Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/132478)

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

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

Since only unary symbols occur, one can reverse all terms and obtains the TRS
1(0(3(2(1(2(2(3(3(3(1(1(1(0(0(0(0(0(x1)))))))))))))))))) → 3(1(3(2(2(0(3(1(0(0(1(1(0(0(1(3(2(0(x1)))))))))))))))))) (82)
3(0(2(1(2(1(3(0(2(3(0(0(1(2(2(0(0(0(x1)))))))))))))))))) → 3(2(0(3(2(2(0(0(0(2(1(3(0(1(1(2(0(0(x1)))))))))))))))))) (83)
1(2(3(1(0(2(3(0(0(2(1(2(3(3(2(1(0(0(x1)))))))))))))))))) → 1(0(1(1(2(3(0(3(0(3(0(2(2(0(1(3(2(2(x1)))))))))))))))))) (84)
0(1(0(3(1(1(1(2(1(3(3(0(0(2(0(0(1(0(x1)))))))))))))))))) → 0(0(0(1(1(0(3(1(3(0(1(0(1(3(2(2(1(0(x1)))))))))))))))))) (85)
0(0(1(0(3(1(3(3(1(3(3(1(2(1(1(0(1(0(x1)))))))))))))))))) → 0(2(3(1(3(1(3(0(3(1(1(0(1(3(1(1(0(0(x1)))))))))))))))))) (86)
2(1(0(0(3(3(0(0(3(3(3(3(2(1(1(0(1(0(x1)))))))))))))))))) → 2(3(3(0(0(3(1(3(1(3(0(0(2(3(0(1(0(1(x1)))))))))))))))))) (87)
1(2(2(2(0(1(0(0(1(2(3(1(2(1(0(1(1(0(x1)))))))))))))))))) → 1(1(2(2(2(1(3(1(1(0(0(1(1(0(2(2(0(0(x1)))))))))))))))))) (88)
3(1(2(3(3(2(3(0(1(1(0(2(3(2(0(2(1(0(x1)))))))))))))))))) → 3(1(1(0(2(2(3(1(3(1(3(2(3(2(0(2(0(0(x1)))))))))))))))))) (89)
3(2(1(0(0(0(2(0(3(1(2(0(1(2(2(2(1(0(x1)))))))))))))))))) → 3(2(1(3(0(2(2(0(2(0(1(1(0(2(0(2(1(0(x1)))))))))))))))))) (90)
0(0(2(3(2(0(3(0(2(0(2(0(2(0(0(0(2(0(x1)))))))))))))))))) → 0(0(0(0(2(2(3(2(2(0(0(0(0(0(3(2(2(0(x1)))))))))))))))))) (91)
3(3(3(1(0(1(3(2(1(0(3(1(2(2(1(0(2(0(x1)))))))))))))))))) → 3(2(3(0(3(1(2(1(3(2(0(0(1(3(2(1(1(0(x1)))))))))))))))))) (92)
3(3(3(0(3(0(2(0(0(0(3(1(0(3(2(0(2(0(x1)))))))))))))))))) → 3(3(3(3(2(2(0(0(0(3(0(3(2(1(0(0(0(0(x1)))))))))))))))))) (93)
0(3(0(1(1(0(3(2(2(2(0(2(0(1(0(1(2(0(x1)))))))))))))))))) → 0(0(1(3(2(2(2(2(2(0(1(1(3(0(1(0(0(0(x1)))))))))))))))))) (94)
3(3(3(2(1(0(2(1(0(0(2(3(0(1(2(2(2(0(x1)))))))))))))))))) → 3(2(0(1(3(0(1(3(2(0(2(3(1(2(2(2(0(0(x1)))))))))))))))))) (95)
0(2(3(3(0(1(0(0(0(0(2(0(2(3(0(2(3(0(x1)))))))))))))))))) → 0(0(2(0(3(3(0(3(1(3(0(2(0(2(0(0(2(0(x1)))))))))))))))))) (96)
1(2(3(1(1(2(3(3(2(1(0(1(1(0(1(2(3(0(x1)))))))))))))))))) → 1(0(3(0(1(2(2(0(3(2(1(3(1(1(3(1(2(1(x1)))))))))))))))))) (97)
