Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/140639)

The rewrite relation of the following TRS is considered.

0(0(0(0(0(0(1(1(1(2(2(2(1(x1))))))))))))) → 0(1(1(0(0(1(1(1(0(1(1(1(1(1(1(2(1(x1))))))))))))))))) (1)
0(0(0(1(1(1(2(0(2(0(1(1(1(x1))))))))))))) → 1(1(1(1(2(1(1(0(1(1(1(0(1(1(1(0(1(x1))))))))))))))))) (2)
0(0(1(0(0(1(2(2(1(0(1(0(1(x1))))))))))))) → 0(0(1(1(1(0(0(0(1(0(0(1(1(2(1(1(1(x1))))))))))))))))) (3)
0(0(1(0(1(0(1(1(2(0(1(2(1(x1))))))))))))) → 0(0(0(0(1(1(1(2(1(0(0(0(0(0(1(0(1(x1))))))))))))))))) (4)
0(0(1(1(1(1(0(0(0(0(2(2(2(x1))))))))))))) → 0(1(0(0(1(1(1(0(0(1(1(0(2(0(0(1(1(x1))))))))))))))))) (5)
0(0(1(1(2(2(0(0(0(1(2(1(2(x1))))))))))))) → 0(0(0(1(1(0(1(0(0(0(1(2(1(0(2(1(2(x1))))))))))))))))) (6)
0(0(1(2(1(1(1(1(0(2(0(1(2(x1))))))))))))) → 1(0(0(1(1(1(2(0(0(0(1(1(2(0(0(1(2(x1))))))))))))))))) (7)
0(0(1(2(2(2(0(1(0(0(0(0(1(x1))))))))))))) → 0(1(0(1(1(1(2(0(1(0(1(1(0(0(1(1(1(x1))))))))))))))))) (8)
0(0(2(1(0(0(1(0(1(0(1(0(0(x1))))))))))))) → 0(0(0(0(1(0(1(0(1(0(1(0(0(1(1(1(1(x1))))))))))))))))) (9)
0(1(0(0(1(0(0(2(0(1(0(1(2(x1))))))))))))) → 0(0(0(0(1(0(0(0(0(0(1(0(0(2(1(1(1(x1))))))))))))))))) (10)
0(1(0(1(0(2(0(1(1(0(1(0(2(x1))))))))))))) → 0(1(2(0(1(0(0(0(1(0(0(0(1(0(1(0(2(x1))))))))))))))))) (11)
0(1(0(2(2(0(0(1(0(1(0(0(1(x1))))))))))))) → 0(1(1(1(1(2(0(1(1(0(1(0(1(0(1(0(1(x1))))))))))))))))) (12)
0(1(1(0(1(0(2(1(1(0(0(2(1(x1))))))))))))) → 1(1(2(0(0(0(0(0(1(0(1(1(2(2(0(0(1(x1))))))))))))))))) (13)
0(1(1(0(1(1(1(1(0(2(0(1(0(x1))))))))))))) → 0(1(0(0(0(0(0(0(1(0(1(1(2(0(0(0(1(x1))))))))))))))))) (14)
0(1(1(1(2(1(0(0(1(1(2(2(1(x1))))))))))))) → 1(1(0(1(1(2(0(1(1(1(1(0(1(1(2(0(1(x1))))))))))))))))) (15)
0(1(2(0(1(1(2(0(0(1(0(1(0(x1))))))))))))) → 0(1(2(0(0(0(0(0(0(1(2(1(0(0(1(0(0(x1))))))))))))))))) (16)
0(1(2(0(2(1(1(2(2(2(1(0(1(x1))))))))))))) → 0(1(2(0(0(2(1(1(1(2(0(1(1(2(0(1(1(x1))))))))))))))))) (17)
0(1(2(2(0(0(2(0(0(1(0(0(0(x1))))))))))))) → 1(1(2(0(0(1(0(1(2(0(1(0(1(0(0(0(0(x1))))))))))))))))) (18)
0(2(1(0(0(0(1(1(2(1(0(1(1(x1))))))))))))) → 1(1(1(1(0(0(1(1(0(1(2(1(0(1(0(0(1(x1))))))))))))))))) (19)
0(2(1(0(1(0(1(0(0(0(2(0(1(x1))))))))))))) → 0(0(0(0(0(1(0(0(1(1(2(0(1(1(0(0(1(x1))))))))))))))))) (20)
0(2(2(0(0(0(0(1(1(1(0(1(1(x1))))))))))))) → 1(1(1(1(1(1(0(1(0(0(1(1(1(0(1(1(1(x1))))))))))))))))) (21)
1(0(0(0(0(0(1(0(2(0(1(2(0(x1))))))))))))) → 1(0(1(1(1(0(1(1(1(0(0(1(2(2(0(0(0(x1))))))))))))))))) (22)
1(0(0(1(0(1(2(0(1(0(1(2(0(x1))))))))))))) → 0(0(0(1(1(0(1(0(1(2(0(1(0(1(0(1(0(x1))))))))))))))))) (23)
1(0(0(1(1(0(0(2(2(0(1(0(1(x1))))))))))))) → 0(0(0(1(1(1(1(1(0(0(0(0(0(2(1(1(1(x1))))))))))))))))) (24)
