Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/139004)

The relative rewrite relation R/S is considered where R is the following TRS

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

and S is the following TRS.

0(1(2(3(x1)))) → 0(1(2(3(x1)))) (81)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{0(☐), 3(☐), 1(☐), 2(☐)}

We obtain the transformed TRS

There are 143 ruless (increase limit for explicit display).

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 572 ruless (increase limit for explicit display).

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1 + 9
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[20(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 2
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1 + 9
[33(x1)] = 1 · x1 + 3
[30(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 3
[12(x1)] = 1 · x1 + 1
[22(x1)] = 1 · x1
all of the following rules can be deleted.

There are 532 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[01(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 7
[03(x1)] = 1 · x1 + 2
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[00(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 7
[23(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 5
[11(x1)] = 1 · x1 + 3
[22(x1)] = 1 · x1 + 6
[21(x1)] = 1 · x1 + 4
all of the following rules can be deleted.
01(13(32(20(03(30(02(20(00(03(30(03(33(31(10(01(13(30(x1)))))))))))))))))) → 02(23(33(33(30(00(00(01(12(20(00(03(31(11(10(03(33(30(x1)))))))))))))))))) (206)
01(13(32(20(03(30(02(20(00(03(30(03(33(31(10(01(13(33(x1)))))))))))))))))) → 02(23(33(33(30(00(00(01(12(20(00(03(31(11(10(03(33(33(x1)))))))))))))))))) (207)
01(13(32(20(03(30(02(20(00(03(30(03(33(31(10(01(13(31(x1)))))))))))))))))) → 02(23(33(33(30(00(00(01(12(20(00(03(31(11(10(03(33(31(x1)))))))))))))))))) (208)
01(13(32(20(03(30(02(20(00(03(30(03(33(31(10(01(13(32(x1)))))))))))))))))) → 02(23(33(33(30(00(00(01(12(20(00(03(31(11(10(03(33(32(x1)))))))))))))))))) (209)
12(23(30(02(20(02(23(32(20(00(01(10(03(30(00(01(13(30(x1)))))))))))))))))) → 12(23(30(00(01(10(00(00(03(32(23(32(20(00(01(12(23(30(x1)))))))))))))))))) (266)
12(23(30(02(20(02(23(32(20(00(01(10(03(30(00(01(13(33(x1)))))))))))))))))) → 12(23(30(00(01(10(00(00(03(32(23(32(20(00(01(12(23(33(x1)))))))))))))))))) (267)
12(23(30(02(20(02(23(32(20(00(01(10(03(30(00(01(13(31(x1)))))))))))))))))) → 12(23(30(00(01(10(00(00(03(32(23(32(20(00(01(12(23(31(x1)))))))))))))))))) (268)
12(23(30(02(20(02(23(32(20(00(01(10(03(30(00(01(13(32(x1)))))))))))))))))) → 12(23(30(00(01(10(00(00(03(32(23(32(20(00(01(12(23(32(x1)))))))))))))))))) (269)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[03(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 1
[20(x1)] = 1 · x1 + 1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1 + 4
[10(x1)] = 1 · x1
[33(x1)] = 1 · x1
[23(x1)] = 1 · x1 + 3
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 2
[11(x1)] = 1 · x1 + 3
[02(x1)] = 1 · x1
all of the following rules can be deleted.
03(30(00(01(12(20(03(31(13(30(01(13(31(10(01(13(33(30(x1)))))))))))))))))) → 03(31(10(01(13(30(03(33(30(00(01(10(01(12(23(31(13(30(x1)))))))))))))))))) (214)
03(30(00(01(12(20(03(31(13(30(01(13(31(10(01(13(33(33(x1)))))))))))))))))) → 03(31(10(01(13(30(03(33(30(00(01(10(01(12(23(31(13(33(x1)))))))))))))))))) (215)