1(1(3(1(0(2(1(0(0(1(0(1(0(3(1(2(3(0(x1)))))))))))))))))) → 0(1(1(3(0(2(0(1(1(1(3(1(3(0(1(0(2(0(x1)))))))))))))))))) (98)
1(0(0(1(2(0(1(0(3(3(0(0(2(1(2(2(3(0(x1)))))))))))))))))) → 0(3(1(0(0(0(0(2(2(3(2(1(0(2(1(1(3(0(x1)))))))))))))))))) (99)
0(0(2(0(2(1(0(2(1(3(0(1(0(1(3(3(3(0(x1)))))))))))))))))) → 1(2(2(2(0(1(3(3(1(0(1(3(0(0(0(0(3(0(x1)))))))))))))))))) (100)
1(2(1(1(2(3(2(0(0(3(0(2(1(2(3(1(1(1(x1)))))))))))))))))) → 1(0(2(1(3(3(0(1(1(1(2(2(3(2(0(2(1(1(x1)))))))))))))))))) (101)
0(2(3(0(1(1(1(3(2(1(3(2(3(0(2(3(1(1(x1)))))))))))))))))) → 0(3(1(3(3(2(0(1(2(1(3(1(3(0(2(2(1(1(x1)))))))))))))))))) (102)
0(1(1(2(3(3(1(0(2(3(3(1(2(2(1(0(2(1(x1)))))))))))))))))) → 2(2(1(1(0(1(2(3(0(3(2(3(1(3(1(0(2(1(x1)))))))))))))))))) (103)
1(0(2(0(2(3(2(3(0(1(1(2(2(3(1(0(2(1(x1)))))))))))))))))) → 0(1(3(3(0(2(2(0(0(2(2(2(1(1(3(1(2(1(x1)))))))))))))))))) (104)
3(0(0(2(1(2(3(3(0(3(2(0(3(3(2(1(2(1(x1)))))))))))))))))) → 3(3(2(3(1(0(2(1(0(0(2(2(2(3(0(3(3(1(x1)))))))))))))))))) (105)
1(1(1(2(1(2(3(0(0(1(3(3(3(1(0(3(2(1(x1)))))))))))))))))) → 0(0(1(3(2(1(0(1(3(3(1(1(3(2(1(3(2(1(x1)))))))))))))))))) (106)
2(1(1(2(1(1(2(0(2(1(0(0(0(2(0(3(2(1(x1)))))))))))))))))) → 2(2(2(3(1(1(1(1(2(0(0(0(1(0(2(2(0(1(x1)))))))))))))))))) (107)
3(1(3(2(3(0(2(1(2(0(3(3(3(3(0(3(2(1(x1)))))))))))))))))) → 3(1(3(2(3(0(2(1(3(1(3(3(3(2(2(3(0(0(x1)))))))))))))))))) (108)
2(3(1(2(0(1(1(1(2(1(2(0(1(3(1(2(3(1(x1)))))))))))))))))) → 2(0(1(1(3(2(1(2(0(3(1(1(1(2(2(1(3(1(x1)))))))))))))))))) (109)
1(3(1(2(3(3(0(1(1(2(2(0(1(1(0(3(3(1(x1)))))))))))))))))) → 1(1(1(0(0(3(2(3(3(2(0(1(1(1(3(2(3(1(x1)))))))))))))))))) (110)
3(1(2(0(1(2(0(3(0(2(1(1(2(3(1(0(0(2(x1)))))))))))))))))) → 3(2(1(0(0(2(2(1(3(2(2(1(3(1(1(0(0(0(x1)))))))))))))))))) (111)
1(1(2(3(2(3(1(1(2(0(0(0(2(3(3(0(0(2(x1)))))))))))))))))) → 1(2(0(0(1(1(2(3(2(2(3(0(0(1(3(0(3(2(x1)))))))))))))))))) (112)
3(1(2(1(2(0(2(3(0(1(2(3(2(0(2(1(0(2(x1)))))))))))))))))) → 1(3(0(2(2(1(0(3(1(3(0(2(2(0(1(2(2(2(x1)))))))))))))))))) (113)
0(2(2(1(0(1(2(2(1(0(0(0(1(0(3(2(0(2(x1)))))))))))))))))) → 1(3(0(0(2(1(1(0(1(0(0(0(0(2(2(2(2(2(x1)))))))))))))))))) (114)
