Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/128515)

The rewrite relation of the following TRS is considered.

0(0(0(0(0(1(3(2(0(3(3(2(3(2(2(2(1(1(x1)))))))))))))))))) → 0(0(3(3(2(0(2(0(2(3(1(0(3(1(2(1(2(0(x1)))))))))))))))))) (1)
0(0(0(0(2(1(3(2(2(2(3(3(2(0(3(0(0(2(x1)))))))))))))))))) → 0(3(2(3(1(0(0(2(3(0(2(3(0(0(2(0(2(2(x1)))))))))))))))))) (2)
0(0(0(1(1(1(1(1(0(0(2(2(1(0(0(1(3(1(x1)))))))))))))))))) → 0(0(0(1(2(1(3(0(1(0(1(0(2(0(1(1(1(1(x1)))))))))))))))))) (3)
0(0(1(2(2(3(2(3(1(0(0(1(3(2(1(1(1(0(x1)))))))))))))))))) → 0(0(1(1(0(1(0(2(3(2(3(1(1(1(3(2(2(0(x1)))))))))))))))))) (4)
0(0(1(3(3(2(3(2(2(3(0(0(2(3(2(2(3(0(x1)))))))))))))))))) → 0(1(2(3(3(2(3(2(3(0(2(2(0(2(0(3(3(0(x1)))))))))))))))))) (5)
0(0(2(3(1(1(2(2(0(1(2(3(1(2(3(2(2(3(x1)))))))))))))))))) → 3(1(1(2(3(0(2(2(2(1(2(3(0(2(3(0(2(1(x1)))))))))))))))))) (6)
0(0(3(1(0(2(0(3(2(1(3(3(3(3(2(0(0(1(x1)))))))))))))))))) → 0(0(2(0(3(0(0(3(3(0(2(1(1(2(3(3(1(3(x1)))))))))))))))))) (7)
0(0(3(1(1(1(0(1(0(1(0(1(0(2(2(3(2(2(x1)))))))))))))))))) → 3(3(1(2(1(0(0(1(1(0(1(0(2(0(2(1(0(2(x1)))))))))))))))))) (8)
0(0(3(1(3(0(0(3(2(2(0(0(3(1(3(3(2(3(x1)))))))))))))))))) → 0(3(0(0(3(2(0(1(1(3(3(0(0(2(3(3(2(3(x1)))))))))))))))))) (9)
0(0(3(3(0(3(3(1(2(2(3(3(0(1(0(3(0(1(x1)))))))))))))))))) → 0(1(0(0(1(3(0(1(3(3(2(0(2(3(3(3(0(3(x1)))))))))))))))))) (10)
0(1(1(0(3(0(2(3(2(1(1(2(2(0(1(1(3(2(x1)))))))))))))))))) → 0(1(1(3(0(2(3(2(1(3(0(2(0(2(1(2(1(1(x1)))))))))))))))))) (11)
0(1(1(1(3(3(3(3(3(0(3(2(0(0(0(0(0(1(x1)))))))))))))))))) → 0(1(3(1(0(3(1(3(0(3(0(3(0(3(0(2(0(1(x1)))))))))))))))))) (12)
0(1(2(0(3(3(3(2(0(3(0(0(0(3(0(0(3(3(x1)))))))))))))))))) → 0(0(2(0(3(2(1(0(0(0(0(3(3(3(0(3(3(3(x1)))))))))))))))))) (13)
0(1(3(0(1(1(0(3(2(3(0(3(2(0(1(3(2(1(x1)))))))))))))))))) → 0(1(3(3(0(2(2(1(1(0(1(0(2(3(0(3(3(1(x1)))))))))))))))))) (14)
0(2(0(3(0(3(3(2(2(0(2(1(1(2(0(3(2(1(x1)))))))))))))))))) → 0(2(2(0(2(3(0(2(3(3(3(1(0(2(2(1(0(1(x1)))))))))))))))))) (15)
0(3(0(0(0(3(3(3(1(0(3(0(0(3(1(2(2(1(x1)))))))))))))))))) → 0(3(1(1(3(0(3(0(0(2(1(0(0(2(3(3(3(0(x1)))))))))))))))))) (16)
0(3(1(2(2(0(1(2(0(1(3(1(3(0(1(3(1(3(x1)))))))))))))))))) → 0(1(1(1(2(0(1(0(3(1(2(1(3(0(2(3(3(3(x1)))))))))))))))))) (17)
1(0(0(0(2(0(2(1(0(2(2(1(1(3(0(0(3(3(x1)))))))))))))))))) → 1(1(0(2(0(2(2(3(0(1(1(0(0(0(0(2(3(3(x1)))))))))))))))))) (18)
1(0(0(1(2(2(1(0(1(3(2(0(2(0(3(0(0(0(x1)))))))))))))))))) → 1(3(0(0(1(0(3(0(2(0(0(2(1(2(2(0(1(0(x1)))))))))))))))))) (19)
1(0(1(0(1(0(3(3(0(3(1(2(2(1(3(1(3(1(x1)))))))))))))))))) → 1(0(3(1(0(3(2(3(0(1(3(1(1(0(1(2(3(1(x1)))))))))))))))))) (20)
1(1(0(0(1(2(1(2(2(0(2(3(2(3(0(2(2(0(x1)))))))))))))))))) → 1(0(2(0(3(2(1(2(1(0(2(2(3(2(1(0(2(0(x1)))))))))))))))))) (21)
1(1(0(0(1(3(2(3(2(0(1(2(3(1(1(2(2(1(x1)))))))))))))))))) → 1(1(2(2(3(1(1(1(2(2(1(3(0(0(2(1(3(0(x1)))))))))))))))))) (22)