1(0(1(0(1(0(1(1(1(0(1(2(2(x1))))))))))))) → 0(1(0(1(1(1(0(0(1(1(0(1(0(0(1(2(1(x1))))))))))))))))) (25)
1(0(1(1(2(0(1(2(2(0(0(0(0(x1))))))))))))) → 1(0(0(0(0(1(0(1(1(2(2(2(0(0(0(0(0(x1))))))))))))))))) (26)
1(0(1(2(2(0(1(1(0(2(0(1(1(x1))))))))))))) → 0(0(1(0(0(1(1(1(0(2(2(2(0(0(0(1(1(x1))))))))))))))))) (27)
1(1(0(0(0(0(1(1(1(0(0(2(1(x1))))))))))))) → 0(1(0(1(1(0(0(0(0(1(1(0(0(0(0(0(1(x1))))))))))))))))) (28)
1(1(0(0(1(0(1(1(1(0(1(0(2(x1))))))))))))) → 0(1(1(1(1(1(0(1(0(0(0(1(1(1(1(1(2(x1))))))))))))))))) (29)
1(1(0(1(0(0(1(2(1(0(2(1(1(x1))))))))))))) → 0(1(0(1(0(0(1(0(1(0(2(2(0(1(0(0(1(x1))))))))))))))))) (30)
1(1(0(1(1(0(2(0(0(1(0(1(0(x1))))))))))))) → 0(1(1(0(1(1(0(1(0(0(0(1(1(1(0(0(0(x1))))))))))))))))) (31)
1(1(1(1(2(0(1(1(2(0(0(0(0(x1))))))))))))) → 1(1(0(1(1(0(0(1(1(0(1(1(0(1(0(0(1(x1))))))))))))))))) (32)
1(1(1(2(0(0(0(0(1(1(2(1(1(x1))))))))))))) → 1(1(1(1(1(0(0(0(1(1(0(2(1(0(0(0(1(x1))))))))))))))))) (33)
1(1(2(1(0(1(1(1(2(1(0(0(1(x1))))))))))))) → 0(1(2(2(0(0(0(1(1(0(0(0(0(0(0(1(1(x1))))))))))))))))) (34)
1(2(0(0(1(0(1(1(1(0(1(0(0(x1))))))))))))) → 0(1(0(1(1(0(0(1(1(0(0(0(0(0(0(0(1(x1))))))))))))))))) (35)
1(2(0(0(1(1(0(1(1(0(0(0(2(x1))))))))))))) → 0(0(1(1(0(0(1(1(1(1(0(1(1(0(1(1(1(x1))))))))))))))))) (36)
1(2(1(0(1(1(1(0(1(1(0(0(2(x1))))))))))))) → 0(0(1(0(0(0(1(1(0(0(0(2(0(0(0(1(2(x1))))))))))))))))) (37)
1(2(1(2(2(0(1(1(1(0(1(0(0(x1))))))))))))) → 0(1(2(1(0(1(0(0(2(1(1(0(0(1(1(0(0(x1))))))))))))))))) (38)
1(2(2(2(1(1(0(1(1(0(2(0(0(x1))))))))))))) → 1(0(0(0(0(0(1(2(2(1(2(1(0(0(2(0(0(x1))))))))))))))))) (39)
1(2(2(2(1(2(0(0(0(1(2(1(0(x1))))))))))))) → 1(1(1(0(0(2(1(1(0(2(0(1(1(2(0(1(0(x1))))))))))))))))) (40)
2(0(0(0(0(1(2(1(2(0(1(2(0(x1))))))))))))) → 2(0(0(1(1(0(1(0(0(2(1(0(1(1(2(0(0(x1))))))))))))))))) (41)
2(0(0(1(1(2(0(1(0(1(1(0(1(x1))))))))))))) → 0(1(0(0(0(2(0(0(0(1(0(1(0(1(1(0(1(x1))))))))))))))))) (42)
2(0(1(0(0(1(0(0(1(2(1(0(0(x1))))))))))))) → 1(1(1(0(0(0(1(0(0(1(0(0(0(1(0(1(0(x1))))))))))))))))) (43)
2(0(1(0(0(2(1(1(0(2(1(2(2(x1))))))))))))) → 2(1(1(0(0(1(1(0(1(1(1(0(2(1(2(2(2(x1))))))))))))))))) (44)
2(0(1(0(2(0(1(1(1(1(0(0(1(x1))))))))))))) → 0(2(1(1(1(0(1(0(1(0(1(1(1(1(1(0(1(x1))))))))))))))))) (45)
2(0(1(1(0(0(0(1(0(0(2(2(1(x1))))))))))))) → 2(0(0(0(1(1(2(1(0(0(0(0(1(2(0(0(1(x1))))))))))))))))) (46)
2(0(1(1(0(0(1(1(1(1(2(0(0(x1))))))))))))) → 2(0(0(1(1(0(0(0(0(1(0(1(0(0(0(0(0(x1))))))))))))))))) (47)
2(1(2(2(0(0(0(1(0(1(0(1(0(x1))))))))))))) → 2(1(1(1(0(1(0(0(0(0(0(1(2(1(1(1(0(x1))))))))))))))))) (48)
2(2(0(0(0(1(1(0(1(1(1(0(0(x1))))))))))))) → 1(0(1(1(0(0(1(1(0(0(0(0(0(0(1(1(1(x1))))))))))))))))) (49)
