Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/91210)

The rewrite relation of the following TRS is considered.

0(0(0(0(1(1(2(0(0(0(2(1(2(1(1(1(0(0(1(2(1(1(1(x1))))))))))))))))))))))) → 0(2(1(1(0(0(2(1(2(1(0(0(0(1(0(0(2(0(2(2(0(0(0(1(0(0(2(x1))))))))))))))))))))))))))) (1)
0(0(0(2(0(1(0(1(1(1(0(0(2(2(1(0(2(1(1(0(1(1(2(x1))))))))))))))))))))))) → 0(2(1(2(1(0(0(2(1(0(2(2(0(0(2(1(1(1(0(1(2(1(0(0(0(0(1(x1))))))))))))))))))))))))))) (2)
0(0(1(2(2(0(1(1(1(2(0(1(2(1(1(2(0(2(0(2(1(0(2(x1))))))))))))))))))))))) → 0(2(1(0(1(2(0(0(0(1(1(2(0(1(1(2(0(1(0(0(0(0(0(2(0(0(0(x1))))))))))))))))))))))))))) (3)
0(0(2(2(0(1(2(2(0(2(2(0(2(2(0(2(1(0(0(2(0(0(2(x1))))))))))))))))))))))) → 0(2(0(2(0(0(2(1(1(1(1(0(2(1(0(0(2(0(0(0(0(0(0(0(2(0(1(x1))))))))))))))))))))))))))) (4)
0(1(0(0(0(1(1(2(2(0(2(1(1(1(1(1(1(2(0(0(0(2(2(x1))))))))))))))))))))))) → 0(0(2(1(2(2(0(0(1(2(2(1(2(1(0(1(0(1(1(1(1(2(1(0(0(0(0(x1))))))))))))))))))))))))))) (5)
0(1(1(0(2(0(2(1(0(2(1(2(0(1(2(1(0(1(1(0(0(0(1(x1))))))))))))))))))))))) → 0(1(1(0(0(0(1(0(2(0(0(1(2(0(0(1(0(1(0(1(0(0(1(0(0(2(0(x1))))))))))))))))))))))))))) (6)
0(1(1(1(1(2(0(2(1(0(0(2(2(2(1(0(2(1(0(1(0(2(2(x1))))))))))))))))))))))) → 0(2(1(0(0(0(0(2(1(2(2(1(0(0(1(2(0(0(0(2(0(1(2(0(0(0(0(x1))))))))))))))))))))))))))) (7)
0(1(1(2(2(0(0(0(1(0(1(1(1(2(1(1(1(0(0(1(1(2(2(x1))))))))))))))))))))))) → 0(0(0(0(0(1(2(2(0(1(1(1(0(2(2(2(2(2(2(0(0(1(0(0(2(1(0(x1))))))))))))))))))))))))))) (8)
0(1(2(2(0(1(2(0(0(2(0(2(2(1(1(1(0(0(0(0(1(2(2(x1))))))))))))))))))))))) → 0(0(1(0(1(2(1(0(2(1(0(0(2(0(0(2(2(2(0(2(2(0(0(1(1(0(1(x1))))))))))))))))))))))))))) (9)
0(2(0(1(0(0(2(2(1(1(0(0(2(2(1(2(1(1(1(0(2(1(2(x1))))))))))))))))))))))) → 0(2(0(2(0(1(0(0(0(1(0(0(2(2(2(1(0(1(2(0(0(1(2(0(0(0(0(x1))))))))))))))))))))))))))) (10)
0(2(0(1(0(2(0(1(1(1(2(2(0(1(0(1(0(1(0(1(0(1(1(x1))))))))))))))))))))))) → 0(0(0(2(1(0(0(2(0(1(0(2(0(1(0(2(1(0(2(0(0(1(2(2(2(0(1(x1))))))))))))))))))))))))))) (11)
0(2(1(1(1(1(0(0(0(2(0(1(2(1(0(1(2(1(2(0(2(0(2(x1))))))))))))))))))))))) → 0(1(2(2(0(1(0(2(2(1(0(1(0(1(0(0(2(1(0(0(2(0(2(1(2(0(0(x1))))))))))))))))))))))))))) (12)
0(2(1(1(2(2(0(2(2(1(2(1(0(1(1(0(0(1(0(2(0(1(0(x1))))))))))))))))))))))) → 0(0(0(2(0(2(1(1(0(0(0(1(0(2(1(2(0(0(2(0(1(2(1(0(0(2(1(x1))))))))))))))))))))))))))) (13)
1(0(0(0(2(2(2(1(1(0(2(0(2(0(0(0(1(0(1(0(1(2(0(x1))))))))))))))))))))))) → 1(0(0(0(1(2(1(0(0(2(0(0(0(0(2(2(2(2(1(0(1(0(0(2(0(0(1(x1))))))))))))))))))))))))))) (14)
