(0(3(2(2(3(2(0(3(1(0(0(0(0(0(x1)))))))))))))))))) → 0(2(0(3(1(0(2(1(3(3(0(3(0(0(2(0(0(0(x1)))))))))))))))))) (17)
0(2(2(3(1(0(1(2(3(3(3(1(1(1(0(3(2(0(x1)))))))))))))))))) → 3(1(2(0(0(3(1(0(3(2(2(3(1(3(1(1(2(0(x1)))))))))))))))))) (18)
0(2(3(0(0(0(1(3(0(0(0(1(1(2(3(3(2(1(x1)))))))))))))))))) → 3(2(3(0(0(3(0(2(1(0(0(3(1(2(1(0(0(1(x1)))))))))))))))))) (19)
0(3(0(2(1(0(1(2(0(1(2(3(1(3(1(2(0(1(x1)))))))))))))))))) → 0(0(1(0(3(1(3(2(1(3(0(1(1(2(2(0(2(1(x1)))))))))))))))))) (20)
0(3(1(0(1(3(0(1(2(3(2(0(0(3(0(0(0(2(x1)))))))))))))))))) → 0(3(3(0(0(1(0(0(2(0(3(1(2(0(0(3(1(2(x1)))))))))))))))))) (21)
0(3(2(0(1(3(0(3(2(0(1(2(1(0(2(1(0(2(x1)))))))))))))))))) → 0(0(2(0(1(2(3(2(3(2(0(0(1(1(3(1(0(2(x1)))))))))))))))))) (22)
0(3(2(0(3(0(2(1(0(2(2(0(3(0(2(2(2(0(x1)))))))))))))))))) → 0(3(0(2(2(2(0(2(0(0(2(2(2(3(0(3(1(0(x1)))))))))))))))))) (23)
0(3(3(1(2(3(2(2(0(3(3(1(1(0(1(2(3(0(x1)))))))))))))))))) → 0(3(2(0(3(1(3(1(0(1(1(2(2(2(3(3(3(0(x1)))))))))))))))))) (24)
1(0(0(0(0(3(3(3(3(0(3(3(0(0(1(0(1(1(x1)))))))))))))))))) → 1(0(0(0(0(3(0(0(3(0(3(3(3(1(0(1(3(1(x1)))))))))))))))))) (25)
1(0(0(1(0(1(0(0(2(3(2(3(1(0(3(2(2(0(x1)))))))))))))))))) → 1(0(0(0(2(2(2(2(3(1(0(1(3(0(0(3(1(0(x1)))))))))))))))))) (26)
1(0(1(0(1(1(1(3(3(1(2(3(3(1(3(0(1(1(x1)))))))))))))))))) → 1(1(1(1(0(1(3(0(2(3(1(3(3(0(3(1(1(1(x1)))))))))))))))))) (27)
1(0(2(1(2(0(1(1(0(1(1(3(0(1(2(1(3(1(x1)))))))))))))))))) → 1(1(0(2(3(1(1(0(2(1(0(1(2(1(0(3(1(1(x1)))))))))))))))))) (28)
1(1(0(1(2(1(0(1(1(3(1(3(3(0(1(3(1(1(x1)))))))))))))))))) → 1(3(1(2(3(0(1(1(1(0(3(0(3(1(1(1(1(1(x1)))))))))))))))))) (29)
1(1(0(1(2(2(3(1(2(1(3(3(3(0(0(1(3(0(x1)))))))))))))))))) → 1(0(0(3(2(2(3(3(3(1(3(1(1(1(0(2(1(0(x1)))))))))))))))))) (30)
1(1(1(1(2(3(2(2(0(2(2(0(1(1(3(0(1(1(x1)))))))))))))))))) → 1(1(1(1(2(1(0(3(2(3(1(2(0(2(1(0(2(1(x1)))))))))))))))))) (31)
1(1(1(2(2(3(1(3(0(1(3(0(1(3(1(3(1(2(x1)))))))))))))))))) → 1(1(1(1(3(1(1(0(0(3(2(3(3(1(2(3(1(2(x1)))))))))))))))))) (32)
1(1(2(3(2(1(3(1(2(1(2(0(3(0(0(1(2(2(x1)))))))))))))))))) → 1(1(0(2(2(2(0(1(0(3(2(3(1(2(2(1(3(1(x1)))))))))))))))))) (33)
1(2(1(2(1(2(0(1(2(0(1(3(2(0(0(1(2(0(x1)))))))))))))))))) → 1(2(1(1(0(0(0(3(2(1(2(2(2(1(0(1(2(0(x1)))))))))))))))))) (34)
1(2(1(2(2(0(1(3(2(2(1(1(3(1(3(0(1(2(x1)))))))))))))))))) → 1(3(1(1(0(2(3(1(2(1(2(2(3(2(1(0(2(1(x1)))))))))))))))))) (35)
1(2(1(2(2(3(1(0(3(0(1(3(2(0(3(0(0(2(x1)))))))))))))))))) → 1(2(1(0(3(0(3(2(1(0(3(2(1(2(3(0(0(2(x1)))))))))))))))))) (36)
1(2(3(3(1(2(3(1(3(3(2(0(1(0(0(0(3(2(x1)))))))))))))))))) → 1(2(3(1(1(3(2(0(3(0(3(2(0(2(3(0(3(1(x1)))))))))))))))))) (37)
1(3(0(0(2(0(0(2(1(3(2(1(1(3(1(2(0(2(x1)))))))))))))))))) → 1(3(1(1(0(3(2(2(3(1(0(0(2(0(2(1(0(2(x1)))))))))))))))))) (38)