1(1(0(1(0(1(1(2(2(1(1(1(1(3(3(1(3(0(x1)))))))))))))))))) → 1(1(0(0(1(1(1(3(3(1(2(1(2(1(3(1(1(0(x1)))))))))))))))))) (23)
1(1(0(2(3(3(0(0(3(0(0(3(3(1(1(3(3(3(x1)))))))))))))))))) → 1(1(0(2(3(3(3(0(1(0(3(0(1(3(3(0(3(3(x1)))))))))))))))))) (24)
1(1(1(1(3(2(1(0(0(3(3(0(0(3(2(1(3(3(x1)))))))))))))))))) → 1(1(1(3(3(2(3(0(3(0(1(3(1(1(0(0(2(3(x1)))))))))))))))))) (25)
1(1(2(0(0(1(0(3(3(1(0(3(2(1(0(1(2(0(x1)))))))))))))))))) → 1(0(1(3(2(3(1(1(3(0(2(1(0(0(2(1(0(0(x1)))))))))))))))))) (26)
1(1(2(0(3(3(0(1(2(3(2(3(1(0(0(2(1(3(x1)))))))))))))))))) → 1(2(1(0(3(1(3(1(0(2(1(0(2(3(0(2(3(3(x1)))))))))))))))))) (27)
1(1(2(1(1(1(3(1(2(2(2(3(2(3(1(0(3(1(x1)))))))))))))))))) → 1(1(3(1(2(3(3(3(0(2(1(2(1(2(1(2(1(1(x1)))))))))))))))))) (28)
1(1(2(2(3(0(0(3(0(3(3(2(1(3(2(3(0(3(x1)))))))))))))))))) → 1(1(0(3(3(2(3(0(1(3(2(0(2(3(0(3(2(3(x1)))))))))))))))))) (29)
1(1(3(2(0(0(1(0(0(3(1(0(1(3(0(1(1(2(x1)))))))))))))))))) → 1(1(3(1(0(0(2(1(1(0(0(3(1(1(0(3(0(2(x1)))))))))))))))))) (30)
1(2(0(1(2(3(1(2(0(3(0(0(0(3(2(0(1(0(x1)))))))))))))))))) → 1(0(1(1(3(0(2(0(3(1(0(3(2(0(2(2(0(0(x1)))))))))))))))))) (31)
1(2(2(3(1(2(2(1(0(3(2(2(1(0(0(0(0(1(x1)))))))))))))))))) → 1(1(2(2(2(3(0(0(2(3(1(1(0(2(0(0(2(1(x1)))))))))))))))))) (32)
1(2(3(1(2(2(0(1(2(3(2(2(0(3(0(3(3(1(x1)))))))))))))))))) → 2(3(0(3(0(2(3(0(2(1(1(2(3(1(2(3(2(1(x1)))))))))))))))))) (33)
1(2(3(2(1(1(0(1(2(0(3(1(3(2(2(3(3(1(x1)))))))))))))))))) → 1(2(3(1(1(2(2(1(3(2(3(0(2(3(0(3(1(1(x1)))))))))))))))))) (34)
1(3(0(0(3(2(2(1(0(0(0(2(3(1(3(1(1(3(x1)))))))))))))))))) → 1(0(0(3(2(1(2(1(0(1(3(3(3(0(3(0(2(1(x1)))))))))))))))))) (35)
1(3(1(0(2(2(2(2(2(2(3(0(1(2(2(1(3(3(x1)))))))))))))))))) → 1(3(0(2(2(2(2(1(3(1(2(0(2(1(2(2(3(3(x1)))))))))))))))))) (36)
1(3(2(3(2(1(0(1(0(1(1(0(0(0(3(2(1(0(x1)))))))))))))))))) → 1(3(2(3(0(1(1(1(3(1(0(0(0(0(2(1(2(0(x1)))))))))))))))))) (37)
1(3(3(1(3(1(1(3(2(3(2(0(3(3(1(3(3(2(x1)))))))))))))))))) → 1(3(3(3(2(1(3(2(1(1(3(1(3(3(0(2(3(3(x1)))))))))))))))))) (38)
1(3(3(2(2(0(3(1(3(0(1(2(1(2(0(3(2(1(x1)))))))))))))))))) → 1(3(2(0(1(0(1(3(2(2(0(2(3(3(3(1(2(1(x1)))))))))))))))))) (39)
2(0(0(0(2(0(3(2(2(0(3(3(1(3(3(3(1(2(x1)))))))))))))))))) → 2(3(3(2(3(1(3(2(0(0(3(3(0(1(2(0(0(2(x1)))))))))))))))))) (40)
2(0(0(1(3(1(0(2(2(2(2(3(2(3(2(2(2(0(x1)))))))))))))))))) → 3(1(0(2(1(2(2(3(0(2(3(0(2(2(2(2(2(0(x1)))))))))))))))))) (41)
2(0(1(2(1(3(0(3(3(1(0(2(0(2(2(1(2(2(x1)))))))))))))))))) → 2(2(2(1(0(2(1(1(0(3(3(2(3(0(1(0(2(2(x1)))))))))))))))))) (42)
2(0(2(2(1(0(1(0(1(2(2(0(0(3(1(1(1(1(x1)))))))))))))))))) → 2(1(0(1(1(0(0(1(2(3(2(0(1(1(2(0(2(1(x1)))))))))))))))))) (43)
2(0(3(3(3(1(1(3(2(3(2(0(0(2(3(2(2(0(x1)))))))))))))))))) → 2(0(1(0(1(3(2(2(3(2(3(3(3(2(3(0(2(0(x1)))))))))))))))))) (44)
2(1(0(0(3(2(1(1(2(0(1(1(2(2(2(0(2(1(x1)))))))))))))))))) → 2(0(1(1(2(1(3(0(2(2(2(1(0(2(2(0(1(1(x1)))))))))))))))))) (45)
