Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/142142)

The rewrite relation of the following TRS is considered.

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

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(0(2(1(0(0(0(0(1(1(0(0(0(x1))))))))))))) → 0(0(1(1(0(1(1(0(0(1(0(0(0(1(1(1(1(x1))))))))))))))))) (45)
1(1(0(0(1(1(2(2(1(1(0(0(0(x1))))))))))))) → 0(1(0(0(0(1(1(0(1(0(0(0(2(0(1(1(0(x1))))))))))))))))) (46)
0(1(0(0(1(0(0(1(2(0(0(1(0(x1))))))))))))) → 0(0(1(1(1(0(0(1(0(1(1(0(0(0(1(1(0(x1))))))))))))))))) (47)
2(1(0(2(0(0(0(0(1(1(0(1(0(x1))))))))))))) → 2(0(0(1(1(1(0(1(1(0(1(0(0(0(1(1(1(x1))))))))))))))))) (48)
0(0(0(1(0(1(2(0(1(1(0(1(0(x1))))))))))))) → 0(1(1(1(0(1(0(0(1(0(0(1(1(0(1(1(0(x1))))))))))))))))) (49)
1(2(0(0(0(2(0(1(0(0(1(1(0(x1))))))))))))) → 1(1(1(2(1(1(0(0(1(1(0(1(0(1(1(0(0(x1))))))))))))))))) (50)
1(1(1(2(0(0(0(0(0(1(2(1(0(x1))))))))))))) → 1(0(0(0(1(1(0(0(0(1(0(2(1(1(0(1(0(x1))))))))))))))))) (51)
1(0(0(0(0(0(0(1(2(0(0(2(0(x1))))))))))))) → 1(0(0(0(0(1(0(0(0(1(1(1(0(1(0(0(1(x1))))))))))))))))) (52)
1(1(1(0(2(2(1(2(1(0(1(2(0(x1))))))))))))) → 1(1(1(1(2(1(0(0(1(1(2(0(0(1(1(2(0(x1))))))))))))))))) (53)
1(0(0(0(2(1(0(1(2(0(1(2(0(x1))))))))))))) → 0(1(0(0(0(2(0(1(0(0(1(1(1(1(2(0(0(x1))))))))))))))))) (54)
1(1(0(1(2(0(0(0(0(1(1(2(0(x1))))))))))))) → 0(0(0(0(0(1(1(1(2(0(1(0(1(0(1(0(0(x1))))))))))))))))) (55)
1(0(1(0(1(0(1(1(0(0(2(2(0(x1))))))))))))) → 0(0(1(1(0(1(1(0(1(1(0(0(0(1(2(0(0(x1))))))))))))))))) (56)
1(1(2(0(0(1(1(1(0(0(0(0(1(x1))))))))))))) → 0(0(1(0(1(1(0(1(1(1(1(1(1(1(1(1(1(x1))))))))))))))))) (57)
2(0(1(0(0(2(1(1(1(0(0(0(1(x1))))))))))))) → 2(1(0(2(0(0(0(0(0(1(0(0(1(0(1(1(0(x1))))))))))))))))) (58)
0(1(0(0(0(1(0(2(1(0(0(0(1(x1))))))))))))) → 1(0(1(1(1(0(1(1(0(1(1(0(0(0(1(1(1(x1))))))))))))))))) (59)
1(0(1(0(2(0(0(1(2(0(0(0(1(x1))))))))))))) → 0(0(0(1(0(0(0(0(0(1(0(1(1(0(0(1(2(x1))))))))))))))))) (60)
0(1(1(2(0(1(2(1(0(1(0(0(1(x1))))))))))))) → 0(1(1(1(1(1(1(2(0(0(0(0(0(1(1(1(0(x1))))))))))))))))) (61)
1(0(1(1(0(1(1(2(0(1(0(0(1(x1))))))))))))) → 1(1(0(0(1(1(1(0(1(0(1(0(0(1(1(1(1(x1))))))))))))))))) (62)
2(1(1(2(2(0(0(0(2(1(0(0(1(x1))))))))))))) → 2(1(2(1(0(1(1(1(0(2(2(0(1(0(0(0(1(x1))))))))))))))))) (63)
1(2(0(2(0(1(1(1(2(1(0(0(1(x1))))))))))))) → 0(0(1(2(2(1(0(1(0(2(1(0(0(1(0(0(1(x1))))))))))))))))) (64)
2(0(1(2(0(1(0(1(0(2(0(0(1(x1))))))))))))) → 2(0(0(1(1(0(1(0(1(1(0(2(0(0(0(1(0(x1))))))))))))))))) (65)
