Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/128280)

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

0(0(0(1(3(3(0(1(0(1(0(2(1(2(0(2(2(0(x1)))))))))))))))))) → 0(0(0(0(1(0(2(2(1(1(2(2(3(0(0(3(1(0(x1)))))))))))))))))) (1)
0(0(1(0(2(1(0(0(3(0(1(3(0(0(0(3(3(1(x1)))))))))))))))))) → 0(0(0(3(0(1(0(3(0(0(0(0(1(3(2(1(3(1(x1)))))))))))))))))) (2)
0(0(2(0(2(0(0(2(1(2(2(1(1(1(0(2(3(2(x1)))))))))))))))))) → 0(1(0(2(2(1(0(0(0(0(2(2(2(2(1(3(1(2(x1)))))))))))))))))) (3)
0(0(2(0(3(0(0(2(0(0(1(0(2(2(1(2(2(0(x1)))))))))))))))))) → 0(0(3(1(2(2(2(2(0(0(0(2(0(1(0(0(2(0(x1)))))))))))))))))) (4)
0(0(3(3(3(0(2(3(1(0(1(3(3(0(0(0(1(0(x1)))))))))))))))))) → 0(1(0(1(0(3(3(3(0(0(0(3(0(3(1(2(3(0(x1)))))))))))))))))) (5)
0(1(2(3(2(0(2(3(1(0(0(3(1(1(1(1(1(2(x1)))))))))))))))))) → 0(2(0(1(3(2(1(0(3(0(1(2(1(1(3(1(1(2(x1)))))))))))))))))) (6)
0(1(3(0(1(2(0(0(2(3(1(3(1(2(3(1(0(2(x1)))))))))))))))))) → 0(3(0(1(0(0(3(2(3(1(2(1(1(1(3(0(2(2(x1)))))))))))))))))) (7)
0(2(0(0(2(3(3(3(1(0(1(3(1(3(2(0(2(3(x1)))))))))))))))))) → 0(2(1(3(0(2(3(0(3(3(1(3(2(3(2(0(0(1(x1)))))))))))))))))) (8)
0(2(0(1(0(3(1(0(1(1(0(1(2(1(2(0(2(0(x1)))))))))))))))))) → 0(1(0(0(2(1(1(0(1(1(3(2(2(1(0(0(2(0(x1)))))))))))))))))) (9)
0(2(1(0(1(3(0(3(2(0(3(1(0(1(0(3(0(3(x1)))))))))))))))))) → 0(3(0(0(0(0(1(1(3(2(0(1(3(2(1(3(0(3(x1)))))))))))))))))) (10)
0(2(2(3(3(3(0(1(3(0(2(1(2(1(3(0(2(3(x1)))))))))))))))))) → 0(2(3(2(0(3(2(1(1(3(3(0(3(2(0(1(2(3(x1)))))))))))))))))) (11)
0(2(3(1(0(0(3(1(0(2(1(2(0(3(3(0(2(3(x1)))))))))))))))))) → 0(2(0(0(0(1(1(3(3(0(3(2(3(2(0(1(2(3(x1)))))))))))))))))) (12)
0(2(3(3(0(3(0(0(0(0(2(3(3(0(1(3(0(0(x1)))))))))))))))))) → 0(0(3(0(3(3(0(3(2(2(0(0(3(0(1(3(0(0(x1)))))))))))))))))) (13)
0(2(3(3(3(2(2(3(0(1(0(2(0(3(0(0(0(3(x1)))))))))))))))))) → 0(3(3(2(2(2(0(0(3(3(0(0(1(0(3(2(0(3(x1)))))))))))))))))) (14)
0(3(0(0(2(0(2(1(1(3(3(3(1(0(1(0(0(3(x1)))))))))))))))))) → 0(0(0(1(1(0(1(2(3(0(3(3(2(0(3(1(0(3(x1)))))))))))))))))) (15)
0(3(1(1(2(2(3(0(0(2(1(3(3(3(2(2(3(2(x1)))))))))))))))))) → 0(3(0(1(3(2(1(3(0(3(2(2(3(2(2(1(3(2(x1)))))))))))))))))) (16)
0(3(2(2(0(0(0(0(1(0(1(3(3(3(0(0(0(3(x1)))))))))))))))))) → 0(3(1(2(0(0(0(0(2(0(3(0(1(0(3(3(0(3(x1)))))))))))))))))) (17)
1(0(0(0(2(2(3(3(1(2(0(2(2(3(1(2(1(2(x1)))))))))))))))))) → 2(0(3(0(3(2(2(2(2(1(0(1(2(1(3(1(0(2(x1)))))))))))))))))) (18)
1(0(0(2(0(0(2(1(1(2(3(3(1(2(2(0(1(1(x1)))))))))))))))))) → 2(2(0(0(0(0(3(2(1(3(1(2(1(2(1(0(1(1(x1)))))))))))))))))) (19)
1(0(0(2(1(0(0(3(0(0(0(1(3(3(3(1(0(3(x1)))))))))))))))))) → 1(0(1(0(0(0(1(3(1(0(2(3(0(3(0(3(0(3(x1)))))))))))))))))) (20)
1(0(1(3(1(0(1(3(3(1(1(0(1(1(0(2(3(3(x1)))))))))))))))))) → 1(3(2(0(1(3(1(1(1(3(0(0(1(1(0(1(3(3(x1)))))))))))))))))) (21)
1(0(2(0(1(1(2(3(0(0(2(0(2(1(0(1(0(3(x1)))))))))))))))))) → 2(2(1(0(0(0(3(1(1(1(2(1(2(0(0(0(0(3(x1)))))))))))))))))) (22)
1(0(2(2(3(1(0(3(1(0(2(3(1(2(3(0(2(3(x1)))))))))))))))))) → 1(3(2(0(3(2(1(2(0(0(3(3(1(2(1(0(2(3(x1)))))))))))))))))) (23)
1(0(3(1(0(1(1(0(1(1(0(0(2(0(2(0(1(2(x1)))))))))))))))))) → 1(1(2(3(0(1(2(0(1(0(1(0(0(0(0(1(1(2(x1)))))))))))))))))) (24)
1(1(1(2(0(0(2(1(2(1(2(2(2(1(2(3(1(0(x1)))))))))))))))))) → 1(1(1(2(2(2(2(1(1(0(2(0(1(3(2(2(1(0(x1)))))))))))))))))) (25)
1(1(2(3(1(3(3(3(3(0(0(2(1(3(3(3(0(1(x1)))))))))))))))))) → 1(1(1(3(0(3(2(0(3(3(1(3(3(2(0(3(1(3(x1)))))))))))))))))) (26)
1(1(2(3(3(0(1(1(2(1(1(0(2(1(0(2(1(1(x1)))))))))))))))))) → 1(1(0(1(2(1(1(3(1(0(1(1(3(2(0(2(2(1(x1)))))))))))))))))) (27)
1(1(3(1(2(3(1(3(0(3(1(1(2(0(0(2(0(1(x1)))))))))))))))))) → 1(1(1(1(3(1(1(3(2(3(2(0(3(0(0(2(0(1(x1)))))))))))))))))) (28)
1(2(0(2(1(1(2(0(2(3(1(2(1(2(0(2(0(0(x1)))))))))))))))))) → 0(2(1(0(1(2(2(0(1(1(3(1(2(2(2(2(0(0(x1)))))))))))))))))) (29)
1(2(0(3(1(1(2(0(2(3(3(3(1(1(1(2(2(2(x1)))))))))))))))))) → 1(0(3(0(1(1(2(1(3(1(2(3(1(3(2(2(2(2(x1)))))))))))))))))) (30)
1(2(2(0(1(2(2(3(0(0(3(2(3(0(1(1(0(0(x1)))))))))))))))))) → 0(0(0(1(2(2(1(3(1(1(3(2(3(2(2(0(0(0(x1)))))))))))))))))) (31)
1(2(3(1(1(3(1(2(2(2(1(3(2(3(2(1(2(1(x1)))))))))))))))))) → 1(2(3(2(1(3(2(1(1(3(1(3(2(2(2(2(1(1(x1)))))))))))))))))) (32)
1(3(2(0(2(3(0(2(3(0(2(2(3(1(0(1(2(0(x1)))))))))))))))))) → 1(3(2(0(0(0(2(2(3(2(3(2(0(3(1(1(2(0(x1)))))))))))))))))) (33)
2(0(0(0(1(3(3(3(3(2(0(2(2(0(1(2(3(2(x1)))))))))))))))))) → 2(2(3(3(2(2(1(1(0(0(0(0(0(3(2(3(3(2(x1)))))))))))))))))) (34)
2(0(0(3(2(1(0(2(0(3(0(3(0(2(3(0(2(0(x1)))))))))))))))))) → 2(3(2(1(0(2(0(2(0(0(0(0(3(0(3(3(2(0(x1)))))))))))))))))) (35)
2(0(2(0(0(3(2(3(2(3(3(1(1(2(2(1(1(2(x1)))))))))))))))))) → 2(2(2(2(1(3(0(2(1(1(2(3(1(3(0(3(0(2(x1)))))))))))))))))) (36)