2(1(2(0(0(3(0(3(1(2(0(3(1(1(2(2(1(1(x1)))))))))))))))))) → 2(3(3(2(3(2(0(2(0(2(0(1(0(1(1(1(1(1(x1)))))))))))))))))) (46)
2(1(2(2(0(1(2(0(2(1(2(2(3(0(2(2(0(0(x1)))))))))))))))))) → 2(3(0(1(0(2(1(0(2(0(2(2(2(1(2(2(2(0(x1)))))))))))))))))) (47)
2(1(3(0(0(0(0(2(2(2(2(1(3(2(2(3(2(2(x1)))))))))))))))))) → 2(0(2(3(2(1(2(0(2(1(0(2(0(2(3(2(3(2(x1)))))))))))))))))) (48)
2(2(0(0(0(0(0(0(2(2(1(1(3(0(2(1(1(1(x1)))))))))))))))))) → 2(1(0(2(3(0(2(1(0(0(0(2(2(1(0(1(0(1(x1)))))))))))))))))) (49)
2(2(0(0(1(0(0(2(1(3(1(1(0(2(2(2(0(3(x1)))))))))))))))))) → 2(2(0(0(2(2(0(2(0(1(1(1(1(2(3(0(0(3(x1)))))))))))))))))) (50)
2(2(0(1(2(3(1(1(2(1(1(2(3(2(0(1(2(3(x1)))))))))))))))))) → 2(2(1(2(3(1(2(1(3(2(1(0(2(1(3(0(2(1(x1)))))))))))))))))) (51)
2(2(1(0(0(2(1(1(1(2(0(2(2(2(0(3(2(3(x1)))))))))))))))))) → 2(2(1(0(2(2(0(1(1(2(2(1(0(2(3(0(2(3(x1)))))))))))))))))) (52)
2(2(1(2(0(0(0(0(2(2(0(1(2(2(0(0(1(2(x1)))))))))))))))))) → 2(2(0(2(0(0(0(0(1(2(2(1(2(0(0(2(1(2(x1)))))))))))))))))) (53)
2(2(2(0(0(3(2(2(3(2(0(0(3(3(0(0(1(1(x1)))))))))))))))))) → 3(1(2(0(0(0(2(0(2(3(0(1(2(2(3(2(3(0(x1)))))))))))))))))) (54)
2(2(3(3(1(1(3(0(0(0(1(0(1(2(3(2(0(0(x1)))))))))))))))))) → 3(0(2(3(0(2(0(1(1(0(2(2(3(1(0(3(1(0(x1)))))))))))))))))) (55)
2(2(3(3(3(3(2(0(0(3(2(3(2(0(3(0(2(2(x1)))))))))))))))))) → 2(2(2(3(0(2(3(2(3(3(0(3(0(2(3(0(2(3(x1)))))))))))))))))) (56)
2(3(1(2(0(1(3(2(1(2(3(2(2(3(2(0(3(3(x1)))))))))))))))))) → 2(1(2(2(2(3(3(2(3(2(3(0(1(1(2(3(0(3(x1)))))))))))))))))) (57)
2(3(1(3(1(3(2(0(0(3(1(2(2(0(2(2(2(1(x1)))))))))))))))))) → 2(1(2(1(1(0(2(3(3(3(3(2(1(0(2(0(2(2(x1)))))))))))))))))) (58)
2(3(1(3(2(0(0(3(1(1(0(3(3(2(2(0(1(3(x1)))))))))))))))))) → 2(1(2(3(3(1(1(0(0(2(3(3(1(2(3(3(0(0(x1)))))))))))))))))) (59)
2(3(2(1(0(3(3(2(0(1(1(3(3(1(0(2(1(0(x1)))))))))))))))))) → 2(0(2(3(3(1(1(1(1(0(2(1(3(3(2(0(3(0(x1)))))))))))))))))) (60)
2(3(2(1(3(0(3(2(1(3(1(2(3(3(2(0(0(1(x1)))))))))))))))))) → 2(0(2(1(2(3(2(3(0(1(1(2(3(1(3(3(0(3(x1)))))))))))))))))) (61)
2(3(2(2(0(0(1(3(1(1(2(2(0(0(0(1(3(3(x1)))))))))))))))))) → 2(0(2(1(3(0(0(2(3(3(0(2(0(1(1(3(2(1(x1)))))))))))))))))) (62)
2(3(2(3(1(0(0(2(1(3(1(2(2(1(3(2(2(2(x1)))))))))))))))))) → 2(0(2(0(1(1(2(3(1(2(3(3(3(1(2(2(2(2(x1)))))))))))))))))) (63)
2(3(3(1(1(1(1(3(1(1(1(2(2(0(3(2(2(3(x1)))))))))))))))))) → 2(1(1(3(2(3(1(2(3(1(1(1(3(0(2(2(1(3(x1)))))))))))))))))) (64)
3(0(0(0(0(3(2(2(2(0(0(3(3(2(2(3(0(1(x1)))))))))))))))))) → 3(0(3(0(0(0(1(3(2(0(2(2(2(3(3(0(2(0(x1)))))))))))))))))) (65)
3(0(0(0(2(2(0(3(2(2(3(3(3(0(0(1(0(3(x1)))))))))))))))))) → 3(3(0(3(0(2(0(2(3(3(0(0(0(2(3(2(0(1(x1)))))))))))))))))) (66)
3(0(1(3(0(1(0(0(3(3(2(2(2(3(1(1(1(3(x1)))))))))))))))))) → 3(0(0(3(2(1(3(0(3(0(2(2(1(1(3(1(1(3(x1)))))))))))))))))) (67)
3(0(3(0(1(3(2(2(2(3(0(0(0(3(1(3(1(1(x1)))))))))))))))))) → 3(0(1(1(3(3(0(2(0(2(0(3(3(1(2(1(3(0(x1)))))))))))))))))) (68)