1(0(0(1(2(1(1(2(1(0(1(0(1(x1))))))))))))) → 0(0(1(1(0(0(0(0(2(2(0(0(1(0(0(1(1(x1))))))))))))))))) (66)
0(0(1(0(0(1(2(1(0(2(1(0(1(x1))))))))))))) → 1(0(1(2(0(1(1(1(1(0(1(1(0(0(0(0(0(x1))))))))))))))))) (67)
2(0(1(0(1(0(1(0(1(2(2(0(1(x1))))))))))))) → 2(1(2(0(0(0(1(1(0(0(0(0(0(0(1(1(1(x1))))))))))))))))) (68)
0(0(0(0(1(2(2(2(0(1(0(1(1(x1))))))))))))) → 1(0(0(2(0(0(1(0(1(0(1(0(1(1(0(1(1(x1))))))))))))))))) (69)
2(1(2(1(2(1(0(0(2(2(0(1(1(x1))))))))))))) → 2(0(1(0(1(1(2(2(0(1(1(0(0(0(2(1(1(x1))))))))))))))))) (70)
2(0(0(1(1(2(0(1(0(1(1(1(1(x1))))))))))))) → 2(0(1(2(0(0(1(0(1(1(0(1(0(0(1(1(1(x1))))))))))))))))) (71)
0(1(0(2(0(2(0(1(1(2(1(1(1(x1))))))))))))) → 0(1(0(1(2(1(0(0(2(2(0(1(0(0(1(1(1(x1))))))))))))))))) (72)
2(0(1(1(0(1(1(0(1(1(2(1(1(x1))))))))))))) → 2(1(0(0(1(0(0(0(0(0(1(0(1(0(2(1(1(x1))))))))))))))))) (73)
0(0(1(1(0(1(1(1(1(2(2(1(1(x1))))))))))))) → 0(0(0(1(1(1(0(1(0(1(1(2(1(1(1(0(1(x1))))))))))))))))) (74)
0(0(0(1(1(0(0(1(2(2(2(1(1(x1))))))))))))) → 0(0(0(1(1(0(1(2(0(1(1(1(1(1(0(1(0(x1))))))))))))))))) (75)
1(1(1(1(0(1(1(1(0(1(0(2(1(x1))))))))))))) → 1(0(1(1(0(1(1(0(0(1(0(1(1(1(0(0(1(x1))))))))))))))))) (76)
1(2(1(1(0(1(0(0(0(0(1(2(1(x1))))))))))))) → 0(0(0(1(2(0(0(1(1(0(0(0(1(0(0(1(0(x1))))))))))))))))) (77)
1(1(0(2(0(1(0(0(1(0(1(2(1(x1))))))))))))) → 0(1(1(0(0(1(0(0(2(2(0(0(0(1(0(0(1(x1))))))))))))))))) (78)
0(1(0(1(1(1(1(1(1(0(2(2(1(x1))))))))))))) → 0(1(1(0(0(0(1(0(0(1(0(0(2(2(0(0(1(x1))))))))))))))))) (79)
1(1(0(1(2(1(0(1(1(0(0(1(2(x1))))))))))))) → 1(0(1(1(0(0(1(2(0(0(0(0(1(0(0(0(1(x1))))))))))))))))) (80)
2(1(0(1(0(2(0(0(2(1(0(1(2(x1))))))))))))) → 2(1(0(2(2(2(0(1(0(0(0(0(1(0(0(1(2(x1))))))))))))))))) (81)
1(0(0(0(0(1(2(2(2(2(0(1(2(x1))))))))))))) → 0(1(0(2(1(0(0(1(2(1(1(0(0(0(1(1(1(x1))))))))))))))))) (82)
0(1(1(2(1(1(1(0(1(0(1(1(2(x1))))))))))))) → 0(0(1(0(1(1(0(0(1(1(0(0(1(1(2(1(1(x1))))))))))))))))) (83)
1(0(0(0(1(2(0(1(2(0(1(1(2(x1))))))))))))) → 0(1(0(0(1(0(0(1(0(2(1(1(1(1(0(1(0(x1))))))))))))))))) (84)
1(0(2(0(0(1(1(1(1(1(1(1(2(x1))))))))))))) → 0(1(0(2(0(1(0(0(1(0(1(1(1(0(1(1(1(x1))))))))))))))))) (85)
1(0(1(1(1(1(1(2(0(0(1(2(2(x1))))))))))))) → 0(1(1(1(0(1(1(0(1(0(1(0(0(0(2(2(2(x1))))))))))))))))) (86)
2(0(1(0(0(2(1(2(0(0(1(2(2(x1))))))))))))) → 2(1(1(1(0(0(2(1(0(0(1(2(0(0(0(1(2(x1))))))))))))))))) (87)
0(1(0(0(0(0(1(0(0(0(2(2(2(x1))))))))))))) → 0(1(0(1(0(0(0(1(1(0(1(1(1(0(0(2(0(x1))))))))))))))))) (88)

