Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/142146)

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

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

and S is the following TRS.

2(1(2(2(1(2(0(2(0(0(2(1(0(x1))))))))))))) 1(1(2(0(2(2(2(0(2(2(2(2(2(2(1(2(0(x1))))))))))))))))) (73)

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(1(1(2(2(1(2(1(0(0(0(0(x1))))))))))))) 0(2(1(2(2(2(2(2(2(2(2(2(1(0(0(0(0(x1))))))))))))))))) (74)
1(2(0(2(1(1(2(0(1(2(0(0(0(x1))))))))))))) 2(2(1(2(2(2(2(2(2(2(2(1(0(1(0(0(0(x1))))))))))))))))) (75)
0(1(1(0(2(2(0(1(2(2(0(0(0(x1))))))))))))) 1(1(2(0(0(0(0(0(0(0(0(0(0(0(2(1(0(x1))))))))))))))))) (76)
1(0(1(2(0(2(0(0(1(0(1(0(0(x1))))))))))))) 0(2(2(2(1(0(0(0(0(1(2(1(0(0(0(0(0(x1))))))))))))))))) (77)
2(1(2(0(2(1(2(0(1(1(1(0(0(x1))))))))))))) 1(2(1(2(0(2(2(1(0(0(0(0(0(1(0(0(0(x1))))))))))))))))) (78)
2(2(0(1(1(2(0(1(1(0(2(0(0(x1))))))))))))) 2(0(2(2(0(1(2(2(2(2(2(2(0(2(0(0(0(x1))))))))))))))))) (79)
0(0(1(2(2(0(1(2(1(1(2(0(0(x1))))))))))))) 0(2(2(2(2(2(0(2(2(2(1(0(0(0(0(0(1(x1))))))))))))))))) (80)
0(0(2(0(1(2(2(0(0(2(2(0(0(x1))))))))))))) 0(0(2(2(1(0(0(0(0(0(0(0(0(0(1(0(2(x1))))))))))))))))) (81)
1(1(0(1(2(1(1(1(1(0(0(1(0(x1))))))))))))) 0(0(2(2(2(2(0(2(2(0(0(0(1(1(0(1(0(x1))))))))))))))))) (82)
2(1(0(0(1(1(1(1(0(1(0(1(0(x1))))))))))))) 2(2(2(1(0(0(0(1(2(0(2(2(2(2(2(0(0(x1))))))))))))))))) (83)
2(0(1(2(2(0(2(0(1(1(0(1(0(x1))))))))))))) 2(2(2(0(2(2(2(2(0(2(2(2(2(0(2(2(0(x1))))))))))))))))) (84)
2(0(1(1(2(2(2(1(1(1(0(1(0(x1))))))))))))) 0(1(0(0(0(0(0(0(1(2(0(0(0(0(1(0(0(x1))))))))))))))))) (85)
0(1(0(1(1(0(2(2(1(0(1(1(0(x1))))))))))))) 0(0(2(1(0(0(0(1(2(2(1(0(0(0(0(0(0(x1))))))))))))))))) (86)
2(1(1(2(1(0(0(1(2(0(1(1(0(x1))))))))))))) 2(2(2(2(2(2(2(2(2(0(2(1(0(1(0(0(1(x1))))))))))))))))) (87)
1(2(2(2(2(1(1(2(2(2(1(1(0(x1))))))))))))) 2(0(2(2(2(2(2(0(2(2(2(0(2(2(2(2(1(x1))))))))))))))))) (88)
1(2(1(2(1(0(0(1(0(0(2(1(0(x1))))))))))))) 2(0(0(1(0(2(2(2(2(2(2(2(0(2(1(0(0(x1))))))))))))))))) (89)
2(0(2(1(1(2(2(2(1(0(2(1(0(x1))))))))))))) 2(2(0(0(2(0(2(1(0(0(0(0(0(0(0(0(0(x1))))))))))))))))) (90)
1(1(1(0(1(2(0(2(0(2(2(1(0(x1))))))))))))) 1(0(0(0(0(0(0(0(0(0(2(0(2(2(1(0(0(x1))))))))))))))))) (91)
0(2(2(0(0(1(2(2(0(1(0(2(0(x1))))))))))))) 0(2(1(0(0(0(0(0(0(0(0(2(1(0(0(0(0(x1))))))))))))))))) (92)