03(30(00(01(12(20(03(31(13(30(01(13(31(10(01(13(33(31(x1)))))))))))))))))) → 03(31(10(01(13(30(03(33(30(00(01(10(01(12(23(31(13(31(x1)))))))))))))))))) (216)
03(30(00(01(12(20(03(31(13(30(01(13(31(10(01(13(33(32(x1)))))))))))))))))) → 03(31(10(01(13(30(03(33(30(00(01(10(01(12(23(31(13(32(x1)))))))))))))))))) (217)
01(10(01(13(32(23(32(22(20(02(20(01(13(30(00(02(22(21(10(x1))))))))))))))))))) → 02(20(02(23(32(21(10(01(12(23(32(20(00(01(10(02(23(31(10(x1))))))))))))))))))) (482)
01(10(01(13(32(23(32(22(20(02(20(01(13(30(00(02(22(21(13(x1))))))))))))))))))) → 02(20(02(23(32(21(10(01(12(23(32(20(00(01(10(02(23(31(13(x1))))))))))))))))))) (483)
01(10(01(13(32(23(32(22(20(02(20(01(13(30(00(02(22(21(11(x1))))))))))))))))))) → 02(20(02(23(32(21(10(01(12(23(32(20(00(01(10(02(23(31(11(x1))))))))))))))))))) (484)
01(10(01(13(32(23(32(22(20(02(20(01(13(30(00(02(22(21(12(x1))))))))))))))))))) → 02(20(02(23(32(21(10(01(12(23(32(20(00(01(10(02(23(31(12(x1))))))))))))))))))) (485)
31(10(01(13(32(23(32(22(20(02(20(01(13(30(00(02(22(21(10(x1))))))))))))))))))) → 32(20(02(23(32(21(10(01(12(23(32(20(00(01(10(02(23(31(10(x1))))))))))))))))))) (486)
31(10(01(13(32(23(32(22(20(02(20(01(13(30(00(02(22(21(13(x1))))))))))))))))))) → 32(20(02(23(32(21(10(01(12(23(32(20(00(01(10(02(23(31(13(x1))))))))))))))))))) (487)
31(10(01(13(32(23(32(22(20(02(20(01(13(30(00(02(22(21(11(x1))))))))))))))))))) → 32(20(02(23(32(21(10(01(12(23(32(20(00(01(10(02(23(31(11(x1))))))))))))))))))) (488)
31(10(01(13(32(23(32(22(20(02(20(01(13(30(00(02(22(21(12(x1))))))))))))))))))) → 32(20(02(23(32(21(10(01(12(23(32(20(00(01(10(02(23(31(12(x1))))))))))))))))))) (489)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 1
[20(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1
[11(x1)] = 1 · x1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[33(x1)] = 1 · x1
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
all of the following rules can be deleted.
13(30(01(12(20(01(10(03(30(03(31(12(22(21(13(31(13(30(x1)))))))))))))))))) → 11(11(13(32(20(01(10(03(32(23(31(10(01(10(03(32(23(30(x1)))))))))))))))))) (278)
13(30(01(12(20(01(10(03(30(03(31(12(22(21(13(31(13(33(x1)))))))))))))))))) → 11(11(13(32(20(01(10(03(32(23(31(10(01(10(03(32(23(33(x1)))))))))))))))))) (279)
13(30(01(12(20(01(10(03(30(03(31(12(22(21(13(31(13(31(x1)))))))))))))))))) → 11(11(13(32(20(01(10(03(32(23(31(10(01(10(03(32(23(31(x1)))))))))))))))))) (280)
13(30(01(12(20(01(10(03(30(03(31(12(22(21(13(31(13(32(x1)))))))))))))))))) → 11(11(13(32(20(01(10(03(32(23(31(10(01(10(03(32(23(32(x1)))))))))))))))))) (281)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 2
[00(x1)] = 1 · x1 + 2
[10(x1)] = 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[03(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 1
[11(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
01(12(20(00(01(10(01(13(30(03(33(33(31(13(32(23(30(03(31(x1))))))))))))))))))) → 02(21(11(13(33(30(03(30(00(01(12(23(33(31(10(03(33(30(01(x1))))))))))))))))))) (532)
01(12(20(00(01(10(01(13(30(03(33(33(31(13(32(23(30(03(32(x1))))))))))))))))))) → 02(21(11(13(33(30(03(30(00(01(12(23(33(31(10(03(33(30(02(x1))))))))))))))))))) (533)