1.1 Closure Under Flat Contexts

Using the flat contexts

{1(☐), 0(☐), 2(☐)}

We obtain the transformed TRS
0(1(0(0(1(0(0(1(2(0(0(1(0(x1))))))))))))) → 0(0(1(1(1(0(0(1(0(1(1(0(0(0(1(1(0(x1))))))))))))))))) (47)
2(1(0(2(0(0(0(0(1(1(0(1(0(x1))))))))))))) → 2(0(0(1(1(1(0(1(1(0(1(0(0(0(1(1(1(x1))))))))))))))))) (48)
0(0(0(1(0(1(2(0(1(1(0(1(0(x1))))))))))))) → 0(1(1(1(0(1(0(0(1(0(0(1(1(0(1(1(0(x1))))))))))))))))) (49)
1(2(0(0(0(2(0(1(0(0(1(1(0(x1))))))))))))) → 1(1(1(2(1(1(0(0(1(1(0(1(0(1(1(0(0(x1))))))))))))))))) (50)
1(1(1(2(0(0(0(0(0(1(2(1(0(x1))))))))))))) → 1(0(0(0(1(1(0(0(0(1(0(2(1(1(0(1(0(x1))))))))))))))))) (51)
1(0(0(0(0(0(0(1(2(0(0(2(0(x1))))))))))))) → 1(0(0(0(0(1(0(0(0(1(1(1(0(1(0(0(1(x1))))))))))))))))) (52)
1(1(1(0(2(2(1(2(1(0(1(2(0(x1))))))))))))) → 1(1(1(1(2(1(0(0(1(1(2(0(0(1(1(2(0(x1))))))))))))))))) (53)
2(0(1(0(0(2(1(1(1(0(0(0(1(x1))))))))))))) → 2(1(0(2(0(0(0(0(0(1(0(0(1(0(1(1(0(x1))))))))))))))))) (58)
0(1(1(2(0(1(2(1(0(1(0(0(1(x1))))))))))))) → 0(1(1(1(1(1(1(2(0(0(0(0(0(1(1(1(0(x1))))))))))))))))) (61)
1(0(1(1(0(1(1(2(0(1(0(0(1(x1))))))))))))) → 1(1(0(0(1(1(1(0(1(0(1(0(0(1(1(1(1(x1))))))))))))))))) (62)
2(1(1(2(2(0(0(0(2(1(0(0(1(x1))))))))))))) → 2(1(2(1(0(1(1(1(0(2(2(0(1(0(0(0(1(x1))))))))))))))))) (63)
2(0(1(2(0(1(0(1(0(2(0(0(1(x1))))))))))))) → 2(0(0(1(1(0(1(0(1(1(0(2(0(0(0(1(0(x1))))))))))))))))) (65)
2(0(1(0(1(0(1(0(1(2(2(0(1(x1))))))))))))) → 2(1(2(0(0(0(1(1(0(0(0(0(0(0(1(1(1(x1))))))))))))))))) (68)
2(1(2(1(2(1(0(0(2(2(0(1(1(x1))))))))))))) → 2(0(1(0(1(1(2(2(0(1(1(0(0(0(2(1(1(x1))))))))))))))))) (70)
2(0(0(1(1(2(0(1(0(1(1(1(1(x1))))))))))))) → 2(0(1(2(0(0(1(0(1(1(0(1(0(0(1(1(1(x1))))))))))))))))) (71)
0(1(0(2(0(2(0(1(1(2(1(1(1(x1))))))))))))) → 0(1(0(1(2(1(0(0(2(2(0(1(0(0(1(1(1(x1))))))))))))))))) (72)
2(0(1(1(0(1(1(0(1(1(2(1(1(x1))))))))))))) → 2(1(0(0(1(0(0(0(0(0(1(0(1(0(2(1(1(x1))))))))))))))))) (73)
0(0(1(1(0(1(1(1(1(2(2(1(1(x1))))))))))))) → 0(0(0(1(1(1(0(1(0(1(1(2(1(1(1(0(1(x1))))))))))))))))) (74)
0(0(0(1(1(0(0(1(2(2(2(1(1(x1))))))))))))) → 0(0(0(1(1(0(1(2(0(1(1(1(1(1(0(1(0(x1))))))))))))))))) (75)
1(1(1(1(0(1(1(1(0(1(0(2(1(x1))))))))))))) → 1(0(1(1(0(1(1(0(0(1(0(1(1(1(0(0(1(x1))))))))))))))))) (76)
