Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/132782)

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

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

We obtain the transformed TRS

There are 122 ruless (increase limit for explicit display).

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 488 ruless (increase limit for explicit display).

1.1.1 Rule Removal

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

There are 394 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 34
[22(x1)] = 1 · x1 + 49
[20(x1)] = 1 · x1 + 6
[01(x1)] = 1 · x1 + 76
[11(x1)] = 1 · x1 + 24
[10(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 130
[31(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 97
[13(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 65
[23(x1)] = 1 · x1 + 18
[32(x1)] = 1 · x1
[33(x1)] = 1 · x1
[30(x1)] = 1 · x1 + 32
all of the following rules can be deleted.
00(02(22(20(01(11(10(03(31(10(02(22(21(13(31(12(21(10(x1)))))))))))))))))) → 00(01(12(22(20(02(20(00(03(31(12(23(31(11(11(12(21(10(x1)))))))))))))))))) (142)
00(02(22(20(01(11(10(03(31(10(02(22(21(13(31(12(21(11(x1)))))))))))))))))) → 00(01(12(22(20(02(20(00(03(31(12(23(31(11(11(12(21(11(x1)))))))))))))))))) (143)
00(02(22(20(01(11(10(03(31(10(02(22(21(13(31(12(21(12(x1)))))))))))))))))) → 00(01(12(22(20(02(20(00(03(31(12(23(31(11(11(12(21(12(x1)))))))))))))))))) (144)
00(02(22(20(01(11(10(03(31(10(02(22(21(13(31(12(21(13(x1)))))))))))))))))) → 00(01(12(22(20(02(20(00(03(31(12(23(31(11(11(12(21(13(x1)))))))))))))))))) (145)
00(03(32(23(31(10(03(33(30(00(03(30(03(30(03(31(11(13(x1)))))))))))))))))) → 00(00(03(33(31(13(33(31(12(23(33(31(10(00(00(03(30(03(x1)))))))))))))))))) (161)
01(12(22(22(21(11(10(00(00(00(01(10(00(01(11(11(10(00(x1)))))))))))))))))) → 01(11(11(12(20(01(10(00(02(20(01(10(01(11(12(20(00(00(x1)))))))))))))))))) (166)
01(12(22(22(21(11(10(00(00(00(01(10(00(01(11(11(10(01(x1)))))))))))))))))) → 01(11(11(12(20(01(10(00(02(20(01(10(01(11(12(20(00(01(x1)))))))))))))))))) (167)
01(12(22(22(21(11(10(00(00(00(01(10(00(01(11(11(10(02(x1)))))))))))))))))) → 01(11(11(12(20(01(10(00(02(20(01(10(01(11(12(20(00(02(x1)))))))))))))))))) (168)
01(12(22(22(21(11(10(00(00(00(01(10(00(01(11(11(10(03(x1)))))))))))))))))) → 01(11(11(12(20(01(10(00(02(20(01(10(01(11(12(20(00(03(x1)))))))))))))))))) (169)
10(00(01(12(22(23(31(11(10(03(33(32(21(11(13(30(01(10(x1)))))))))))))))))) → 10(01(12(22(23(33(33(30(02(23(31(11(10(01(10(01(11(10(x1)))))))))))))))))) (198)
10(00(01(12(22(23(31(11(10(03(33(32(21(11(13(30(01(11(x1)))))))))))))))))) → 10(01(12(22(23(33(33(30(02(23(31(11(10(01(10(01(11(11(x1)))))))))))))))))) (199)
10(00(01(12(22(23(31(11(10(03(33(32(21(11(13(30(01(12(x1)))))))))))))))))) → 10(01(12(22(23(33(33(30(02(23(31(11(10(01(10(01(11(12(x1)))))))))))))))))) (200)
10(00(01(12(22(23(31(11(10(03(33(32(21(11(13(30(01(13(x1)))))))))))))))))) → 10(01(12(22(23(33(33(30(02(23(31(11(10(01(10(01(11(13(x1)))))))))))))))))) (201)