1(3(0(2(1(1(0(0(1(3(0(3(2(3(2(3(0(1(x1)))))))))))))))))) → 1(3(1(1(0(3(1(3(3(0(3(2(2(0(0(0(2(1(x1)))))))))))))))))) (39)
1(3(1(2(1(2(0(1(2(1(1(2(3(0(1(0(1(1(x1)))))))))))))))))) → 1(2(3(0(2(2(0(1(1(1(1(0(1(2(1(1(3(1(x1)))))))))))))))))) (40)
2(0(2(1(3(1(1(3(0(0(1(0(2(2(0(1(3(3(x1)))))))))))))))))) → 2(0(0(2(2(1(3(3(2(1(1(0(3(1(0(0(1(3(x1)))))))))))))))))) (41)
2(1(0(1(2(2(1(3(3(3(2(0(2(1(2(1(2(3(x1)))))))))))))))))) → 2(1(2(2(1(1(0(0(3(1(3(3(2(2(1(2(2(3(x1)))))))))))))))))) (42)
2(1(1(2(3(2(2(0(1(0(3(3(3(2(3(3(1(2(x1)))))))))))))))))) → 2(1(1(0(3(2(2(2(2(2(3(3(1(3(0(3(3(1(x1)))))))))))))))))) (43)
2(1(1(3(2(2(0(2(2(2(3(0(1(2(1(2(3(2(x1)))))))))))))))))) → 2(0(2(3(2(3(2(2(1(1(1(0(2(2(2(3(1(2(x1)))))))))))))))))) (44)
2(1(2(0(0(1(3(0(2(0(1(3(3(3(1(0(2(2(x1)))))))))))))))))) → 2(1(1(0(2(2(3(1(3(3(0(2(0(0(1(0(3(2(x1)))))))))))))))))) (45)
2(1(2(0(2(1(2(2(3(0(3(3(2(2(2(2(2(2(x1)))))))))))))))))) → 2(1(0(2(3(0(2(3(1(2(2(3(2(2(2(2(2(2(x1)))))))))))))))))) (46)
2(1(2(1(0(1(3(1(2(0(0(2(2(0(1(1(2(2(x1)))))))))))))))))) → 2(1(1(3(0(2(1(1(1(2(2(0(0(1(2(2(0(2(x1)))))))))))))))))) (47)
2(1(3(2(1(2(0(1(2(1(3(1(2(0(1(0(3(2(x1)))))))))))))))))) → 2(3(0(2(2(3(2(1(0(2(1(1(1(3(1(0(2(1(x1)))))))))))))))))) (48)
2(1(3(2(3(0(1(0(1(1(3(3(2(2(1(2(1(2(x1)))))))))))))))))) → 2(0(1(1(1(2(3(2(3(1(3(1(1(2(0(2(3(2(x1)))))))))))))))))) (49)
2(2(1(3(0(2(2(0(3(2(0(1(3(1(2(2(2(3(x1)))))))))))))))))) → 2(2(2(0(2(2(2(1(0(3(2(3(1(2(0(3(1(3(x1)))))))))))))))))) (50)
2(3(3(2(1(2(3(1(0(0(0(2(1(0(3(2(0(1(x1)))))))))))))))))) → 2(3(0(0(2(2(3(3(1(0(0(3(1(0(2(2(1(1(x1)))))))))))))))))) (51)
3(0(0(0(3(3(0(0(3(2(0(0(1(3(3(3(3(1(x1)))))))))))))))))) → 3(0(0(3(1(0(0(0(3(2(1(3(3(0(3(0(3(3(x1)))))))))))))))))) (52)
3(0(1(0(1(3(2(1(2(1(3(3(1(1(1(3(2(1(x1)))))))))))))))))) → 3(1(1(1(1(2(3(2(2(1(0(3(1(3(1(0(1(3(x1)))))))))))))))))) (53)
3(0(1(0(2(3(0(3(0(0(1(3(3(1(1(2(3(3(x1)))))))))))))))))) → 3(1(3(3(1(0(0(1(0(3(0(3(0(2(3(1(2(3(x1)))))))))))))))))) (54)
3(0(1(1(3(0(0(0(2(2(1(2(3(3(2(1(2(0(x1)))))))))))))))))) → 0(3(2(2(1(3(0(2(0(3(3(1(2(2(1(1(0(0(x1)))))))))))))))))) (55)
3(0(1(2(0(1(3(0(2(1(2(0(1(2(0(2(3(3(x1)))))))))))))))))) → 3(0(2(3(0(2(1(1(3(2(0(1(2(1(0(3(0(2(x1)))))))))))))))))) (56)
3(0(1(2(3(2(2(0(0(2(0(1(3(0(3(0(3(2(x1)))))))))))))))))) → 0(3(1(0(3(0(0(2(0(0(3(2(3(1(2(3(2(2(x1)))))))))))))))))) (57)
3(0(1(3(3(3(0(1(1(2(3(3(3(1(1(3(2(2(x1)))))))))))))))))) → 3(0(3(1(3(1(2(2(0(3(2(3(3(1(1(1(3(3(x1)))))))))))))))))) (58)
3(0(3(0(1(1(0(2(3(3(0(1(2(1(3(2(0(1(x1)))))))))))))))))) → 3(0(3(1(2(3(3(1(1(0(0(2(1(2(0(0(3(1(x1)))))))))))))))))) (59)