1(0(0(1(2(2(2(1(2(0(0(2(1(0(2(1(1(2(1(1(2(0(2(x1))))))))))))))))))))))) → 1(0(0(2(1(0(2(0(0(0(0(0(1(1(1(2(1(0(2(0(2(0(0(0(2(0(2(x1))))))))))))))))))))))))))) (15)
1(0(0(2(0(1(2(0(2(2(0(0(1(2(2(0(1(0(2(2(0(1(1(x1))))))))))))))))))))))) → 1(0(0(1(0(1(0(0(2(0(0(1(0(2(1(2(1(0(2(0(0(0(2(2(2(0(0(x1))))))))))))))))))))))))))) (16)
1(0(0(2(1(2(0(1(0(1(2(0(1(1(1(0(2(2(1(0(1(2(2(x1))))))))))))))))))))))) → 1(0(2(1(2(1(0(2(2(0(0(2(1(0(1(1(0(2(1(0(2(0(1(0(2(2(2(x1))))))))))))))))))))))))))) (17)
1(0(2(2(0(0(1(1(2(0(1(0(1(0(0(2(2(1(1(2(1(1(2(x1))))))))))))))))))))))) → 1(0(1(0(2(2(1(1(0(2(0(2(1(0(1(0(1(0(1(1(2(1(2(1(0(0(2(x1))))))))))))))))))))))))))) (18)
1(1(0(2(2(2(1(1(2(0(0(2(2(0(0(0(0(1(0(0(1(2(1(x1))))))))))))))))))))))) → 1(0(1(0(2(0(0(1(0(1(0(1(0(0(1(0(0(0(0(0(0(2(2(0(2(1(0(x1))))))))))))))))))))))))))) (19)
1(1(1(0(1(2(2(0(2(0(2(1(0(0(1(0(0(2(0(2(1(1(2(x1))))))))))))))))))))))) → 2(0(0(0(1(0(0(1(0(1(1(0(0(0(2(2(0(0(0(0(0(0(0(2(0(0(2(x1))))))))))))))))))))))))))) (20)
1(1(2(1(0(0(1(2(0(2(1(0(1(2(2(1(1(0(1(2(1(2(2(x1))))))))))))))))))))))) → 1(0(1(0(2(2(2(2(0(1(2(0(0(0(0(0(0(2(0(0(1(1(0(2(2(0(0(x1))))))))))))))))))))))))))) (21)
1(2(0(2(2(0(1(1(0(0(1(1(2(0(0(0(2(2(2(1(1(0(0(x1))))))))))))))))))))))) → 1(1(0(1(2(1(0(0(2(1(0(0(1(0(2(0(1(1(2(0(1(0(2(0(2(1(0(x1))))))))))))))))))))))))))) (22)
1(2(1(0(0(1(0(1(1(2(0(2(1(1(1(1(1(2(0(0(2(1(0(x1))))))))))))))))))))))) → 1(2(1(0(0(0(0(0(0(2(1(0(1(1(0(0(1(1(2(2(2(1(0(2(0(2(0(x1))))))))))))))))))))))))))) (23)
2(0(0(1(0(1(1(2(0(1(1(2(1(0(0(0(1(1(2(1(2(1(0(x1))))))))))))))))))))))) → 2(1(2(1(0(1(0(0(1(0(0(0(0(0(2(1(2(0(1(0(0(0(0(2(1(0(1(x1))))))))))))))))))))))))))) (24)
2(0(0(2(0(0(1(1(1(2(0(2(0(0(2(0(0(2(1(0(2(2(0(x1))))))))))))))))))))))) → 0(0(2(1(1(0(2(0(0(0(2(1(2(0(0(0(2(1(0(0(0(0(0(2(0(0(1(x1))))))))))))))))))))))))))) (25)
2(0(1(0(0(0(1(1(2(2(1(2(2(0(0(1(1(2(1(1(0(2(1(x1))))))))))))))))))))))) → 2(2(2(0(2(0(0(1(0(0(0(0(0(0(0(2(2(0(0(1(2(2(0(0(0(2(0(x1))))))))))))))))))))))))))) (26)
2(0(1(1(1(0(0(0(2(2(2(0(2(0(2(2(0(1(0(1(0(2(0(x1))))))))))))))))))))))) → 0(1(0(2(0(0(2(1(1(0(1(2(1(0(0(1(0(2(0(0(0(0(2(1(1(0(1(x1))))))))))))))))))))))))))) (27)
2(0(1(1(2(1(0(0(1(1(2(1(0(1(1(1(0(2(2(1(0(2(0(x1))))))))))))))))))))))) → 2(2(0(2(1(1(0(1(0(0(1(2(0(0(0(2(0(2(1(0(0(0(0(2(2(0(0(x1))))))))))))))))))))))))))) (28)
2(0(1(1(2(1(1(1(0(2(1(0(1(1(2(2(1(0(2(0(2(0(0(x1))))))))))))))))))))))) → 0(0(1(2(1(0(1(0(1(1(0(0(2(0(0(2(1(0(2(1(0(1(2(1(2(0(0(x1))))))))))))))))))))))))))) (29)