0(1(0(1(1(1(1(1(1(0(2(2(1(x1))))))))))))) → 0(1(1(0(0(0(1(0(0(1(0(0(2(2(0(0(1(x1))))))))))))))))) (79)
1(1(0(1(2(1(0(1(1(0(0(1(2(x1))))))))))))) → 1(0(1(1(0(0(1(2(0(0(0(0(1(0(0(0(1(x1))))))))))))))))) (80)
2(1(0(1(0(2(0(0(2(1(0(1(2(x1))))))))))))) → 2(1(0(2(2(2(0(1(0(0(0(0(1(0(0(1(2(x1))))))))))))))))) (81)
0(1(1(2(1(1(1(0(1(0(1(1(2(x1))))))))))))) → 0(0(1(0(1(1(0(0(1(1(0(0(1(1(2(1(1(x1))))))))))))))))) (83)
2(0(1(0(0(2(1(2(0(0(1(2(2(x1))))))))))))) → 2(1(1(1(0(0(2(1(0(0(1(2(0(0(0(1(2(x1))))))))))))))))) (87)
0(1(0(0(0(0(1(0(0(0(2(2(2(x1))))))))))))) → 0(1(0(1(0(0(0(1(1(0(1(1(1(0(0(2(0(x1))))))))))))))))) (88)
1(1(0(2(1(0(0(0(0(1(1(0(0(0(x1)))))))))))))) → 1(0(0(1(1(0(1(1(0(0(1(0(0(0(1(1(1(1(x1)))))))))))))))))) (89)
0(1(0(2(1(0(0(0(0(1(1(0(0(0(x1)))))))))))))) → 0(0(0(1(1(0(1(1(0(0(1(0(0(0(1(1(1(1(x1)))))))))))))))))) (90)
2(1(0(2(1(0(0(0(0(1(1(0(0(0(x1)))))))))))))) → 2(0(0(1(1(0(1(1(0(0(1(0(0(0(1(1(1(1(x1)))))))))))))))))) (91)
1(1(1(0(0(1(1(2(2(1(1(0(0(0(x1)))))))))))))) → 1(0(1(0(0(0(1(1(0(1(0(0(0(2(0(1(1(0(x1)))))))))))))))))) (92)
0(1(1(0(0(1(1(2(2(1(1(0(0(0(x1)))))))))))))) → 0(0(1(0(0(0(1(1(0(1(0(0(0(2(0(1(1(0(x1)))))))))))))))))) (93)
2(1(1(0(0(1(1(2(2(1(1(0(0(0(x1)))))))))))))) → 2(0(1(0(0(0(1(1(0(1(0(0(0(2(0(1(1(0(x1)))))))))))))))))) (94)
1(1(0(0(0(2(1(0(1(2(0(1(2(0(x1)))))))))))))) → 1(0(1(0(0(0(2(0(1(0(0(1(1(1(1(2(0(0(x1)))))))))))))))))) (95)
0(1(0(0(0(2(1(0(1(2(0(1(2(0(x1)))))))))))))) → 0(0(1(0(0(0(2(0(1(0(0(1(1(1(1(2(0(0(x1)))))))))))))))))) (96)
2(1(0(0(0(2(1(0(1(2(0(1(2(0(x1)))))))))))))) → 2(0(1(0(0(0(2(0(1(0(0(1(1(1(1(2(0(0(x1)))))))))))))))))) (97)
1(1(1(0(1(2(0(0(0(0(1(1(2(0(x1)))))))))))))) → 1(0(0(0(0(0(1(1(1(2(0(1(0(1(0(1(0(0(x1)))))))))))))))))) (98)
0(1(1(0(1(2(0(0(0(0(1(1(2(0(x1)))))))))))))) → 0(0(0(0(0(0(1(1(1(2(0(1(0(1(0(1(0(0(x1)))))))))))))))))) (99)
2(1(1(0(1(2(0(0(0(0(1(1(2(0(x1)))))))))))))) → 2(0(0(0(0(0(1(1(1(2(0(1(0(1(0(1(0(0(x1)))))))))))))))))) (100)
1(1(0(1(0(1(0(1(1(0(0(2(2(0(x1)))))))))))))) → 1(0(0(1(1(0(1(1(0(1(1(0(0(0(1(2(0(0(x1)))))))))))))))))) (101)
0(1(0(1(0(1(0(1(1(0(0(2(2(0(x1)))))))))))))) → 0(0(0(1(1(0(1(1(0(1(1(0(0(0(1(2(0(0(x1)))))))))))))))))) (102)