3(1(3(2(2(1(1(1(3(3(1(0(0(3(3(3(0(2(x1)))))))))))))))))) → 3(3(3(0(0(1(3(1(3(1(1(1(3(3(2(2(2(0(x1)))))))))))))))))) (115)
0(2(3(3(3(3(0(3(2(2(3(3(1(2(0(0(1(2(x1)))))))))))))))))) → 0(0(2(2(3(0(3(2(3(3(0(2(2(3(1(3(1(3(x1)))))))))))))))))) (116)
0(1(3(2(0(2(3(0(2(0(1(2(3(0(1(0(1(2(x1)))))))))))))))))) → 0(0(2(0(2(1(3(1(3(0(2(2(1(3(0(1(0(2(x1)))))))))))))))))) (117)
1(2(2(2(3(2(3(0(0(3(2(0(2(3(0(1(1(2(x1)))))))))))))))))) → 0(0(2(2(2(1(0(3(2(2(2(3(1(1(3(2(3(0(x1)))))))))))))))))) (118)
0(1(3(1(0(0(0(1(0(3(0(2(2(3(3(1(1(2(x1)))))))))))))))))) → 0(3(2(0(1(1(1(0(3(1(0(2(2(0(1(3(0(3(x1)))))))))))))))))) (119)
1(2(2(1(0(3(3(2(0(2(2(0(2(0(0(2(1(2(x1)))))))))))))))))) → 1(3(2(0(2(2(2(0(1(0(2(0(0(2(2(2(1(3(x1)))))))))))))))))) (120)
0(1(2(0(0(3(0(1(2(3(0(0(3(0(0(2(1(2(x1)))))))))))))))))) → 0(1(3(0(1(2(0(3(0(0(3(2(2(0(0(0(1(2(x1)))))))))))))))))) (121)
1(2(3(3(2(1(0(3(3(1(0(1(0(3(1(2(1(2(x1)))))))))))))))))) → 1(1(0(3(1(3(2(1(1(3(2(0(2(1(2(3(0(3(x1)))))))))))))))))) (122)
0(2(0(2(3(2(3(1(1(2(2(2(3(3(1(2(1(2(x1)))))))))))))))))) → 2(2(3(0(1(2(3(2(1(1(3(0(1(2(3(2(2(2(x1)))))))))))))))))) (123)
0(2(3(3(1(1(1(2(2(3(3(1(0(1(2(2(1(2(x1)))))))))))))))))) → 0(3(2(1(2(0(2(3(1(3(1(1(1(1(3(2(2(2(x1)))))))))))))))))) (124)
2(3(3(0(0(0(0(2(0(3(3(1(1(1(2(2(1(2(x1)))))))))))))))))) → 2(0(0(1(1(0(3(0(3(2(2(0(1(2(3(1(3(2(x1)))))))))))))))))) (125)
0(2(2(2(2(2(0(2(3(3(3(1(2(1(2(3(1(2(x1)))))))))))))))))) → 2(2(2(2(2(1(0(2(2(3(1(3(0(2(1(3(2(3(x1)))))))))))))))))) (126)
3(3(3(2(3(2(1(3(1(2(3(3(3(1(1(0(2(2(x1)))))))))))))))))) → 3(3(2(1(3(1(3(1(2(3(3(2(1(3(2(3(0(2(x1)))))))))))))))))) (127)
3(3(1(1(2(3(1(2(3(2(0(2(3(3(1(0(2(2(x1)))))))))))))))))) → 3(1(1(1(2(2(3(3(2(2(2(3(0(3(0(1(3(2(x1)))))))))))))))))) (128)
1(0(2(0(0(3(3(0(1(2(1(1(1(0(3(0(2(2(x1)))))))))))))))))) → 1(2(1(3(0(3(2(1(3(0(0(0(0(1(0(1(2(2(x1)))))))))))))))))) (129)
3(0(1(1(2(1(3(2(0(2(3(3(3(3(3(2(2(2(x1)))))))))))))))))) → 3(3(0(3(3(2(1(1(2(2(1(3(3(0(3(2(2(2(x1)))))))))))))))))) (130)
2(2(1(2(1(0(0(2(3(0(2(0(2(0(1(1(3(2(x1)))))))))))))))))) → 2(2(0(1(1(1(2(2(3(0(1(3(0(0(0(2(2(2(x1)))))))))))))))))) (131)
1(2(0(1(0(3(3(3(0(2(3(2(1(2(1(1(3(2(x1)))))))))))))))))) → 0(3(1(1(2(1(3(2(3(2(2(0(1(0(3(1(3(2(x1)))))))))))))))))) (132)