2(0(2(0(1(2(3(0(1(2(1(2(3(3(0(2(0(0(x1)))))))))))))))))) → 2(2(3(1(3(3(0(0(1(0(0(2(2(2(2(1(0(0(x1)))))))))))))))))) (37)
2(0(2(1(0(0(3(2(3(1(2(3(1(3(3(3(2(2(x1)))))))))))))))))) → 2(2(3(3(1(3(3(2(0(3(0(0(1(2(1(3(2(2(x1)))))))))))))))))) (38)
2(1(0(1(2(3(0(2(2(3(0(1(1(0(0(2(1(2(x1)))))))))))))))))) → 2(2(0(0(0(1(0(3(0(2(1(3(2(2(1(2(1(1(x1)))))))))))))))))) (39)
2(1(0(3(3(1(0(2(1(2(0(2(1(1(3(1(0(2(x1)))))))))))))))))) → 1(1(3(0(0(0(0(2(3(1(1(2(2(1(3(2(1(2(x1)))))))))))))))))) (40)
2(1(1(0(2(2(2(1(2(3(1(0(2(1(3(1(1(2(x1)))))))))))))))))) → 2(1(1(2(2(0(1(3(2(2(2(1(1(3(2(0(1(1(x1)))))))))))))))))) (41)
2(1(1(2(0(2(1(0(0(2(1(2(3(3(1(0(0(3(x1)))))))))))))))))) → 2(2(2(1(3(2(0(0(2(0(1(0(1(1(3(1(0(3(x1)))))))))))))))))) (42)
2(1(1(2(1(1(3(1(3(0(2(0(0(3(2(2(2(1(x1)))))))))))))))))) → 2(1(3(1(0(1(1(3(0(0(1(2(3(2(2(2(2(1(x1)))))))))))))))))) (43)
2(1(1(2(2(1(3(0(0(0(3(3(3(2(0(2(0(3(x1)))))))))))))))))) → 2(0(3(3(2(0(3(2(0(1(1(0(1(2(3(2(0(3(x1)))))))))))))))))) (44)
2(1(2(1(1(1(3(3(1(1(3(3(3(1(2(0(2(2(x1)))))))))))))))))) → 2(3(2(2(1(3(1(3(3(2(1(1(1(3(1(1(0(2(x1)))))))))))))))))) (45)
2(1(2(3(2(0(1(2(3(0(2(2(0(2(0(1(0(2(x1)))))))))))))))))) → 2(2(1(2(2(2(0(2(0(1(0(3(0(3(0(1(2(2(x1)))))))))))))))))) (46)
2(1(3(2(0(2(1(1(2(3(2(0(2(3(1(0(2(1(x1)))))))))))))))))) → 2(1(3(3(0(1(2(0(0(1(2(2(2(2(3(2(1(1(x1)))))))))))))))))) (47)
2(1(3(3(2(3(1(2(3(1(2(0(1(2(2(1(1(1(x1)))))))))))))))))) → 2(2(1(3(0(3(1(3(3(2(2(2(1(1(1(1(2(1(x1)))))))))))))))))) (48)
2(2(0(2(2(1(2(0(2(3(3(2(1(2(0(2(1(3(x1)))))))))))))))))) → 2(2(3(0(3(0(1(2(2(2(2(1(2(2(1(0(2(3(x1)))))))))))))))))) (49)
2(2(0(2(3(0(2(2(0(2(0(1(1(2(2(1(1(2(x1)))))))))))))))))) → 2(2(2(0(2(1(2(0(0(3(2(1(1(1(0(2(2(2(x1)))))))))))))))))) (50)
2(2(1(0(2(3(0(2(0(2(0(1(0(2(0(2(2(2(x1)))))))))))))))))) → 2(2(2(2(0(0(0(1(1(3(2(2(0(2(0(0(2(2(x1)))))))))))))))))) (51)
2(2(1(1(1(0(2(3(2(2(3(3(3(0(1(1(3(3(x1)))))))))))))))))) → 2(1(1(3(2(3(0(2(1(0(2(1(3(3(2(1(3(3(x1)))))))))))))))))) (52)
2(2(1(1(1(1(3(0(2(3(2(1(2(3(3(3(2(3(x1)))))))))))))))))) → 2(2(2(2(3(1(1(1(2(1(1(3(0(3(3(2(3(3(x1)))))))))))))))))) (53)
2(2(2(2(3(1(0(2(0(2(0(2(0(1(2(2(1(2(x1)))))))))))))))))) → 2(2(0(2(2(1(2(0(0(2(2(0(3(2(1(1(2(2(x1)))))))))))))))))) (54)
2(2(2(2(3(3(1(1(2(0(2(1(3(0(1(0(2(3(x1)))))))))))))))))) → 2(2(0(0(3(1(3(1(3(2(1(2(2(2(1(0(2(3(x1)))))))))))))))))) (55)
2(2(3(1(0(1(0(2(0(2(2(3(1(1(2(2(2(0(x1)))))))))))))))))) → 2(2(1(2(0(0(2(1(1(3(0(2(2(1(3(2(2(0(x1)))))))))))))))))) (56)
2(2(3(2(1(0(3(0(2(3(2(1(2(2(1(1(2(1(x1)))))))))))))))))) → 2(2(3(0(1(3(0(3(2(2(1(1(2(2(2(2(1(1(x1)))))))))))))))))) (57)
2(3(0(2(1(2(1(3(1(1(0(1(3(1(2(0(2(1(x1)))))))))))))))))) → 2(1(1(0(3(0(0(1(3(1(2(2(3(1(2(1(2(1(x1)))))))))))))))))) (58)
2(3(1(2(0(2(1(2(2(1(1(2(3(0(2(0(2(2(x1)))))))))))))))))) → 2(2(2(2(3(2(2(2(1(3(2(1(1(0(0(1(0(2(x1)))))))))))))))))) (59)
2(3(1(3(2(1(2(2(3(3(0(1(0(2(1(2(2(2(x1)))))))))))))))))) → 2(3(0(0(2(2(1(3(1(2(3(1(3(1(2(2(2(2(x1)))))))))))))))))) (60)
2(3(2(2(3(3(2(2(0(2(2(3(0(1(2(3(2(0(x1)))))))))))))))))) → 2(2(3(1(2(2(2(3(2(3(0(3(2(2(2(0(3(0(x1)))))))))))))))))) (61)
2(3(3(1(2(1(2(1(3(1(0(2(0(2(1(1(1(0(x1)))))))))))))))))) → 2(3(0(3(1(1(1(2(3(2(1(0(2(1(1(2(1(0(x1)))))))))))))))))) (62)
3(0(1(0(1(1(1(0(0(1(3(2(3(2(3(0(2(2(x1)))))))))))))))))) → 3(0(3(3(2(1(0(1(3(1(0(0(1(1(0(2(2(2(x1)))))))))))))))))) (63)
3(0(1(2(3(3(3(0(0(3(2(3(2(3(1(2(1(0(x1)))))))))))))))))) → 3(3(1(1(2(2(0(0(3(3(2(0(3(3(2(1(3(0(x1)))))))))))))))))) (64)
3(0(2(0(2(1(3(3(2(0(0(2(0(2(3(2(2(0(x1)))))))))))))))))) → 3(0(2(0(2(3(3(0(2(3(2(0(0(1(2(2(2(0(x1)))))))))))))))))) (65)
3(0(2(3(3(0(0(3(0(2(1(1(0(3(1(2(3(2(x1)))))))))))))))))) → 3(0(3(3(0(2(1(3(0(1(2(0(3(2(0(1(3(2(x1)))))))))))))))))) (66)
3(0(3(0(2(3(0(0(2(1(3(2(3(3(0(0(2(0(x1)))))))))))))))))) → 3(0(0(3(0(3(0(0(3(2(0(1(2(3(3(2(2(0(x1)))))))))))))))))) (67)
3(1(1(3(2(1(3(3(3(3(2(1(3(1(0(1(1(3(x1)))))))))))))))))) → 3(0(3(3(1(1(3(1(3(1(3(1(1(1(3(2(2(3(x1)))))))))))))))))) (68)
3(1(2(0(2(1(2(1(3(1(0(2(2(1(2(0(1(2(x1)))))))))))))))))) → 3(1(1(2(1(0(1(0(1(0(2(2(2(3(2(2(1(2(x1)))))))))))))))))) (69)
3(2(2(1(3(2(3(3(0(1(1(1(0(2(3(1(2(2(x1)))))))))))))))))) → 3(2(2(2(1(1(3(3(1(0(1(0(3(2(2(3(1(2(x1)))))))))))))))))) (70)
3(2(2(3(3(1(1(1(3(2(3(3(0(1(2(0(1(2(x1)))))))))))))))))) → 3(0(1(3(1(3(2(1(3(0(1(3(2(1(3(2(2(2(x1)))))))))))))))))) (71)
3(2(3(3(3(0(1(0(3(2(1(2(3(1(2(3(2(0(x1)))))))))))))))))) → 3(3(2(3(2(0(3(1(1(2(2(3(1(3(2(0(3(0(x1)))))))))))))))))) (72)
3(3(0(0(3(1(0(3(2(3(3(3(1(1(0(1(3(1(x1)))))))))))))))))) → 3(3(0(1(1(2(3(3(0(3(3(1(0(3(0(1(3(1(x1)))))))))))))))))) (73)