2(1(0(1(0(1(0(1(1(0(0(2(2(0(x1)))))))))))))) → 2(0(0(1(1(0(1(1(0(1(1(0(0(0(1(2(0(0(x1)))))))))))))))))) (103)
1(1(1(2(0(0(1(1(1(0(0(0(0(1(x1)))))))))))))) → 1(0(0(1(0(1(1(0(1(1(1(1(1(1(1(1(1(1(x1)))))))))))))))))) (104)
0(1(1(2(0(0(1(1(1(0(0(0(0(1(x1)))))))))))))) → 0(0(0(1(0(1(1(0(1(1(1(1(1(1(1(1(1(1(x1)))))))))))))))))) (105)
2(1(1(2(0(0(1(1(1(0(0(0(0(1(x1)))))))))))))) → 2(0(0(1(0(1(1(0(1(1(1(1(1(1(1(1(1(1(x1)))))))))))))))))) (106)
1(0(1(0(0(0(1(0(2(1(0(0(0(1(x1)))))))))))))) → 1(1(0(1(1(1(0(1(1(0(1(1(0(0(0(1(1(1(x1)))))))))))))))))) (107)
0(0(1(0(0(0(1(0(2(1(0(0(0(1(x1)))))))))))))) → 0(1(0(1(1(1(0(1(1(0(1(1(0(0(0(1(1(1(x1)))))))))))))))))) (108)
2(0(1(0(0(0(1(0(2(1(0(0(0(1(x1)))))))))))))) → 2(1(0(1(1(1(0(1(1(0(1(1(0(0(0(1(1(1(x1)))))))))))))))))) (109)
1(1(0(1(0(2(0(0(1(2(0(0(0(1(x1)))))))))))))) → 1(0(0(0(1(0(0(0(0(0(1(0(1(1(0(0(1(2(x1)))))))))))))))))) (110)
0(1(0(1(0(2(0(0(1(2(0(0(0(1(x1)))))))))))))) → 0(0(0(0(1(0(0(0(0(0(1(0(1(1(0(0(1(2(x1)))))))))))))))))) (111)
2(1(0(1(0(2(0(0(1(2(0(0(0(1(x1)))))))))))))) → 2(0(0(0(1(0(0(0(0(0(1(0(1(1(0(0(1(2(x1)))))))))))))))))) (112)
1(1(2(0(2(0(1(1(1(2(1(0(0(1(x1)))))))))))))) → 1(0(0(1(2(2(1(0(1(0(2(1(0(0(1(0(0(1(x1)))))))))))))))))) (113)
0(1(2(0(2(0(1(1(1(2(1(0(0(1(x1)))))))))))))) → 0(0(0(1(2(2(1(0(1(0(2(1(0(0(1(0(0(1(x1)))))))))))))))))) (114)
2(1(2(0(2(0(1(1(1(2(1(0(0(1(x1)))))))))))))) → 2(0(0(1(2(2(1(0(1(0(2(1(0(0(1(0(0(1(x1)))))))))))))))))) (115)
1(1(0(0(1(2(1(1(2(1(0(1(0(1(x1)))))))))))))) → 1(0(0(1(1(0(0(0(0(2(2(0(0(1(0(0(1(1(x1)))))))))))))))))) (116)
0(1(0(0(1(2(1(1(2(1(0(1(0(1(x1)))))))))))))) → 0(0(0(1(1(0(0(0(0(2(2(0(0(1(0(0(1(1(x1)))))))))))))))))) (117)
2(1(0(0(1(2(1(1(2(1(0(1(0(1(x1)))))))))))))) → 2(0(0(1(1(0(0(0(0(2(2(0(0(1(0(0(1(1(x1)))))))))))))))))) (118)
1(0(0(1(0(0(1(2(1(0(2(1(0(1(x1)))))))))))))) → 1(1(0(1(2(0(1(1(1(1(0(1(1(0(0(0(0(0(x1)))))))))))))))))) (119)
0(0(0(1(0(0(1(2(1(0(2(1(0(1(x1)))))))))))))) → 0(1(0(1(2(0(1(1(1(1(0(1(1(0(0(0(0(0(x1)))))))))))))))))) (120)
2(0(0(1(0(0(1(2(1(0(2(1(0(1(x1)))))))))))))) → 2(1(0(1(2(0(1(1(1(1(0(1(1(0(0(0(0(0(x1)))))))))))))))))) (121)
1(0(0(0(0(1(2(2(2(0(1(0(1(1(x1)))))))))))))) → 1(1(0(0(2(0(0(1(0(1(0(1(0(1(1(0(1(1(x1)))))))))))))))))) (122)
