Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/139036)

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

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

We obtain the transformed TRS

There are 137 ruless (increase limit for explicit display).

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 548 ruless (increase limit for explicit display).

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
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1 + 5
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 5
[23(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 5
[03(x1)] = 1 · x1
[33(x1)] = 1 · x1
[30(x1)] = 1 · x1 + 1
[11(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
all of the following rules can be deleted.

There are 496 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[02(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[22(x1)] = 1 · x1
[13(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[33(x1)] = 1 · x1
[30(x1)] = 1 · x1
all of the following rules can be deleted.
22(22(22(20(03(32(22(20(03(31(10(01(13(33(31(10(02(20(x1)))))))))))))))))) 22(22(22(21(10(02(22(21(10(03(31(10(03(30(03(33(32(20(x1)))))))))))))))))) (322)
22(22(22(20(03(32(22(20(03(31(10(01(13(33(31(10(02(22(x1)))))))))))))))))) 22(22(22(21(10(02(22(21(10(03(31(10(03(30(03(33(32(22(x1)))))))))))))))))) (323)
22(22(22(20(03(32(22(20(03(31(10(01(13(33(31(10(02(21(x1)))))))))))))))))) 22(22(22(21(10(02(22(21(10(03(31(10(03(30(03(33(32(21(x1)))))))))))))))))) (324)
22(22(22(20(03(32(22(20(03(31(10(01(13(33(31(10(02(23(x1)))))))))))))))))) 22(22(22(21(10(02(22(21(10(03(31(10(03(30(03(33(32(23(x1)))))))))))))))))) (325)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[02(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[22(x1)] = 1 · x1
[13(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1
[32(x1)] = 1 · x1
[30(x1)] = 1 · x1
[03(x1)] = 1 · x1
all of the following rules can be deleted.
13(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(00(x1))))))))))))))))))) 12(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(00(x1))))))))))))))))))) (602)
13(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(02(x1))))))))))))))))))) 12(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(02(x1))))))))))))))))))) (603)
13(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(01(x1))))))))))))))))))) 12(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(01(x1))))))))))))))))))) (604)
13(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(03(x1))))))))))))))))))) 12(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(03(x1))))))))))))))))))) (605)
13(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(20(x1))))))))))))))))))) 12(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(20(x1))))))))))))))))))) (666)
13(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(22(x1))))))))))))))))))) 12(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(22(x1))))))))))))))))))) (667)
13(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(21(x1))))))))))))))))))) 12(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(21(x1))))))))))))))))))) (668)
13(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(23(x1))))))))))))))))))) 12(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(23(x1))))))))))))))))))) (669)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 2
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[02(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 2
[20(x1)] = 1 · x1 + 2
[00(x1)] = 1 · x1
[22(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1 + 1
[32(x1)] = 1 · x1
[30(x1)] = 1 · x1
[03(x1)] = 1 · x1
all of the following rules can be deleted.
21(10(01(12(21(11(10(02(23(31(11(12(21(10(01(11(12(20(x1)))))))))))))))))) 21(10(00(01(12(22(21(11(10(01(11(11(12(21(13(31(12(20(x1)))))))))))))))))) (294)
21(10(01(12(21(11(10(02(23(31(11(12(21(10(01(11(12(22(x1)))))))))))))))))) 21(10(00(01(12(22(21(11(10(01(11(11(12(21(13(31(12(22(x1)))))))))))))))))) (295)
21(10(01(12(21(11(10(02(23(31(11(12(21(10(01(11(12(21(x1)))))))))))))))))) 21(10(00(01(12(22(21(11(10(01(11(11(12(21(13(31(12(21(x1)))))))))))))))))) (296)
21(10(01(12(21(11(10(02(23(31(11(12(21(10(01(11(12(23(x1)))))))))))))))))) 21(10(00(01(12(22(21(11(10(01(11(11(12(21(13(31(12(23(x1)))))))))))))))))) (297)
30(01(13(33(31(11(13(31(12(21(11(13(32(23(31(10(01(10(x1)))))))))))))))))) 31(11(13(33(31(13(33(30(01(12(21(11(13(32(21(10(01(10(x1)))))))))))))))))) (354)
30(01(13(33(31(11(13(31(12(21(11(13(32(23(31(10(01(12(x1)))))))))))))))))) 31(11(13(33(31(13(33(30(01(12(21(11(13(32(21(10(01(12(x1)))))))))))))))))) (355)
30(01(13(33(31(11(13(31(12(21(11(13(32(23(31(10(01(11(x1)))))))))))))))))) 31(11(13(33(31(13(33(30(01(12(21(11(13(32(21(10(01(11(x1)))))))))))))))))) (356)
30(01(13(33(31(11(13(31(12(21(11(13(32(23(31(10(01(13(x1)))))))))))))))))) 31(11(13(33(31(13(33(30(01(12(21(11(13(32(21(10(01(13(x1)))))))))))))))))) (357)
33(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(00(x1))))))))))))))))))) 32(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(00(x1))))))))))))))))))) (606)
33(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(02(x1))))))))))))))))))) 32(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(02(x1))))))))))))))))))) (607)
33(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(01(x1))))))))))))))))))) 32(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(01(x1))))))))))))))))))) (608)
33(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(03(x1))))))))))))))))))) 32(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(03(x1))))))))))))))))))) (609)
33(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(20(x1))))))))))))))))))) 32(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(20(x1))))))))))))))))))) (670)
33(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(22(x1))))))))))))))))))) 32(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(22(x1))))))))))))))))))) (671)
33(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(21(x1))))))))))))))))))) 32(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(21(x1))))))))))))))))))) (672)
33(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(23(x1))))))))))))))))))) 32(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(23(x1))))))))))))))))))) (673)