3(0(3(0(3(2(0(3(1(1(3(0(3(3(3(3(1(3(x1)))))))))))))))))) → 3(0(3(3(3(0(3(2(1(3(0(1(3(3(0(3(1(3(x1)))))))))))))))))) (69)
3(1(1(3(3(2(2(1(2(2(2(1(3(2(2(3(2(2(x1)))))))))))))))))) → 3(2(3(3(2(1(1(2(1(2(3(2(1(3(2(2(2(2(x1)))))))))))))))))) (70)
3(1(1(3(3(2(2(2(1(1(1(0(0(3(0(1(3(0(x1)))))))))))))))))) → 0(1(3(3(0(2(1(3(3(2(1(1(3(0(2(1(1(0(x1)))))))))))))))))) (71)
3(1(2(0(3(1(0(0(3(1(1(2(0(1(2(1(1(1(x1)))))))))))))))))) → 3(1(1(0(2(1(1(2(0(1(3(1(0(3(1(0(2(1(x1)))))))))))))))))) (72)
3(1(3(3(2(2(0(3(1(1(2(3(2(2(3(3(2(0(x1)))))))))))))))))) → 3(2(2(1(2(3(1(1(3(3(0(2(2(3(2(3(3(0(x1)))))))))))))))))) (73)
3(1(3(3(3(2(2(3(2(3(2(3(2(0(3(1(3(0(x1)))))))))))))))))) → 3(2(3(1(2(1(2(3(3(2(3(3(3(3(2(0(3(0(x1)))))))))))))))))) (74)
3(2(0(0(0(0(0(2(2(1(2(3(1(2(3(0(2(1(x1)))))))))))))))))) → 3(2(0(0(2(2(0(1(3(3(1(0(2(0(2(0(2(1(x1)))))))))))))))))) (75)
3(2(2(0(2(2(3(2(3(2(0(0(3(2(0(3(2(1(x1)))))))))))))))))) → 0(0(2(3(3(3(0(2(2(2(3(3(2(2(0(2(2(1(x1)))))))))))))))))) (76)
3(2(2(0(3(0(3(1(2(2(1(2(1(0(0(1(2(3(x1)))))))))))))))))) → 3(2(1(0(2(2(1(3(1(2(3(2(3(0(0(2(0(1(x1)))))))))))))))))) (77)
3(2(2(3(2(2(0(2(0(3(3(0(3(1(2(1(2(2(x1)))))))))))))))))) → 3(3(0(2(1(0(2(2(3(2(3(3(2(0(2(2(1(2(x1)))))))))))))))))) (78)
3(2(2(3(2(3(3(3(1(0(1(2(2(0(1(0(3(1(x1)))))))))))))))))) → 3(2(1(0(2(0(0(1(1(3(3(2(2(2(3(3(3(1(x1)))))))))))))))))) (79)
3(3(2(2(1(0(1(3(1(3(0(3(2(3(3(0(1(2(x1)))))))))))))))))) → 3(3(0(1(2(3(3(3(0(2(1(1(3(0(2(3(1(2(x1)))))))))))))))))) (80)

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(1(2(2(2(3(2(3(3(0(2(3(1(0(0(0(0(0(x1)))))))))))))))))) → 0(2(1(2(1(3(0(1(3(2(0(2(0(2(3(3(0(0(x1)))))))))))))))))) (81)
2(0(0(3(0(2(3(3(2(2(2(3(1(2(0(0(0(0(x1)))))))))))))))))) → 2(2(0(2(0(0(3(2(0(3(2(0(0(1(3(2(3(0(x1)))))))))))))))))) (82)
1(3(1(0(0(1(2(2(0(0(1(1(1(1(1(0(0(0(x1)))))))))))))))))) → 1(1(1(1(0(2(0(1(0(1(0(3(1(2(1(0(0(0(x1)))))))))))))))))) (83)
0(1(1(1(2(3(1(0(0(1(3(2(3(2(2(1(0(0(x1)))))))))))))))))) → 0(2(2(3(1(1(1(3(2(3(2(0(1(0(1(1(0(0(x1)))))))))))))))))) (84)
0(3(2(2(3(2(0(0(3(2(2(3(2(3(3(1(0(0(x1)))))))))))))))))) → 0(3(3(0(2(0(2(2(0(3(2(3(2(3(3(2(1(0(x1)))))))))))))))))) (85)
3(2(2(3(2(1(3(2(1(0(2(2(1(1(3(2(0(0(x1)))))))))))))))))) → 1(2(0(3(2(0(3(2(1(2(2(2(0(3(2(1(1(3(x1)))))))))))))))))) (86)
1(0(0(2(3(3(3(3(1(2(3(0(2(0(1(3(0(0(x1)))))))))))))))))) → 3(1(3(3(2(1(1(2(0(3(3(0(0(3(0(2(0(0(x1)))))))))))))))))) (87)
2(2(3(2(2(0(1(0(1(0(1(0(1(1(1(3(0(0(x1)))))))))))))))))) → 2(0(1(2(0(2(0(1(0(1(1(0(0(1(2(1(3(3(x1)))))))))))))))))) (88)
3(2(3(3(1(3(0(0(2(2(3(0(0(3(1(3(0(0(x1)))))))))))))))))) → 3(2(3(3(2(0(0(3(3(1(1(0(2(3(0(0(3(0(x1)))))))))))))))))) (89)
1(0(3(0(1(0(3(3(2(2(1(3(3(0(3(3(0(0(x1)))))))))))))))))) → 3(0(3(3(3(2(0(2(3(3(1(0(3(1(0(0(1(0(x1)))))))))))))))))) (90)