0(2(1(2(2(3(0(1(2(2(2(3(2(1(2(3(3(2(x1)))))))))))))))))) → 1(3(2(2(2(1(3(0(2(2(2(3(1(2(0(3(2(2(x1)))))))))))))))))) (133)
2(3(2(3(1(2(1(2(1(1(3(2(0(2(3(3(3(2(x1)))))))))))))))))) → 2(3(1(2(2(3(0(3(2(1(3(1(3(3(2(1(2(2(x1)))))))))))))))))) (134)
3(1(0(3(3(3(1(0(1(1(0(3(3(0(0(0(0(3(x1)))))))))))))))))) → 3(0(0(3(3(0(1(1(3(0(0(0(1(3(1(0(3(3(x1)))))))))))))))))) (135)
0(1(1(1(1(1(3(3(1(0(2(1(3(1(2(0(0(3(x1)))))))))))))))))) → 2(3(1(1(0(0(0(1(3(2(1(3(0(1(1(1(1(3(x1)))))))))))))))))) (136)
1(2(1(1(2(1(3(0(3(2(2(3(3(0(3(2(0(3(x1)))))))))))))))))) → 0(3(2(0(3(2(1(0(1(3(1(3(3(2(2(2(1(3(x1)))))))))))))))))) (137)
0(3(3(3(3(0(2(3(0(2(0(0(0(2(2(3(0(3(x1)))))))))))))))))) → 0(3(3(3(2(0(3(2(2(3(0(0(3(0(0(0(2(3(x1)))))))))))))))))) (138)
0(0(0(0(3(3(0(3(0(2(2(3(1(0(3(3(0(3(x1)))))))))))))))))) → 0(0(0(3(2(3(2(3(0(3(0(0(1(3(0(3(0(3(x1)))))))))))))))))) (139)
0(0(2(1(2(3(0(2(2(1(0(0(1(1(0(1(1(3(x1)))))))))))))))))) → 1(0(1(0(1(0(2(2(2(1(0(3(1(2(0(0(1(3(x1)))))))))))))))))) (140)
2(0(2(2(3(3(0(3(3(0(2(3(2(3(1(1(1(3(x1)))))))))))))))))) → 2(2(0(3(1(2(2(3(1(3(3(3(2(0(3(0(1(3(x1)))))))))))))))))) (141)
1(2(1(2(2(1(3(3(0(0(2(1(1(0(0(0(2(3(x1)))))))))))))))))) → 0(3(1(1(1(1(0(3(3(0(0(2(1(0(2(2(2(2(x1)))))))))))))))))) (142)
3(1(2(0(3(0(3(3(3(1(3(1(1(1(1(0(2(3(x1)))))))))))))))))) → 3(1(3(2(1(0(3(0(1(3(1(0(1(1(3(3(2(3(x1)))))))))))))))))) (143)
3(1(1(0(1(2(3(1(3(3(2(3(0(3(1(0(2(3(x1)))))))))))))))))) → 3(1(1(3(2(2(3(3(0(1(0(1(3(0(3(1(3(2(x1)))))))))))))))))) (144)
1(2(3(1(2(3(3(3(1(0(0(3(2(3(1(0(2(3(x1)))))))))))))))))) → 2(3(2(1(3(3(2(0(0(1(3(3(0(2(1(1(3(3(x1)))))))))))))))))) (145)
0(2(1(1(0(1(2(1(0(0(3(2(0(3(3(0(2(3(x1)))))))))))))))))) → 1(1(3(0(2(1(3(2(0(3(1(0(2(0(0(0(2(3(x1)))))))))))))))))) (146)
1(2(0(3(3(1(2(1(3(1(3(3(1(0(2(1(2(3(x1)))))))))))))))))) → 1(3(1(1(3(0(0(3(2(1(1(3(2(2(3(2(1(3(x1)))))))))))))))))) (147)
0(2(2(3(0(2(2(1(3(3(1(2(2(1(2(1(2(3(x1)))))))))))))))))) → 0(3(1(3(2(2(1(1(2(2(2(3(2(2(1(3(0(2(x1)))))))))))))))))) (148)
3(3(2(0(1(0(2(0(3(3(1(1(1(1(3(1(2(3(x1)))))))))))))))))) → 3(1(0(3(2(3(1(3(2(0(0(1(3(1(2(1(1(3(x1)))))))))))))))))) (149)