3(3(1(0(1(1(0(2(3(1(3(3(1(0(0(2(2(1(x1)))))))))))))))))) → 3(1(3(2(1(0(3(0(1(1(3(0(1(3(2(0(2(1(x1)))))))))))))))))) (74)
3(3(1(0(1(3(3(0(0(3(3(1(2(3(0(2(3(3(x1)))))))))))))))))) → 3(3(0(3(2(0(3(3(2(0(1(1(0(3(3(1(3(3(x1)))))))))))))))))) (75)
3(3(1(0(3(1(3(1(3(1(2(1(2(1(2(3(2(3(x1)))))))))))))))))) → 3(3(1(2(3(3(1(1(1(1(3(2(2(3(2(0(1(3(x1)))))))))))))))))) (76)
3(3(2(3(0(0(0(1(3(3(3(2(2(0(2(2(3(3(x1)))))))))))))))))) → 3(3(0(1(3(2(3(2(2(2(3(0(3(0(0(2(3(3(x1)))))))))))))))))) (77)
3(3(3(0(0(1(3(0(3(0(0(2(2(3(1(1(3(3(x1)))))))))))))))))) → 3(3(0(3(1(3(3(2(0(3(0(1(0(2(0(1(3(3(x1)))))))))))))))))) (78)
3(3(3(1(0(3(1(3(3(3(1(1(0(1(3(0(1(0(x1)))))))))))))))))) → 3(3(1(0(0(1(1(3(0(3(1(1(3(3(3(1(3(0(x1)))))))))))))))))) (79)
3(3(3(3(1(0(1(2(2(2(3(3(1(0(0(2(0(0(x1)))))))))))))))))) → 3(3(0(3(2(3(2(2(1(1(0(1(3(3(2(0(0(0(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
0(2(2(0(2(1(2(0(1(0(1(0(3(3(1(0(0(0(x1)))))))))))))))))) → 0(1(3(0(0(3(2(2(1(1(2(2(0(1(0(0(0(0(x1)))))))))))))))))) (82)
1(3(3(0(0(0(3(1(0(3(0(0(1(2(0(1(0(0(x1)))))))))))))))))) → 1(3(1(2(3(1(0(0(0(0(3(0(1(0(3(0(0(0(x1)))))))))))))))))) (83)
2(3(2(0(1(1(1(2(2(1(2(0(0(2(0(2(0(0(x1)))))))))))))))))) → 2(1(3(1(2(2(2(2(0(0(0(0(1(2(2(0(1(0(x1)))))))))))))))))) (84)
0(2(2(1(2(2(0(1(0(0(2(0(0(3(0(2(0(0(x1)))))))))))))))))) → 0(2(0(0(1(0(2(0(0(0(2(2(2(2(1(3(0(0(x1)))))))))))))))))) (85)
0(1(0(0(0(3(3(1(0(1(3(2(0(3(3(3(0(0(x1)))))))))))))))))) → 0(3(2(1(3(0(3(0(0(0(3(3(3(0(1(0(1(0(x1)))))))))))))))))) (86)
2(1(1(1(1(1(3(0(0(1(3(2(0(2(3(2(1(0(x1)))))))))))))))))) → 2(1(1(3(1(1(2(1(0(3(0(1(2(3(1(0(2(0(x1)))))))))))))))))) (87)
2(0(1(3(2(1(3(1(3(2(0(0(2(1(0(3(1(0(x1)))))))))))))))))) → 2(2(0(3(1(1(1(2(1(3(2(3(0(0(1(0(3(0(x1)))))))))))))))))) (88)
3(2(0(2(3(1(3(1(0(1(3(3(3(2(0(0(2(0(x1)))))))))))))))))) → 1(0(0(2(3(2(3(1(3(3(0(3(2(0(3(1(2(0(x1)))))))))))))))))) (89)
0(2(0(2(1(2(1(0(1(1(0(1(3(0(1(0(2(0(x1)))))))))))))))))) → 0(2(0(0(1(2(2(3(1(1(0(1(1(2(0(0(1(0(x1)))))))))))))))))) (90)
3(0(3(0(1(0(1(3(0(2(3(0(3(1(0(1(2(0(x1)))))))))))))))))) → 3(0(3(1(2(3(1(0(2(3(1(1(0(0(0(0(3(0(x1)))))))))))))))))) (91)
3(2(0(3(1(2(1(2(0(3(1(0(3(3(3(2(2(0(x1)))))))))))))))))) → 3(2(1(0(2(3(0(3(3(1(1(2(3(0(2(3(2(0(x1)))))))))))))))))) (92)
3(2(0(3(3(0(2(1(2(0(1(3(0(0(1(3(2(0(x1)))))))))))))))))) → 3(2(1(0(2(3(2(3(0(3(3(1(1(0(0(0(2(0(x1)))))))))))))))))) (93)
0(0(3(1(0(3(3(2(0(0(0(0(3(0(3(3(2(0(x1)))))))))))))))))) → 0(0(3(1(0(3(0(0(2(2(3(0(3(3(0(3(0(0(x1)))))))))))))))))) (94)
3(0(0(0(3(0(2(0(1(0(3(2(2(3(3(3(2(0(x1)))))))))))))))))) → 3(0(2(3(0(1(0(0(3(3(0(0(2(2(2(3(3(0(x1)))))))))))))))))) (95)
3(0(0(1(0(1(3(3(3(1(1(2(0(2(0(0(3(0(x1)))))))))))))))))) → 3(0(1(3(0(2(3(3(0(3(2(1(0(1(1(0(0(0(x1)))))))))))))))))) (96)
2(3(2(2(3(3(3(1(2(0(0(3(2(2(1(1(3(0(x1)))))))))))))))))) → 2(3(1(2(2(3(2(2(3(0(3(1(2(3(1(0(3(0(x1)))))))))))))))))) (97)
3(0(0(0(3(3(3(1(0(1(0(0(0(0(2(2(3(0(x1)))))))))))))))))) → 3(0(3(3(0(1(0(3(0(2(0(0(0(0(2(1(3(0(x1)))))))))))))))))) (98)
2(1(2(1(3(2(2(0(2(1(3(3(2(2(0(0(0(1(x1)))))))))))))))))) → 2(0(1(3(1(2(1(0(1(2(2(2(2(3(0(3(0(2(x1)))))))))))))))))) (99)
1(1(0(2(2(1(3(3(2(1(1(2(0(0(2(0(0(1(x1)))))))))))))))))) → 1(1(0(1(2(1(2(1(3(1(2(3(0(0(0(0(2(2(x1)))))))))))))))))) (100)
3(0(1(3(3(3(1(0(0(0(3(0(0(1(2(0(0(1(x1)))))))))))))))))) → 3(0(3(0(3(0(3(2(0(1(3(1(0(0(0(1(0(1(x1)))))))))))))))))) (101)
3(3(2(0(1(1(0(1(1(3(3(1(0(1(3(1(0(1(x1)))))))))))))))))) → 3(3(1(0(1(1(0(0(3(1(1(1(3(1(0(2(3(1(x1)))))))))))))))))) (102)
3(0(1(0(1(2(0(2(0(0(3(2(1(1(0(2(0(1(x1)))))))))))))))))) → 3(0(0(0(0(2(1(2(1(1(1(3(0(0(0(1(2(2(x1)))))))))))))))))) (103)
3(2(0(3(2(1(3(2(0(1(3(0(1(3(2(2(0(1(x1)))))))))))))))))) → 3(2(0(1(2(1(3(3(0(0(2(1(2(3(0(2(3(1(x1)))))))))))))))))) (104)
2(1(0(2(0(2(0(0(1(1(0(1(1(0(1(3(0(1(x1)))))))))))))))))) → 2(1(1(0(0(0(0(1(0(1(0(2(1(0(3(2(1(1(x1)))))))))))))))))) (105)
0(1(3(2(1(2(2(2(1(2(1(2(0(0(2(1(1(1(x1)))))))))))))))))) → 0(1(2(2(3(1(0(2(0(1(1(2(2(2(2(1(1(1(x1)))))))))))))))))) (106)
1(0(3(3(3(1(2(0(0(3(3(3(3(1(3(2(1(1(x1)))))))))))))))))) → 3(1(3(0(2(3(3(1(3(3(0(2(3(0(3(1(1(1(x1)))))))))))))))))) (107)
1(1(2(0(1(2(0(1(1(2(1(1(0(3(3(2(1(1(x1)))))))))))))))))) → 1(2(2(0(2(3(1(1(0(1(3(1(1(2(1(0(1(1(x1)))))))))))))))))) (108)
1(0(2(0(0(2(1(1(3(0(3(1(3(2(1(3(1(1(x1)))))))))))))))))) → 1(0(2(0(0(3(0(2(3(2(3(1(1(3(1(1(1(1(x1)))))))))))))))))) (109)
0(0(2(0(2(1(2(1(3(2(0(2(1(1(2(0(2(1(x1)))))))))))))))))) → 0(0(2(2(2(2(1(3(1(1(0(2(2(1(0(1(2(0(x1)))))))))))))))))) (110)
2(2(2(1(1(1(3(3(3(2(0(2(1(1(3(0(2(1(x1)))))))))))))))))) → 2(2(2(2(3(1(3(2(1(3(1(2(1(1(0(3(0(1(x1)))))))))))))))))) (111)