2(0(1(2(0(0(0(0(2(0(1(0(1(2(0(0(1(1(2(1(1(1(0(x1))))))))))))))))))))))) → 2(0(2(2(1(2(0(1(1(2(2(1(0(0(1(0(0(0(0(0(2(1(0(0(2(0(0(x1))))))))))))))))))))))))))) (30)
2(1(0(1(2(0(2(1(1(2(0(1(2(0(2(2(0(0(2(1(2(1(2(x1))))))))))))))))))))))) → 0(2(0(0(0(1(0(2(0(0(2(0(0(1(0(1(1(2(0(0(2(2(0(0(2(0(2(x1))))))))))))))))))))))))))) (31)
2(1(1(2(0(1(2(0(0(0(2(1(0(1(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))) → 0(2(1(0(0(0(0(1(0(1(2(2(0(2(1(0(0(0(1(0(2(2(1(0(0(1(1(x1))))))))))))))))))))))))))) (32)
2(1(1(2(0(2(2(1(0(2(1(2(2(0(0(0(1(0(2(0(1(0(0(x1))))))))))))))))))))))) → 2(2(1(0(2(0(0(2(0(0(1(1(0(0(0(2(0(0(0(1(2(1(0(1(0(0(0(x1))))))))))))))))))))))))))) (33)
2(1(1(2(1(1(2(2(2(0(2(0(0(0(1(0(1(2(2(1(0(0(0(x1))))))))))))))))))))))) → 0(0(2(1(2(2(1(0(0(2(0(0(2(0(1(0(0(0(0(0(2(0(0(1(1(0(1(x1))))))))))))))))))))))))))) (34)
2(1(2(1(1(0(2(1(0(2(1(0(0(2(1(1(1(1(1(2(1(2(1(x1))))))))))))))))))))))) → 0(0(0(2(2(0(1(1(2(1(0(2(0(0(0(1(0(2(1(0(1(0(0(2(1(1(1(x1))))))))))))))))))))))))))) (35)
2(2(0(0(2(0(0(1(0(0(1(2(2(1(1(1(0(0(1(2(2(0(1(x1))))))))))))))))))))))) → 0(1(0(1(2(1(0(2(2(1(0(0(0(2(0(2(1(0(0(2(1(0(1(2(2(1(1(x1))))))))))))))))))))))))))) (36)
2(2(1(0(0(0(2(1(0(1(1(2(0(1(0(1(0(0(1(0(2(2(2(x1))))))))))))))))))))))) → 2(0(0(0(2(2(1(2(0(1(0(1(0(0(0(0(0(0(1(0(1(0(0(0(1(1(2(x1))))))))))))))))))))))))))) (37)
2(2(1(1(0(0(2(1(2(1(2(0(0(2(0(2(1(0(2(2(1(0(1(x1))))))))))))))))))))))) → 2(1(0(1(1(0(0(2(0(0(0(2(1(0(2(2(0(0(0(1(1(1(0(0(0(2(1(x1))))))))))))))))))))))))))) (38)
2(2(1(2(0(1(0(2(1(2(1(1(0(2(1(1(0(1(2(0(0(0(1(x1))))))))))))))))))))))) → 2(2(0(0(0(2(2(0(0(0(1(0(2(2(2(1(0(1(0(1(1(0(2(0(0(0(0(x1))))))))))))))))))))))))))) (39)
2(2(2(1(1(1(1(0(1(1(0(2(2(0(0(0(2(0(0(0(1(1(0(x1))))))))))))))))))))))) → 0(1(0(1(2(1(0(0(1(0(0(1(2(1(0(0(0(1(2(1(2(2(1(2(0(0(1(x1))))))))))))))))))))))))))) (40)

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(☐)}

We obtain the transformed TRS
0(0(0(0(1(1(2(0(0(0(2(1(2(1(1(1(0(0(1(2(1(1(1(x1))))))))))))))))))))))) → 0(2(1(1(0(0(2(1(2(1(0(0(0(1(0(0(2(0(2(2(0(0(0(1(0(0(2(x1))))))))))))))))))))))))))) (1)
0(0(0(2(0(1(0(1(1(1(0(0(2(2(1(0(2(1(1(0(1(1(2(x1))))))))))))))))))))))) → 0(2(1(2(1(0(0(2(1(0(2(2(0(0(2(1(1(1(0(1(2(1(0(0(0(0(1(x1))))))))))))))))))))))))))) (2)
0(0(1(2(2(0(1(1(1(2(0(1(2(1(1(2(0(2(0(2(1(0(2(x1))))))))))))))))))))))) → 0(2(1(0(1(2(0(0(0(1(1(2(0(1(1(2(0(1(0(0(0(0(0(2(0(0(0(x1))))))))))))))))))))))))))) (3)
0(0(2(2(0(1(2(2(0(2(2(0(2(2(0(2(1(0(0(2(0(0(2(x1))))))))))))))))))))))) → 0(2(0(2(0(0(2(1(1(1(1(0(2(1(0(0(2(0(0(0(0(0(0(0(2(0(1(x1))))))))))))))))))))))))))) (4)