0(0(0(0(0(1(2(2(2(0(1(0(1(1(x1)))))))))))))) → 0(1(0(0(2(0(0(1(0(1(0(1(0(1(1(0(1(1(x1)))))))))))))))))) (123)
2(0(0(0(0(1(2(2(2(0(1(0(1(1(x1)))))))))))))) → 2(1(0(0(2(0(0(1(0(1(0(1(0(1(1(0(1(1(x1)))))))))))))))))) (124)
1(1(2(1(1(0(1(0(0(0(0(1(2(1(x1)))))))))))))) → 1(0(0(0(1(2(0(0(1(1(0(0(0(1(0(0(1(0(x1)))))))))))))))))) (125)
0(1(2(1(1(0(1(0(0(0(0(1(2(1(x1)))))))))))))) → 0(0(0(0(1(2(0(0(1(1(0(0(0(1(0(0(1(0(x1)))))))))))))))))) (126)
2(1(2(1(1(0(1(0(0(0(0(1(2(1(x1)))))))))))))) → 2(0(0(0(1(2(0(0(1(1(0(0(0(1(0(0(1(0(x1)))))))))))))))))) (127)
1(1(1(0(2(0(1(0(0(1(0(1(2(1(x1)))))))))))))) → 1(0(1(1(0(0(1(0(0(2(2(0(0(0(1(0(0(1(x1)))))))))))))))))) (128)
0(1(1(0(2(0(1(0(0(1(0(1(2(1(x1)))))))))))))) → 0(0(1(1(0(0(1(0(0(2(2(0(0(0(1(0(0(1(x1)))))))))))))))))) (129)
2(1(1(0(2(0(1(0(0(1(0(1(2(1(x1)))))))))))))) → 2(0(1(1(0(0(1(0(0(2(2(0(0(0(1(0(0(1(x1)))))))))))))))))) (130)
1(1(0(0(0(0(1(2(2(2(2(0(1(2(x1)))))))))))))) → 1(0(1(0(2(1(0(0(1(2(1(1(0(0(0(1(1(1(x1)))))))))))))))))) (131)
0(1(0(0(0(0(1(2(2(2(2(0(1(2(x1)))))))))))))) → 0(0(1(0(2(1(0(0(1(2(1(1(0(0(0(1(1(1(x1)))))))))))))))))) (132)
2(1(0(0(0(0(1(2(2(2(2(0(1(2(x1)))))))))))))) → 2(0(1(0(2(1(0(0(1(2(1(1(0(0(0(1(1(1(x1)))))))))))))))))) (133)
1(1(0(0(0(1(2(0(1(2(0(1(1(2(x1)))))))))))))) → 1(0(1(0(0(1(0(0(1(0(2(1(1(1(1(0(1(0(x1)))))))))))))))))) (134)
0(1(0(0(0(1(2(0(1(2(0(1(1(2(x1)))))))))))))) → 0(0(1(0(0(1(0(0(1(0(2(1(1(1(1(0(1(0(x1)))))))))))))))))) (135)
2(1(0(0(0(1(2(0(1(2(0(1(1(2(x1)))))))))))))) → 2(0(1(0(0(1(0(0(1(0(2(1(1(1(1(0(1(0(x1)))))))))))))))))) (136)
1(1(0(2(0(0(1(1(1(1(1(1(1(2(x1)))))))))))))) → 1(0(1(0(2(0(1(0(0(1(0(1(1(1(0(1(1(1(x1)))))))))))))))))) (137)
0(1(0(2(0(0(1(1(1(1(1(1(1(2(x1)))))))))))))) → 0(0(1(0(2(0(1(0(0(1(0(1(1(1(0(1(1(1(x1)))))))))))))))))) (138)
2(1(0(2(0(0(1(1(1(1(1(1(1(2(x1)))))))))))))) → 2(0(1(0(2(0(1(0(0(1(0(1(1(1(0(1(1(1(x1)))))))))))))))))) (139)
1(1(0(1(1(1(1(1(2(0(0(1(2(2(x1)))))))))))))) → 1(0(1(1(1(0(1(1(0(1(0(1(0(0(0(2(2(2(x1)))))))))))))))))) (140)
0(1(0(1(1(1(1(1(2(0(0(1(2(2(x1)))))))))))))) → 0(0(1(1(1(0(1(1(0(1(0(1(0(0(0(2(2(2(x1)))))))))))))))))) (141)
2(1(0(1(1(1(1(1(2(0(0(1(2(2(x1)))))))))))))) → 2(0(1(1(1(0(1(1(0(1(0(1(0(0(0(2(2(2(x1)))))))))))))))))) (142)