0(2(2(0(1(2(2(0(1(1(0(2(0(x1))))))))))))) 1(0(0(0(1(2(2(2(2(2(2(2(2(0(0(0(2(x1))))))))))))))))) (93)
1(1(0(2(0(1(2(1(1(1(0(2(0(x1))))))))))))) 0(1(0(0(0(0(1(0(0(1(0(2(2(2(0(2(0(x1))))))))))))))))) (94)
1(1(0(0(0(1(0(2(1(0(1(2(0(x1))))))))))))) 0(1(2(0(2(2(0(2(2(2(2(2(2(2(2(1(2(x1))))))))))))))))) (95)
2(0(0(1(2(2(2(1(2(0(1(2(0(x1))))))))))))) 2(1(0(2(0(1(0(0(0(0(0(0(0(0(0(0(0(x1))))))))))))))))) (96)
0(1(1(0(2(0(2(2(1(2(1(2(0(x1))))))))))))) 0(2(0(0(0(0(0(0(0(2(1(0(0(0(1(0(2(x1))))))))))))))))) (97)
1(0(0(1(0(1(1(1(1(0(0(0(1(x1))))))))))))) 0(2(1(0(0(0(0(0(0(0(0(0(0(1(0(2(2(x1))))))))))))))))) (98)
0(2(1(0(1(2(1(2(0(1(0(0(1(x1))))))))))))) 0(2(2(0(0(0(0(0(1(0(0(0(0(0(2(0(0(x1))))))))))))))))) (99)
1(1(1(2(1(2(2(0(0(2(0(0(1(x1))))))))))))) 0(0(2(0(2(1(0(0(0(0(0(0(1(2(0(1(0(x1))))))))))))))))) (100)
2(1(0(0(1(2(2(1(0(2(0(0(1(x1))))))))))))) 2(0(0(2(0(0(0(0(0(0(0(0(0(0(0(0(0(x1))))))))))))))))) (101)
1(2(0(0(2(1(0(1(1(2(0(0(1(x1))))))))))))) 1(0(2(2(2(0(2(2(2(1(0(0(0(2(0(0(1(x1))))))))))))))))) (102)
1(1(2(0(1(1(0(0(2(2(0(0(1(x1))))))))))))) 1(0(0(2(0(2(2(0(2(2(2(2(2(2(0(2(1(x1))))))))))))))))) (103)
0(1(2(1(0(2(0(2(1(2(1(0(1(x1))))))))))))) 0(2(1(2(2(2(2(1(0(0(0(0(0(1(0(0(2(x1))))))))))))))))) (104)
1(0(1(0(0(0(1(1(1(0(2(0(1(x1))))))))))))) 0(2(1(2(2(1(0(0(0(0(0(1(0(0(0(0(1(x1))))))))))))))))) (105)
1(1(1(1(1(0(1(0(2(0(2(0(1(x1))))))))))))) 1(0(0(1(2(0(2(0(2(2(2(2(2(2(0(1(1(x1))))))))))))))))) (106)
1(0(2(0(1(2(0(0(1(1(2(0(1(x1))))))))))))) 1(1(1(2(2(1(1(0(0(0(0(0(0(0(0(1(0(x1))))))))))))))))) (107)
0(1(2(0(0(1(0(1(2(1(2(0(1(x1))))))))))))) 0(0(2(0(0(1(2(2(2(1(0(0(0(0(0(1(2(x1))))))))))))))))) (108)
2(1(0(2(0(1(1(1(2(1(2(0(1(x1))))))))))))) 2(0(2(0(0(2(0(0(0(0(1(0(0(0(0(1(1(x1))))))))))))))))) (109)
0(2(2(1(0(0(0(2(2(1(2(0(1(x1))))))))))))) 2(0(1(0(2(2(2(2(2(2(2(2(2(2(2(0(0(x1))))))))))))))))) (110)
2(1(0(1(2(1(1(0(1(0(0(1(1(x1))))))))))))) 2(0(2(0(1(0(0(2(2(1(0(0(0(0(0(0(1(x1))))))))))))))))) (111)
0(0(2(0(1(2(2(0(1(1(0(1(1(x1))))))))))))) 1(0(0(0(2(1(2(2(2(2(1(0(0(0(0(1(0(x1))))))))))))))))) (112)
2(1(2(0(1(2(0(1(0(2(0(1(1(x1))))))))))))) 2(1(2(2(2(2(2(2(1(1(2(2(2(2(2(2(1(x1))))))))))))))))) (113)
2(1(2(2(0(0(1(1(0(2(0(1(1(x1))))))))))))) 2(0(0(0(0(0(2(2(1(0(0(0(0(0(1(2(0(x1))))))))))))))))) (114)