3(0(3(3(0(0(3(0(1(1(1(3(2(3(1(0(3(1(x1)))))))))))))))))) → 3(0(3(1(1(3(2(1(0(0(0(1(3(3(3(0(3(1(x1)))))))))))))))))) (60)
3(1(3(1(3(2(0(0(3(2(3(0(1(0(1(3(1(1(x1)))))))))))))))))) → 3(1(2(1(0(0(0(3(3(1(3(1(1(0(3(1(2(3(x1)))))))))))))))))) (61)
3(2(0(1(0(2(3(3(2(0(2(0(1(2(3(2(3(2(x1)))))))))))))))))) → 3(2(2(1(2(0(2(1(3(3(3(3(0(0(2(0(2(2(x1)))))))))))))))))) (62)
3(2(0(1(2(3(0(0(1(3(0(1(2(1(1(3(2(3(x1)))))))))))))))))) → 0(0(2(0(3(1(2(1(2(3(0(3(2(3(1(1(1(3(x1)))))))))))))))))) (63)
3(2(0(3(1(3(3(2(1(2(2(2(3(0(1(2(0(2(x1)))))))))))))))))) → 3(2(1(3(2(0(2(2(2(0(2(3(1(2(3(0(3(1(x1)))))))))))))))))) (64)
3(2(1(2(3(0(2(1(3(2(0(1(2(3(3(3(2(3(x1)))))))))))))))))) → 3(0(2(3(1(3(1(3(2(1(3(2(2(2(0(2(3(3(x1)))))))))))))))))) (65)
3(2(2(2(0(1(3(2(0(2(0(2(3(0(1(2(1(2(x1)))))))))))))))))) → 3(2(2(2(0(2(2(3(0(2(1(2(2(0(3(1(1(0(x1)))))))))))))))))) (66)
3(2(3(0(1(0(3(0(2(3(2(0(1(1(0(3(3(2(x1)))))))))))))))))) → 0(2(3(2(3(0(3(1(3(1(0(0(2(0(2(3(3(1(x1)))))))))))))))))) (67)
3(2(3(1(0(2(0(1(2(1(0(1(2(3(1(1(3(0(x1)))))))))))))))))) → 3(1(1(3(1(1(3(1(3(2(0(2(0(2(1(0(2(0(x1)))))))))))))))))) (68)
3(3(0(1(1(2(0(1(0(0(3(0(3(3(2(2(3(0(x1)))))))))))))))))) → 3(1(1(3(3(0(0(3(2(3(3(1(0(2(0(0(2(0(x1)))))))))))))))))) (69)
3(3(0(1(2(0(1(2(3(0(0(0(0(2(3(3(3(2(x1)))))))))))))))))) → 0(2(0(0(3(3(0(2(3(3(3(1(2(0(0(2(3(1(x1)))))))))))))))))) (70)
3(3(0(1(3(1(1(0(1(3(2(1(3(0(3(3(2(0(x1)))))))))))))))))) → 3(0(2(2(3(1(1(3(3(0(1(0(3(1(3(3(1(0(x1)))))))))))))))))) (71)
3(3(0(2(2(3(0(2(1(2(3(2(1(0(1(2(1(2(x1)))))))))))))))))) → 3(0(2(2(2(2(1(1(2(3(1(2(0(3(3(1(0(2(x1)))))))))))))))))) (72)
3(3(0(3(2(0(3(3(1(3(1(1(0(3(3(0(0(1(x1)))))))))))))))))) → 3(1(0(3(1(3(1(1(0(3(3(3(0(2(0(3(0(3(x1)))))))))))))))))) (73)
3(3(1(0(0(1(2(3(3(0(1(2(1(1(2(1(1(1(x1)))))))))))))))))) → 1(1(1(1(3(0(3(1(0(2(3(1(2(3(2(0(1(1(x1)))))))))))))))))) (74)
3(3(1(1(1(0(1(0(3(3(1(3(1(3(2(0(3(2(x1)))))))))))))))))) → 3(3(1(0(3(2(1(1(1(3(3(3(0(2(1(0(3(1(x1)))))))))))))))))) (75)
3(3(1(2(2(3(0(3(0(1(2(1(0(1(2(0(0(2(x1)))))))))))))))))) → 0(2(0(2(1(0(0(3(3(1(2(3(2(1(0(2(1(3(x1)))))))))))))))))) (76)
3(3(1(2(3(3(0(2(1(0(1(0(0(1(0(1(3(2(x1)))))))))))))))))) → 0(1(0(2(0(3(0(3(1(3(1(2(1(2(0(3(1(3(x1)))))))))))))))))) (77)
3(3(2(0(0(0(1(1(2(3(1(0(3(2(3(2(2(2(x1)))))))))))))))))) → 3(1(1(1(0(2(0(2(3(3(0(2(3(0(3(2(2(2(x1)))))))))))))))))) (78)
3(3(3(0(1(1(0(1(3(2(1(1(3(0(0(3(3(3(x1)))))))))))))))))) → 3(1(3(0(1(3(1(2(3(0(1(0(0(3(1(3(3(3(x1)))))))))))))))))) (79)
3(3(3(0(1(3(0(1(1(0(0(1(0(1(0(1(0(1(x1)))))))))))))))))) → 3(3(1(1(1(0(0(0(1(0(0(3(1(1(0(1(0(3(x1)))))))))))))))))) (80)

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

There are 119 ruless (increase limit for explicit display).