2(2(0(1(1(0(0(1(0(1(1(1(1(x1))))))))))))) → 2(1(0(0(0(0(0(1(1(1(0(1(0(1(1(0(1(x1))))))))))))))))) (50)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
1(2(2(2(1(1(1(0(0(0(0(0(0(x1))))))))))))) → 1(2(1(1(1(1(1(1(0(1(1(1(0(0(1(1(0(x1))))))))))))))))) (51)
1(1(1(0(2(0(2(1(1(1(0(0(0(x1))))))))))))) → 1(0(1(1(1(0(1(1(1(0(1(1(2(1(1(1(1(x1))))))))))))))))) (52)
1(0(1(0(1(2(2(1(0(0(1(0(0(x1))))))))))))) → 1(1(1(2(1(1(0(0(1(0(0(0(1(1(1(0(0(x1))))))))))))))))) (53)
1(2(1(0(2(1(1(0(1(0(1(0(0(x1))))))))))))) → 1(0(1(0(0(0(0(0(1(2(1(1(1(0(0(0(0(x1))))))))))))))))) (54)
2(2(2(0(0(0(0(1(1(1(1(0(0(x1))))))))))))) → 1(1(0(0(2(0(1(1(0(0(1(1(1(0(0(1(0(x1))))))))))))))))) (55)
2(1(2(1(0(0(0(2(2(1(1(0(0(x1))))))))))))) → 2(1(2(0(1(2(1(0(0(0(1(0(1(1(0(0(0(x1))))))))))))))))) (56)
2(1(0(2(0(1(1(1(1(2(1(0(0(x1))))))))))))) → 2(1(0(0(2(1(1(0(0(0(2(1(1(1(0(0(1(x1))))))))))))))))) (57)
1(0(0(0(0(1(0(2(2(2(1(0(0(x1))))))))))))) → 1(1(1(0(0(1(1(0(1(0(2(1(1(1(0(1(0(x1))))))))))))))))) (58)
0(0(1(0(1(0(1(0(0(1(2(0(0(x1))))))))))))) → 1(1(1(1(0(0(1(0(1(0(1(0(1(0(0(0(0(x1))))))))))))))))) (59)
2(1(0(1(0(2(0(0(1(0(0(1(0(x1))))))))))))) → 1(1(1(2(0(0(1(0(0(0(0(0(1(0(0(0(0(x1))))))))))))))))) (60)
2(0(1(0(1(1(0(2(0(1(0(1(0(x1))))))))))))) → 2(0(1(0(1(0(0(0(1(0(0(0(1(0(2(1(0(x1))))))))))))))))) (61)
1(0(0(1(0(1(0(0(2(2(0(1(0(x1))))))))))))) → 1(0(1(0(1(0(1(0(1(1(0(2(1(1(1(1(0(x1))))))))))))))))) (62)
1(2(0(0(1(1(2(0(1(0(1(1(0(x1))))))))))))) → 1(0(0(2(2(1(1(0(1(0(0(0(0(0(2(1(1(x1))))))))))))))))) (63)
0(1(0(2(0(1(1(1(1(0(1(1(0(x1))))))))))))) → 1(0(0(0(2(1(1(0(1(0(0(0(0(0(0(1(0(x1))))))))))))))))) (64)
1(2(2(1(1(0(0(1(2(1(1(1(0(x1))))))))))))) → 1(0(2(1(1(0(1(1(1(1(0(2(1(1(0(1(1(x1))))))))))))))))) (65)
0(1(0(1(0(0(2(1(1(0(2(1(0(x1))))))))))))) → 0(0(1(0(0(1(2(1(0(0(0(0(0(0(2(1(0(x1))))))))))))))))) (66)
1(0(1(2(2(2(1(1(2(0(2(1(0(x1))))))))))))) → 1(1(0(2(1(1(0(2(1(1(1(2(0(0(2(1(0(x1))))))))))))))))) (67)
0(0(0(1(0(0(2(0(0(2(2(1(0(x1))))))))))))) → 0(0(0(0(1(0(1(0(2(1(0(1(0(0(2(1(1(x1))))))))))))))))) (68)
1(1(0(1(2(1(1(0(0(0(1(2(0(x1))))))))))))) → 1(0(0(1(0(1(2(1(0(1(1(0(0(1(1(1(1(x1))))))))))))))))) (69)
1(0(2(0(0(0(1(0(1(0(1(2(0(x1))))))))))))) → 1(0(0(1(1(0(2(1(1(0(0(1(0(0(0(0(0(x1))))))))))))))))) (70)
1(1(0(1(1(1(0(0(0(0(2(2(0(x1))))))))))))) → 1(1(1(0(1(1(1(0(0(1(0(1(1(1(1(1(1(x1))))))))))))))))) (71)
0(2(1(0(2(0(1(0(0(0(0(0(1(x1))))))))))))) → 0(0(0(2(2(1(0(0(1(1(1(0(1(1(1(0(1(x1))))))))))))))))) (72)
0(2(1(0(1(0(2(1(0(1(0(0(1(x1))))))))))))) → 0(1(0(1(0(1(0(2(1(0(1(0(1(1(0(0(0(x1))))))))))))))))) (73)