0(0(1(1(0(3(2(3(0(0(3(2(2(1(0(2(2(1(x1)))))))))))))))))) → 0(0(0(2(2(3(2(3(1(1(3(1(2(2(1(0(0(0(x1)))))))))))))))))) (112)
1(2(1(2(3(2(3(1(2(2(2(1(3(1(1(3(2(1(x1)))))))))))))))))) → 1(1(2(2(2(2(3(1(3(1(1(2(3(1(2(3(2(1(x1)))))))))))))))))) (113)
0(2(1(0(1(3(2(2(0(3(2(0(3(2(0(2(3(1(x1)))))))))))))))))) → 0(2(1(1(3(0(2(3(2(3(2(2(0(0(0(2(3(1(x1)))))))))))))))))) (114)
2(3(2(1(0(2(2(0(2(3(3(3(3(1(0(0(0(2(x1)))))))))))))))))) → 2(3(3(2(3(0(0(0(0(0(1(1(2(2(3(3(2(2(x1)))))))))))))))))) (115)
0(2(0(3(2(0(3(0(3(0(2(0(1(2(3(0(0(2(x1)))))))))))))))))) → 0(2(3(3(0(3(0(0(0(0(2(0(2(0(1(2(3(2(x1)))))))))))))))))) (116)
2(1(1(2(2(1(1(3(3(2(3(2(3(0(0(2(0(2(x1)))))))))))))))))) → 2(0(3(0(3(1(3(2(1(1(2(0(3(1(2(2(2(2(x1)))))))))))))))))) (117)
0(0(2(0(3(3(2(1(2(1(0(3(2(1(0(2(0(2(x1)))))))))))))))))) → 0(0(1(2(2(2(2(0(0(1(0(0(3(3(1(3(2(2(x1)))))))))))))))))) (118)
2(2(3(3(3(1(3(2(1(3(2(3(0(0(1(2(0(2(x1)))))))))))))))))) → 2(2(3(1(2(1(0(0(3(0(2(3(3(1(3(3(2(2(x1)))))))))))))))))) (119)
2(1(2(0(0(1(1(0(3(2(2(0(3(2(1(0(1(2(x1)))))))))))))))))) → 1(1(2(1(2(2(3(1(2(0(3(0(1(0(0(0(2(2(x1)))))))))))))))))) (120)
2(0(1(3(1(1(2(0(2(1(2(0(1(3(3(0(1(2(x1)))))))))))))))))) → 2(1(2(3(1(2(2(1(1(3(2(0(0(0(0(3(1(1(x1)))))))))))))))))) (121)
2(1(1(3(1(2(0(1(3(2(1(2(2(2(0(1(1(2(x1)))))))))))))))))) → 1(1(0(2(3(1(1(2(2(2(3(1(0(2(2(1(1(2(x1)))))))))))))))))) (122)
3(0(0(1(3(3(2(1(2(0(0(1(2(0(2(1(1(2(x1)))))))))))))))))) → 3(0(1(3(1(1(0(1(0(2(0(0(2(3(1(2(2(2(x1)))))))))))))))))) (123)
1(2(2(2(3(0(0(2(0(3(1(3(1(1(2(1(1(2(x1)))))))))))))))))) → 1(2(2(2(2(3(2(1(0(0(3(1(1(0(1(3(1(2(x1)))))))))))))))))) (124)
3(0(2(0(2(3(3(3(0(0(0(3(1(2(2(1(1(2(x1)))))))))))))))))) → 3(0(2(3(2(1(0(1(1(0(2(3(0(2(3(3(0(2(x1)))))))))))))))))) (125)
2(2(0(2(1(3(3(3(1(1(3(3(1(1(1(2(1(2(x1)))))))))))))))))) → 2(0(1(1(3(1(1(1(2(3(3(1(3(1(2(2(3(2(x1)))))))))))))))))) (126)
2(0(1(0(2(0(2(2(0(3(2(1(0(2(3(2(1(2(x1)))))))))))))))))) → 2(2(1(0(3(0(3(0(1(0(2(0(2(2(2(1(2(2(x1)))))))))))))))))) (127)
1(2(0(1(3(2(0(2(3(2(1(1(2(0(2(3(1(2(x1)))))))))))))))))) → 1(1(2(3(2(2(2(2(1(0(0(2(1(0(3(3(1(2(x1)))))))))))))))))) (128)
1(1(1(2(2(1(0(2(1(3(2(1(3(2(3(3(1(2(x1)))))))))))))))))) → 1(2(1(1(1(1(2(2(2(3(3(1(3(0(3(1(2(2(x1)))))))))))))))))) (129)
3(1(2(0(2(1(2(3(3(2(0(2(1(2(2(0(2(2(x1)))))))))))))))))) → 3(2(0(1(2(2(1(2(2(2(2(1(0(3(0(3(2(2(x1)))))))))))))))))) (130)
2(1(1(2(2(1(1(0(2(0(2(2(0(3(2(0(2(2(x1)))))))))))))))))) → 2(2(2(0(1(1(1(2(3(0(0(2(1(2(0(2(2(2(x1)))))))))))))))))) (131)
2(2(2(0(2(0(1(0(2(0(2(0(3(2(0(1(2(2(x1)))))))))))))))))) → 2(2(0(0(2(0(2(2(3(1(1(0(0(0(2(2(2(2(x1)))))))))))))))))) (132)
3(3(1(1(0(3(3(3(2(2(3(2(0(1(1(1(2(2(x1)))))))))))))))))) → 3(3(1(2(3(3(1(2(0(1(2(0(3(2(3(1(1(2(x1)))))))))))))))))) (133)
3(2(3(3(3(2(1(2(3(2(0(3(1(1(1(1(2(2(x1)))))))))))))))))) → 3(3(2(3(3(0(3(1(1(2(1(1(1(3(2(2(2(2(x1)))))))))))))))))) (134)
2(1(2(2(1(0(2(0(2(0(2(0(1(3(2(2(2(2(x1)))))))))))))))))) → 2(2(1(1(2(3(0(2(2(0(0(2(1(2(2(0(2(2(x1)))))))))))))))))) (135)
3(2(0(1(0(3(1(2(0(2(1(1(3(3(2(2(2(2(x1)))))))))))))))))) → 3(2(0(1(2(2(2(1(2(3(1(3(1(3(0(0(2(2(x1)))))))))))))))))) (136)
0(2(2(2(1(1(3(2(2(0(2(0(1(0(1(3(2(2(x1)))))))))))))))))) → 0(2(2(3(1(2(2(0(3(1(1(2(0(0(2(1(2(2(x1)))))))))))))))))) (137)
1(2(1(1(2(2(1(2(3(2(0(3(0(1(2(3(2(2(x1)))))))))))))))))) → 1(1(2(2(2(2(1(1(2(2(3(0(3(1(0(3(2(2(x1)))))))))))))))))) (138)
1(2(0(2(1(3(1(0(1(1(3(1(2(1(2(0(3(2(x1)))))))))))))))))) → 1(2(1(2(1(3(2(2(1(3(1(0(0(3(0(1(1(2(x1)))))))))))))))))) (139)
2(2(0(2(0(3(2(1(1(2(2(1(2(0(2(1(3(2(x1)))))))))))))))))) → 2(0(1(0(0(1(1(2(3(1(2(2(2(3(2(2(2(2(x1)))))))))))))))))) (140)
2(2(2(1(2(0(1(0(3(3(2(2(1(2(3(1(3(2(x1)))))))))))))))))) → 2(2(2(2(1(3(1(3(2(1(3(1(2(2(0(0(3(2(x1)))))))))))))))))) (141)
0(2(3(2(1(0(3(2(2(0(2(2(3(3(2(2(3(2(x1)))))))))))))))))) → 0(3(0(2(2(2(3(0(3(2(3(2(2(2(1(3(2(2(x1)))))))))))))))))) (142)
0(1(1(1(2(0(2(0(1(3(1(2(1(2(1(3(3(2(x1)))))))))))))))))) → 0(1(2(1(1(2(0(1(2(3(2(1(1(1(3(0(3(2(x1)))))))))))))))))) (143)
2(2(0(3(2(3(2(3(1(0(0(1(1(1(0(1(0(3(x1)))))))))))))))))) → 2(2(2(0(1(1(0(0(1(3(1(0(1(2(3(3(0(3(x1)))))))))))))))))) (144)
0(1(2(1(3(2(3(2(3(0(0(3(3(3(2(1(0(3(x1)))))))))))))))))) → 0(3(1(2(3(3(0(2(3(3(0(0(2(2(1(1(3(3(x1)))))))))))))))))) (145)
0(2(2(3(2(0(2(0(0(2(3(3(1(2(0(2(0(3(x1)))))))))))))))))) → 0(2(2(2(1(0(0(2(3(2(0(3(3(2(0(2(0(3(x1)))))))))))))))))) (146)
2(3(2(1(3(0(1(1(2(0(3(0(0(3(3(2(0(3(x1)))))))))))))))))) → 2(3(1(0(2(3(0(2(1(0(3(1(2(0(3(3(0(3(x1)))))))))))))))))) (147)
0(2(0(0(3(3(2(3(1(2(0(0(3(2(0(3(0(3(x1)))))))))))))))))) → 0(2(2(3(3(2(1(0(2(3(0(0(3(0(3(0(0(3(x1)))))))))))))))))) (148)