31(12(20(00(01(10(01(13(30(03(33(33(31(13(32(23(30(03(31(x1))))))))))))))))))) → 32(21(11(13(33(30(03(30(00(01(12(23(33(31(10(03(33(30(01(x1))))))))))))))))))) (536)
31(12(20(00(01(10(01(13(30(03(33(33(31(13(32(23(30(03(32(x1))))))))))))))))))) → 32(21(11(13(33(30(03(30(00(01(12(23(33(31(10(03(33(30(02(x1))))))))))))))))))) (537)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[12(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 2
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1
[11(x1)] = 1 · x1
[23(x1)] = 1 · x1
[01(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 2
[30(x1)] = 1 · x1
[32(x1)] = 1 · x1
[33(x1)] = 1 · x1
[00(x1)] = 1 · x1
all of the following rules can be deleted.
12(21(10(02(22(20(02(21(13(31(12(21(11(11(10(02(21(12(20(x1))))))))))))))))))) → 11(12(21(12(21(12(22(20(02(20(02(23(31(10(01(11(11(12(20(x1))))))))))))))))))) (634)
12(21(10(02(22(20(02(21(13(31(12(21(11(11(10(02(21(12(23(x1))))))))))))))))))) → 11(12(21(12(21(12(22(20(02(20(02(23(31(10(01(11(11(12(23(x1))))))))))))))))))) (635)
12(21(10(02(22(20(02(21(13(31(12(21(11(11(10(02(21(12(21(x1))))))))))))))))))) → 11(12(21(12(21(12(22(20(02(20(02(23(31(10(01(11(11(12(21(x1))))))))))))))))))) (636)
12(21(10(02(22(20(02(21(13(31(12(21(11(11(10(02(21(12(22(x1))))))))))))))))))) → 11(12(21(12(21(12(22(20(02(20(02(23(31(10(01(11(11(12(22(x1))))))))))))))))))) (637)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1
[20(x1)] = 1 · x1
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[23(x1)] = 1 · x1
[01(x1)] = 1 · x1
[03(x1)] = 1 · x1
[30(x1)] = 1 · x1
[32(x1)] = 1 · x1
[33(x1)] = 1 · x1
[00(x1)] = 1 · x1
all of the following rules can be deleted.
22(21(10(02(22(20(02(21(13(31(12(21(11(11(10(02(21(12(20(x1))))))))))))))))))) → 21(12(21(12(21(12(22(20(02(20(02(23(31(10(01(11(11(12(20(x1))))))))))))))))))) (638)
22(21(10(02(22(20(02(21(13(31(12(21(11(11(10(02(21(12(23(x1))))))))))))))))))) → 21(12(21(12(21(12(22(20(02(20(02(23(31(10(01(11(11(12(23(x1))))))))))))))))))) (639)
22(21(10(02(22(20(02(21(13(31(12(21(11(11(10(02(21(12(21(x1))))))))))))))))))) → 21(12(21(12(21(12(22(20(02(20(02(23(31(10(01(11(11(12(21(x1))))))))))))))))))) (640)
22(21(10(02(22(20(02(21(13(31(12(21(11(11(10(02(21(12(22(x1))))))))))))))))))) → 21(12(21(12(21(12(22(20(02(20(02(23(31(10(01(11(11(12(22(x1))))))))))))))))))) (641)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[03(x1)] = 1 · x1
[30(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[33(x1)] = 1 · x1
[00(x1)] = 1 · x1
[21(x1)] = 1 · x1
all of the following rules can be deleted.
03(30(03(31(13(31(13(32(22(23(32(22(22(20(01(13(31(12(20(x1))))))))))))))))))) → 01(11(12(23(33(30(00(03(33(33(32(21(12(21(12(23(32(22(20(x1))))))))))))))))))) (674)
03(30(03(31(13(31(13(32(22(23(32(22(22(20(01(13(31(12(23(x1))))))))))))))))))) → 01(11(12(23(33(30(00(03(33(33(32(21(12(21(12(23(32(22(23(x1))))))))))))))))))) (675)
03(30(03(31(13(31(13(32(22(23(32(22(22(20(01(13(31(12(21(x1))))))))))))))))))) → 01(11(12(23(33(30(00(03(33(33(32(21(12(21(12(23(32(22(21(x1))))))))))))))))))) (676)
03(30(03(31(13(31(13(32(22(23(32(22(22(20(01(13(31(12(22(x1))))))))))))))))))) → 01(11(12(23(33(30(00(03(33(33(32(21(12(21(12(23(32(22(22(x1))))))))))))))))))) (677)

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.