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 476 ruless (increase limit for explicit display).

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1 + 19
[01(x1)] = 1 · x1 + 23
[10(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 12
[03(x1)] = 1 · x1 + 18
[32(x1)] = 1 · x1
[02(x1)] = 1 · x1
[23(x1)] = 1 · x1 + 6
[30(x1)] = 1 · x1 + 19
[21(x1)] = 1 · x1 + 2
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 3
[33(x1)] = 1 · x1 + 13
[11(x1)] = 1 · x1
all of the following rules can be deleted.

There are 456 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1 + 1
[20(x1)] = 1 · x1
[23(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1 + 1
[10(x1)] = 1 · x1 + 2
[01(x1)] = 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 2
[33(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 1
[12(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
00(00(00(03(32(20(03(32(23(32(22(21(10(00(00(01(13(30(x1)))))))))))))))))) → 00(03(32(21(10(03(30(00(00(02(23(32(22(20(00(01(13(30(x1)))))))))))))))))) (137)
00(00(00(03(32(20(03(32(23(32(22(21(10(00(00(01(13(31(x1)))))))))))))))))) → 00(03(32(21(10(03(30(00(00(02(23(32(22(20(00(01(13(31(x1)))))))))))))))))) (138)
00(00(00(03(32(20(03(32(23(32(22(21(10(00(00(01(13(32(x1)))))))))))))))))) → 00(03(32(21(10(03(30(00(00(02(23(32(22(20(00(01(13(32(x1)))))))))))))))))) (139)
00(00(00(03(32(20(03(32(23(32(22(21(10(00(00(01(13(33(x1)))))))))))))))))) → 00(03(32(21(10(03(30(00(00(02(23(32(22(20(00(01(13(33(x1)))))))))))))))))) (140)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 1
[23(x1)] = 1 · x1 + 1
[32(x1)] = 1 · x1
[31(x1)] = 1 · x1
[03(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 4
[21(x1)] = 1 · x1
[33(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
10(00(01(10(01(10(00(02(23(32(23(31(10(03(32(22(20(00(x1)))))))))))))))))) → 10(00(00(02(22(22(22(23(31(10(01(13(30(00(03(31(10(00(x1)))))))))))))))))) (213)
10(00(01(10(01(10(00(02(23(32(23(31(10(03(32(22(20(01(x1)))))))))))))))))) → 10(00(00(02(22(22(22(23(31(10(01(13(30(00(03(31(10(01(x1)))))))))))))))))) (214)
10(00(01(10(01(10(00(02(23(32(23(31(10(03(32(22(20(02(x1)))))))))))))))))) → 10(00(00(02(22(22(22(23(31(10(01(13(30(00(03(31(10(02(x1)))))))))))))))))) (215)
10(00(01(10(01(10(00(02(23(32(23(31(10(03(32(22(20(03(x1)))))))))))))))))) → 10(00(00(02(22(22(22(23(31(10(01(13(30(00(03(31(10(03(x1)))))))))))))))))) (216)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1
[33(x1)] = 1 · x1
[30(x1)] = 1 · x1
[23(x1)] = 1 · x1
[03(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
all of the following rules can be deleted.