3(1(1(0(1(3(1(2(3(3(3(3(1(2(3(1(1(3(x1)))))))))))))))))) → 3(2(2(3(1(1(1(3(1(3(1(3(1(1(3(3(0(3(x1)))))))))))))))))) (149)
2(1(0(2(1(2(2(0(1(3(1(2(1(2(0(2(1(3(x1)))))))))))))))))) → 2(1(2(2(3(2(2(2(0(1(0(1(0(1(2(1(1(3(x1)))))))))))))))))) (150)
2(2(1(3(2(0(1(1(1(0(3(3(2(3(1(2(2(3(x1)))))))))))))))))) → 2(1(3(2(2(3(0(1(0(1(3(3(1(1(2(2(2(3(x1)))))))))))))))))) (151)
2(1(0(2(1(0(3(3(2(3(1(1(1(3(3(2(2(3(x1)))))))))))))))))) → 2(2(2(3(1(2(3(1(0(3(1(2(3(1(3(1(0(3(x1)))))))))))))))))) (152)
0(2(3(2(1(3(2(1(2(3(0(1(0(3(3(3(2(3(x1)))))))))))))))))) → 0(3(0(2(3(1(3(2(2(1(1(3(0(2(3(2(3(3(x1)))))))))))))))))) (153)
1(3(1(0(1(1(3(3(3(2(3(0(1(3(0(0(3(3(x1)))))))))))))))))) → 1(3(1(0(3(0(1(3(3(0(3(3(2(1(1(0(3(3(x1)))))))))))))))))) (154)
1(2(2(0(0(1(3(3(1(3(2(0(1(1(0(1(3(3(x1)))))))))))))))))) → 1(2(0(2(3(1(0(3(1(1(0(3(0(1(2(3(1(3(x1)))))))))))))))))) (155)
3(3(2(0(3(2(1(3(3(0(0(3(3(1(0(1(3(3(x1)))))))))))))))))) → 3(3(1(3(3(0(1(1(0(2(3(3(0(2(3(0(3(3(x1)))))))))))))))))) (156)
3(2(3(2(1(2(1(2(1(3(1(3(1(3(0(1(3(3(x1)))))))))))))))))) → 3(1(0(2(3(2(2(3(1(1(1(1(3(3(2(1(3(3(x1)))))))))))))))))) (157)
3(3(2(2(0(2(2(3(3(3(1(0(0(0(3(2(3(3(x1)))))))))))))))))) → 3(3(2(0(0(3(0(3(2(2(2(3(2(3(1(0(3(3(x1)))))))))))))))))) (158)
3(3(1(1(3(2(2(0(0(3(0(3(1(0(0(3(3(3(x1)))))))))))))))))) → 3(3(1(0(2(0(1(0(3(0(2(3(3(1(3(0(3(3(x1)))))))))))))))))) (159)
0(1(0(3(1(0(1(1(3(3(3(1(3(0(1(3(3(3(x1)))))))))))))))))) → 0(3(1(3(3(3(1(1(3(0(3(1(1(0(0(1(3(3(x1)))))))))))))))))) (160)
0(0(2(0(0(1(3(3(2(2(2(1(0(1(3(3(3(3(x1)))))))))))))))))) → 0(0(0(2(3(3(1(0(1(1(2(2(3(2(3(0(3(3(x1)))))))))))))))))) (161)

1.1 Closure Under Flat Contexts

Using the flat contexts

{0(☐), 2(☐), 1(☐), 3(☐)}

We obtain the transformed TRS
0(2(2(0(2(1(2(0(1(0(1(0(3(3(1(0(0(0(x1)))))))))))))))))) → 0(1(3(0(0(3(2(2(1(1(2(2(0(1(0(0(0(0(x1)))))))))))))))))) (82)
1(3(3(0(0(0(3(1(0(3(0(0(1(2(0(1(0(0(x1)))))))))))))))))) → 1(3(1(2(3(1(0(0(0(0(3(0(1(0(3(0(0(0(x1)))))))))))))))))) (83)
2(3(2(0(1(1(1(2(2(1(2(0(0(2(0(2(0(0(x1)))))))))))))))))) → 2(1(3(1(2(2(2(2(0(0(0(0(1(2(2(0(1(0(x1)))))))))))))))))) (84)
0(2(2(1(2(2(0(1(0(0(2(0(0(3(0(2(0(0(x1)))))))))))))))))) → 0(2(0(0(1(0(2(0(0(0(2(2(2(2(1(3(0(0(x1)))))))))))))))))) (85)
0(1(0(0(0(3(3(1(0(1(3(2(0(3(3(3(0(0(x1)))))))))))))))))) → 0(3(2(1(3(0(3(0(0(0(3(3(3(0(1(0(1(0(x1)))))))))))))))))) (86)
2(1(1(1(1(1(3(0(0(1(3(2(0(2(3(2(1(0(x1)))))))))))))))))) → 2(1(1(3(1(1(2(1(0(3(0(1(2(3(1(0(2(0(x1)))))))))))))))))) (87)
2(0(1(3(2(1(3(1(3(2(0(0(2(1(0(3(1(0(x1)))))))))))))))))) → 2(2(0(3(1(1(1(2(1(3(2(3(0(0(1(0(3(0(x1)))))))))))))))))) (88)
0(2(0(2(1(2(1(0(1(1(0(1(3(0(1(0(2(0(x1)))))))))))))))))) → 0(2(0(0(1(2(2(3(1(1(0(1(1(2(0(0(1(0(x1)))))))))))))))))) (90)
3(0(3(0(1(0(1(3(0(2(3(0(3(1(0(1(2(0(x1)))))))))))))))))) → 3(0(3(1(2(3(1(0(2(3(1(1(0(0(0(0(3(0(x1)))))))))))))))))) (91)
3(2(0(3(1(2(1(2(0(3(1(0(3(3(3(2(2(0(x1)))))))))))))))))) → 3(2(1(0(2(3(0(3(3(1(1(2(3(0(2(3(2(0(x1)))))))))))))))))) (92)
3(2(0(3(3(0(2(1(2(0(1(3(0(0(1(3(2(0(x1)))))))))))))))))) → 3(2(1(0(2(3(2(3(0(3(3(1(1(0(0(0(2(0(x1)))))))))))))))))) (93)
0(0(3(1(0(3(3(2(0(0(0(0(3(0(3(3(2(0(x1)))))))))))))))))) → 0(0(3(1(0(3(0(0(2(2(3(0(3(3(0(3(0(0(x1)))))))))))))))))) (94)
3(0(0(0(3(0(2(0(1(0(3(2(2(3(3(3(2(0(x1)))))))))))))))))) → 3(0(2(3(0(1(0(0(3(3(0(0(2(2(2(3(3(0(x1)))))))))))))))))) (95)
3(0(0(1(0(1(3(3(3(1(1(2(0(2(0(0(3(0(x1)))))))))))))))))) → 3(0(1(3(0(2(3(3(0(3(2(1(0(1(1(0(0(0(x1)))))))))))))))))) (96)
2(3(2(2(3(3(3(1(2(0(0(3(2(2(1(1(3(0(x1)))))))))))))))))) → 2(3(1(2(2(3(2(2(3(0(3(1(2(3(1(0(3(0(x1)))))))))))))))))) (97)
3(0(0(0(3(3(3(1(0(1(0(0(0(0(2(2(3(0(x1)))))))))))))))))) → 3(0(3(3(0(1(0(3(0(2(0(0(0(0(2(1(3(0(x1)))))))))))))))))) (98)
2(1(2(1(3(2(2(0(2(1(3(3(2(2(0(0(0(1(x1)))))))))))))))))) → 2(0(1(3(1(2(1(0(1(2(2(2(2(3(0(3(0(2(x1)))))))))))))))))) (99)
1(1(0(2(2(1(3(3(2(1(1(2(0(0(2(0(0(1(x1)))))))))))))))))) → 1(1(0(1(2(1(2(1(3(1(2(3(0(0(0(0(2(2(x1)))))))))))))))))) (100)
3(0(1(3(3(3(1(0(0(0(3(0(0(1(2(0(0(1(x1)))))))))))))))))) → 3(0(3(0(3(0(3(2(0(1(3(1(0(0(0(1(0(1(x1)))))))))))))))))) (101)
3(3(2(0(1(1(0(1(1(3(3(1(0(1(3(1(0(1(x1)))))))))))))))))) → 3(3(1(0(1(1(0(0(3(1(1(1(3(1(0(2(3(1(x1)))))))))))))))))) (102)
3(0(1(0(1(2(0(2(0(0(3(2(1(1(0(2(0(1(x1)))))))))))))))))) → 3(0(0(0(0(2(1(2(1(1(1(3(0(0(0(1(2(2(x1)))))))))))))))))) (103)
3(2(0(3(2(1(3(2(0(1(3(0(1(3(2(2(0(1(x1)))))))))))))))))) → 3(2(0(1(2(1(3(3(0(0(2(1(2(3(0(2(3(1(x1)))))))))))))))))) (104)
2(1(0(2(0(2(0(0(1(1(0(1(1(0(1(3(0(1(x1)))))))))))))))))) → 2(1(1(0(0(0(0(1(0(1(0(2(1(0(3(2(1(1(x1)))))))))))))))))) (105)