1(0(1(0(2(2(0(0(1(1(0(0(1(x1))))))))))))) → 1(1(1(2(0(0(0(0(0(1(1(1(1(1(0(0(0(x1))))))))))))))))) (74)
2(2(1(0(1(1(1(0(1(0(1(0(1(x1))))))))))))) → 1(2(1(0(0(1(0(1(1(0(0(1(1(1(0(1(0(x1))))))))))))))))) (75)
0(0(0(0(2(2(1(0(2(1(1(0(1(x1))))))))))))) → 0(0(0(0(0(2(2(2(1(1(0(1(0(0(0(0(1(x1))))))))))))))))) (76)
1(1(0(2(0(1(1(0(2(2(1(0(1(x1))))))))))))) → 1(1(0(0(0(2(2(2(0(1(1(1(0(0(1(0(0(x1))))))))))))))))) (77)
1(2(0(0(1(1(1(0(0(0(0(1(1(x1))))))))))))) → 1(0(0(0(0(0(1(1(0(0(0(0(1(1(0(1(0(x1))))))))))))))))) (78)
2(0(1(0(1(1(1(0(1(0(0(1(1(x1))))))))))))) → 2(1(1(1(1(1(0(0(0(1(0(1(1(1(1(1(0(x1))))))))))))))))) (79)
1(1(2(0(1(2(1(0(0(1(0(1(1(x1))))))))))))) → 1(0(0(1(0(2(2(0(1(0(1(0(0(1(0(1(0(x1))))))))))))))))) (80)
0(1(0(1(0(0(2(0(1(1(0(1(1(x1))))))))))))) → 0(0(0(1(1(1(0(0(0(1(0(1(1(0(1(1(0(x1))))))))))))))))) (81)
0(0(0(0(2(1(1(0(2(1(1(1(1(x1))))))))))))) → 1(0(0(1(0(1(1(0(1(1(0(0(1(1(0(1(1(x1))))))))))))))))) (82)
1(1(2(1(1(0(0(0(0(2(1(1(1(x1))))))))))))) → 1(0(0(0(1(2(0(1(1(0(0(0(1(1(1(1(1(x1))))))))))))))))) (83)
1(0(0(1(2(1(1(1(0(1(2(1(1(x1))))))))))))) → 1(1(0(0(0(0(0(0(1(1(0(0(0(2(2(1(0(x1))))))))))))))))) (84)
0(0(1(0(1(1(1(0(1(0(0(2(1(x1))))))))))))) → 1(0(0(0(0(0(0(0(1(1(0(0(1(1(0(1(0(x1))))))))))))))))) (85)
2(0(0(0(1(1(0(1(1(0(0(2(1(x1))))))))))))) → 1(1(1(0(1(1(0(1(1(1(1(0(0(1(1(0(0(x1))))))))))))))))) (86)
2(0(0(1(1(0(1(1(1(0(1(2(1(x1))))))))))))) → 2(1(0(0(0(2(0(0(0(1(1(0(0(0(1(0(0(x1))))))))))))))))) (87)
0(0(1(0(1(1(1(0(2(2(1(2(1(x1))))))))))))) → 0(0(1(1(0(0(1(1(2(0(0(1(0(1(2(1(0(x1))))))))))))))))) (88)
0(0(2(0(1(1(0(1(1(2(2(2(1(x1))))))))))))) → 0(0(2(0(0(1(2(1(2(2(1(0(0(0(0(0(1(x1))))))))))))))))) (89)
0(1(2(1(0(0(0(2(1(2(2(2(1(x1))))))))))))) → 0(1(0(2(1(1(0(2(0(1(1(2(0(0(1(1(1(x1))))))))))))))))) (90)
0(2(1(0(2(1(2(1(0(0(0(0(2(x1))))))))))))) → 0(0(2(1(1(0(1(2(0(0(1(0(1(1(0(0(2(x1))))))))))))))))) (91)
1(0(1(1(0(1(0(2(1(1(0(0(2(x1))))))))))))) → 1(0(1(1(0(1(0(1(0(0(0(2(0(0(0(1(0(x1))))))))))))))))) (92)
0(0(1(2(1(0(0(1(0(0(1(0(2(x1))))))))))))) → 0(1(0(1(0(0(0(1(0(0(1(0(0(0(1(1(1(x1))))))))))))))))) (93)
2(2(1(2(0(1(1(2(0(0(1(0(2(x1))))))))))))) → 2(2(2(1(2(0(1(1(1(0(1(1(0(0(1(1(2(x1))))))))))))))))) (94)
1(0(0(1(1(1(1(0(2(0(1(0(2(x1))))))))))))) → 1(0(1(1(1(1(1(0(1(0(1(0(1(1(1(2(0(x1))))))))))))))))) (95)
1(2(2(0(0(1(0(0(0(1(1(0(2(x1))))))))))))) → 1(0(0(2(1(0(0(0(0(1(2(1(1(0(0(0(2(x1))))))))))))))))) (96)
0(0(2(1(1(1(1(0(0(1(1(0(2(x1))))))))))))) → 0(0(0(0(0(1(0(1(0(0(0(0(1(1(0(0(2(x1))))))))))))))))) (97)
0(1(0(1(0(1(0(0(0(2(2(1(2(x1))))))))))))) → 0(1(1(1(2(1(0(0(0(0(0(1(0(1(1(1(2(x1))))))))))))))))) (98)