0(1(0(0(0(1(1(2(2(0(2(1(1(1(1(1(1(2(0(0(0(2(2(x1))))))))))))))))))))))) → 0(0(2(1(2(2(0(0(1(2(2(1(2(1(0(1(0(1(1(1(1(2(1(0(0(0(0(x1))))))))))))))))))))))))))) (5)
0(1(1(0(2(0(2(1(0(2(1(2(0(1(2(1(0(1(1(0(0(0(1(x1))))))))))))))))))))))) → 0(1(1(0(0(0(1(0(2(0(0(1(2(0(0(1(0(1(0(1(0(0(1(0(0(2(0(x1))))))))))))))))))))))))))) (6)
0(1(1(1(1(2(0(2(1(0(0(2(2(2(1(0(2(1(0(1(0(2(2(x1))))))))))))))))))))))) → 0(2(1(0(0(0(0(2(1(2(2(1(0(0(1(2(0(0(0(2(0(1(2(0(0(0(0(x1))))))))))))))))))))))))))) (7)
0(1(1(2(2(0(0(0(1(0(1(1(1(2(1(1(1(0(0(1(1(2(2(x1))))))))))))))))))))))) → 0(0(0(0(0(1(2(2(0(1(1(1(0(2(2(2(2(2(2(0(0(1(0(0(2(1(0(x1))))))))))))))))))))))))))) (8)
0(1(2(2(0(1(2(0(0(2(0(2(2(1(1(1(0(0(0(0(1(2(2(x1))))))))))))))))))))))) → 0(0(1(0(1(2(1(0(2(1(0(0(2(0(0(2(2(2(0(2(2(0(0(1(1(0(1(x1))))))))))))))))))))))))))) (9)
0(2(0(1(0(0(2(2(1(1(0(0(2(2(1(2(1(1(1(0(2(1(2(x1))))))))))))))))))))))) → 0(2(0(2(0(1(0(0(0(1(0(0(2(2(2(1(0(1(2(0(0(1(2(0(0(0(0(x1))))))))))))))))))))))))))) (10)
0(2(0(1(0(2(0(1(1(1(2(2(0(1(0(1(0(1(0(1(0(1(1(x1))))))))))))))))))))))) → 0(0(0(2(1(0(0(2(0(1(0(2(0(1(0(2(1(0(2(0(0(1(2(2(2(0(1(x1))))))))))))))))))))))))))) (11)
0(2(1(1(1(1(0(0(0(2(0(1(2(1(0(1(2(1(2(0(2(0(2(x1))))))))))))))))))))))) → 0(1(2(2(0(1(0(2(2(1(0(1(0(1(0(0(2(1(0(0(2(0(2(1(2(0(0(x1))))))))))))))))))))))))))) (12)
0(2(1(1(2(2(0(2(2(1(2(1(0(1(1(0(0(1(0(2(0(1(0(x1))))))))))))))))))))))) → 0(0(0(2(0(2(1(1(0(0(0(1(0(2(1(2(0(0(2(0(1(2(1(0(0(2(1(x1))))))))))))))))))))))))))) (13)
1(0(0(0(2(2(2(1(1(0(2(0(2(0(0(0(1(0(1(0(1(2(0(x1))))))))))))))))))))))) → 1(0(0(0(1(2(1(0(0(2(0(0(0(0(2(2(2(2(1(0(1(0(0(2(0(0(1(x1))))))))))))))))))))))))))) (14)
1(0(0(1(2(2(2(1(2(0(0(2(1(0(2(1(1(2(1(1(2(0(2(x1))))))))))))))))))))))) → 1(0(0(2(1(0(2(0(0(0(0(0(1(1(1(2(1(0(2(0(2(0(0(0(2(0(2(x1))))))))))))))))))))))))))) (15)
1(0(0(2(0(1(2(0(2(2(0(0(1(2(2(0(1(0(2(2(0(1(1(x1))))))))))))))))))))))) → 1(0(0(1(0(1(0(0(2(0(0(1(0(2(1(2(1(0(2(0(0(0(2(2(2(0(0(x1))))))))))))))))))))))))))) (16)
1(0(0(2(1(2(0(1(0(1(2(0(1(1(1(0(2(2(1(0(1(2(2(x1))))))))))))))))))))))) → 1(0(2(1(2(1(0(2(2(0(0(2(1(0(1(1(0(2(1(0(2(0(1(0(2(2(2(x1))))))))))))))))))))))))))) (17)
1(0(2(2(0(0(1(1(2(0(1(0(1(0(0(2(2(1(1(2(1(1(2(x1))))))))))))))))))))))) → 1(0(1(0(2(2(1(1(0(2(0(2(1(0(1(0(1(0(1(1(2(1(2(1(0(0(2(x1))))))))))))))))))))))))))) (18)