11(10(01(12(21(10(01(11(13(31(13(33(30(01(13(31(11(10(x1)))))))))))))))))) → 13(31(12(23(30(01(11(11(10(03(30(03(31(11(11(11(11(10(x1)))))))))))))))))) (225)
11(10(01(12(21(10(01(11(13(31(13(33(30(01(13(31(11(11(x1)))))))))))))))))) → 13(31(12(23(30(01(11(11(10(03(30(03(31(11(11(11(11(11(x1)))))))))))))))))) (226)
11(10(01(12(21(10(01(11(13(31(13(33(30(01(13(31(11(12(x1)))))))))))))))))) → 13(31(12(23(30(01(11(11(10(03(30(03(31(11(11(11(11(12(x1)))))))))))))))))) (227)
11(10(01(12(21(10(01(11(13(31(13(33(30(01(13(31(11(13(x1)))))))))))))))))) → 13(31(12(23(30(01(11(11(10(03(30(03(31(11(11(11(11(13(x1)))))))))))))))))) (228)

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 + 1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[33(x1)] = 1 · x1
[11(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[21(x1)] = 1 · x1
all of the following rules can be deleted.
10(02(22(23(31(10(01(12(23(33(33(31(11(11(10(03(32(20(00(x1))))))))))))))))))) → 13(31(12(20(00(03(31(10(03(32(22(23(31(13(31(11(12(20(00(x1))))))))))))))))))) (453)
10(02(22(23(31(10(01(12(23(33(33(31(11(11(10(03(32(20(01(x1))))))))))))))))))) → 13(31(12(20(00(03(31(10(03(32(22(23(31(13(31(11(12(20(01(x1))))))))))))))))))) (454)
10(02(22(23(31(10(01(12(23(33(33(31(11(11(10(03(32(20(02(x1))))))))))))))))))) → 13(31(12(20(00(03(31(10(03(32(22(23(31(13(31(11(12(20(02(x1))))))))))))))))))) (455)
10(02(22(23(31(10(01(12(23(33(33(31(11(11(10(03(32(20(03(x1))))))))))))))))))) → 13(31(12(20(00(03(31(10(03(32(22(23(31(13(31(11(12(20(03(x1))))))))))))))))))) (456)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[23(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 3
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
all of the following rules can be deleted.
23(33(31(12(23(33(30(02(21(10(01(10(00(01(10(01(13(32(20(x1))))))))))))))))))) → 20(01(10(02(20(03(30(03(31(13(31(12(21(12(20(03(31(13(30(x1))))))))))))))))))) (601)
33(33(31(12(23(33(30(02(21(10(01(10(00(01(10(01(13(32(20(x1))))))))))))))))))) → 30(01(10(02(20(03(30(03(31(13(31(12(21(12(20(03(31(13(30(x1))))))))))))))))))) (605)
33(33(31(12(23(33(30(02(21(10(01(10(00(01(10(01(13(32(23(x1))))))))))))))))))) → 30(01(10(02(20(03(30(03(31(13(31(12(21(12(20(03(31(13(33(x1))))))))))))))))))) (608)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[23(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
[00(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1 + 1
[20(x1)] = 1 · x1
[03(x1)] = 1 · x1
all of the following rules can be deleted.
23(33(31(12(23(33(30(02(21(10(01(10(00(01(10(01(13(32(23(x1))))))))))))))))))) → 20(01(10(02(20(03(30(03(31(13(31(12(21(12(20(03(31(13(33(x1))))))))))))))))))) (604)

1.1.1.1.1.1.1.1.1.1 R is empty

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