Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/139174)

The rewrite relation of the following TRS is considered.

0(0(0(0(1(1(3(2(3(3(0(1(1(2(0(1(3(1(x1)))))))))))))))))) → 0(0(0(1(1(1(0(1(3(3(1(0(3(2(0(2(3(1(x1)))))))))))))))))) (1)
0(0(0(2(3(0(0(0(0(0(0(1(1(0(3(0(0(0(x1)))))))))))))))))) → 0(0(3(1(0(0(0(0(2(0(0(0(3(1(0(0(0(0(x1)))))))))))))))))) (2)
0(0(1(1(2(1(3(1(3(1(1(1(2(2(1(0(2(2(x1)))))))))))))))))) → 2(1(1(1(1(0(3(1(1(1(2(0(3(1(0(2(2(2(x1)))))))))))))))))) (3)
0(0(3(3(0(2(3(0(3(0(2(1(0(0(0(3(3(0(x1)))))))))))))))))) → 0(0(3(1(0(3(0(0(3(3(0(3(3(0(0(2(0(2(x1)))))))))))))))))) (4)
0(2(0(1(3(0(0(1(3(0(2(0(1(3(1(1(3(1(x1)))))))))))))))))) → 0(2(0(1(3(1(1(2(1(0(0(3(3(0(0(1(3(1(x1)))))))))))))))))) (5)
0(2(0(2(1(1(2(1(2(0(0(3(2(2(3(0(0(3(x1)))))))))))))))))) → 0(3(2(2(0(2(2(2(2(0(3(0(1(0(1(1(0(3(x1)))))))))))))))))) (6)
0(2(1(2(1(1(2(3(2(1(0(0(1(0(0(3(0(3(x1)))))))))))))))))) → 0(3(2(0(0(1(2(1(0(2(1(1(3(1(0(0(2(3(x1)))))))))))))))))) (7)
0(2(3(1(2(3(2(1(0(1(0(0(3(1(3(1(2(3(x1)))))))))))))))))) → 0(2(3(3(1(2(1(1(1(0(2(3(1(0(2(0(3(3(x1)))))))))))))))))) (8)
0(3(0(0(2(1(0(0(2(1(2(2(3(2(0(0(2(0(x1)))))))))))))))))) → 0(2(0(0(0(2(0(3(0(2(2(0(1(1(3(2(2(0(x1)))))))))))))))))) (9)
0(3(2(0(3(0(0(1(2(1(2(0(1(2(1(3(0(0(x1)))))))))))))))))) → 0(2(2(0(3(3(1(0(0(3(0(2(1(1(0(1(2(0(x1)))))))))))))))))) (10)
0(3(3(0(0(0(1(3(2(2(3(2(1(3(0(0(2(1(x1)))))))))))))))))) → 0(2(2(0(0(3(0(2(3(1(0(3(1(0(3(2(3(1(x1)))))))))))))))))) (11)
1(0(0(1(0(0(3(3(0(0(2(1(2(1(3(3(1(3(x1)))))))))))))))))) → 1(0(3(0(0(3(3(0(0(2(1(0(3(1(2(3(1(1(x1)))))))))))))))))) (12)
1(0(3(1(2(0(3(3(2(1(2(0(0(1(0(2(1(3(x1)))))))))))))))))) → 1(0(3(3(1(0(2(0(3(1(0(3(2(1(2(2(1(0(x1)))))))))))))))))) (13)
1(1(2(0(3(0(1(3(0(2(3(3(3(2(2(3(0(3(x1)))))))))))))))))) → 3(3(0(3(1(0(3(2(3(2(2(0(3(2(1(1(0(3(x1)))))))))))))))))) (14)
1(1(2(0(3(0(2(1(3(3(0(3(3(3(3(3(1(2(x1)))))))))))))))))) → 1(0(3(3(0(3(3(1(0(2(3(1(1(3(2(3(2(3(x1)))))))))))))))))) (15)
1(1(2(1(0(0(3(0(3(1(3(0(1(3(1(3(3(3(x1)))))))))))))))))) → 3(0(1(1(1(1(3(0(3(1(1(0(3(0(2(3(3(3(x1)))))))))))))))))) (16)
1(1(3(2(1(0(0(3(3(1(1(3(2(0(1(2(1(1(x1)))))))))))))))))) → 1(1(2(3(3(0(0(3(3(1(1(1(1(1(0(2(2(1(x1)))))))))))))))))) (17)
1(2(2(3(3(2(0(1(3(1(2(0(3(0(3(2(0(2(x1)))))))))))))))))) → 2(3(1(0(3(1(0(2(3(3(2(2(0(0(3(1(2(2(x1)))))))))))))))))) (18)
1(2(3(1(3(0(1(2(2(0(2(0(1(1(1(1(0(1(x1)))))))))))))))))) → 1(2(2(0(1(1(1(1(0(0(3(2(1(0(2(3(1(1(x1)))))))))))))))))) (19)
1(3(0(1(3(1(2(1(2(1(2(2(2(1(3(0(1(3(x1)))))))))))))))))) → 2(3(2(1(1(2(2(1(3(1(0(2(1(1(1(0(3(3(x1)))))))))))))))))) (20)
1(3(0(3(0(1(3(1(2(1(0(2(1(1(3(0(1(1(x1)))))))))))))))))) → 1(0(1(0(3(1(3(1(0(1(1(3(2(3(0(2(1(1(x1)))))))))))))))))) (21)
1(3(2(1(0(1(2(1(3(1(1(3(3(3(0(2(1(3(x1)))))))))))))))))) → 1(0(3(1(1(2(2(3(2(3(1(1(3(1(1(0(3(3(x1)))))))))))))))))) (22)
1(3(3(0(3(2(1(1(1(0(1(3(2(3(1(1(2(1(x1)))))))))))))))))) → 1(3(1(3(0(1(1(3(2(3(2(2(0(3(1(1(1(1(x1)))))))))))))))))) (23)
1(3(3(1(3(2(2(2(2(2(0(1(2(2(0(3(0(3(x1)))))))))))))))))) → 1(3(2(0(1(0(2(2(2(2(2(1(0(3(3(3(2(3(x1)))))))))))))))))) (24)
1(3(3(3(1(2(0(1(2(0(0(2(1(3(3(3(0(2(x1)))))))))))))))))) → 1(3(1(1(0(3(3(0(2(0(1(3(0(3(2(3(2(2(x1)))))))))))))))))) (25)
2(0(1(2(0(0(0(2(2(1(3(2(1(0(1(0(2(0(x1)))))))))))))))))) → 2(0(2(0(1(0(1(1(0(1(0(3(2(0(2(2(2(0(x1)))))))))))))))))) (26)