10(03(31(13(30(03(32(20(00(01(11(11(13(30(03(32(20(00(x1)))))))))))))))))) → 10(01(11(13(33(31(12(23(30(01(10(00(03(32(20(03(30(00(x1)))))))))))))))))) (206)
10(03(31(13(30(03(32(20(00(01(11(11(13(30(03(32(20(01(x1)))))))))))))))))) → 10(01(11(13(33(31(12(23(30(01(10(00(03(32(20(03(30(01(x1)))))))))))))))))) (207)
10(03(31(13(30(03(32(20(00(01(11(11(13(30(03(32(20(02(x1)))))))))))))))))) → 10(01(11(13(33(31(12(23(30(01(10(00(03(32(20(03(30(02(x1)))))))))))))))))) (208)
10(03(31(13(30(03(32(20(00(01(11(11(13(30(03(32(20(03(x1)))))))))))))))))) → 10(01(11(13(33(31(12(23(30(01(10(00(03(32(20(03(30(03(x1)))))))))))))))))) (209)
11(12(23(32(20(03(31(11(12(22(22(21(10(02(23(30(01(10(x1)))))))))))))))))) → 12(20(03(32(22(23(31(10(01(10(02(21(13(31(12(22(21(10(x1)))))))))))))))))) (230)
11(12(23(32(20(03(31(11(12(22(22(21(10(02(23(30(01(11(x1)))))))))))))))))) → 12(20(03(32(22(23(31(10(01(10(02(21(13(31(12(22(21(11(x1)))))))))))))))))) (231)
11(12(23(32(20(03(31(11(12(22(22(21(10(02(23(30(01(12(x1)))))))))))))))))) → 12(20(03(32(22(23(31(10(01(10(02(21(13(31(12(22(21(12(x1)))))))))))))))))) (232)
11(12(23(32(20(03(31(11(12(22(22(21(10(02(23(30(01(13(x1)))))))))))))))))) → 12(20(03(32(22(23(31(10(01(10(02(21(13(31(12(22(21(13(x1)))))))))))))))))) (233)
12(20(00(01(11(13(30(02(22(22(21(11(11(10(01(13(30(00(x1)))))))))))))))))) → 13(31(10(00(01(10(01(10(02(23(32(22(21(12(21(11(10(00(x1)))))))))))))))))) (234)
12(20(00(01(11(13(30(02(22(22(21(11(11(10(01(13(30(01(x1)))))))))))))))))) → 13(31(10(00(01(10(01(10(02(23(32(22(21(12(21(11(10(01(x1)))))))))))))))))) (235)
12(20(00(01(11(13(30(02(22(22(21(11(11(10(01(13(30(02(x1)))))))))))))))))) → 13(31(10(00(01(10(01(10(02(23(32(22(21(12(21(11(10(02(x1)))))))))))))))))) (236)
12(20(00(01(11(13(30(02(22(22(21(11(11(10(01(13(30(03(x1)))))))))))))))))) → 13(31(10(00(01(10(01(10(02(23(32(22(21(12(21(11(10(03(x1)))))))))))))))))) (237)
13(33(33(32(22(23(30(03(30(02(21(12(22(23(33(33(32(20(x1)))))))))))))))))) → 11(10(03(33(32(23(33(32(22(22(23(30(03(33(32(23(32(20(x1)))))))))))))))))) (254)
13(33(33(32(22(23(30(03(30(02(21(12(22(23(33(33(32(21(x1)))))))))))))))))) → 11(10(03(33(32(23(33(32(22(22(23(30(03(33(32(23(32(21(x1)))))))))))))))))) (255)
13(33(33(32(22(23(30(03(30(02(21(12(22(23(33(33(32(22(x1)))))))))))))))))) → 11(10(03(33(32(23(33(32(22(22(23(30(03(33(32(23(32(22(x1)))))))))))))))))) (256)
13(33(33(32(22(23(30(03(30(02(21(12(22(23(33(33(32(23(x1)))))))))))))))))) → 11(10(03(33(32(23(33(32(22(22(23(30(03(33(32(23(32(23(x1)))))))))))))))))) (257)
20(03(31(12(22(23(31(12(21(10(00(00(01(11(11(12(20(00(x1)))))))))))))))))) → 21(12(21(11(10(01(13(33(31(12(20(00(02(20(01(12(20(00(x1)))))))))))))))))) (258)
20(03(31(12(22(23(31(12(21(10(00(00(01(11(11(12(20(01(x1)))))))))))))))))) → 21(12(21(11(10(01(13(33(31(12(20(00(02(20(01(12(20(01(x1)))))))))))))))))) (259)
20(03(31(12(22(23(31(12(21(10(00(00(01(11(11(12(20(02(x1)))))))))))))))))) → 21(12(21(11(10(01(13(33(31(12(20(00(02(20(01(12(20(02(x1)))))))))))))))))) (260)