0(0(1(1(1(0(1(1(0(0(0(2(2(x1))))))))))))) → 1(1(1(0(0(0(0(0(0(1(1(0(0(1(1(0(1(x1))))))))))))))))) (99)
1(1(1(1(0(1(0(0(1(1(0(2(2(x1))))))))))))) → 1(0(1(1(0(1(0(1(1(1(0(0(0(0(0(1(2(x1))))))))))))))))) (100)

1.1 Closure Under Flat Contexts

Using the flat contexts

{1(☐), 2(☐), 0(☐)}

We obtain the transformed TRS
1(2(2(2(1(1(1(0(0(0(0(0(0(x1))))))))))))) → 1(2(1(1(1(1(1(1(0(1(1(1(0(0(1(1(0(x1))))))))))))))))) (51)
1(1(1(0(2(0(2(1(1(1(0(0(0(x1))))))))))))) → 1(0(1(1(1(0(1(1(1(0(1(1(2(1(1(1(1(x1))))))))))))))))) (52)
1(0(1(0(1(2(2(1(0(0(1(0(0(x1))))))))))))) → 1(1(1(2(1(1(0(0(1(0(0(0(1(1(1(0(0(x1))))))))))))))))) (53)
1(2(1(0(2(1(1(0(1(0(1(0(0(x1))))))))))))) → 1(0(1(0(0(0(0(0(1(2(1(1(1(0(0(0(0(x1))))))))))))))))) (54)
2(1(2(1(0(0(0(2(2(1(1(0(0(x1))))))))))))) → 2(1(2(0(1(2(1(0(0(0(1(0(1(1(0(0(0(x1))))))))))))))))) (56)
2(1(0(2(0(1(1(1(1(2(1(0(0(x1))))))))))))) → 2(1(0(0(2(1(1(0(0(0(2(1(1(1(0(0(1(x1))))))))))))))))) (57)
1(0(0(0(0(1(0(2(2(2(1(0(0(x1))))))))))))) → 1(1(1(0(0(1(1(0(1(0(2(1(1(1(0(1(0(x1))))))))))))))))) (58)
2(0(1(0(1(1(0(2(0(1(0(1(0(x1))))))))))))) → 2(0(1(0(1(0(0(0(1(0(0(0(1(0(2(1(0(x1))))))))))))))))) (61)
1(0(0(1(0(1(0(0(2(2(0(1(0(x1))))))))))))) → 1(0(1(0(1(0(1(0(1(1(0(2(1(1(1(1(0(x1))))))))))))))))) (62)
1(2(0(0(1(1(2(0(1(0(1(1(0(x1))))))))))))) → 1(0(0(2(2(1(1(0(1(0(0(0(0(0(2(1(1(x1))))))))))))))))) (63)
1(2(2(1(1(0(0(1(2(1(1(1(0(x1))))))))))))) → 1(0(2(1(1(0(1(1(1(1(0(2(1(1(0(1(1(x1))))))))))))))))) (65)
0(1(0(1(0(0(2(1(1(0(2(1(0(x1))))))))))))) → 0(0(1(0(0(1(2(1(0(0(0(0(0(0(2(1(0(x1))))))))))))))))) (66)
1(0(1(2(2(2(1(1(2(0(2(1(0(x1))))))))))))) → 1(1(0(2(1(1(0(2(1(1(1(2(0(0(2(1(0(x1))))))))))))))))) (67)
0(0(0(1(0(0(2(0(0(2(2(1(0(x1))))))))))))) → 0(0(0(0(1(0(1(0(2(1(0(1(0(0(2(1(1(x1))))))))))))))))) (68)
1(1(0(1(2(1(1(0(0(0(1(2(0(x1))))))))))))) → 1(0(0(1(0(1(2(1(0(1(1(0(0(1(1(1(1(x1))))))))))))))))) (69)
1(0(2(0(0(0(1(0(1(0(1(2(0(x1))))))))))))) → 1(0(0(1(1(0(2(1(1(0(0(1(0(0(0(0(0(x1))))))))))))))))) (70)
1(1(0(1(1(1(0(0(0(0(2(2(0(x1))))))))))))) → 1(1(1(0(1(1(1(0(0(1(0(1(1(1(1(1(1(x1))))))))))))))))) (71)
0(2(1(0(2(0(1(0(0(0(0(0(1(x1))))))))))))) → 0(0(0(2(2(1(0(0(1(1(1(0(1(1(1(0(1(x1))))))))))))))))) (72)
0(2(1(0(1(0(2(1(0(1(0(0(1(x1))))))))))))) → 0(1(0(1(0(1(0(2(1(0(1(0(1(1(0(0(0(x1))))))))))))))))) (73)
1(0(1(0(2(2(0(0(1(1(0(0(1(x1))))))))))))) → 1(1(1(2(0(0(0(0(0(1(1(1(1(1(0(0(0(x1))))))))))))))))) (74)
0(0(0(0(2(2(1(0(2(1(1(0(1(x1))))))))))))) → 0(0(0(0(0(2(2(2(1(1(0(1(0(0(0(0(1(x1))))))))))))))))) (76)
1(1(0(2(0(1(1(0(2(2(1(0(1(x1))))))))))))) → 1(1(0(0(0(2(2(2(0(1(1(1(0(0(1(0(0(x1))))))))))))))))) (77)