2(0(3(2(0(3(1(0(0(0(1(3(0(0(3(0(2(2(x1)))))))))))))))))) → 2(3(0(0(0(0(0(0(0(2(3(1(3(1(0(3(2(2(x1)))))))))))))))))) (27)
2(1(0(0(3(0(0(0(1(3(0(1(3(0(1(3(2(1(x1)))))))))))))))))) → 3(0(0(0(1(1(2(3(3(0(0(3(0(1(0(2(1(1(x1)))))))))))))))))) (28)
2(1(0(1(1(2(2(3(3(2(1(2(1(2(2(3(0(2(x1)))))))))))))))))) → 2(3(2(1(1(0(1(2(2(2(0(3(2(2(1(1(3(2(x1)))))))))))))))))) (29)
2(1(0(2(0(1(0(3(2(1(3(1(1(3(2(2(1(0(x1)))))))))))))))))) → 2(3(1(2(3(2(2(1(2(0(0(1(0(3(1(1(1(0(x1)))))))))))))))))) (30)
2(1(1(1(0(1(0(0(3(3(0(3(2(0(1(2(1(3(x1)))))))))))))))))) → 2(2(3(0(3(1(0(0(3(0(1(1(1(1(2(1(0(3(x1)))))))))))))))))) (31)
2(1(1(3(0(2(1(3(2(2(3(3(3(1(3(2(3(3(x1)))))))))))))))))) → 2(2(3(3(1(2(1(0(1(1(3(3(2(3(3(2(3(3(x1)))))))))))))))))) (32)
2(1(2(2(1(0(1(3(0(1(1(0(0(3(3(2(2(3(x1)))))))))))))))))) → 2(0(3(1(0(1(3(1(3(2(2(2(2(0(1(1(0(3(x1)))))))))))))))))) (33)
2(1(3(0(3(0(1(0(1(3(3(0(1(3(3(0(2(3(x1)))))))))))))))))) → 2(3(3(0(3(3(0(0(3(2(3(1(1(1(0(3(1(0(x1)))))))))))))))))) (34)
2(1(3(1(1(2(1(2(3(0(3(2(0(1(0(1(2(1(x1)))))))))))))))))) → 3(0(0(3(1(1(1(1(1(2(3(2(0(2(2(1(2(1(x1)))))))))))))))))) (35)
2(2(1(0(1(3(2(2(0(2(2(2(2(0(2(2(1(2(x1)))))))))))))))))) → 2(3(2(2(2(1(2(2(2(0(1(0(2(2(0(2(1(2(x1)))))))))))))))))) (36)
2(2(1(2(0(1(3(0(1(1(1(1(2(2(0(0(2(2(x1)))))))))))))))))) → 2(0(0(1(1(0(2(2(3(1(2(2(1(2(1(1(0(2(x1)))))))))))))))))) (37)
2(2(1(2(2(2(2(2(1(3(2(1(3(0(3(3(1(2(x1)))))))))))))))))) → 2(2(0(2(2(2(3(1(2(1(3(2(3(1(1(2(3(2(x1)))))))))))))))))) (38)
2(2(1(2(2(2(3(2(0(3(3(0(1(2(0(2(3(3(x1)))))))))))))))))) → 2(3(1(3(1(0(2(2(3(0(2(2(2(3(2(2(0(3(x1)))))))))))))))))) (39)
2(2(1(2(3(2(1(1(1(0(0(3(3(1(2(2(1(3(x1)))))))))))))))))) → 2(2(1(1(1(2(1(2(3(2(3(1(3(1(0(2(0(3(x1)))))))))))))))))) (40)
2(2(1(3(0(1(3(0(1(2(0(2(2(1(3(2(2(2(x1)))))))))))))))))) → 1(1(2(2(3(3(3(2(2(0(2(0(1(0(2(2(1(2(x1)))))))))))))))))) (41)
2(2(3(0(2(3(1(2(0(2(2(1(3(0(0(1(3(2(x1)))))))))))))))))) → 3(0(2(1(3(2(2(1(1(2(0(3(0(0(2(2(3(2(x1)))))))))))))))))) (42)
2(2(3(1(0(3(0(2(1(3(2(2(2(3(3(3(1(2(x1)))))))))))))))))) → 1(1(0(3(2(2(3(2(2(1(0(2(3(3(2(3(3(2(x1)))))))))))))))))) (43)
2(2(3(1(1(2(0(2(0(3(0(0(2(1(0(1(3(2(x1)))))))))))))))))) → 2(2(3(2(2(0(3(0(3(1(0(1(1(1(2(2(0(0(x1)))))))))))))))))) (44)
2(2(3(1(3(1(3(0(2(1(3(1(2(3(1(3(3(1(x1)))))))))))))))))) → 2(3(3(1(3(1(3(3(2(3(1(1(2(3(2(1(0(1(x1)))))))))))))))))) (45)
2(3(2(1(2(0(3(0(1(3(3(3(0(2(3(0(0(0(x1)))))))))))))))))) → 3(0(0(3(0(2(3(1(1(3(3(2(2(0(3(0(2(0(x1)))))))))))))))))) (46)
2(3(2(1(3(1(2(0(0(1(3(0(1(2(0(3(1(1(x1)))))))))))))))))) → 1(2(3(1(1(0(1(3(0(0(0(3(2(3(2(2(1(1(x1)))))))))))))))))) (47)
2(3(3(1(3(1(0(0(1(1(3(3(3(1(2(2(1(3(x1)))))))))))))))))) → 2(2(1(3(3(1(1(0(3(1(1(1(3(3(2(3(0(3(x1)))))))))))))))))) (48)
3(0(0(3(1(3(3(2(2(1(1(0(2(1(2(0(2(0(x1)))))))))))))))))) → 3(3(1(0(2(0(2(0(2(0(2(3(1(1(3(1(0(2(x1)))))))))))))))))) (49)
3(0(1(0(0(1(3(0(0(1(3(0(1(2(0(3(3(3(x1)))))))))))))))))) → 1(1(3(0(0(3(1(0(1(0(2(3(0(3(3(0(0(3(x1)))))))))))))))))) (50)
3(0(1(0(2(0(3(1(3(3(1(3(3(0(2(2(3(0(x1)))))))))))))))))) → 3(0(2(1(0(3(0(3(3(0(2(3(3(1(1(2(3(0(x1)))))))))))))))))) (51)
3(0(1(3(2(2(3(2(0(1(0(0(3(0(1(0(2(0(x1)))))))))))))))))) → 2(3(1(2(0(2(0(3(1(0(0(0(3(2(3(1(0(0(x1)))))))))))))))))) (52)
3(0(1(3(2(3(3(2(0(1(1(2(0(1(3(1(3(2(x1)))))))))))))))))) → 3(2(3(3(0(3(1(1(0(2(2(3(1(2(1(1(0(3(x1)))))))))))))))))) (53)