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 240 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

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

There are 216 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 2
[22(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 3
[01(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
all of the following rules can be deleted.
01(10(00(00(00(01(10(00(00(02(22(22(22(x1))))))))))))) → 01(10(01(10(00(00(01(11(10(01(11(11(10(00(02(20(02(x1))))))))))))))))) (220)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 1
[12(x1)] = 1 · x1
[01(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
all of the following rules can be deleted.
11(11(10(02(22(21(12(21(10(01(12(20(00(x1))))))))))))) → 11(11(11(12(21(10(00(01(11(12(20(00(01(11(12(20(00(x1))))))))))))))))) (161)
11(11(10(02(22(21(12(21(10(01(12(20(01(x1))))))))))))) → 11(11(11(12(21(10(00(01(11(12(20(00(01(11(12(20(01(x1))))))))))))))))) (162)
11(11(10(02(22(21(12(21(10(01(12(20(02(x1))))))))))))) → 11(11(11(12(21(10(00(01(11(12(20(00(01(11(12(20(02(x1))))))))))))))))) (163)
01(10(01(11(11(11(11(11(10(02(22(21(10(x1))))))))))))) → 01(11(10(00(00(01(10(00(01(10(00(02(22(20(00(01(10(x1))))))))))))))))) (203)
01(10(01(11(11(11(11(11(10(02(22(21(11(x1))))))))))))) → 01(11(10(00(00(01(10(00(01(10(00(02(22(20(00(01(11(x1))))))))))))))))) (204)
01(10(01(11(11(11(11(11(10(02(22(21(12(x1))))))))))))) → 01(11(10(00(00(01(10(00(01(10(00(02(22(20(00(01(12(x1))))))))))))))))) (205)
21(10(01(10(01(10(01(11(10(00(02(22(20(00(x1)))))))))))))) → 20(00(01(11(10(01(11(10(01(11(10(00(00(01(12(20(00(00(x1)))))))))))))))))) (263)
21(10(01(10(01(10(01(11(10(00(02(22(20(01(x1)))))))))))))) → 20(00(01(11(10(01(11(10(01(11(10(00(00(01(12(20(00(01(x1)))))))))))))))))) (264)
21(10(01(10(01(10(01(11(10(00(02(22(20(02(x1)))))))))))))) → 20(00(01(11(10(01(11(10(01(11(10(00(00(01(12(20(00(02(x1)))))))))))))))))) (265)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[11(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
11(10(01(10(01(10(01(11(10(00(02(22(20(00(x1)))))))))))))) → 10(00(01(11(10(01(11(10(01(11(10(00(00(01(12(20(00(00(x1)))))))))))))))))) (257)
11(10(01(10(01(10(01(11(10(00(02(22(20(01(x1)))))))))))))) → 10(00(01(11(10(01(11(10(01(11(10(00(00(01(12(20(00(01(x1)))))))))))))))))) (258)
11(10(01(10(01(10(01(11(10(00(02(22(20(02(x1)))))))))))))) → 10(00(01(11(10(01(11(10(01(11(10(00(00(01(12(20(00(02(x1)))))))))))))))))) (259)
01(10(01(10(01(10(01(11(10(00(02(22(20(00(x1)))))))))))))) → 00(00(01(11(10(01(11(10(01(11(10(00(00(01(12(20(00(00(x1)))))))))))))))))) (260)
01(10(01(10(01(10(01(11(10(00(02(22(20(01(x1)))))))))))))) → 00(00(01(11(10(01(11(10(01(11(10(00(00(01(12(20(00(01(x1)))))))))))))))))) (261)
01(10(01(10(01(10(01(11(10(00(02(22(20(02(x1)))))))))))))) → 00(00(01(11(10(01(11(10(01(11(10(00(00(01(12(20(00(02(x1)))))))))))))))))) (262)

1.1.1.1.1.1.1.1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

10

Hence, it suffices to show innermost termination in the following.