20(03(31(12(22(23(31(12(21(10(00(00(01(11(11(12(20(03(x1)))))))))))))))))) → 21(12(21(11(10(01(13(33(31(12(20(00(02(20(01(12(20(03(x1)))))))))))))))))) (261)
21(13(32(23(31(11(11(13(33(30(01(11(13(32(22(21(12(20(x1)))))))))))))))))) → 21(11(11(12(23(31(12(20(01(13(32(21(13(32(23(33(31(10(x1)))))))))))))))))) (266)
21(13(32(23(31(11(11(13(33(30(01(11(13(32(22(21(12(21(x1)))))))))))))))))) → 21(11(11(12(23(31(12(20(01(13(32(21(13(32(23(33(31(11(x1)))))))))))))))))) (267)
21(13(32(23(31(11(11(13(33(30(01(11(13(32(22(21(12(23(x1)))))))))))))))))) → 21(11(11(12(23(31(12(20(01(13(32(21(13(32(23(33(31(13(x1)))))))))))))))))) (269)
22(20(03(30(03(33(32(22(22(20(01(13(30(02(22(20(01(13(x1)))))))))))))))))) → 22(20(01(10(02(21(12(23(32(22(20(03(33(32(23(30(00(03(x1)))))))))))))))))) (273)
22(22(23(33(30(02(23(30(00(02(22(20(02(22(21(12(22(20(x1)))))))))))))))))) → 22(22(22(23(30(01(13(32(22(20(03(32(22(20(02(20(02(20(x1)))))))))))))))))) (282)
22(22(23(33(30(02(23(30(00(02(22(20(02(22(21(12(22(21(x1)))))))))))))))))) → 22(22(22(23(30(01(13(32(22(20(03(32(22(20(02(20(02(21(x1)))))))))))))))))) (283)
22(22(23(33(30(02(23(30(00(02(22(20(02(22(21(12(22(22(x1)))))))))))))))))) → 22(22(22(23(30(01(13(32(22(20(03(32(22(20(02(20(02(22(x1)))))))))))))))))) (284)
22(22(23(33(30(02(23(30(00(02(22(20(02(22(21(12(22(23(x1)))))))))))))))))) → 22(22(22(23(30(01(13(32(22(20(03(32(22(20(02(20(02(23(x1)))))))))))))))))) (285)
22(23(33(31(13(32(22(23(32(21(11(13(33(30(03(33(33(30(x1)))))))))))))))))) → 23(33(32(20(03(32(23(32(21(11(12(23(33(31(13(33(33(30(x1)))))))))))))))))) (286)
22(23(33(31(13(32(22(23(32(21(11(13(33(30(03(33(33(31(x1)))))))))))))))))) → 23(33(32(20(03(32(23(32(21(11(12(23(33(31(13(33(33(31(x1)))))))))))))))))) (287)
22(23(33(31(13(32(22(23(32(21(11(13(33(30(03(33(33(32(x1)))))))))))))))))) → 23(33(32(20(03(32(23(32(21(11(12(23(33(31(13(33(33(32(x1)))))))))))))))))) (288)
22(23(33(31(13(32(22(23(32(21(11(13(33(30(03(33(33(33(x1)))))))))))))))))) → 23(33(32(20(03(32(23(32(21(11(12(23(33(31(13(33(33(33(x1)))))))))))))))))) (289)
30(02(22(22(23(30(00(01(11(13(33(30(03(32(21(10(00(00(x1)))))))))))))))))) → 33(30(01(10(01(13(31(10(02(20(03(32(22(23(32(20(00(00(x1)))))))))))))))))) (302)
30(02(22(22(23(30(00(01(11(13(33(30(03(32(21(10(00(01(x1)))))))))))))))))) → 33(30(01(10(01(13(31(10(02(20(03(32(22(23(32(20(00(01(x1)))))))))))))))))) (303)
30(02(22(22(23(30(00(01(11(13(33(30(03(32(21(10(00(02(x1)))))))))))))))))) → 33(30(01(10(01(13(31(10(02(20(03(32(22(23(32(20(00(02(x1)))))))))))))))))) (304)
30(02(22(22(23(30(00(01(11(13(33(30(03(32(21(10(00(03(x1)))))))))))))))))) → 33(30(01(10(01(13(31(10(02(20(03(32(22(23(32(20(00(03(x1)))))))))))))))))) (305)
31(10(00(01(12(23(33(30(02(23(33(30(00(01(11(13(30(00(x1)))))))))))))))))) → 32(23(32(23(31(10(01(11(13(33(30(01(10(00(00(03(30(00(x1)))))))))))))))))) (310)
31(10(00(01(12(23(33(30(02(23(33(30(00(01(11(13(30(01(x1)))))))))))))))))) → 32(23(32(23(31(10(01(11(13(33(30(01(10(00(00(03(30(01(x1)))))))))))))))))) (311)