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
[32(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1
[21(x1)] = 1 · x1
[30(x1)] = 1 · x1
[22(x1)] = 1 · x1
[12(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
[02(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
[11(x1)] = 1 · x1
all of the following rules can be deleted.
31(12(23(32(22(22(22(22(21(10(01(10(02(20(00(00(03(30(x1)))))))))))))))))) 32(23(32(20(01(10(02(21(10(02(22(21(12(22(20(00(03(30(x1)))))))))))))))))) (366)
31(12(23(32(22(22(22(22(21(10(01(10(02(20(00(00(03(32(x1)))))))))))))))))) 32(23(32(20(01(10(02(21(10(02(22(21(12(22(20(00(03(32(x1)))))))))))))))))) (367)
31(12(23(32(22(22(22(22(21(10(01(10(02(20(00(00(03(31(x1)))))))))))))))))) 32(23(32(20(01(10(02(21(10(02(22(21(12(22(20(00(03(31(x1)))))))))))))))))) (368)
31(12(23(32(22(22(22(22(21(10(01(10(02(20(00(00(03(33(x1)))))))))))))))))) 32(23(32(20(01(10(02(21(10(02(22(21(12(22(20(00(03(33(x1)))))))))))))))))) (369)

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
[32(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 2
[13(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1
[30(x1)] = 1 · x1
[22(x1)] = 1 · x1
[03(x1)] = 1 · x1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1
all of the following rules can be deleted.
23(33(32(23(33(31(13(32(23(32(23(33(32(21(13(32(23(30(x1)))))))))))))))))) 23(32(23(32(22(21(13(32(21(13(33(33(33(33(33(32(23(30(x1)))))))))))))))))) (338)
23(33(32(23(33(31(13(32(23(32(23(33(32(21(13(32(23(32(x1)))))))))))))))))) 23(32(23(32(22(21(13(32(21(13(33(33(33(33(33(32(23(32(x1)))))))))))))))))) (339)
23(33(32(23(33(31(13(32(23(32(23(33(32(21(13(32(23(31(x1)))))))))))))))))) 23(32(23(32(22(21(13(32(21(13(33(33(33(33(33(32(23(31(x1)))))))))))))))))) (340)
23(33(32(23(33(31(13(32(23(32(23(33(32(21(13(32(23(33(x1)))))))))))))))))) 23(32(23(32(22(21(13(32(21(13(33(33(33(33(33(32(23(33(x1)))))))))))))))))) (341)
23(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(00(x1))))))))))))))))))) 22(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(00(x1))))))))))))))))))) (598)
23(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(02(x1))))))))))))))))))) 22(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(02(x1))))))))))))))))))) (599)
23(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(01(x1))))))))))))))))))) 22(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(01(x1))))))))))))))))))) (600)
23(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(03(x1))))))))))))))))))) 22(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(03(x1))))))))))))))))))) (601)
03(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(20(x1))))))))))))))))))) 02(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(20(x1))))))))))))))))))) (658)
03(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(22(x1))))))))))))))))))) 02(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(22(x1))))))))))))))))))) (659)
03(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(21(x1))))))))))))))))))) 02(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(21(x1))))))))))))))))))) (660)
03(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(23(x1))))))))))))))))))) 02(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(23(x1))))))))))))))))))) (661)
23(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(20(x1))))))))))))))))))) 22(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(20(x1))))))))))))))))))) (662)
23(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(22(x1))))))))))))))))))) 22(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(22(x1))))))))))))))))))) (663)
23(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(21(x1))))))))))))))))))) 22(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(21(x1))))))))))))))))))) (664)
23(31(13(32(22(22(22(23(32(20(03(30(00(03(32(22(22(22(23(x1))))))))))))))))))) 22(22(21(10(03(32(22(22(23(30(00(02(23(33(33(32(22(22(23(x1))))))))))))))))))) (665)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1
all of the following rules can be deleted.
03(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(00(x1))))))))))))))))))) 02(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(00(x1))))))))))))))))))) (594)
03(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(02(x1))))))))))))))))))) 02(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(02(x1))))))))))))))))))) (595)
03(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(01(x1))))))))))))))))))) 02(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(01(x1))))))))))))))))))) (596)
03(31(10(00(02(20(01(11(12(21(13(32(22(22(22(20(00(00(03(x1))))))))))))))))))) 02(23(32(21(10(02(23(31(10(00(01(11(12(22(22(20(00(00(03(x1))))))))))))))))))) (597)

1.1.1.1.1.1.1.1.1.1 R is empty

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