0(3(0(2(0(1(2(1(2(0(2(3(2(0(1(2(2(3(x1)))))))))))))))))) → 2(3(2(2(2(2(2(0(1(1(1(0(3(0(3(0(0(2(x1)))))))))))))))))) (150)
2(1(3(2(0(2(3(1(0(0(1(3(3(0(1(2(2(3(x1)))))))))))))))))) → 2(0(0(2(2(2(1(1(1(0(3(2(3(1(3(0(3(3(x1)))))))))))))))))) (151)
1(1(2(3(0(2(3(1(2(3(3(0(0(2(0(3(2(3(x1)))))))))))))))))) → 3(0(1(3(2(2(1(1(0(3(3(2(2(3(0(2(0(3(x1)))))))))))))))))) (152)
3(0(0(0(3(1(2(0(3(2(1(1(2(1(1(0(3(3(x1)))))))))))))))))) → 3(0(2(1(1(1(0(0(3(2(1(3(2(0(1(0(3(3(x1)))))))))))))))))) (153)
3(2(3(3(1(3(2(3(3(0(1(2(2(1(1(0(3(3(x1)))))))))))))))))) → 3(2(2(3(1(1(3(3(3(2(1(3(0(0(1(3(2(3(x1)))))))))))))))))) (154)
0(1(3(3(0(1(1(1(0(0(2(0(2(1(2(1(3(3(x1)))))))))))))))))) → 0(1(1(3(0(0(0(3(1(1(2(2(2(0(1(1(3(3(x1)))))))))))))))))) (155)
0(0(1(0(0(2(1(2(0(2(3(2(1(0(1(2(3(3(x1)))))))))))))))))) → 0(1(2(2(2(0(0(2(1(3(0(2(1(1(0(0(3(3(x1)))))))))))))))))) (156)
3(1(3(0(2(3(3(3(3(3(3(1(3(0(1(2(3(3(x1)))))))))))))))))) → 3(1(3(0(3(2(3(3(0(1(3(2(3(3(1(3(3(3(x1)))))))))))))))))) (157)
1(1(2(1(0(1(0(2(1(1(0(1(2(1(2(2(3(3(x1)))))))))))))))))) → 1(1(3(2(0(1(1(0(2(1(1(2(1(0(1(2(2(3(x1)))))))))))))))))) (158)
3(3(3(2(0(2(1(0(3(1(2(3(0(3(3(2(3(3(x1)))))))))))))))))) → 3(3(3(0(2(0(3(2(1(1(2(3(0(3(3(2(3(3(x1)))))))))))))))))) (159)
0(2(3(3(0(3(2(1(0(0(0(2(0(3(0(3(3(3(x1)))))))))))))))))) → 1(3(0(0(0(3(2(3(3(2(2(3(0(0(0(3(0(3(x1)))))))))))))))))) (160)
2(1(0(3(1(2(0(3(0(1(0(2(0(0(1(3(3(3(x1)))))))))))))))))) → 2(0(0(2(1(0(2(1(3(1(3(3(1(0(0(3(0(3(x1)))))))))))))))))) (161)

1.1 Closure Under Flat Contexts

Using the flat contexts

{1(☐), 0(☐), 3(☐), 2(☐)}

We obtain the transformed TRS

There are 149 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 596 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 25
[21(x1)] = 1 · x1 + 101
[12(x1)] = 1 · x1 + 38
[13(x1)] = 1 · x1
[23(x1)] = 1 · x1 + 188
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 38
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 111
[32(x1)] = 1 · x1 + 12
[11(x1)] = 1 · x1 + 1
[31(x1)] = 1 · x1 + 65
[10(x1)] = 1 · x1
[33(x1)] = 1 · x1 + 86
all of the following rules can be deleted.

There are 596 ruless (increase limit for explicit display).

1.1.1.1.1 R is empty

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