0(1(3(2(1(2(2(2(1(2(1(2(0(0(2(1(1(1(x1)))))))))))))))))) → 0(1(2(2(3(1(0(2(0(1(1(2(2(2(2(1(1(1(x1)))))))))))))))))) (106)
1(1(2(0(1(2(0(1(1(2(1(1(0(3(3(2(1(1(x1)))))))))))))))))) → 1(2(2(0(2(3(1(1(0(1(3(1(1(2(1(0(1(1(x1)))))))))))))))))) (108)
1(0(2(0(0(2(1(1(3(0(3(1(3(2(1(3(1(1(x1)))))))))))))))))) → 1(0(2(0(0(3(0(2(3(2(3(1(1(3(1(1(1(1(x1)))))))))))))))))) (109)
0(0(2(0(2(1(2(1(3(2(0(2(1(1(2(0(2(1(x1)))))))))))))))))) → 0(0(2(2(2(2(1(3(1(1(0(2(2(1(0(1(2(0(x1)))))))))))))))))) (110)
2(2(2(1(1(1(3(3(3(2(0(2(1(1(3(0(2(1(x1)))))))))))))))))) → 2(2(2(2(3(1(3(2(1(3(1(2(1(1(0(3(0(1(x1)))))))))))))))))) (111)
0(0(1(1(0(3(2(3(0(0(3(2(2(1(0(2(2(1(x1)))))))))))))))))) → 0(0(0(2(2(3(2(3(1(1(3(1(2(2(1(0(0(0(x1)))))))))))))))))) (112)
1(2(1(2(3(2(3(1(2(2(2(1(3(1(1(3(2(1(x1)))))))))))))))))) → 1(1(2(2(2(2(3(1(3(1(1(2(3(1(2(3(2(1(x1)))))))))))))))))) (113)
0(2(1(0(1(3(2(2(0(3(2(0(3(2(0(2(3(1(x1)))))))))))))))))) → 0(2(1(1(3(0(2(3(2(3(2(2(0(0(0(2(3(1(x1)))))))))))))))))) (114)
2(3(2(1(0(2(2(0(2(3(3(3(3(1(0(0(0(2(x1)))))))))))))))))) → 2(3(3(2(3(0(0(0(0(0(1(1(2(2(3(3(2(2(x1)))))))))))))))))) (115)
0(2(0(3(2(0(3(0(3(0(2(0(1(2(3(0(0(2(x1)))))))))))))))))) → 0(2(3(3(0(3(0(0(0(0(2(0(2(0(1(2(3(2(x1)))))))))))))))))) (116)
2(1(1(2(2(1(1(3(3(2(3(2(3(0(0(2(0(2(x1)))))))))))))))))) → 2(0(3(0(3(1(3(2(1(1(2(0(3(1(2(2(2(2(x1)))))))))))))))))) (117)
0(0(2(0(3(3(2(1(2(1(0(3(2(1(0(2(0(2(x1)))))))))))))))))) → 0(0(1(2(2(2(2(0(0(1(0(0(3(3(1(3(2(2(x1)))))))))))))))))) (118)
2(2(3(3(3(1(3(2(1(3(2(3(0(0(1(2(0(2(x1)))))))))))))))))) → 2(2(3(1(2(1(0(0(3(0(2(3(3(1(3(3(2(2(x1)))))))))))))))))) (119)
2(0(1(3(1(1(2(0(2(1(2(0(1(3(3(0(1(2(x1)))))))))))))))))) → 2(1(2(3(1(2(2(1(1(3(2(0(0(0(0(3(1(1(x1)))))))))))))))))) (121)
3(0(0(1(3(3(2(1(2(0(0(1(2(0(2(1(1(2(x1)))))))))))))))))) → 3(0(1(3(1(1(0(1(0(2(0(0(2(3(1(2(2(2(x1)))))))))))))))))) (123)
1(2(2(2(3(0(0(2(0(3(1(3(1(1(2(1(1(2(x1)))))))))))))))))) → 1(2(2(2(2(3(2(1(0(0(3(1(1(0(1(3(1(2(x1)))))))))))))))))) (124)
3(0(2(0(2(3(3(3(0(0(0(3(1(2(2(1(1(2(x1)))))))))))))))))) → 3(0(2(3(2(1(0(1(1(0(2(3(0(2(3(3(0(2(x1)))))))))))))))))) (125)
2(2(0(2(1(3(3(3(1(1(3(3(1(1(1(2(1(2(x1)))))))))))))))))) → 2(0(1(1(3(1(1(1(2(3(3(1(3(1(2(2(3(2(x1)))))))))))))))))) (126)
2(0(1(0(2(0(2(2(0(3(2(1(0(2(3(2(1(2(x1)))))))))))))))))) → 2(2(1(0(3(0(3(0(1(0(2(0(2(2(2(1(2(2(x1)))))))))))))))))) (127)
1(2(0(1(3(2(0(2(3(2(1(1(2(0(2(3(1(2(x1)))))))))))))))))) → 1(1(2(3(2(2(2(2(1(0(0(2(1(0(3(3(1(2(x1)))))))))))))))))) (128)
1(1(1(2(2(1(0(2(1(3(2(1(3(2(3(3(1(2(x1)))))))))))))))))) → 1(2(1(1(1(1(2(2(2(3(3(1(3(0(3(1(2(2(x1)))))))))))))))))) (129)
3(1(2(0(2(1(2(3(3(2(0(2(1(2(2(0(2(2(x1)))))))))))))))))) → 3(2(0(1(2(2(1(2(2(2(2(1(0(3(0(3(2(2(x1)))))))))))))))))) (130)
2(1(1(2(2(1(1(0(2(0(2(2(0(3(2(0(2(2(x1)))))))))))))))))) → 2(2(2(0(1(1(1(2(3(0(0(2(1(2(0(2(2(2(x1)))))))))))))))))) (131)
2(2(2(0(2(0(1(0(2(0(2(0(3(2(0(1(2(2(x1)))))))))))))))))) → 2(2(0(0(2(0(2(2(3(1(1(0(0(0(2(2(2(2(x1)))))))))))))))))) (132)
3(3(1(1(0(3(3(3(2(2(3(2(0(1(1(1(2(2(x1)))))))))))))))))) → 3(3(1(2(3(3(1(2(0(1(2(0(3(2(3(1(1(2(x1)))))))))))))))))) (133)
3(2(3(3(3(2(1(2(3(2(0(3(1(1(1(1(2(2(x1)))))))))))))))))) → 3(3(2(3(3(0(3(1(1(2(1(1(1(3(2(2(2(2(x1)))))))))))))))))) (134)
2(1(2(2(1(0(2(0(2(0(2(0(1(3(2(2(2(2(x1)))))))))))))))))) → 2(2(1(1(2(3(0(2(2(0(0(2(1(2(2(0(2(2(x1)))))))))))))))))) (135)
3(2(0(1(0(3(1(2(0(2(1(1(3(3(2(2(2(2(x1)))))))))))))))))) → 3(2(0(1(2(2(2(1(2(3(1(3(1(3(0(0(2(2(x1)))))))))))))))))) (136)
0(2(2(2(1(1(3(2(2(0(2(0(1(0(1(3(2(2(x1)))))))))))))))))) → 0(2(2(3(1(2(2(0(3(1(1(2(0(0(2(1(2(2(x1)))))))))))))))))) (137)
1(2(1(1(2(2(1(2(3(2(0(3(0(1(2(3(2(2(x1)))))))))))))))))) → 1(1(2(2(2(2(1(1(2(2(3(0(3(1(0(3(2(2(x1)))))))))))))))))) (138)
1(2(0(2(1(3(1(0(1(1(3(1(2(1(2(0(3(2(x1)))))))))))))))))) → 1(2(1(2(1(3(2(2(1(3(1(0(0(3(0(1(1(2(x1)))))))))))))))))) (139)
2(2(0(2(0(3(2(1(1(2(2(1(2(0(2(1(3(2(x1)))))))))))))))))) → 2(0(1(0(0(1(1(2(3(1(2(2(2(3(2(2(2(2(x1)))))))))))))))))) (140)
2(2(2(1(2(0(1(0(3(3(2(2(1(2(3(1(3(2(x1)))))))))))))))))) → 2(2(2(2(1(3(1(3(2(1(3(1(2(2(0(0(3(2(x1)))))))))))))))))) (141)
0(2(3(2(1(0(3(2(2(0(2(2(3(3(2(2(3(2(x1)))))))))))))))))) → 0(3(0(2(2(2(3(0(3(2(3(2(2(2(1(3(2(2(x1)))))))))))))))))) (142)
0(1(1(1(2(0(2(0(1(3(1(2(1(2(1(3(3(2(x1)))))))))))))))))) → 0(1(2(1(1(2(0(1(2(3(2(1(1(1(3(0(3(2(x1)))))))))))))))))) (143)
2(2(0(3(2(3(2(3(1(0(0(1(1(1(0(1(0(3(x1)))))))))))))))))) → 2(2(2(0(1(1(0(0(1(3(1(0(1(2(3(3(0(3(x1)))))))))))))))))) (144)
0(1(2(1(3(2(3(2(3(0(0(3(3(3(2(1(0(3(x1)))))))))))))))))) → 0(3(1(2(3(3(0(2(3(3(0(0(2(2(1(1(3(3(x1)))))))))))))))))) (145)