1(1(0(2(2(2(1(1(2(0(0(2(2(0(0(0(0(1(0(0(1(2(1(x1))))))))))))))))))))))) → 1(0(1(0(2(0(0(1(0(1(0(1(0(0(1(0(0(0(0(0(0(2(2(0(2(1(0(x1))))))))))))))))))))))))))) (19)
1(1(2(1(0(0(1(2(0(2(1(0(1(2(2(1(1(0(1(2(1(2(2(x1))))))))))))))))))))))) → 1(0(1(0(2(2(2(2(0(1(2(0(0(0(0(0(0(2(0(0(1(1(0(2(2(0(0(x1))))))))))))))))))))))))))) (21)
1(2(0(2(2(0(1(1(0(0(1(1(2(0(0(0(2(2(2(1(1(0(0(x1))))))))))))))))))))))) → 1(1(0(1(2(1(0(0(2(1(0(0(1(0(2(0(1(1(2(0(1(0(2(0(2(1(0(x1))))))))))))))))))))))))))) (22)
1(2(1(0(0(1(0(1(1(2(0(2(1(1(1(1(1(2(0(0(2(1(0(x1))))))))))))))))))))))) → 1(2(1(0(0(0(0(0(0(2(1(0(1(1(0(0(1(1(2(2(2(1(0(2(0(2(0(x1))))))))))))))))))))))))))) (23)
2(0(0(1(0(1(1(2(0(1(1(2(1(0(0(0(1(1(2(1(2(1(0(x1))))))))))))))))))))))) → 2(1(2(1(0(1(0(0(1(0(0(0(0(0(2(1(2(0(1(0(0(0(0(2(1(0(1(x1))))))))))))))))))))))))))) (24)
2(0(1(0(0(0(1(1(2(2(1(2(2(0(0(1(1(2(1(1(0(2(1(x1))))))))))))))))))))))) → 2(2(2(0(2(0(0(1(0(0(0(0(0(0(0(2(2(0(0(1(2(2(0(0(0(2(0(x1))))))))))))))))))))))))))) (26)
2(0(1(1(2(1(0(0(1(1(2(1(0(1(1(1(0(2(2(1(0(2(0(x1))))))))))))))))))))))) → 2(2(0(2(1(1(0(1(0(0(1(2(0(0(0(2(0(2(1(0(0(0(0(2(2(0(0(x1))))))))))))))))))))))))))) (28)
2(0(1(2(0(0(0(0(2(0(1(0(1(2(0(0(1(1(2(1(1(1(0(x1))))))))))))))))))))))) → 2(0(2(2(1(2(0(1(1(2(2(1(0(0(1(0(0(0(0(0(2(1(0(0(2(0(0(x1))))))))))))))))))))))))))) (30)
2(1(1(2(0(2(2(1(0(2(1(2(2(0(0(0(1(0(2(0(1(0(0(x1))))))))))))))))))))))) → 2(2(1(0(2(0(0(2(0(0(1(1(0(0(0(2(0(0(0(1(2(1(0(1(0(0(0(x1))))))))))))))))))))))))))) (33)
2(2(1(0(0(0(2(1(0(1(1(2(0(1(0(1(0(0(1(0(2(2(2(x1))))))))))))))))))))))) → 2(0(0(0(2(2(1(2(0(1(0(1(0(0(0(0(0(0(1(0(1(0(0(0(1(1(2(x1))))))))))))))))))))))))))) (37)
2(2(1(1(0(0(2(1(2(1(2(0(0(2(0(2(1(0(2(2(1(0(1(x1))))))))))))))))))))))) → 2(1(0(1(1(0(0(2(0(0(0(2(1(0(2(2(0(0(0(1(1(1(0(0(0(2(1(x1))))))))))))))))))))))))))) (38)
2(2(1(2(0(1(0(2(1(2(1(1(0(2(1(1(0(1(2(0(0(0(1(x1))))))))))))))))))))))) → 2(2(0(0(0(2(2(0(0(0(1(0(2(2(2(1(0(1(0(1(1(0(2(0(0(0(0(x1))))))))))))))))))))))))))) (39)
0(1(1(1(0(1(2(2(0(2(0(2(1(0(0(1(0(0(2(0(2(1(1(2(x1)))))))))))))))))))))))) → 0(2(0(0(0(1(0(0(1(0(1(1(0(0(0(2(2(0(0(0(0(0(0(0(2(0(0(2(x1)))))))))))))))))))))))))))) (41)
1(1(1(1(0(1(2(2(0(2(0(2(1(0(0(1(0(0(2(0(2(1(1(2(x1)))))))))))))))))))))))) → 1(2(0(0(0(1(0(0(1(0(1(1(0(0(0(2(2(0(0(0(0(0(0(0(2(0(0(2(x1)))))))))))))))))))))))))))) (42)
2(1(1(1(0(1(2(2(0(2(0(2(1(0(0(1(0(0(2(0(2(1(1(2(x1)))))))))))))))))))))))) → 2(2(0(0(0(1(0(0(1(0(1(1(0(0(0(2(2(0(0(0(0(0(0(0(2(0(0(2(x1)))))))))))))))))))))))))))) (43)