3(0(2(1(3(3(0(1(1(3(0(1(2(3(1(2(1(3(x1)))))))))))))))))) → 0(1(1(2(3(2(3(1(0(2(1(1(1(0(3(3(3(3(x1)))))))))))))))))) (54)
3(0(3(0(1(3(0(2(0(1(1(0(1(0(2(3(0(3(x1)))))))))))))))))) → 3(0(0(2(1(1(0(1(0(3(2(1(0(0(3(3(0(3(x1)))))))))))))))))) (55)
3(0(3(2(0(3(1(1(2(1(3(3(2(1(0(3(2(1(x1)))))))))))))))))) → 2(3(3(0(1(1(1(0(2(1(2(3(3(0(3(3(2(1(x1)))))))))))))))))) (56)
3(0(3(2(2(3(2(0(3(0(2(2(1(2(0(2(2(0(x1)))))))))))))))))) → 3(2(1(0(0(3(0(2(3(2(2(0(2(2(3(2(2(0(x1)))))))))))))))))) (57)
3(1(1(3(1(3(3(0(3(1(3(0(2(1(2(2(1(3(x1)))))))))))))))))) → 3(3(3(2(0(2(1(1(2(3(1(1(3(0(3(1(1(3(x1)))))))))))))))))) (58)
3(1(2(0(3(0(1(2(0(3(3(2(3(2(2(1(2(3(x1)))))))))))))))))) → 3(1(3(2(1(1(3(2(0(0(0(3(2(2(2(3(2(3(x1)))))))))))))))))) (59)
3(1(2(3(0(2(2(1(1(3(1(0(0(1(0(2(3(0(x1)))))))))))))))))) → 3(1(0(0(0(3(1(1(0(3(1(2(1(2(2(3(2(0(x1)))))))))))))))))) (60)
3(1(3(3(0(2(2(2(0(1(2(0(1(0(1(2(2(0(x1)))))))))))))))))) → 3(2(3(2(0(3(1(2(1(1(1(2(0(0(0(2(2(0(x1)))))))))))))))))) (61)
3(2(0(1(0(0(0(3(1(3(1(3(3(2(0(1(1(1(x1)))))))))))))))))) → 3(2(0(3(0(1(1(1(0(1(1(0(3(0(3(2(3(1(x1)))))))))))))))))) (62)
3(2(0(1(2(1(2(3(2(3(2(0(2(1(2(1(2(0(x1)))))))))))))))))) → 2(1(1(3(2(0(2(3(1(1(2(2(0(2(3(2(2(0(x1)))))))))))))))))) (63)
3(2(0(2(2(0(0(2(3(0(2(3(0(1(2(2(0(2(x1)))))))))))))))))) → 2(0(3(2(2(2(0(3(2(0(2(1(3(2(0(0(0(2(x1)))))))))))))))))) (64)
3(2(0(2(2(3(0(0(1(3(0(2(0(0(2(1(3(0(x1)))))))))))))))))) → 2(1(0(0(0(0(3(2(3(0(3(1(3(0(0(2(2(2(x1)))))))))))))))))) (65)
3(2(1(2(3(3(1(2(1(1(2(3(0(2(3(1(2(3(x1)))))))))))))))))) → 2(0(2(3(1(1(3(3(1(2(2(2(3(3(1(1(2(3(x1)))))))))))))))))) (66)
3(2(1(3(2(2(3(3(3(2(1(2(0(1(0(1(1(2(x1)))))))))))))))))) → 2(3(1(1(0(2(2(0(3(3(3(1(2(3(1(2(1(2(x1)))))))))))))))))) (67)
3(2(2(3(1(2(2(2(1(3(3(3(0(2(1(3(2(1(x1)))))))))))))))))) → 3(2(2(1(2(2(2(3(2(3(3(1(1(3(3(0(2(1(x1)))))))))))))))))) (68)
3(2(3(1(1(1(3(2(2(1(2(1(0(0(3(1(3(3(x1)))))))))))))))))) → 3(2(3(2(0(3(1(1(2(3(2(0(1(3(1(1(1(3(x1)))))))))))))))))) (69)
3(2(3(2(1(3(0(1(2(1(2(2(1(0(1(1(0(0(x1)))))))))))))))))) → 3(2(2(0(1(1(1(1(1(0(0(0(3(2(2(1(3(2(x1)))))))))))))))))) (70)
3(2(3(3(0(2(3(2(0(2(3(2(2(1(3(3(0(3(x1)))))))))))))))))) → 3(3(2(2(3(3(2(3(0(0(3(2(2(1(0(3(2(3(x1)))))))))))))))))) (71)
3(3(0(1(0(1(3(3(3(2(2(3(3(0(3(3(0(3(x1)))))))))))))))))) → 3(3(3(0(2(3(1(0(1(3(2(3(3(3(3(0(0(3(x1)))))))))))))))))) (72)
3(3(0(2(3(3(0(0(2(1(0(3(0(1(0(1(0(2(x1)))))))))))))))))) → 2(0(3(3(3(0(2(0(3(1(1(1(0(0(0(0(3(2(x1)))))))))))))))))) (73)
3(3(0(3(1(3(3(1(2(0(3(2(2(2(3(2(3(0(x1)))))))))))))))))) → 3(2(3(1(0(3(2(1(2(3(3(3(0(0(3(3(2(2(x1)))))))))))))))))) (74)
3(3(1(1(3(0(2(1(3(2(0(1(2(3(3(3(2(3(x1)))))))))))))))))) → 3(1(0(3(1(1(0(3(3(2(2(2(2(3(1(3(3(3(x1)))))))))))))))))) (75)
3(3(2(0(3(1(2(2(3(1(3(2(3(2(2(1(2(0(x1)))))))))))))))))) → 3(2(1(3(2(3(2(0(3(2(2(3(2(3(1(1(2(0(x1)))))))))))))))))) (76)
3(3(2(0(3(2(1(2(2(1(2(0(1(2(3(3(2(0(x1)))))))))))))))))) → 2(0(3(3(1(2(1(2(1(0(2(3(3(2(3(2(2(0(x1)))))))))))))))))) (77)
3(3(2(1(1(1(1(3(0(0(3(1(2(0(2(2(3(0(x1)))))))))))))))))) → 3(3(1(1(0(1(0(2(0(1(1(2(3(3(2(2(3(0(x1)))))))))))))))))) (78)
3(3(3(0(1(3(3(1(2(3(0(2(1(3(2(1(0(0(x1)))))))))))))))))) → 3(1(1(1(3(0(1(3(0(2(3(2(3(0(3(2(3(0(x1)))))))))))))))))) (79)