0(2(2(3(2(0(2(0(0(2(3(3(1(2(0(2(0(3(x1)))))))))))))))))) → 0(2(2(2(1(0(0(2(3(2(0(3(3(2(0(2(0(3(x1)))))))))))))))))) (146)
2(3(2(1(3(0(1(1(2(0(3(0(0(3(3(2(0(3(x1)))))))))))))))))) → 2(3(1(0(2(3(0(2(1(0(3(1(2(0(3(3(0(3(x1)))))))))))))))))) (147)
0(2(0(0(3(3(2(3(1(2(0(0(3(2(0(3(0(3(x1)))))))))))))))))) → 0(2(2(3(3(2(1(0(2(3(0(0(3(0(3(0(0(3(x1)))))))))))))))))) (148)
3(1(1(0(1(3(1(2(3(3(3(3(1(2(3(1(1(3(x1)))))))))))))))))) → 3(2(2(3(1(1(1(3(1(3(1(3(1(1(3(3(0(3(x1)))))))))))))))))) (149)
2(1(0(2(1(2(2(0(1(3(1(2(1(2(0(2(1(3(x1)))))))))))))))))) → 2(1(2(2(3(2(2(2(0(1(0(1(0(1(2(1(1(3(x1)))))))))))))))))) (150)
2(2(1(3(2(0(1(1(1(0(3(3(2(3(1(2(2(3(x1)))))))))))))))))) → 2(1(3(2(2(3(0(1(0(1(3(3(1(1(2(2(2(3(x1)))))))))))))))))) (151)
2(1(0(2(1(0(3(3(2(3(1(1(1(3(3(2(2(3(x1)))))))))))))))))) → 2(2(2(3(1(2(3(1(0(3(1(2(3(1(3(1(0(3(x1)))))))))))))))))) (152)
0(2(3(2(1(3(2(1(2(3(0(1(0(3(3(3(2(3(x1)))))))))))))))))) → 0(3(0(2(3(1(3(2(2(1(1(3(0(2(3(2(3(3(x1)))))))))))))))))) (153)
1(3(1(0(1(1(3(3(3(2(3(0(1(3(0(0(3(3(x1)))))))))))))))))) → 1(3(1(0(3(0(1(3(3(0(3(3(2(1(1(0(3(3(x1)))))))))))))))))) (154)
1(2(2(0(0(1(3(3(1(3(2(0(1(1(0(1(3(3(x1)))))))))))))))))) → 1(2(0(2(3(1(0(3(1(1(0(3(0(1(2(3(1(3(x1)))))))))))))))))) (155)
3(3(2(0(3(2(1(3(3(0(0(3(3(1(0(1(3(3(x1)))))))))))))))))) → 3(3(1(3(3(0(1(1(0(2(3(3(0(2(3(0(3(3(x1)))))))))))))))))) (156)
3(2(3(2(1(2(1(2(1(3(1(3(1(3(0(1(3(3(x1)))))))))))))))))) → 3(1(0(2(3(2(2(3(1(1(1(1(3(3(2(1(3(3(x1)))))))))))))))))) (157)
3(3(2(2(0(2(2(3(3(3(1(0(0(0(3(2(3(3(x1)))))))))))))))))) → 3(3(2(0(0(3(0(3(2(2(2(3(2(3(1(0(3(3(x1)))))))))))))))))) (158)
3(3(1(1(3(2(2(0(0(3(0(3(1(0(0(3(3(3(x1)))))))))))))))))) → 3(3(1(0(2(0(1(0(3(0(2(3(3(1(3(0(3(3(x1)))))))))))))))))) (159)
0(1(0(3(1(0(1(1(3(3(3(1(3(0(1(3(3(3(x1)))))))))))))))))) → 0(3(1(3(3(3(1(1(3(0(3(1(1(0(0(1(3(3(x1)))))))))))))))))) (160)
0(0(2(0(0(1(3(3(2(2(2(1(0(1(3(3(3(3(x1)))))))))))))))))) → 0(0(0(2(3(3(1(0(1(1(2(2(3(2(3(0(3(3(x1)))))))))))))))))) (161)
0(3(2(0(2(3(1(3(1(0(1(3(3(3(2(0(0(2(0(x1))))))))))))))))))) → 0(1(0(0(2(3(2(3(1(3(3(0(3(2(0(3(1(2(0(x1))))))))))))))))))) (162)
2(3(2(0(2(3(1(3(1(0(1(3(3(3(2(0(0(2(0(x1))))))))))))))))))) → 2(1(0(0(2(3(2(3(1(3(3(0(3(2(0(3(1(2(0(x1))))))))))))))))))) (163)
1(3(2(0(2(3(1(3(1(0(1(3(3(3(2(0(0(2(0(x1))))))))))))))))))) → 1(1(0(0(2(3(2(3(1(3(3(0(3(2(0(3(1(2(0(x1))))))))))))))))))) (164)
3(3(2(0(2(3(1(3(1(0(1(3(3(3(2(0(0(2(0(x1))))))))))))))))))) → 3(1(0(0(2(3(2(3(1(3(3(0(3(2(0(3(1(2(0(x1))))))))))))))))))) (165)
0(1(0(3(3(3(1(2(0(0(3(3(3(3(1(3(2(1(1(x1))))))))))))))))))) → 0(3(1(3(0(2(3(3(1(3(3(0(2(3(0(3(1(1(1(x1))))))))))))))))))) (166)
2(1(0(3(3(3(1(2(0(0(3(3(3(3(1(3(2(1(1(x1))))))))))))))))))) → 2(3(1(3(0(2(3(3(1(3(3(0(2(3(0(3(1(1(1(x1))))))))))))))))))) (167)
1(1(0(3(3(3(1(2(0(0(3(3(3(3(1(3(2(1(1(x1))))))))))))))))))) → 1(3(1(3(0(2(3(3(1(3(3(0(2(3(0(3(1(1(1(x1))))))))))))))))))) (168)
3(1(0(3(3(3(1(2(0(0(3(3(3(3(1(3(2(1(1(x1))))))))))))))))))) → 3(3(1(3(0(2(3(3(1(3(3(0(2(3(0(3(1(1(1(x1))))))))))))))))))) (169)
0(2(1(2(0(0(1(1(0(3(2(2(0(3(2(1(0(1(2(x1))))))))))))))))))) → 0(1(1(2(1(2(2(3(1(2(0(3(0(1(0(0(0(2(2(x1))))))))))))))))))) (170)
2(2(1(2(0(0(1(1(0(3(2(2(0(3(2(1(0(1(2(x1))))))))))))))))))) → 2(1(1(2(1(2(2(3(1(2(0(3(0(1(0(0(0(2(2(x1))))))))))))))))))) (171)
1(2(1(2(0(0(1(1(0(3(2(2(0(3(2(1(0(1(2(x1))))))))))))))))))) → 1(1(1(2(1(2(2(3(1(2(0(3(0(1(0(0(0(2(2(x1))))))))))))))))))) (172)
3(2(1(2(0(0(1(1(0(3(2(2(0(3(2(1(0(1(2(x1))))))))))))))))))) → 3(1(1(2(1(2(2(3(1(2(0(3(0(1(0(0(0(2(2(x1))))))))))))))))))) (173)
0(2(1(1(3(1(2(0(1(3(2(1(2(2(2(0(1(1(2(x1))))))))))))))))))) → 0(1(1(0(2(3(1(1(2(2(2(3(1(0(2(2(1(1(2(x1))))))))))))))))))) (174)
2(2(1(1(3(1(2(0(1(3(2(1(2(2(2(0(1(1(2(x1))))))))))))))))))) → 2(1(1(0(2(3(1(1(2(2(2(3(1(0(2(2(1(1(2(x1))))))))))))))))))) (175)
1(2(1(1(3(1(2(0(1(3(2(1(2(2(2(0(1(1(2(x1))))))))))))))))))) → 1(1(1(0(2(3(1(1(2(2(2(3(1(0(2(2(1(1(2(x1))))))))))))))))))) (176)
3(2(1(1(3(1(2(0(1(3(2(1(2(2(2(0(1(1(2(x1))))))))))))))))))) → 3(1(1(0(2(3(1(1(2(2(2(3(1(0(2(2(1(1(2(x1))))))))))))))))))) (177)

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 368 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 55
[21(x1)] = 1 · x1 + 29
[12(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 35
[10(x1)] = 1 · x1 + 17
[03(x1)] = 1 · x1 + 12
[33(x1)] = 1 · x1 + 20
[31(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 24
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[32(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 15
[23(x1)] = 1 · x1 + 11
all of the following rules can be deleted.