0(2(0(0(2(0(0(1(1(1(2(0(2(0(0(2(0(0(2(1(0(2(2(0(x1)))))))))))))))))))))))) → 0(0(0(2(1(1(0(2(0(0(0(2(1(2(0(0(0(2(1(0(0(0(0(0(2(0(0(1(x1)))))))))))))))))))))))))))) (44)
1(2(0(0(2(0(0(1(1(1(2(0(2(0(0(2(0(0(2(1(0(2(2(0(x1)))))))))))))))))))))))) → 1(0(0(2(1(1(0(2(0(0(0(2(1(2(0(0(0(2(1(0(0(0(0(0(2(0(0(1(x1)))))))))))))))))))))))))))) (45)
2(2(0(0(2(0(0(1(1(1(2(0(2(0(0(2(0(0(2(1(0(2(2(0(x1)))))))))))))))))))))))) → 2(0(0(2(1(1(0(2(0(0(0(2(1(2(0(0(0(2(1(0(0(0(0(0(2(0(0(1(x1)))))))))))))))))))))))))))) (46)
0(2(0(1(1(1(0(0(0(2(2(2(0(2(0(2(2(0(1(0(1(0(2(0(x1)))))))))))))))))))))))) → 0(0(1(0(2(0(0(2(1(1(0(1(2(1(0(0(1(0(2(0(0(0(0(2(1(1(0(1(x1)))))))))))))))))))))))))))) (47)
1(2(0(1(1(1(0(0(0(2(2(2(0(2(0(2(2(0(1(0(1(0(2(0(x1)))))))))))))))))))))))) → 1(0(1(0(2(0(0(2(1(1(0(1(2(1(0(0(1(0(2(0(0(0(0(2(1(1(0(1(x1)))))))))))))))))))))))))))) (48)
2(2(0(1(1(1(0(0(0(2(2(2(0(2(0(2(2(0(1(0(1(0(2(0(x1)))))))))))))))))))))))) → 2(0(1(0(2(0(0(2(1(1(0(1(2(1(0(0(1(0(2(0(0(0(0(2(1(1(0(1(x1)))))))))))))))))))))))))))) (49)
0(2(0(1(1(2(1(1(1(0(2(1(0(1(1(2(2(1(0(2(0(2(0(0(x1)))))))))))))))))))))))) → 0(0(0(1(2(1(0(1(0(1(1(0(0(2(0(0(2(1(0(2(1(0(1(2(1(2(0(0(x1)))))))))))))))))))))))))))) (50)
1(2(0(1(1(2(1(1(1(0(2(1(0(1(1(2(2(1(0(2(0(2(0(0(x1)))))))))))))))))))))))) → 1(0(0(1(2(1(0(1(0(1(1(0(0(2(0(0(2(1(0(2(1(0(1(2(1(2(0(0(x1)))))))))))))))))))))))))))) (51)
2(2(0(1(1(2(1(1(1(0(2(1(0(1(1(2(2(1(0(2(0(2(0(0(x1)))))))))))))))))))))))) → 2(0(0(1(2(1(0(1(0(1(1(0(0(2(0(0(2(1(0(2(1(0(1(2(1(2(0(0(x1)))))))))))))))))))))))))))) (52)
0(2(1(0(1(2(0(2(1(1(2(0(1(2(0(2(2(0(0(2(1(2(1(2(x1)))))))))))))))))))))))) → 0(0(2(0(0(0(1(0(2(0(0(2(0(0(1(0(1(1(2(0(0(2(2(0(0(2(0(2(x1)))))))))))))))))))))))))))) (53)
1(2(1(0(1(2(0(2(1(1(2(0(1(2(0(2(2(0(0(2(1(2(1(2(x1)))))))))))))))))))))))) → 1(0(2(0(0(0(1(0(2(0(0(2(0(0(1(0(1(1(2(0(0(2(2(0(0(2(0(2(x1)))))))))))))))))))))))))))) (54)
2(2(1(0(1(2(0(2(1(1(2(0(1(2(0(2(2(0(0(2(1(2(1(2(x1)))))))))))))))))))))))) → 2(0(2(0(0(0(1(0(2(0(0(2(0(0(1(0(1(1(2(0(0(2(2(0(0(2(0(2(x1)))))))))))))))))))))))))))) (55)
0(2(1(1(2(0(1(2(0(0(0(2(1(0(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))) → 0(0(2(1(0(0(0(0(1(0(1(2(2(0(2(1(0(0(0(1(0(2(2(1(0(0(1(1(x1)))))))))))))))))))))))))))) (56)
1(2(1(1(2(0(1(2(0(0(0(2(1(0(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))) → 1(0(2(1(0(0(0(0(1(0(1(2(2(0(2(1(0(0(0(1(0(2(2(1(0(0(1(1(x1)))))))))))))))))))))))))))) (57)
2(2(1(1(2(0(1(2(0(0(0(2(1(0(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))) → 2(0(2(1(0(0(0(0(1(0(1(2(2(0(2(1(0(0(0(1(0(2(2(1(0(0(1(1(x1)))))))))))))))))))))))))))) (58)