3(3(3(1(3(0(3(3(1(3(0(1(0(0(2(1(2(0(x1)))))))))))))))))) → 3(3(1(1(0(1(3(3(0(3(3(0(2(2(3(1(0(0(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(3(1(0(2(1(1(0(3(3(2(3(1(1(0(0(0(0(x1)))))))))))))))))) → 1(3(2(0(2(3(0(1(3(3(1(0(1(1(1(0(0(0(x1)))))))))))))))))) (81)
0(0(0(3(0(1(1(0(0(0(0(0(0(3(2(0(0(0(x1)))))))))))))))))) → 0(0(0(0(1(3(0(0(0(2(0(0(0(0(1(3(0(0(x1)))))))))))))))))) (82)
2(2(0(1(2(2(1(1(1(3(1(3(1(2(1(1(0(0(x1)))))))))))))))))) → 2(2(2(0(1(3(0(2(1(1(1(3(0(1(1(1(1(2(x1)))))))))))))))))) (83)
0(3(3(0(0(0(1(2(0(3(0(3(2(0(3(3(0(0(x1)))))))))))))))))) → 2(0(2(0(0(3(3(0(3(3(0(0(3(0(1(3(0(0(x1)))))))))))))))))) (84)
1(3(1(1(3(1(0(2(0(3(1(0(0(3(1(0(2(0(x1)))))))))))))))))) → 1(3(1(0(0(3(3(0(0(1(2(1(1(3(1(0(2(0(x1)))))))))))))))))) (85)
3(0(0(3(2(2(3(0(0(2(1(2(1(1(2(0(2(0(x1)))))))))))))))))) → 3(0(1(1(0(1(0(3(0(2(2(2(2(0(2(2(3(0(x1)))))))))))))))))) (86)
3(0(3(0(0(1(0(0(1(2(3(2(1(1(2(1(2(0(x1)))))))))))))))))) → 3(2(0(0(1(3(1(1(2(0(1(2(1(0(0(2(3(0(x1)))))))))))))))))) (87)
3(2(1(3(1(3(0(0(1(0(1(2(3(2(1(3(2(0(x1)))))))))))))))))) → 3(3(0(2(0(1(3(2(0(1(1(1(2(1(3(3(2(0(x1)))))))))))))))))) (88)
0(2(0(0(2(3(2(2(1(2(0(0(1(2(0(0(3(0(x1)))))))))))))))))) → 0(2(2(3(1(1(0(2(2(0(3(0(2(0(0(0(2(0(x1)))))))))))))))))) (89)
0(0(3(1(2(1(0(2(1(2(1(0(0(3(0(2(3(0(x1)))))))))))))))))) → 0(2(1(0(1(1(2(0(3(0(0(1(3(3(0(2(2(0(x1)))))))))))))))))) (90)
1(2(0(0(3(1(2(3(2(2(3(1(0(0(0(3(3(0(x1)))))))))))))))))) → 1(3(2(3(0(1(3(0(1(3(2(0(3(0(0(2(2(0(x1)))))))))))))))))) (91)
3(1(3(3(1(2(1(2(0(0(3(3(0(0(1(0(0(1(x1)))))))))))))))))) → 1(1(3(2(1(3(0(1(2(0(0(3(3(0(0(3(0(1(x1)))))))))))))))))) (92)
3(1(2(0(1(0(0(2(1(2(3(3(0(2(1(3(0(1(x1)))))))))))))))))) → 0(1(2(2(1(2(3(0(1(3(0(2(0(1(3(3(0(1(x1)))))))))))))))))) (93)
3(0(3(2(2(3(3(3(2(0(3(1(0(3(0(2(1(1(x1)))))))))))))))))) → 3(0(1(1(2(3(0(2(2(3(2(3(0(1(3(0(3(3(x1)))))))))))))))))) (94)
2(1(3(3(3(3(3(0(3(3(1(2(0(3(0(2(1(1(x1)))))))))))))))))) → 3(2(3(2(3(1(1(3(2(0(1(3(3(0(3(3(0(1(x1)))))))))))))))))) (95)
3(3(3(1(3(1(0(3(1(3(0(3(0(0(1(2(1(1(x1)))))))))))))))))) → 3(3(3(2(0(3(0(1(1(3(0(3(1(1(1(1(0(3(x1)))))))))))))))))) (96)
1(1(2(1(0(2(3(1(1(3(3(0(0(1(2(3(1(1(x1)))))))))))))))))) → 1(2(2(0(1(1(1(1(1(3(3(0(0(3(3(2(1(1(x1)))))))))))))))))) (97)
2(0(2(3(0(3(0(2(1(3(1(0(2(3(3(2(2(1(x1)))))))))))))))))) → 2(2(1(3(0(0(2(2(3(3(2(0(1(3(0(1(3(2(x1)))))))))))))))))) (98)
1(0(1(1(1(1(0(2(0(2(2(1(0(3(1(3(2(1(x1)))))))))))))))))) → 1(1(3(2(0(1(2(3(0(0(1(1(1(1(0(2(2(1(x1)))))))))))))))))) (99)
3(1(0(3(1(2(2(2(1(2(1(2(1(3(1(0(3(1(x1)))))))))))))))))) → 3(3(0(1(1(1(2(0(1(3(1(2(2(1(1(2(3(2(x1)))))))))))))))))) (100)
1(1(0(3(1(1(2(0(1(2(1(3(1(0(3(0(3(1(x1)))))))))))))))))) → 1(1(2(0(3(2(3(1(1(0(1(3(1(3(0(1(0(1(x1)))))))))))))))))) (101)
3(1(2(0(3(3(3(1(1(3(1(2(1(0(1(2(3(1(x1)))))))))))))))))) → 3(3(0(1(1(3(1(1(3(2(3(2(2(1(1(3(0(1(x1)))))))))))))))))) (102)
1(2(1(1(3(2(3(1(0(1(1(1(2(3(0(3(3(1(x1)))))))))))))))))) → 1(1(1(1(3(0(2(2(3(2(3(1(1(0(3(1(3(1(x1)))))))))))))))))) (103)
3(0(3(0(2(2(1(0(2(2(2(2(2(3(1(3(3(1(x1)))))))))))))))))) → 3(2(3(3(3(0(1(2(2(2(2(2(0(1(0(2(3(1(x1)))))))))))))))))) (104)
2(0(3(3(3(1(2(0(0(2(1(0(2(1(3(3(3(1(x1)))))))))))))))))) → 2(2(3(2(3(0(3(1(0(2(0(3(3(0(1(1(3(1(x1)))))))))))))))))) (105)
0(2(0(1(0(1(2(3(1(2(2(0(0(0(2(1(0(2(x1)))))))))))))))))) → 0(2(2(2(0(2(3(0(1(0(1(1(0(1(0(2(0(2(x1)))))))))))))))))) (106)
2(2(0(3(0(0(3(1(0(0(0(1(3(0(2(3(0(2(x1)))))))))))))))))) → 2(2(3(0(1(3(1(3(2(0(0(0(0(0(0(0(3(2(x1)))))))))))))))))) (107)