31(10(00(01(12(23(33(30(02(23(33(30(00(01(11(13(30(02(x1)))))))))))))))))) → 32(23(32(23(31(10(01(11(13(33(30(01(10(00(00(03(30(02(x1)))))))))))))))))) (312)
31(10(00(01(12(23(33(30(02(23(33(30(00(01(11(13(30(03(x1)))))))))))))))))) → 32(23(32(23(31(10(01(11(13(33(30(01(10(00(00(03(30(03(x1)))))))))))))))))) (313)
31(11(10(03(30(03(30(01(10(00(02(22(21(13(32(23(30(00(x1)))))))))))))))))) → 32(21(12(23(30(01(10(02(23(33(30(03(31(10(00(01(10(00(x1)))))))))))))))))) (314)
31(11(10(03(30(03(30(01(10(00(02(22(21(13(32(23(30(01(x1)))))))))))))))))) → 32(21(12(23(30(01(10(02(23(33(30(03(31(10(00(01(10(01(x1)))))))))))))))))) (315)
31(11(10(03(30(03(30(01(10(00(02(22(21(13(32(23(30(02(x1)))))))))))))))))) → 32(21(12(23(30(01(10(02(23(33(30(03(31(10(00(01(10(02(x1)))))))))))))))))) (316)
31(11(10(03(30(03(30(01(10(00(02(22(21(13(32(23(30(03(x1)))))))))))))))))) → 32(21(12(23(30(01(10(02(23(33(30(03(31(10(00(01(10(03(x1)))))))))))))))))) (317)
31(12(23(33(30(02(21(12(23(31(12(23(33(33(30(03(33(30(x1)))))))))))))))))) → 32(23(33(30(03(31(13(32(22(22(21(10(03(33(31(13(33(30(x1)))))))))))))))))) (318)
31(12(23(33(30(02(21(12(23(31(12(23(33(33(30(03(33(31(x1)))))))))))))))))) → 32(23(33(30(03(31(13(32(22(22(21(10(03(33(31(13(33(31(x1)))))))))))))))))) (319)
31(12(23(33(30(02(21(12(23(31(12(23(33(33(30(03(33(32(x1)))))))))))))))))) → 32(23(33(30(03(31(13(32(22(22(21(10(03(33(31(13(33(32(x1)))))))))))))))))) (320)
31(12(23(33(30(02(21(12(23(31(12(23(33(33(30(03(33(33(x1)))))))))))))))))) → 32(23(33(30(03(31(13(32(22(22(21(10(03(33(31(13(33(33(x1)))))))))))))))))) (321)
33(30(03(31(11(13(30(00(02(22(23(32(23(30(03(33(33(30(x1)))))))))))))))))) → 33(33(30(03(31(10(00(02(21(13(32(23(30(03(32(23(33(30(x1)))))))))))))))))) (362)
33(30(03(31(11(13(30(00(02(22(23(32(23(30(03(33(33(31(x1)))))))))))))))))) → 33(33(30(03(31(10(00(02(21(13(32(23(30(03(32(23(33(31(x1)))))))))))))))))) (363)
33(30(03(31(11(13(30(00(02(22(23(32(23(30(03(33(33(32(x1)))))))))))))))))) → 33(33(30(03(31(10(00(02(21(13(32(23(30(03(32(23(33(32(x1)))))))))))))))))) (364)
33(30(03(31(11(13(30(00(02(22(23(32(23(30(03(33(33(33(x1)))))))))))))))))) → 33(33(30(03(31(10(00(02(21(13(32(23(30(03(32(23(33(33(x1)))))))))))))))))) (365)
00(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(10(x1))))))))))))))))))) → 01(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(10(x1))))))))))))))))))) (418)
00(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(11(x1))))))))))))))))))) → 01(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(11(x1))))))))))))))))))) (419)
00(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(12(x1))))))))))))))))))) → 01(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(12(x1))))))))))))))))))) (420)
00(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(13(x1))))))))))))))))))) → 01(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(13(x1))))))))))))))))))) (421)