2(3(1(1(0(2(2(1(1(2(3(2(0(3(0(1(1(0(x1)))))))))))))))))) → 1(1(2(1(2(0(2(0(3(1(2(3(2(0(3(1(1(0(x1)))))))))))))))))) (91)
1(0(0(0(0(0(2(3(0(3(3(3(3(3(1(1(1(0(x1)))))))))))))))))) → 1(0(2(0(3(0(3(0(3(0(3(1(3(0(1(3(1(0(x1)))))))))))))))))) (92)
3(3(0(0(3(0(0(0(3(0(2(3(3(3(0(2(1(0(x1)))))))))))))))))) → 3(3(3(0(3(3(3(0(0(0(0(1(2(3(0(2(0(0(x1)))))))))))))))))) (93)
1(2(3(1(0(2(3(0(3(2(3(0(1(1(0(3(1(0(x1)))))))))))))))))) → 1(3(3(0(3(2(0(1(0(1(1(2(2(0(3(3(1(0(x1)))))))))))))))))) (94)
1(2(3(0(2(1(1(2(0(2(2(3(3(0(3(0(2(0(x1)))))))))))))))))) → 1(0(1(2(2(0(1(3(3(3(2(0(3(2(0(2(2(0(x1)))))))))))))))))) (95)
1(2(2(1(3(0(0(3(0(1(3(3(3(0(0(0(3(0(x1)))))))))))))))))) → 0(3(3(3(2(0(0(1(2(0(0(3(0(3(1(1(3(0(x1)))))))))))))))))) (96)
3(1(3(1(0(3(1(3(1(0(2(1(0(2(2(1(3(0(x1)))))))))))))))))) → 3(3(3(2(0(3(1(2(1(3(0(1(0(2(1(1(1(0(x1)))))))))))))))))) (97)
3(3(0(0(3(1(1(2(2(0(1(2(0(2(0(0(0(1(x1)))))))))))))))))) → 3(3(2(0(0(0(0(1(1(0(3(2(2(0(2(0(1(1(x1)))))))))))))))))) (98)
0(0(0(3(0(2(0(2(3(1(0(1(2(2(1(0(0(1(x1)))))))))))))))))) → 0(1(0(2(2(1(2(0(0(2(0(3(0(1(0(0(3(1(x1)))))))))))))))))) (99)
1(3(1(3(1(2(2(1(3(0(3(3(0(1(0(1(0(1(x1)))))))))))))))))) → 1(3(2(1(0(1(1(3(1(0(3(2(3(0(1(3(0(1(x1)))))))))))))))))) (100)
0(2(2(0(3(2(3(2(0(2(2(1(2(1(0(0(1(1(x1)))))))))))))))))) → 0(2(0(1(2(3(2(2(0(1(2(1(2(3(0(2(0(1(x1)))))))))))))))))) (101)
1(2(2(1(1(3(2(1(0(2(3(2(3(1(0(0(1(1(x1)))))))))))))))))) → 0(3(1(2(0(0(3(1(2(2(1(1(1(3(2(2(1(1(x1)))))))))))))))))) (102)
0(3(1(3(3(1(1(1(1(2(2(1(1(0(1(0(1(1(x1)))))))))))))))))) → 0(1(1(3(1(2(1(2(1(3(3(1(1(1(0(0(1(1(x1)))))))))))))))))) (103)
3(3(3(1(1(3(3(0(0(3(0(0(3(3(2(0(1(1(x1)))))))))))))))))) → 3(3(0(3(3(1(0(3(0(1(0(3(3(3(2(0(1(1(x1)))))))))))))))))) (104)
3(3(1(2(3(0(0(3(3(0(0(1(2(3(1(1(1(1(x1)))))))))))))))))) → 3(2(0(0(1(1(3(1(0(3(0(3(2(3(3(1(1(1(x1)))))))))))))))))) (105)
0(2(1(0(1(2(3(0(1(3(3(0(1(0(0(2(1(1(x1)))))))))))))))))) → 0(0(1(2(0(0(1(2(0(3(1(1(3(2(3(1(0(1(x1)))))))))))))))))) (106)
3(1(2(0(0(1(3(2(3(2(1(0(3(3(0(2(1(1(x1)))))))))))))))))) → 3(3(2(0(3(2(0(1(2(0(1(3(1(3(0(1(2(1(x1)))))))))))))))))) (107)
1(3(0(1(3(2(3(2(2(2(1(3(1(1(1(2(1(1(x1)))))))))))))))))) → 1(1(2(1(2(1(2(1(2(0(3(3(3(2(1(3(1(1(x1)))))))))))))))))) (108)
3(0(3(2(3(1(2(3(3(0(3(0(0(3(2(2(1(1(x1)))))))))))))))))) → 3(2(3(0(3(2(0(2(3(1(0(3(2(3(3(0(1(1(x1)))))))))))))))))) (109)
2(1(1(0(3(1(0(1(3(0(0(1(0(0(2(3(1(1(x1)))))))))))))))))) → 2(0(3(0(1(1(3(0(0(1(1(2(0(0(1(3(1(1(x1)))))))))))))))))) (110)
0(1(0(2(3(0(0(0(3(0(2(1(3(2(1(0(2(1(x1)))))))))))))))))) → 0(0(2(2(0(2(3(0(1(3(0(2(0(3(1(1(0(1(x1)))))))))))))))))) (111)
1(0(0(0(0(1(2(2(3(0(1(2(2(1(3(2(2(1(x1)))))))))))))))))) → 1(2(0(0(2(0(1(1(3(2(0(0(3(2(2(2(1(1(x1)))))))))))))))))) (112)
1(3(3(0(3(0(2(2(3(2(1(0(2(2(1(3(2(1(x1)))))))))))))))))) → 1(2(3(2(1(3(2(1(1(2(0(3(2(0(3(0(3(2(x1)))))))))))))))))) (113)