1(2(3(1(0(3(1(0(3(1(0(0(0(3(0(0(1(2(x1)))))))))))))))))) → 1(1(2(0(1(0(3(0(0(3(3(2(1(1(0(0(0(3(x1)))))))))))))))))) (108)
2(0(3(2(2(1(2(1(2(3(3(2(2(1(1(0(1(2(x1)))))))))))))))))) → 2(3(1(1(2(2(3(0(2(2(2(1(0(1(1(2(3(2(x1)))))))))))))))))) (109)
0(1(2(2(3(1(1(3(1(2(3(0(1(0(2(0(1(2(x1)))))))))))))))))) → 0(1(1(1(3(0(1(0(0(2(1(2(2(3(2(1(3(2(x1)))))))))))))))))) (110)
3(1(2(1(0(2(3(0(3(3(0(0(1(0(1(1(1(2(x1)))))))))))))))))) → 3(0(1(2(1(1(1(1(0(3(0(0(1(3(0(3(2(2(x1)))))))))))))))))) (111)
3(3(2(3(1(3(3(3(2(2(3(1(2(0(3(1(1(2(x1)))))))))))))))))) → 3(3(2(3(3(2(3(3(1(1(0(1(2(1(3(3(2(2(x1)))))))))))))))))) (112)
3(2(2(3(3(0(0(1(1(0(3(1(0(1(2(2(1(2(x1)))))))))))))))))) → 3(0(1(1(0(2(2(2(2(3(1(3(1(0(1(3(0(2(x1)))))))))))))))))) (113)
3(2(0(3(3(1(0(3(3(1(0(1(0(3(0(3(1(2(x1)))))))))))))))))) → 0(1(3(0(1(1(1(3(2(3(0(0(3(3(0(3(3(2(x1)))))))))))))))))) (114)
1(2(1(0(1(0(2(3(0(3(2(1(2(1(1(3(1(2(x1)))))))))))))))))) → 1(2(1(2(2(0(2(3(2(1(1(1(1(1(3(0(0(3(x1)))))))))))))))))) (115)
2(1(2(2(0(2(2(2(2(0(2(2(3(1(0(1(2(2(x1)))))))))))))))))) → 2(1(2(0(2(2(0(1(0(2(2(2(1(2(2(2(3(2(x1)))))))))))))))))) (116)
2(2(0(0(2(2(1(1(1(1(0(3(1(0(2(1(2(2(x1)))))))))))))))))) → 2(0(1(1(2(1(2(2(1(3(2(2(0(1(1(0(0(2(x1)))))))))))))))))) (117)
2(1(3(3(0(3(1(2(3(1(2(2(2(2(2(1(2(2(x1)))))))))))))))))) → 2(3(2(1(1(3(2(3(1(2(1(3(2(2(2(0(2(2(x1)))))))))))))))))) (118)
3(3(2(0(2(1(0(3(3(0(2(3(2(2(2(1(2(2(x1)))))))))))))))))) → 3(0(2(2(3(2(2(2(0(3(2(2(0(1(3(1(3(2(x1)))))))))))))))))) (119)
3(1(2(2(1(3(3(0(0(1(1(1(2(3(2(1(2(2(x1)))))))))))))))))) → 3(0(2(0(1(3(1(3(2(3(2(1(2(1(1(1(2(2(x1)))))))))))))))))) (120)
2(2(2(3(1(2(2(0(2(1(0(3(1(0(3(1(2(2(x1)))))))))))))))))) → 2(1(2(2(0(1(0(2(0(2(2(3(3(3(2(2(1(1(x1)))))))))))))))))) (121)
2(3(1(0(0(3(1(2(2(0(2(1(3(2(0(3(2(2(x1)))))))))))))))))) → 2(3(2(2(0(0(3(0(2(1(1(2(2(3(1(2(0(3(x1)))))))))))))))))) (122)
2(1(3(3(3(2(2(2(3(1(2(0(3(0(1(3(2(2(x1)))))))))))))))))) → 2(3(3(2(3(3(2(0(1(2(2(3(2(2(3(0(1(1(x1)))))))))))))))))) (123)
2(3(1(0(1(2(0(0(3(0(2(0(2(1(1(3(2(2(x1)))))))))))))))))) → 0(0(2(2(1(1(1(0(1(3(0(3(0(2(2(3(2(2(x1)))))))))))))))))) (124)
1(3(3(1(3(2(1(3(1(2(0(3(1(3(1(3(2(2(x1)))))))))))))))))) → 1(0(1(2(3(2(1(1(3(2(3(3(1(3(1(3(3(2(x1)))))))))))))))))) (125)
0(0(0(3(2(0(3(3(3(1(0(3(0(2(1(2(3(2(x1)))))))))))))))))) → 0(2(0(3(0(2(2(3(3(1(1(3(2(0(3(0(0(3(x1)))))))))))))))))) (126)
1(1(3(0(2(1(0(3(1(0(0(2(1(3(1(2(3(2(x1)))))))))))))))))) → 1(1(2(2(3(2(3(0(0(0(3(1(0(1(1(3(2(1(x1)))))))))))))))))) (127)
3(1(2(2(1(3(3(3(1(1(0(0(1(3(1(3(3(2(x1)))))))))))))))))) → 3(0(3(2(3(3(1(1(1(3(0(1(1(3(3(1(2(2(x1)))))))))))))))))) (128)
0(2(0(2(1(2(0(1(1(2(2(3(3(1(3(0(0(3(x1)))))))))))))))))) → 2(0(1(3(1(1(3(2(0(2(0(2(0(2(0(1(3(3(x1)))))))))))))))))) (129)
3(3(3(0(2(1(0(3(1(0(0(3(1(0(0(1(0(3(x1)))))))))))))))))) → 3(0(0(3(3(0(3(2(0(1(0(1(3(0(0(3(1(1(x1)))))))))))))))))) (130)
0(3(2(2(0(3(3(1(3(3(1(3(0(2(0(1(0(3(x1)))))))))))))))))) → 0(3(2(1(1(3(3(2(0(3(3(0(3(0(1(2(0(3(x1)))))))))))))))))) (131)
0(2(0(1(0(3(0(0(1(0(2(3(2(2(3(1(0(3(x1)))))))))))))))))) → 0(0(1(3(2(3(0(0(0(1(3(0(2(0(2(1(3(2(x1)))))))))))))))))) (132)
2(3(1(3(1(0(2(1(1(0(2(3(3(2(3(1(0(3(x1)))))))))))))))))) → 3(0(1(1(2(1(3(2(2(0(1(1(3(0(3(3(2(3(x1)))))))))))))))))) (133)
3(1(2(1(3(2(1(0(3(1(1(0(3(3(1(2(0(3(x1)))))))))))))))))) → 3(3(3(3(0(1(1(1(2(0(1(3(2(3(2(1(1(0(x1)))))))))))))))))) (134)