1(2(0(0(1(1(1(0(0(0(0(1(1(x1))))))))))))) → 1(0(0(0(0(0(1(1(0(0(0(0(1(1(0(1(0(x1))))))))))))))))) (78)
2(0(1(0(1(1(1(0(1(0(0(1(1(x1))))))))))))) → 2(1(1(1(1(1(0(0(0(1(0(1(1(1(1(1(0(x1))))))))))))))))) (79)
1(1(2(0(1(2(1(0(0(1(0(1(1(x1))))))))))))) → 1(0(0(1(0(2(2(0(1(0(1(0(0(1(0(1(0(x1))))))))))))))))) (80)
0(1(0(1(0(0(2(0(1(1(0(1(1(x1))))))))))))) → 0(0(0(1(1(1(0(0(0(1(0(1(1(0(1(1(0(x1))))))))))))))))) (81)
1(1(2(1(1(0(0(0(0(2(1(1(1(x1))))))))))))) → 1(0(0(0(1(2(0(1(1(0(0(0(1(1(1(1(1(x1))))))))))))))))) (83)
1(0(0(1(2(1(1(1(0(1(2(1(1(x1))))))))))))) → 1(1(0(0(0(0(0(0(1(1(0(0(0(2(2(1(0(x1))))))))))))))))) (84)
2(0(0(1(1(0(1(1(1(0(1(2(1(x1))))))))))))) → 2(1(0(0(0(2(0(0(0(1(1(0(0(0(1(0(0(x1))))))))))))))))) (87)
0(0(1(0(1(1(1(0(2(2(1(2(1(x1))))))))))))) → 0(0(1(1(0(0(1(1(2(0(0(1(0(1(2(1(0(x1))))))))))))))))) (88)
0(0(2(0(1(1(0(1(1(2(2(2(1(x1))))))))))))) → 0(0(2(0(0(1(2(1(2(2(1(0(0(0(0(0(1(x1))))))))))))))))) (89)
0(1(2(1(0(0(0(2(1(2(2(2(1(x1))))))))))))) → 0(1(0(2(1(1(0(2(0(1(1(2(0(0(1(1(1(x1))))))))))))))))) (90)
0(2(1(0(2(1(2(1(0(0(0(0(2(x1))))))))))))) → 0(0(2(1(1(0(1(2(0(0(1(0(1(1(0(0(2(x1))))))))))))))))) (91)
1(0(1(1(0(1(0(2(1(1(0(0(2(x1))))))))))))) → 1(0(1(1(0(1(0(1(0(0(0(2(0(0(0(1(0(x1))))))))))))))))) (92)
0(0(1(2(1(0(0(1(0(0(1(0(2(x1))))))))))))) → 0(1(0(1(0(0(0(1(0(0(1(0(0(0(1(1(1(x1))))))))))))))))) (93)
2(2(1(2(0(1(1(2(0(0(1(0(2(x1))))))))))))) → 2(2(2(1(2(0(1(1(1(0(1(1(0(0(1(1(2(x1))))))))))))))))) (94)
1(0(0(1(1(1(1(0(2(0(1(0(2(x1))))))))))))) → 1(0(1(1(1(1(1(0(1(0(1(0(1(1(1(2(0(x1))))))))))))))))) (95)
1(2(2(0(0(1(0(0(0(1(1(0(2(x1))))))))))))) → 1(0(0(2(1(0(0(0(0(1(2(1(1(0(0(0(2(x1))))))))))))))))) (96)
0(0(2(1(1(1(1(0(0(1(1(0(2(x1))))))))))))) → 0(0(0(0(0(1(0(1(0(0(0(0(1(1(0(0(2(x1))))))))))))))))) (97)
0(1(0(1(0(1(0(0(0(2(2(1(2(x1))))))))))))) → 0(1(1(1(2(1(0(0(0(0(0(1(0(1(1(1(2(x1))))))))))))))))) (98)
1(1(1(1(0(1(0(0(1(1(0(2(2(x1))))))))))))) → 1(0(1(1(0(1(0(1(1(1(0(0(0(0(0(1(2(x1))))))))))))))))) (100)
1(2(2(2(0(0(0(0(1(1(1(1(0(0(x1)))))))))))))) → 1(1(1(0(0(2(0(1(1(0(0(1(1(1(0(0(1(0(x1)))))))))))))))))) (101)
2(2(2(2(0(0(0(0(1(1(1(1(0(0(x1)))))))))))))) → 2(1(1(0(0(2(0(1(1(0(0(1(1(1(0(0(1(0(x1)))))))))))))))))) (102)
0(2(2(2(0(0(0(0(1(1(1(1(0(0(x1)))))))))))))) → 0(1(1(0(0(2(0(1(1(0(0(1(1(1(0(0(1(0(x1)))))))))))))))))) (103)
1(0(0(1(0(1(0(1(0(0(1(2(0(0(x1)))))))))))))) → 1(1(1(1(1(0(0(1(0(1(0(1(0(1(0(0(0(0(x1)))))))))))))))))) (104)
2(0(0(1(0(1(0(1(0(0(1(2(0(0(x1)))))))))))))) → 2(1(1(1(1(0(0(1(0(1(0(1(0(1(0(0(0(0(x1)))))))))))))))))) (105)