1(3(3(2(2(3(1(3(0(2(1(0(1(1(2(3(2(1(x1)))))))))))))))))) → 1(1(3(0(3(2(0(3(2(3(1(2(2(1(1(3(2(1(x1)))))))))))))))))) (114)
3(1(1(3(1(3(2(0(0(0(1(2(2(3(0(0(3(1(x1)))))))))))))))))) → 1(2(0(3(0(3(3(3(1(0(1(2(1(2(3(0(0(1(x1)))))))))))))))))) (115)
3(3(1(2(2(1(0(3(2(2(2(2(2(2(0(1(3(1(x1)))))))))))))))))) → 3(3(2(2(1(2(0(2(1(3(1(2(2(2(2(0(3(1(x1)))))))))))))))))) (116)
0(1(2(3(0(0(0(1(1(0(1(0(1(2(3(2(3(1(x1)))))))))))))))))) → 0(2(1(2(0(0(0(0(1(3(1(1(1(0(3(2(3(1(x1)))))))))))))))))) (117)
2(3(3(1(3(3(0(2(3(2(3(1(1(3(1(3(3(1(x1)))))))))))))))))) → 3(3(2(0(3(3(1(3(1(1(2(3(1(2(3(3(3(1(x1)))))))))))))))))) (118)
1(2(3(0(2(1(2(1(0(3(1(3(0(2(2(3(3(1(x1)))))))))))))))))) → 1(2(1(3(3(3(2(0(2(2(3(1(0(1(0(2(3(1(x1)))))))))))))))))) (119)
2(1(3(3(3(1(3(3(0(2(2(3(0(2(0(0(0(2(x1)))))))))))))))))) → 2(0(0(2(1(0(3(3(0(0(2(3(1(3(2(3(3(2(x1)))))))))))))))))) (120)
0(2(2(2(3(2(3(2(2(2(2(0(1(3(1(0(0(2(x1)))))))))))))))))) → 0(2(2(2(2(2(0(3(2(0(3(2(2(1(2(0(1(3(x1)))))))))))))))))) (121)
2(2(1(2(2(0(2(0(1(3(3(0(3(1(2(1(0(2(x1)))))))))))))))))) → 2(2(0(1(0(3(2(3(3(0(1(1(2(0(1(2(2(2(x1)))))))))))))))))) (122)
1(1(1(1(3(0(0(2(2(1(0(1(0(1(2(2(0(2(x1)))))))))))))))))) → 1(2(0(2(1(1(0(2(3(2(1(0(0(1(1(0(1(2(x1)))))))))))))))))) (123)
0(2(2(3(2(0(0(2(3(2(3(1(1(3(3(3(0(2(x1)))))))))))))))))) → 0(2(0(3(2(3(3(3(2(3(2(2(3(1(0(1(0(2(x1)))))))))))))))))) (124)
1(2(0(2(2(2(1(1(0(2(1(1(2(3(0(0(1(2(x1)))))))))))))))))) → 1(1(0(2(2(0(1(2(2(2(0(3(1(2(1(1(0(2(x1)))))))))))))))))) (125)
1(1(2(2(1(1(3(0(2(1(3(0(3(0(0(2(1(2(x1)))))))))))))))))) → 1(1(1(1(1(0(1(0(2(0(2(0(2(3(2(3(3(2(x1)))))))))))))))))) (126)
0(0(2(2(0(3(2(2(1(2(0(2(1(0(2(2(1(2(x1)))))))))))))))))) → 0(2(2(2(1(2(2(2(0(2(0(1(2(0(1(0(3(2(x1)))))))))))))))))) (127)
2(2(3(2(2(3(1(2(2(2(2(0(0(0(0(3(1(2(x1)))))))))))))))))) → 2(3(2(3(2(0(2(0(1(2(0(2(1(2(3(2(0(2(x1)))))))))))))))))) (128)
1(1(1(2(0(3(1(1(2(2(0(0(0(0(0(0(2(2(x1)))))))))))))))))) → 1(0(1(0(1(2(2(0(0(0(1(2(0(3(2(0(1(2(x1)))))))))))))))))) (129)
3(0(2(2(2(0(1(1(3(1(2(0(0(1(0(0(2(2(x1)))))))))))))))))) → 3(0(0(3(2(1(1(1(1(0(2(0(2(2(0(0(2(2(x1)))))))))))))))))) (130)
3(2(1(0(2(3(2(1(1(2(1(1(3(2(1(0(2(2(x1)))))))))))))))))) → 1(2(0(3(1(2(0(1(2(3(1(2(1(3(2(1(2(2(x1)))))))))))))))))) (131)
3(2(3(0(2(2(2(0(2(1(1(1(2(0(0(1(2(2(x1)))))))))))))))))) → 3(2(0(3(2(0(1(2(2(1(1(0(2(2(0(1(2(2(x1)))))))))))))))))) (132)
2(1(0(0(2(2(1(0(2(2(0(0(0(0(2(1(2(2(x1)))))))))))))))))) → 2(1(2(0(0(2(1(2(2(1(0(0(0(0(2(0(2(2(x1)))))))))))))))))) (133)
1(1(0(0(3(3(0(0(2(3(2(2(3(0(0(2(2(2(x1)))))))))))))))))) → 0(3(2(3(2(2(1(0(3(2(0(2(0(0(0(2(1(3(x1)))))))))))))))))) (134)
0(0(2(3(2(1(0(1(0(0(0(3(1(1(3(3(2(2(x1)))))))))))))))))) → 0(1(3(0(1(3(2(2(0(1(1(0(2(0(3(2(0(3(x1)))))))))))))))))) (135)