10(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(10(x1))))))))))))))))))) → 11(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(10(x1))))))))))))))))))) (422)
10(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(11(x1))))))))))))))))))) → 11(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(11(x1))))))))))))))))))) (423)
10(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(12(x1))))))))))))))))))) → 11(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(12(x1))))))))))))))))))) (424)
10(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(13(x1))))))))))))))))))) → 11(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(13(x1))))))))))))))))))) (425)
20(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(10(x1))))))))))))))))))) → 21(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(10(x1))))))))))))))))))) (426)
20(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(11(x1))))))))))))))))))) → 21(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(11(x1))))))))))))))))))) (427)
20(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(12(x1))))))))))))))))))) → 21(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(12(x1))))))))))))))))))) (428)
20(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(13(x1))))))))))))))))))) → 21(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(13(x1))))))))))))))))))) (429)
30(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(10(x1))))))))))))))))))) → 31(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(10(x1))))))))))))))))))) (430)
30(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(11(x1))))))))))))))))))) → 31(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(11(x1))))))))))))))))))) (431)
30(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(12(x1))))))))))))))))))) → 31(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(12(x1))))))))))))))))))) (432)
30(03(30(02(22(21(12(22(23(30(03(32(21(11(11(11(10(01(13(x1))))))))))))))))))) → 31(12(22(20(03(33(30(02(23(31(11(12(20(02(21(11(10(01(13(x1))))))))))))))))))) (433)
32(22(21(11(11(12(22(22(20(03(33(30(02(23(30(00(03(32(20(x1))))))))))))))))))) → 33(32(22(21(10(01(12(20(02(21(12(20(00(03(32(23(33(32(20(x1))))))))))))))))))) (590)
32(22(21(11(11(12(22(22(20(03(33(30(02(23(30(00(03(32(21(x1))))))))))))))))))) → 33(32(22(21(10(01(12(20(02(21(12(20(00(03(32(23(33(32(21(x1))))))))))))))))))) (591)
32(22(21(11(11(12(22(22(20(03(33(30(02(23(30(00(03(32(22(x1))))))))))))))))))) → 33(32(22(21(10(01(12(20(02(21(12(20(00(03(32(23(33(32(22(x1))))))))))))))))))) (592)
32(22(21(11(11(12(22(22(20(03(33(30(02(23(30(00(03(32(23(x1))))))))))))))))))) → 33(32(22(21(10(01(12(20(02(21(12(20(00(03(32(23(33(32(23(x1))))))))))))))))))) (593)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[11(x1)] = 1 · x1
[33(x1)] = 1 · x1
[30(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1 + 2
[22(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[10(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[00(x1)] = 1 · x1
all of the following rules can be deleted.