0(0(0(1(0(1(0(1(0(0(1(2(0(0(x1)))))))))))))) → 0(1(1(1(1(0(0(1(0(1(0(1(0(1(0(0(0(0(x1)))))))))))))))))) (106)
1(2(1(0(1(0(2(0(0(1(0(0(1(0(x1)))))))))))))) → 1(1(1(1(2(0(0(1(0(0(0(0(0(1(0(0(0(0(x1)))))))))))))))))) (107)
2(2(1(0(1(0(2(0(0(1(0(0(1(0(x1)))))))))))))) → 2(1(1(1(2(0(0(1(0(0(0(0(0(1(0(0(0(0(x1)))))))))))))))))) (108)
0(2(1(0(1(0(2(0(0(1(0(0(1(0(x1)))))))))))))) → 0(1(1(1(2(0(0(1(0(0(0(0(0(1(0(0(0(0(x1)))))))))))))))))) (109)
1(0(1(0(2(0(1(1(1(1(0(1(1(0(x1)))))))))))))) → 1(1(0(0(0(2(1(1(0(1(0(0(0(0(0(0(1(0(x1)))))))))))))))))) (110)
2(0(1(0(2(0(1(1(1(1(0(1(1(0(x1)))))))))))))) → 2(1(0(0(0(2(1(1(0(1(0(0(0(0(0(0(1(0(x1)))))))))))))))))) (111)
0(0(1(0(2(0(1(1(1(1(0(1(1(0(x1)))))))))))))) → 0(1(0(0(0(2(1(1(0(1(0(0(0(0(0(0(1(0(x1)))))))))))))))))) (112)
1(2(2(1(0(1(1(1(0(1(0(1(0(1(x1)))))))))))))) → 1(1(2(1(0(0(1(0(1(1(0(0(1(1(1(0(1(0(x1)))))))))))))))))) (113)
2(2(2(1(0(1(1(1(0(1(0(1(0(1(x1)))))))))))))) → 2(1(2(1(0(0(1(0(1(1(0(0(1(1(1(0(1(0(x1)))))))))))))))))) (114)
0(2(2(1(0(1(1(1(0(1(0(1(0(1(x1)))))))))))))) → 0(1(2(1(0(0(1(0(1(1(0(0(1(1(1(0(1(0(x1)))))))))))))))))) (115)
1(0(0(0(0(2(1(1(0(2(1(1(1(1(x1)))))))))))))) → 1(1(0(0(1(0(1(1(0(1(1(0(0(1(1(0(1(1(x1)))))))))))))))))) (116)
2(0(0(0(0(2(1(1(0(2(1(1(1(1(x1)))))))))))))) → 2(1(0(0(1(0(1(1(0(1(1(0(0(1(1(0(1(1(x1)))))))))))))))))) (117)
0(0(0(0(0(2(1(1(0(2(1(1(1(1(x1)))))))))))))) → 0(1(0(0(1(0(1(1(0(1(1(0(0(1(1(0(1(1(x1)))))))))))))))))) (118)
1(0(0(1(0(1(1(1(0(1(0(0(2(1(x1)))))))))))))) → 1(1(0(0(0(0(0(0(0(1(1(0(0(1(1(0(1(0(x1)))))))))))))))))) (119)
2(0(0(1(0(1(1(1(0(1(0(0(2(1(x1)))))))))))))) → 2(1(0(0(0(0(0(0(0(1(1(0(0(1(1(0(1(0(x1)))))))))))))))))) (120)
0(0(0(1(0(1(1(1(0(1(0(0(2(1(x1)))))))))))))) → 0(1(0(0(0(0(0(0(0(1(1(0(0(1(1(0(1(0(x1)))))))))))))))))) (121)
1(2(0(0(0(1(1(0(1(1(0(0(2(1(x1)))))))))))))) → 1(1(1(1(0(1(1(0(1(1(1(1(0(0(1(1(0(0(x1)))))))))))))))))) (122)
2(2(0(0(0(1(1(0(1(1(0(0(2(1(x1)))))))))))))) → 2(1(1(1(0(1(1(0(1(1(1(1(0(0(1(1(0(0(x1)))))))))))))))))) (123)
0(2(0(0(0(1(1(0(1(1(0(0(2(1(x1)))))))))))))) → 0(1(1(1(0(1(1(0(1(1(1(1(0(0(1(1(0(0(x1)))))))))))))))))) (124)
1(0(0(1(1(1(0(1(1(0(0(0(2(2(x1)))))))))))))) → 1(1(1(1(0(0(0(0(0(0(1(1(0(0(1(1(0(1(x1)))))))))))))))))) (125)
2(0(0(1(1(1(0(1(1(0(0(0(2(2(x1)))))))))))))) → 2(1(1(1(0(0(0(0(0(0(1(1(0(0(1(1(0(1(x1)))))))))))))))))) (126)
0(0(0(1(1(1(0(1(1(0(0(0(2(2(x1)))))))))))))) → 0(1(1(1(0(0(0(0(0(0(1(1(0(0(1(1(0(1(x1)))))))))))))))))) (127)

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 204 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 61
[21(x1)] = 1 · x1 + 74
[11(x1)] = 1 · x1 + 9