2(2(0(3(0(2(3(2(3(0(0(2(3(3(3(3(2(2(x1)))))))))))))))))) → 3(2(0(3(2(0(3(0(3(3(2(3(2(0(3(2(2(2(x1)))))))))))))))))) (136)
3(3(0(2(3(2(2(3(2(1(2(3(1(0(2(1(3(2(x1)))))))))))))))))) → 3(0(3(2(1(1(0(3(2(3(2(3(3(2(2(2(1(2(x1)))))))))))))))))) (137)
1(2(2(2(0(2(2(1(3(0(0(2(3(1(3(1(3(2(x1)))))))))))))))))) → 2(2(0(2(0(1(2(3(3(3(3(2(0(1(1(2(1(2(x1)))))))))))))))))) (138)
3(1(0(2(2(3(3(0(1(1(3(0(0(2(3(1(3(2(x1)))))))))))))))))) → 0(0(3(3(2(1(3(3(2(0(0(1(1(3(3(2(1(2(x1)))))))))))))))))) (139)
0(1(2(0(1(3(3(1(1(0(2(3(3(0(1(2(3(2(x1)))))))))))))))))) → 0(3(0(2(3(3(1(2(0(1(1(1(1(3(3(2(0(2(x1)))))))))))))))))) (140)
1(0(0(2(3(3(2(1(3(1(2(3(0(3(1(2(3(2(x1)))))))))))))))))) → 3(0(3(3(1(3(2(1(1(0(3(2(3(2(1(2(0(2(x1)))))))))))))))))) (141)
3(3(1(0(0(0(2(2(1(1(3(1(0(0(2(2(3(2(x1)))))))))))))))))) → 1(2(3(1(1(0(2(0(3(3(2(0(0(3(1(2(0(2(x1)))))))))))))))))) (142)
2(2(2(3(1(2(2(1(3(1(2(0(0(1(3(2(3(2(x1)))))))))))))))))) → 2(2(2(2(1(3(3(3(2(1(3(2(1(1(0(2(0(2(x1)))))))))))))))))) (143)
3(2(2(3(0(2(2(1(1(1(3(1(1(1(1(3(3(2(x1)))))))))))))))))) → 3(1(2(2(0(3(1(1(1(3(2(1(3(2(3(1(1(2(x1)))))))))))))))))) (144)
1(0(3(2(2(3(3(0(0(2(2(2(3(0(0(0(0(3(x1)))))))))))))))))) → 0(2(0(3(3(2(2(2(0(2(3(1(0(0(0(3(0(3(x1)))))))))))))))))) (145)
3(0(1(0(0(3(3(3(2(2(3(0(2(2(0(0(0(3(x1)))))))))))))))))) → 1(0(2(3(2(0(0(0(3(3(2(0(2(0(3(0(3(3(x1)))))))))))))))))) (146)
3(1(1(1(3(2(2(2(3(3(0(0(1(0(3(1(0(3(x1)))))))))))))))))) → 3(1(1(3(1(1(2(2(0(3(0(3(1(2(3(0(0(3(x1)))))))))))))))))) (147)
1(1(3(1(3(0(0(0(3(2(2(2(3(1(0(3(0(3(x1)))))))))))))))))) → 0(3(1(2(1(3(3(0(2(0(2(0(3(3(1(1(0(3(x1)))))))))))))))))) (148)
3(1(3(3(3(3(0(3(1(1(3(0(2(3(0(3(0(3(x1)))))))))))))))))) → 3(1(3(0(3(3(1(0(3(1(2(3(0(3(3(3(0(3(x1)))))))))))))))))) (149)
2(2(3(2(2(3(1(2(2(2(1(2(2(3(3(1(1(3(x1)))))))))))))))))) → 2(2(2(2(3(1(2(3(2(1(2(1(1(2(3(3(2(3(x1)))))))))))))))))) (150)
0(3(1(0(3(0(0(1(1(1(2(2(2(3(3(1(1(3(x1)))))))))))))))))) → 0(1(1(2(0(3(1(1(2(3(3(1(2(0(3(3(1(0(x1)))))))))))))))))) (151)
1(1(1(2(1(0(2(1(1(3(0(0(1(3(0(2(1(3(x1)))))))))))))))))) → 1(2(0(1(3(0(1(3(1(0(2(1(1(2(0(1(1(3(x1)))))))))))))))))) (152)
0(2(3(3(2(2(3(2(1(1(3(0(2(2(3(3(1(3(x1)))))))))))))))))) → 0(3(3(2(3(2(2(0(3(3(1(1(3(2(1(2(2(3(x1)))))))))))))))))) (153)
0(3(1(3(0(2(3(2(3(2(3(2(2(3(3(3(1(3(x1)))))))))))))))))) → 0(3(0(2(3(3(3(3(2(3(3(2(1(2(1(3(2(3(x1)))))))))))))))))) (154)
1(2(0(3(2(1(3(2(1(2(2(0(0(0(0(0(2(3(x1)))))))))))))))))) → 1(2(0(2(0(2(0(1(3(3(1(0(2(2(0(0(2(3(x1)))))))))))))))))) (155)
1(2(3(0(2(3(0(0(2(3(2(3(2(2(0(2(2(3(x1)))))))))))))))))) → 1(2(2(0(2(2(3(3(2(2(2(0(3(3(3(2(0(0(x1)))))))))))))))))) (156)
3(2(1(0(0(1(2(1(2(2(1(3(0(3(0(2(2(3(x1)))))))))))))))))) → 1(0(2(0(0(3(2(3(2(1(3(1(2(2(0(1(2(3(x1)))))))))))))))))) (157)
2(2(1(2(1(3(0(3(3(0(2(0(2(2(3(2(2(3(x1)))))))))))))))))) → 2(1(2(2(0(2(3(3(2(3(2(2(0(1(2(0(3(3(x1)))))))))))))))))) (158)