3(0(3(2(0(1(0(1(1(0(2(0(3(1(0(3(0(3(x1)))))))))))))))))) → 3(0(3(3(0(0(1(2(3(0(1(0(1(1(2(0(0(3(x1)))))))))))))))))) (135)
1(2(3(0(1(2(3(3(1(2(1(1(3(0(2(3(0(3(x1)))))))))))))))))) → 1(2(3(3(0(3(3(2(1(2(0(1(1(1(0(3(3(2(x1)))))))))))))))))) (136)
0(2(2(0(2(1(2(2(0(3(0(2(3(2(2(3(0(3(x1)))))))))))))))))) → 0(2(2(3(2(2(0(2(2(3(2(0(3(0(0(1(2(3(x1)))))))))))))))))) (137)
3(1(2(2(1(2(0(3(1(3(0(3(3(1(3(1(1(3(x1)))))))))))))))))) → 3(1(1(3(0(3(1(1(3(2(1(1(2(0(2(3(3(3(x1)))))))))))))))))) (138)
3(2(1(2(2(3(2(3(3(0(2(1(0(3(0(2(1(3(x1)))))))))))))))))) → 3(2(3(2(2(2(3(0(0(0(2(3(1(1(2(3(1(3(x1)))))))))))))))))) (139)
0(3(2(0(1(0(0(1(3(1(1(2(2(0(3(2(1(3(x1)))))))))))))))))) → 0(2(3(2(2(1(2(1(3(0(1(1(3(0(0(0(1(3(x1)))))))))))))))))) (140)
0(2(2(1(0(1(0(2(1(0(2(2(2(0(3(3(1(3(x1)))))))))))))))))) → 0(2(2(0(0(0(2(1(1(1(2(1(3(0(2(3(2(3(x1)))))))))))))))))) (141)
1(1(1(0(2(3(3(1(3(1(3(0(0(0(1(0(2(3(x1)))))))))))))))))) → 1(3(2(3(0(3(0(1(1(0(1(1(1(0(3(0(2(3(x1)))))))))))))))))) (142)
0(2(1(2(1(2(0(2(3(2(3(2(1(2(1(0(2(3(x1)))))))))))))))))) → 0(2(2(3(2(0(2(2(1(1(3(2(0(2(3(1(1(2(x1)))))))))))))))))) (143)
2(0(2(2(1(0(3(2(0(3(2(0(0(2(2(0(2(3(x1)))))))))))))))))) → 2(0(0(0(2(3(1(2(0(2(3(0(2(2(2(3(0(2(x1)))))))))))))))))) (144)
0(3(1(2(0(0(2(0(3(1(0(0(3(2(2(0(2(3(x1)))))))))))))))))) → 2(2(2(0(0(3(1(3(0(3(2(3(0(0(0(0(1(2(x1)))))))))))))))))) (145)
3(2(1(3(2(0(3(2(1(1(2(1(3(3(2(1(2(3(x1)))))))))))))))))) → 3(2(1(1(3(3(2(2(2(1(3(3(1(1(3(2(0(2(x1)))))))))))))))))) (146)
2(1(1(0(1(0(2(1(2(3(3(3(2(2(3(1(2(3(x1)))))))))))))))))) → 2(1(2(1(3(2(1(3(3(3(0(2(2(0(1(1(3(2(x1)))))))))))))))))) (147)
1(2(3(1(2(0(3(3(3(1(2(2(2(1(3(2(2(3(x1)))))))))))))))))) → 1(2(0(3(3(1(1(3(3(2(3(2(2(2(1(2(2(3(x1)))))))))))))))))) (148)
3(3(1(3(0(0(1(2(1(2(2(3(1(1(1(3(2(3(x1)))))))))))))))))) → 3(1(1(1(3(1(0(2(3(2(1(1(3(0(2(3(2(3(x1)))))))))))))))))) (149)
0(0(1(1(0(1(2(2(1(2(1(0(3(1(2(3(2(3(x1)))))))))))))))))) → 2(3(1(2(2(3(0(0(0(1(1(1(1(1(0(2(2(3(x1)))))))))))))))))) (150)
3(0(3(3(1(2(2(3(2(0(2(3(2(0(3(3(2(3(x1)))))))))))))))))) → 3(2(3(0(1(2(2(3(0(0(3(2(3(3(2(2(3(3(x1)))))))))))))))))) (151)
3(0(3(3(0(3(3(2(2(3(3(3(1(0(1(0(3(3(x1)))))))))))))))))) → 3(0(0(3(3(3(3(2(3(1(0(1(3(2(0(3(3(3(x1)))))))))))))))))) (152)
2(0(1(0(1(0(3(0(1(2(0(0(3(3(2(0(3(3(x1)))))))))))))))))) → 2(3(0(0(0(0(1(1(1(3(0(2(0(3(3(3(0(2(x1)))))))))))))))))) (153)
0(3(2(3(2(2(2(3(0(2(1(3(3(1(3(0(3(3(x1)))))))))))))))))) → 2(2(3(3(0(0(3(3(3(2(1(2(3(0(1(3(2(3(x1)))))))))))))))))) (154)
3(2(3(3(3(2(1(0(2(3(1(2(0(3(1(1(3(3(x1)))))))))))))))))) → 3(3(3(1(3(2(2(2(2(3(3(0(1(1(3(0(1(3(x1)))))))))))))))))) (155)
0(2(1(2(2(3(2(3(1(3(2(2(1(3(0(2(3(3(x1)))))))))))))))))) → 0(2(1(1(3(2(3(2(2(3(0(2(3(2(3(1(2(3(x1)))))))))))))))))) (156)
0(2(3(3(2(1(0(2(1(2(2(1(2(3(0(2(3(3(x1)))))))))))))))))) → 0(2(2(3(2(3(3(2(0(1(2(1(2(1(3(3(0(2(x1)))))))))))))))))) (157)
0(3(2(2(0(2(1(3(0(0(3(1(1(1(1(2(3(3(x1)))))))))))))))))) → 0(3(2(2(3(3(2(1(1(0(2(0(1(0(1(1(3(3(x1)))))))))))))))))) (158)
0(0(1(2(3(1(2(0(3(2(1(3(3(1(0(3(3(3(x1)))))))))))))))))) → 0(3(2(3(0(3(2(3(2(0(3(1(0(3(1(1(1(3(x1)))))))))))))))))) (159)
0(2(1(2(0(0(1(0(3(1(3(3(0(3(1(3(3(3(x1)))))))))))))))))) → 0(0(1(3(2(2(0(3(3(0(3(3(1(0(1(1(3(3(x1)))))))))))))))))) (160)

1.1 Closure Under Flat Contexts

Using the flat contexts