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1 + 9
[02(x1)] = 1 · x1 + 1
[20(x1)] = 1 · x1 + 104
all of the following rules can be deleted.

There are 193 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 · x1 + 3
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[00(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 1
[22(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 2
all of the following rules can be deleted.
11(10(02(20(01(11(10(02(22(21(10(01(12(x1))))))))))))) → 11(10(00(00(02(22(22(20(01(11(11(10(00(01(10(00(02(x1))))))))))))))))) (192)
11(10(02(20(01(11(10(02(22(21(10(01(10(x1))))))))))))) → 11(10(00(00(02(22(22(20(01(11(11(10(00(01(10(00(00(x1))))))))))))))))) (193)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 1
[10(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 2
[01(x1)] = 1 · x1
[00(x1)] = 1 · x1
[11(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
[12(x1)] = 1 · x1
all of the following rules can be deleted.
02(21(10(02(20(01(10(00(00(00(00(01(11(x1))))))))))))) → 00(00(02(22(21(10(00(01(11(11(10(01(11(11(10(01(11(x1))))))))))))))))) (179)
02(21(10(02(20(01(10(00(00(00(00(01(12(x1))))))))))))) → 00(00(02(22(21(10(00(01(11(11(10(01(11(11(10(01(12(x1))))))))))))))))) (180)
02(21(10(02(20(01(10(00(00(00(00(01(10(x1))))))))))))) → 00(00(02(22(21(10(00(01(11(11(10(01(11(11(10(01(10(x1))))))))))))))))) (181)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1
all of the following rules can be deleted.
00(02(20(01(11(10(01(11(12(22(22(21(11(x1))))))))))))) → 00(02(20(00(01(12(21(12(22(21(10(00(00(00(00(01(11(x1))))))))))))))))) (218)
00(02(20(01(11(10(01(11(12(22(22(21(12(x1))))))))))))) → 00(02(20(00(01(12(21(12(22(21(10(00(00(00(00(01(12(x1))))))))))))))))) (219)
00(02(20(01(11(10(01(11(12(22(22(21(10(x1))))))))))))) → 00(02(20(00(01(12(21(12(22(21(10(00(00(00(00(01(10(x1))))))))))))))))) (220)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[12(x1)] = 1 · x1
[00(x1)] = 1 · x1
[11(x1)] = 1 · x1
[01(x1)] = 1 · x1
[20(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
02(21(10(02(21(12(21(10(00(00(00(02(21(x1))))))))))))) → 00(02(21(11(10(01(12(20(00(01(10(01(11(10(00(02(21(x1))))))))))))))))) (224)
02(21(10(02(21(12(21(10(00(00(00(02(22(x1))))))))))))) → 00(02(21(11(10(01(12(20(00(01(10(01(11(10(00(02(22(x1))))))))))))))))) (225)
02(21(10(02(21(12(21(10(00(00(00(02(20(x1))))))))))))) → 00(02(21(11(10(01(12(20(00(01(10(01(11(10(00(02(20(x1))))))))))))))))) (226)

1.1.1.1.1.1.1.1.1 R is empty

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