1(3(0(1(0(2(2(1(0(1(3(3(3(2(3(2(2(3(x1)))))))))))))))))) → 1(3(3(3(2(2(2(3(3(1(1(0(0(2(0(1(2(3(x1)))))))))))))))))) (159)
2(1(0(3(3(2(3(0(3(1(3(1(0(1(2(2(3(3(x1)))))))))))))))))) → 2(1(3(2(0(3(1(1(2(0(3(3(3(2(1(0(3(3(x1)))))))))))))))))) (160)

1.1 Closure Under Flat Contexts

Using the flat contexts

{1(☐), 2(☐), 3(☐), 0(☐)}

We obtain the transformed TRS

There are 140 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 560 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 2
[03(x1)] = 1 · x1
[30(x1)] = 1 · x1 + 9
[02(x1)] = 1 · x1 + 1
[23(x1)] = 1 · x1 + 25
[33(x1)] = 1 · x1 + 6
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 31
[31(x1)] = 1 · x1 + 6
[12(x1)] = 1 · x1 + 3
[01(x1)] = 1 · x1
[13(x1)] = 1 · x1 + 2
[10(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 3
[21(x1)] = 1 · x1 + 26
all of the following rules can be deleted.

There are 552 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 1
[11(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1
[30(x1)] = 1 · x1
[03(x1)] = 1 · x1
[33(x1)] = 1 · x1
[10(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
13(31(11(13(31(13(32(20(00(00(01(12(22(23(30(00(03(31(12(x1))))))))))))))))))) → 11(12(20(03(30(03(33(33(31(10(01(12(21(12(23(30(00(01(12(x1))))))))))))))))))) (593)
13(31(11(13(31(13(32(20(00(00(01(12(22(23(30(00(03(31(10(x1))))))))))))))))))) → 11(12(20(03(30(03(33(33(31(10(01(12(21(12(23(30(00(01(10(x1))))))))))))))))))) (594)
13(31(11(13(31(13(32(20(00(00(01(12(22(23(30(00(03(31(13(x1))))))))))))))))))) → 11(12(20(03(30(03(33(33(31(10(01(12(21(12(23(30(00(01(13(x1))))))))))))))))))) (595)
13(31(11(13(31(13(32(20(00(00(01(12(22(23(30(00(03(31(11(x1))))))))))))))))))) → 11(12(20(03(30(03(33(33(31(10(01(12(21(12(23(30(00(01(11(x1))))))))))))))))))) (596)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[11(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
[30(x1)] = 1 · x1
[03(x1)] = 1 · x1
[21(x1)] = 1 · x1
[33(x1)] = 1 · x1
[10(x1)] = 1 · x1
all of the following rules can be deleted.
23(31(11(13(31(13(32(20(00(00(01(12(22(23(30(00(03(31(12(x1))))))))))))))))))) → 21(12(20(03(30(03(33(33(31(10(01(12(21(12(23(30(00(01(12(x1))))))))))))))))))) (597)
23(31(11(13(31(13(32(20(00(00(01(12(22(23(30(00(03(31(10(x1))))))))))))))))))) → 21(12(20(03(30(03(33(33(31(10(01(12(21(12(23(30(00(01(10(x1))))))))))))))))))) (598)
23(31(11(13(31(13(32(20(00(00(01(12(22(23(30(00(03(31(13(x1))))))))))))))))))) → 21(12(20(03(30(03(33(33(31(10(01(12(21(12(23(30(00(01(13(x1))))))))))))))))))) (599)
23(31(11(13(31(13(32(20(00(00(01(12(22(23(30(00(03(31(11(x1))))))))))))))))))) → 21(12(20(03(30(03(33(33(31(10(01(12(21(12(23(30(00(01(11(x1))))))))))))))))))) (600)

1.1.1.1.1.1.1 R is empty

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