{1(☐), 3(☐), 0(☐), 2(☐)}

We obtain the transformed TRS

There are 113 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 452 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

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

There are 410 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
[00(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1 + 2
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 1
[10(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1
all of the following rules can be deleted.
30(00(03(32(22(23(30(00(02(21(12(21(11(12(20(02(20(01(x1)))))))))))))))))) → 30(01(11(10(01(10(03(30(02(22(22(22(20(02(22(23(30(01(x1)))))))))))))))))) (221)
30(00(03(32(22(23(30(00(02(21(12(21(11(12(20(02(20(03(x1)))))))))))))))))) → 30(01(11(10(01(10(03(30(02(22(22(22(20(02(22(23(30(03(x1)))))))))))))))))) (222)
30(00(03(32(22(23(30(00(02(21(12(21(11(12(20(02(20(00(x1)))))))))))))))))) → 30(01(11(10(01(10(03(30(02(22(22(22(20(02(22(23(30(00(x1)))))))))))))))))) (223)
30(00(03(32(22(23(30(00(02(21(12(21(11(12(20(02(20(02(x1)))))))))))))))))) → 30(01(11(10(01(10(03(30(02(22(22(22(20(02(22(23(30(02(x1)))))))))))))))))) (224)
30(03(30(02(22(21(10(02(22(22(22(22(23(31(13(33(31(11(x1)))))))))))))))))) → 32(23(33(33(30(01(12(22(22(22(22(20(01(10(02(23(31(11(x1)))))))))))))))))) (281)
30(03(30(02(22(21(10(02(22(22(22(22(23(31(13(33(31(13(x1)))))))))))))))))) → 32(23(33(33(30(01(12(22(22(22(22(20(01(10(02(23(31(13(x1)))))))))))))))))) (282)
30(03(30(02(22(21(10(02(22(22(22(22(23(31(13(33(31(10(x1)))))))))))))))))) → 32(23(33(33(30(01(12(22(22(22(22(20(01(10(02(23(31(10(x1)))))))))))))))))) (283)
30(03(30(02(22(21(10(02(22(22(22(22(23(31(13(33(31(12(x1)))))))))))))))))) → 32(23(33(33(30(01(12(22(22(22(22(20(01(10(02(23(31(12(x1)))))))))))))))))) (284)
23(31(10(00(03(31(12(22(20(02(21(13(32(20(03(32(22(21(x1)))))))))))))))))) → 23(32(22(20(00(03(30(02(21(11(12(22(23(31(12(20(03(31(x1)))))))))))))))))) (349)
32(21(12(22(23(32(23(33(30(02(21(10(03(30(02(21(13(31(x1)))))))))))))))))) → 32(23(32(22(22(23(30(00(00(02(23(31(11(12(23(31(13(31(x1)))))))))))))))))) (405)
32(21(12(22(23(32(23(33(30(02(21(10(03(30(02(21(13(33(x1)))))))))))))))))) → 32(23(32(22(22(23(30(00(00(02(23(31(11(12(23(31(13(33(x1)))))))))))))))))) (406)
32(21(12(22(23(32(23(33(30(02(21(10(03(30(02(21(13(30(x1)))))))))))))))))) → 32(23(32(22(22(23(30(00(00(02(23(31(11(12(23(31(13(30(x1)))))))))))))))))) (407)
32(21(12(22(23(32(23(33(30(02(21(10(03(30(02(21(13(32(x1)))))))))))))))))) → 32(23(32(22(22(23(30(00(00(02(23(31(11(12(23(31(13(32(x1)))))))))))))))))) (408)
20(02(22(21(10(03(32(20(03(32(20(00(02(22(20(02(23(33(x1)))))))))))))))))) → 20(00(00(02(23(31(12(20(02(23(30(02(22(22(23(30(02(23(x1)))))))))))))))))) (426)
20(02(22(21(10(03(32(20(03(32(20(00(02(22(20(02(23(30(x1)))))))))))))))))) → 20(00(00(02(23(31(12(20(02(23(30(02(22(22(23(30(02(20(x1)))))))))))))))))) (427)
20(02(22(21(10(03(32(20(03(32(20(00(02(22(20(02(23(32(x1)))))))))))))))))) → 20(00(00(02(23(31(12(20(02(23(30(02(22(22(23(30(02(22(x1)))))))))))))))))) (428)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1
[03(x1)] = 1 · x1
[30(x1)] = 1 · x1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[00(x1)] = 1 · x1
all of the following rules can be deleted.