0(2(1(1(2(1(1(2(2(2(0(2(0(0(0(1(0(1(2(2(1(0(0(0(x1)))))))))))))))))))))))) → 0(0(0(2(1(2(2(1(0(0(2(0(0(2(0(1(0(0(0(0(0(2(0(0(1(1(0(1(x1)))))))))))))))))))))))))))) (59)
1(2(1(1(2(1(1(2(2(2(0(2(0(0(0(1(0(1(2(2(1(0(0(0(x1)))))))))))))))))))))))) → 1(0(0(2(1(2(2(1(0(0(2(0(0(2(0(1(0(0(0(0(0(2(0(0(1(1(0(1(x1)))))))))))))))))))))))))))) (60)
2(2(1(1(2(1(1(2(2(2(0(2(0(0(0(1(0(1(2(2(1(0(0(0(x1)))))))))))))))))))))))) → 2(0(0(2(1(2(2(1(0(0(2(0(0(2(0(1(0(0(0(0(0(2(0(0(1(1(0(1(x1)))))))))))))))))))))))))))) (61)
0(2(1(2(1(1(0(2(1(0(2(1(0(0(2(1(1(1(1(1(2(1(2(1(x1)))))))))))))))))))))))) → 0(0(0(0(2(2(0(1(1(2(1(0(2(0(0(0(1(0(2(1(0(1(0(0(2(1(1(1(x1)))))))))))))))))))))))))))) (62)
1(2(1(2(1(1(0(2(1(0(2(1(0(0(2(1(1(1(1(1(2(1(2(1(x1)))))))))))))))))))))))) → 1(0(0(0(2(2(0(1(1(2(1(0(2(0(0(0(1(0(2(1(0(1(0(0(2(1(1(1(x1)))))))))))))))))))))))))))) (63)
2(2(1(2(1(1(0(2(1(0(2(1(0(0(2(1(1(1(1(1(2(1(2(1(x1)))))))))))))))))))))))) → 2(0(0(0(2(2(0(1(1(2(1(0(2(0(0(0(1(0(2(1(0(1(0(0(2(1(1(1(x1)))))))))))))))))))))))))))) (64)
0(2(2(0(0(2(0(0(1(0(0(1(2(2(1(1(1(0(0(1(2(2(0(1(x1)))))))))))))))))))))))) → 0(0(1(0(1(2(1(0(2(2(1(0(0(0(2(0(2(1(0(0(2(1(0(1(2(2(1(1(x1)))))))))))))))))))))))))))) (65)
1(2(2(0(0(2(0(0(1(0(0(1(2(2(1(1(1(0(0(1(2(2(0(1(x1)))))))))))))))))))))))) → 1(0(1(0(1(2(1(0(2(2(1(0(0(0(2(0(2(1(0(0(2(1(0(1(2(2(1(1(x1)))))))))))))))))))))))))))) (66)
2(2(2(0(0(2(0(0(1(0(0(1(2(2(1(1(1(0(0(1(2(2(0(1(x1)))))))))))))))))))))))) → 2(0(1(0(1(2(1(0(2(2(1(0(0(0(2(0(2(1(0(0(2(1(0(1(2(2(1(1(x1)))))))))))))))))))))))))))) (67)
0(2(2(2(1(1(1(1(0(1(1(0(2(2(0(0(0(2(0(0(0(1(1(0(x1)))))))))))))))))))))))) → 0(0(1(0(1(2(1(0(0(1(0(0(1(2(1(0(0(0(1(2(1(2(2(1(2(0(0(1(x1)))))))))))))))))))))))))))) (68)
1(2(2(2(1(1(1(1(0(1(1(0(2(2(0(0(0(2(0(0(0(1(1(0(x1)))))))))))))))))))))))) → 1(0(1(0(1(2(1(0(0(1(0(0(1(2(1(0(0(0(1(2(1(2(2(1(2(0(0(1(x1)))))))))))))))))))))))))))) (69)
2(2(2(2(1(1(1(1(0(1(1(0(2(2(0(0(0(2(0(0(0(1(1(0(x1)))))))))))))))))))))))) → 2(0(1(0(1(2(1(0(0(1(0(0(1(2(1(0(0(0(1(2(1(2(2(1(2(0(0(1(x1)))))))))))))))))))))))))))) (70)

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 180 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 + 2
[11(x1)] = 1 · x1 + 3
[12(x1)] = 1 · x1 + 3
[20(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1 + 2
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 2
all of the following rules can be deleted.