1.1.1.1.1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
20#(01(10(00(02(21(11(11(10(00(00(01(10(x1))))))))))))) → 21#(10(02(20(00(00(00(00(01(10(00(01(10(01(11(10(00(x1))))))))))))))))) (383)
20#(01(10(00(02(21(11(11(10(00(00(01(10(x1))))))))))))) → 20#(00(00(00(00(01(10(00(01(10(01(11(10(00(x1)))))))))))))) (384)
20#(01(10(00(02(21(11(11(10(00(00(01(11(x1))))))))))))) → 21#(10(02(20(00(00(00(00(01(10(00(01(10(01(11(10(01(x1))))))))))))))))) (385)
20#(01(10(00(02(21(11(11(10(00(00(01(11(x1))))))))))))) → 20#(00(00(00(00(01(10(00(01(10(01(11(10(01(x1)))))))))))))) (386)
21#(11(12(22(20(00(00(02(21(10(00(01(10(x1))))))))))))) → 21#(12(21(10(01(11(11(10(02(22(20(01(10(00(00(01(10(x1))))))))))))))))) (387)
21#(11(12(22(20(00(00(02(21(10(00(01(10(x1))))))))))))) → 21#(10(01(11(11(10(02(22(20(01(10(00(00(01(10(x1))))))))))))))) (388)
21#(11(12(22(20(00(00(02(21(10(00(01(10(x1))))))))))))) → 20#(01(10(00(00(01(10(x1))))))) (389)
21#(11(12(22(20(00(00(02(21(10(00(01(11(x1))))))))))))) → 21#(12(21(10(01(11(11(10(02(22(20(01(10(00(00(01(11(x1))))))))))))))))) (390)
21#(11(12(22(20(00(00(02(21(10(00(01(11(x1))))))))))))) → 21#(10(01(11(11(10(02(22(20(01(10(00(00(01(11(x1))))))))))))))) (391)
21#(11(12(22(20(00(00(02(21(10(00(01(11(x1))))))))))))) → 20#(01(10(00(00(01(11(x1))))))) (392)
21#(11(12(22(20(00(00(02(21(10(00(01(12(x1))))))))))))) → 21#(12(21(10(01(11(11(10(02(22(20(01(10(00(00(01(12(x1))))))))))))))))) (393)
21#(11(12(22(20(00(00(02(21(10(00(01(12(x1))))))))))))) → 21#(10(01(11(11(10(02(22(20(01(10(00(00(01(12(x1))))))))))))))) (394)
21#(11(12(22(20(00(00(02(21(10(00(01(12(x1))))))))))))) → 20#(01(10(00(00(01(12(x1))))))) (395)
20#(00(01(11(12(20(01(10(01(11(11(11(10(x1))))))))))))) → 20#(01(12(20(00(01(10(01(11(10(01(10(00(01(11(11(10(x1))))))))))))))))) (396)
20#(00(01(11(12(20(01(10(01(11(11(11(10(x1))))))))))))) → 20#(00(01(10(01(11(10(01(10(00(01(11(11(10(x1)))))))))))))) (397)
20#(00(01(11(12(20(01(10(01(11(11(11(11(x1))))))))))))) → 20#(01(12(20(00(01(10(01(11(10(01(10(00(01(11(11(11(x1))))))))))))))))) (398)
20#(00(01(11(12(20(01(10(01(11(11(11(11(x1))))))))))))) → 20#(00(01(10(01(11(10(01(10(00(01(11(11(11(x1)))))))))))))) (399)
20#(00(01(11(12(20(01(10(01(11(11(11(12(x1))))))))))))) → 20#(01(12(20(00(01(10(01(11(10(01(10(00(01(11(11(12(x1))))))))))))))))) (400)
20#(00(01(11(12(20(01(10(01(11(11(11(12(x1))))))))))))) → 20#(00(01(10(01(11(10(01(10(00(01(11(11(12(x1)))))))))))))) (401)

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.