21(13(33(33(32(22(22(23(31(12(20(03(30(01(13(32(22(20(x1)))))))))))))))))) → 23(33(32(23(33(32(20(01(12(22(23(32(22(23(30(01(11(10(x1)))))))))))))))))) (355)
21(13(33(33(32(22(22(23(31(12(20(03(30(01(13(32(22(22(x1)))))))))))))))))) → 23(33(32(23(33(32(20(01(12(22(23(32(22(23(30(01(11(12(x1)))))))))))))))))) (356)
02(21(12(22(23(32(23(31(13(32(22(21(13(30(02(23(33(31(x1)))))))))))))))))) → 02(21(11(13(32(23(32(22(23(30(02(23(32(23(31(12(23(31(x1)))))))))))))))))) (461)
02(21(12(22(23(32(23(31(13(32(22(21(13(30(02(23(33(33(x1)))))))))))))))))) → 02(21(11(13(32(23(32(22(23(30(02(23(32(23(31(12(23(33(x1)))))))))))))))))) (462)
02(21(12(22(23(32(23(31(13(32(22(21(13(30(02(23(33(30(x1)))))))))))))))))) → 02(21(11(13(32(23(32(22(23(30(02(23(32(23(31(12(23(30(x1)))))))))))))))))) (463)
02(21(12(22(23(32(23(31(13(32(22(21(13(30(02(23(33(32(x1)))))))))))))))))) → 02(21(11(13(32(23(32(22(23(30(02(23(32(23(31(12(23(32(x1)))))))))))))))))) (464)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1
[31(x1)] = 1 · x1
[11(x1)] = 1 · x1 + 1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[33(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
all of the following rules can be deleted.
03(32(22(20(02(21(13(30(00(03(31(11(11(11(12(23(33(31(x1)))))))))))))))))) → 03(32(22(23(33(32(21(11(10(02(20(01(10(01(11(13(33(31(x1)))))))))))))))))) (469)
03(32(22(20(02(21(13(30(00(03(31(11(11(11(12(23(33(33(x1)))))))))))))))))) → 03(32(22(23(33(32(21(11(10(02(20(01(10(01(11(13(33(33(x1)))))))))))))))))) (470)
03(32(22(20(02(21(13(30(00(03(31(11(11(11(12(23(33(30(x1)))))))))))))))))) → 03(32(22(23(33(32(21(11(10(02(20(01(10(01(11(13(33(30(x1)))))))))))))))))) (471)
03(32(22(20(02(21(13(30(00(03(31(11(11(11(12(23(33(32(x1)))))))))))))))))) → 03(32(22(23(33(32(21(11(10(02(20(01(10(01(11(13(33(32(x1)))))))))))))))))) (472)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[10(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[22(x1)] = 1 · x1
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1
[20(x1)] = 1 · x1
all of the following rules can be deleted.
10(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(31(x1))))))))))))))))))) → 12(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(31(x1))))))))))))))))))) (641)
10(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(33(x1))))))))))))))))))) → 12(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(33(x1))))))))))))))))))) (642)
10(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(30(x1))))))))))))))))))) → 12(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(30(x1))))))))))))))))))) (643)
10(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(32(x1))))))))))))))))))) → 12(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(32(x1))))))))))))))))))) (644)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[30(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[22(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[00(x1)] = 1 · x1
[12(x1)] = 1 · x1
[01(x1)] = 1 · x1
[20(x1)] = 1 · x1
all of the following rules can be deleted.
30(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(31(x1))))))))))))))))))) → 32(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(31(x1))))))))))))))))))) (645)
30(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(33(x1))))))))))))))))))) → 32(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(33(x1))))))))))))))))))) (646)
30(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(30(x1))))))))))))))))))) → 32(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(30(x1))))))))))))))))))) (647)
30(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(32(x1))))))))))))))))))) → 32(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(32(x1))))))))))))))))))) (648)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1
[01(x1)] = 1 · x1
[20(x1)] = 1 · x1
all of the following rules can be deleted.
00(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(31(x1))))))))))))))))))) → 02(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(31(x1))))))))))))))))))) (649)
00(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(33(x1))))))))))))))))))) → 02(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(33(x1))))))))))))))))))) (650)
00(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(30(x1))))))))))))))))))) → 02(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(30(x1))))))))))))))))))) (651)
00(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(32(x1))))))))))))))))))) → 02(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(32(x1))))))))))))))))))) (652)

1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[20(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[22(x1)] = 1 · x1
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[00(x1)] = 1 · x1
[12(x1)] = 1 · x1
[01(x1)] = 1 · x1
all of the following rules can be deleted.
20(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(31(x1))))))))))))))))))) → 22(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(31(x1))))))))))))))))))) (653)
20(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(33(x1))))))))))))))))))) → 22(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(33(x1))))))))))))))))))) (654)
20(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(30(x1))))))))))))))))))) → 22(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(30(x1))))))))))))))))))) (655)
20(03(32(23(32(22(22(23(30(02(21(13(33(31(13(30(03(33(32(x1))))))))))))))))))) → 22(22(23(33(30(00(03(33(33(32(21(12(23(30(01(13(32(23(32(x1))))))))))))))))))) (656)

1.1.1.1.1.1.1.1.1.1.1.1 R is empty

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