There are 344 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[30(x1)] = 1 · x1 + 1
[00(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1 + 1
[31(x1)] = 1 · x1 + 1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1 + 2
[20(x1)] = 1 · x1
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 1
[11(x1)] = 1 · x1
all of the following rules can be deleted.
30(00(00(03(33(33(31(10(01(10(00(00(00(02(22(23(30(00(x1)))))))))))))))))) → 30(03(33(30(01(10(03(30(02(20(00(00(00(02(21(13(30(00(x1)))))))))))))))))) (238)
30(00(00(03(33(33(31(10(01(10(00(00(00(02(22(23(30(02(x1)))))))))))))))))) → 30(03(33(30(01(10(03(30(02(20(00(00(00(02(21(13(30(02(x1)))))))))))))))))) (239)
30(00(00(03(33(33(31(10(01(10(00(00(00(02(22(23(30(01(x1)))))))))))))))))) → 30(03(33(30(01(10(03(30(02(20(00(00(00(02(21(13(30(01(x1)))))))))))))))))) (240)
30(00(00(03(33(33(31(10(01(10(00(00(00(02(22(23(30(03(x1)))))))))))))))))) → 30(03(33(30(01(10(03(30(02(20(00(00(00(02(21(13(30(03(x1)))))))))))))))))) (241)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1 + 1
[30(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[00(x1)] = 1 · x1
[33(x1)] = 1 · x1 + 1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1
[11(x1)] = 1 · x1
[13(x1)] = 1 · x1
all of the following rules can be deleted.
02(20(03(32(20(03(30(03(30(02(20(01(12(23(30(00(02(20(x1)))))))))))))))))) → 02(23(33(30(03(30(00(00(00(02(20(02(20(01(12(23(32(20(x1)))))))))))))))))) (306)
02(20(03(32(20(03(30(03(30(02(20(01(12(23(30(00(02(22(x1)))))))))))))))))) → 02(23(33(30(03(30(00(00(00(02(20(02(20(01(12(23(32(22(x1)))))))))))))))))) (307)
02(20(03(32(20(03(30(03(30(02(20(01(12(23(30(00(02(21(x1)))))))))))))))))) → 02(23(33(30(03(30(00(00(00(02(20(02(20(01(12(23(32(21(x1)))))))))))))))))) (308)
02(20(03(32(20(03(30(03(30(02(20(01(12(23(30(00(02(23(x1)))))))))))))))))) → 02(23(33(30(03(30(00(00(00(02(20(02(20(01(12(23(32(23(x1)))))))))))))))))) (309)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 2
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[30(x1)] = 1 · x1
[13(x1)] = 1 · x1
[12(x1)] = 1 · x1
[33(x1)] = 1 · x1
[21(x1)] = 1 · x1
[02(x1)] = 1 · x1
all of the following rules can be deleted.
22(20(03(32(23(32(23(31(10(00(01(11(11(10(01(10(03(30(x1)))))))))))))))))) → 22(22(20(01(11(10(00(01(13(31(10(01(12(23(33(30(03(30(x1)))))))))))))))))) (410)
22(20(03(32(23(32(23(31(10(00(01(11(11(10(01(10(03(32(x1)))))))))))))))))) → 22(22(20(01(11(10(00(01(13(31(10(01(12(23(33(30(03(32(x1)))))))))))))))))) (411)
22(20(03(32(23(32(23(31(10(00(01(11(11(10(01(10(03(31(x1)))))))))))))))))) → 22(22(20(01(11(10(00(01(13(31(10(01(12(23(33(30(03(31(x1)))))))))))))))))) (412)
22(20(03(32(23(32(23(31(10(00(01(11(11(10(01(10(03(33(x1)))))))))))))))))) → 22(22(20(01(11(10(00(01(13(31(10(01(12(23(33(30(03(33(x1)))))))))))))))))) (413)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1
[30(x1)] = 1 · x1
[23(x1)] = 1 · x1
[32(x1)] = 1 · x1
[11(x1)] = 1 · x1
[33(x1)] = 1 · x1
[00(x1)] = 1 · x1
[03(x1)] = 1 · x1
all of the following rules can be deleted.
21(10(02(21(12(22(20(01(13(31(12(21(12(20(02(21(13(30(x1)))))))))))))))))) → 21(12(22(23(32(22(22(20(01(10(01(10(01(12(21(11(13(30(x1)))))))))))))))))) (434)
21(10(02(21(12(22(20(01(13(31(12(21(12(20(02(21(13(32(x1)))))))))))))))))) → 21(12(22(23(32(22(22(20(01(10(01(10(01(12(21(11(13(32(x1)))))))))))))))))) (435)
21(10(02(21(12(22(20(01(13(31(12(21(12(20(02(21(13(31(x1)))))))))))))))))) → 21(12(22(23(32(22(22(20(01(10(01(10(01(12(21(11(13(31(x1)))))))))))))))))) (436)
21(10(02(21(12(22(20(01(13(31(12(21(12(20(02(21(13(33(x1)))))))))))))))))) → 21(12(22(23(32(22(22(20(01(10(01(10(01(12(21(11(13(33(x1)))))))))))))))))) (437)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[33(x1)] = 1 · x1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
13(31(10(01(11(13(33(33(32(23(30(01(13(30(00(03(33(30(x1)))))))))))))))))) → 13(31(10(03(30(01(13(33(30(03(33(32(21(11(10(03(33(30(x1)))))))))))))))))) (450)
13(31(10(01(11(13(33(33(32(23(30(01(13(30(00(03(33(32(x1)))))))))))))))))) → 13(31(10(03(30(01(13(33(30(03(33(32(21(11(10(03(33(32(x1)))))))))))))))))) (451)
13(31(10(01(11(13(33(33(32(23(30(01(13(30(00(03(33(31(x1)))))))))))))))))) → 13(31(10(03(30(01(13(33(30(03(33(32(21(11(10(03(33(31(x1)))))))))))))))))) (452)
13(31(10(01(11(13(33(33(32(23(30(01(13(30(00(03(33(33(x1)))))))))))))))))) → 13(31(10(03(30(01(13(33(30(03(33(32(21(11(10(03(33(33(x1)))))))))))))))))) (453)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1
[30(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[11(x1)] = 1 · x1
all of the following rules can be deleted.
32(23(32(21(12(21(12(21(13(31(13(31(13(30(01(13(33(30(x1)))))))))))))))))) → 31(10(02(23(32(22(23(31(11(11(11(13(33(32(21(13(33(30(x1)))))))))))))))))) (462)
32(23(32(21(12(21(12(21(13(31(13(31(13(30(01(13(33(32(x1)))))))))))))))))) → 31(10(02(23(32(22(23(31(11(11(11(13(33(32(21(13(33(32(x1)))))))))))))))))) (463)
32(23(32(21(12(21(12(21(13(31(13(31(13(30(01(13(33(31(x1)))))))))))))))))) → 31(10(02(23(32(22(23(31(11(11(11(13(33(32(21(13(33(31(x1)))))))))))))))))) (464)
32(23(32(21(12(21(12(21(13(31(13(31(13(30(01(13(33(33(x1)))))))))))))))))) → 31(10(02(23(32(22(23(31(11(11(11(13(33(32(21(13(33(33(x1)))))))))))))))))) (465)

1.1.1.1.1.1.1.1.1.1.1 R is empty

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