21(13(32(23(31(11(11(13(33(30(01(11(13(32(22(21(12(22(x1)))))))))))))))))) → 21(11(11(12(23(31(12(20(01(13(32(21(13(32(23(33(31(12(x1)))))))))))))))))) (268)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1 + 1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[11(x1)] = 1 · x1
[23(x1)] = 1 · x1
[01(x1)] = 1 · x1
[10(x1)] = 1 · x1
[03(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1
[22(x1)] = 1 · x1
[00(x1)] = 1 · x1
all of the following rules can be deleted.
33(33(31(13(33(32(21(13(30(02(21(11(13(32(23(30(01(10(x1)))))))))))))))))) → 30(03(33(31(12(23(31(13(33(32(23(31(13(31(12(20(01(10(x1)))))))))))))))))) (390)
33(33(31(13(33(32(21(13(30(02(21(11(13(32(23(30(01(11(x1)))))))))))))))))) → 30(03(33(31(12(23(31(13(33(32(23(31(13(31(12(20(01(11(x1)))))))))))))))))) (391)
33(33(31(13(33(32(21(13(30(02(21(11(13(32(23(30(01(12(x1)))))))))))))))))) → 30(03(33(31(12(23(31(13(33(32(23(31(13(31(12(20(01(12(x1)))))))))))))))))) (392)
33(33(31(13(33(32(21(13(30(02(21(11(13(32(23(30(01(13(x1)))))))))))))))))) → 30(03(33(31(12(23(31(13(33(32(23(31(13(31(12(20(01(13(x1)))))))))))))))))) (393)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 · x1 + 1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1
[03(x1)] = 1 · x1
[33(x1)] = 1 · x1
[30(x1)] = 1 · x1
[23(x1)] = 1 · x1
[00(x1)] = 1 · x1
[32(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
all of the following rules can be deleted.
02(22(21(11(11(12(22(22(20(03(33(30(02(23(30(00(03(32(20(x1))))))))))))))))))) → 03(32(22(21(10(01(12(20(02(21(12(20(00(03(32(23(33(32(20(x1))))))))))))))))))) (578)
02(22(21(11(11(12(22(22(20(03(33(30(02(23(30(00(03(32(21(x1))))))))))))))))))) → 03(32(22(21(10(01(12(20(02(21(12(20(00(03(32(23(33(32(21(x1))))))))))))))))))) (579)
02(22(21(11(11(12(22(22(20(03(33(30(02(23(30(00(03(32(22(x1))))))))))))))))))) → 03(32(22(21(10(01(12(20(02(21(12(20(00(03(32(23(33(32(22(x1))))))))))))))))))) (580)
02(22(21(11(11(12(22(22(20(03(33(30(02(23(30(00(03(32(23(x1))))))))))))))))))) → 03(32(22(21(10(01(12(20(02(21(12(20(00(03(32(23(33(32(23(x1))))))))))))))))))) (581)

1.1.1.1.1.1.1.1 R is empty

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