There are 170 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
[20(x1)] = 1 · x1 + 2
[01(x1)] = 1 · x1 + 1
[10(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 4
[22(x1)] = 1 · x1 + 3
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 3
all of the following rules can be deleted.
00(00(02(20(01(10(01(11(11(10(00(02(22(21(10(02(21(11(10(01(11(12(21(x1))))))))))))))))))))))) → 02(21(12(21(10(00(02(21(10(02(22(20(00(02(21(11(11(10(01(12(21(10(00(00(00(01(11(x1))))))))))))))))))))))))))) (75)
00(02(22(20(01(12(22(20(02(22(20(02(22(20(02(21(10(00(02(20(00(02(21(x1))))))))))))))))))))))) → 02(20(02(20(00(02(21(11(11(11(10(02(21(10(00(02(20(00(00(00(00(00(00(02(20(01(11(x1))))))))))))))))))))))))))) (81)
01(12(22(20(01(12(20(00(02(20(02(22(21(11(11(10(00(00(00(01(12(22(21(x1))))))))))))))))))))))) → 00(01(10(01(12(21(10(02(21(10(00(02(20(00(02(22(22(20(02(22(20(00(01(11(10(01(11(x1))))))))))))))))))))))))))) (96)
02(20(01(10(00(02(22(21(11(10(00(02(22(21(12(21(11(11(10(02(21(12(21(x1))))))))))))))))))))))) → 02(20(02(20(01(10(00(00(01(10(00(02(22(22(21(10(01(12(20(00(01(12(20(00(00(00(01(x1))))))))))))))))))))))))))) (99)

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
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1
all of the following rules can be deleted.
22(21(10(00(00(02(21(10(01(11(12(20(01(10(01(10(00(01(10(02(22(22(20(x1))))))))))))))))))))))) → 20(00(00(02(22(21(12(20(01(10(01(10(00(00(00(00(00(01(10(01(10(00(00(01(11(12(20(x1))))))))))))))))))))))))))) (152)
22(21(10(00(00(02(21(10(01(11(12(20(01(10(01(10(00(01(10(02(22(22(21(x1))))))))))))))))))))))) → 20(00(00(02(22(21(12(20(01(10(01(10(00(00(00(00(00(01(10(01(10(00(00(01(11(12(21(x1))))))))))))))))))))))))))) (153)
22(21(10(00(00(02(21(10(01(11(12(20(01(10(01(10(00(01(10(02(22(22(22(x1))))))))))))))))))))))) → 20(00(00(02(22(21(12(20(01(10(01(10(00(00(00(00(00(01(10(01(10(00(00(01(11(12(22(x1))))))))))))))))))))))))))) (154)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[22(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
all of the following rules can be deleted.
22(21(11(10(00(02(21(12(21(12(20(00(02(20(02(21(10(02(22(21(10(01(10(x1))))))))))))))))))))))) → 21(10(01(11(10(00(02(20(00(00(02(21(10(02(22(20(00(00(01(11(11(10(00(00(02(21(10(x1))))))))))))))))))))))))))) (155)
22(21(11(10(00(02(21(12(21(12(20(00(02(20(02(21(10(02(22(21(10(01(11(x1))))))))))))))))))))))) → 21(10(01(11(10(00(02(20(00(00(02(21(10(02(22(20(00(00(01(11(11(10(00(00(02(21(11(x1))))))))))))))))))))))))))) (156)
22(21(11(10(00(02(21(12(21(12(20(00(02(20(02(21(10(02(22(21(10(01(12(x1))))))))))))))))))))))) → 21(10(01(11(10(00(02(20(00(00(02(21(10(02(22(20(00(00(01(11(11(10(00(00(02(21(12(x1))))))))))))))))))))))))))) (157)

1.1.1.1.1.1.1 R is empty

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