2(0(1(1(0(1(1(0(0(0(1(1(1(x1))))))))))))) 2(1(2(2(2(0(2(2(0(2(2(2(2(2(0(0(0(x1))))))))))))))))) (115)
2(0(2(0(1(2(0(2(0(0(1(1(1(x1))))))))))))) 2(2(2(2(0(2(2(0(2(2(2(2(1(0(2(0(1(x1))))))))))))))))) (116)
2(1(2(0(0(1(2(2(2(2(1(1(1(x1))))))))))))) 2(1(1(2(2(1(0(0(0(0(1(0(0(0(0(0(0(x1))))))))))))))))) (117)
0(2(0(0(1(0(2(0(2(0(2(1(1(x1))))))))))))) 1(2(2(2(2(0(2(2(2(1(0(0(0(0(1(0(2(x1))))))))))))))))) (118)
2(2(0(2(0(1(0(0(0(1(2(1(1(x1))))))))))))) 2(2(2(2(2(1(0(0(0(0(0(0(0(1(2(0(1(x1))))))))))))))))) (119)
0(1(1(1(1(0(2(0(0(2(2(1(1(x1))))))))))))) 0(0(1(2(2(2(2(1(0(1(2(1(0(0(0(0(2(x1))))))))))))))))) (120)
0(0(2(2(1(2(2(1(1(0(0(2(1(x1))))))))))))) 2(2(1(0(0(0(0(0(0(0(0(0(2(0(2(0(1(x1))))))))))))))))) (121)
1(0(0(2(2(2(0(1(2(0(0(2(1(x1))))))))))))) 1(0(1(2(2(0(2(0(2(2(2(1(0(0(0(0(0(x1))))))))))))))))) (122)
1(1(0(1(1(1(1(0(0(1(0(2(1(x1))))))))))))) 0(1(1(2(1(0(0(0(0(0(1(0(2(0(0(0(0(x1))))))))))))))))) (123)
0(0(1(0(1(2(2(1(2(1(0(2(1(x1))))))))))))) 0(0(1(0(1(2(2(2(2(2(0(2(1(0(0(0(1(x1))))))))))))))))) (124)
1(0(0(1(2(2(0(0(2(1(1(2(1(x1))))))))))))) 0(2(2(2(2(1(0(0(1(0(0(0(0(0(0(1(1(x1))))))))))))))))) (125)
1(0(0(1(2(0(2(1(0(0(2(2(1(x1))))))))))))) 2(2(2(2(2(2(2(2(0(2(2(2(2(2(0(1(0(x1))))))))))))))))) (126)
1(1(2(1(0(2(2(2(1(0(2(2(1(x1))))))))))))) 1(0(0(0(2(1(2(2(0(2(2(2(2(2(0(0(0(x1))))))))))))))))) (127)
2(0(1(0(0(2(1(1(0(0(0(0(2(x1))))))))))))) 2(2(2(2(0(2(0(2(2(2(2(1(0(0(0(1(0(x1))))))))))))))))) (128)
1(1(1(0(1(0(1(2(1(2(0(0(2(x1))))))))))))) 0(0(2(0(2(2(2(1(1(0(0(0(0(0(0(0(1(x1))))))))))))))))) (129)
1(2(1(0(2(0(0(0(1(0(1(0(2(x1))))))))))))) 0(0(0(0(0(0(0(0(1(0(0(1(0(0(1(0(2(x1))))))))))))))))) (130)
2(0(1(1(2(0(0(1(1(0(2(0(2(x1))))))))))))) 2(0(1(0(0(0(1(0(0(0(0(1(0(0(0(2(2(x1))))))))))))))))) (131)
1(0(2(1(2(0(2(2(0(2(2(0(2(x1))))))))))))) 0(2(2(1(0(0(0(0(0(0(0(0(0(2(2(2(2(x1))))))))))))))))) (132)
1(2(2(2(1(2(1(2(1(2(0(1(2(x1))))))))))))) 1(2(1(0(0(0(0(0(2(0(2(1(0(0(0(1(2(x1))))))))))))))))) (133)
2(0(1(0(1(0(2(1(1(2(1(1(2(x1))))))))))))) 2(1(2(2(2(2(2(2(0(2(1(0(0(2(0(2(2(x1))))))))))))))))) (134)
0(1(0(2(0(0(2(0(2(2(1(1(2(x1))))))))))))) 0(0(2(2(0(2(2(2(2(2(2(2(1(0(1(0(2(x1))))))))))))))))) (135)
2(2(1(0(2(0(2(2(0(0(2(1(2(x1))))))))))))) 2(0(2(2(2(2(2(2(1(0(0(2(0(0(0(0(2(x1))))))))))))))))) (136)
2(1(0(0(1(1(2(2(2(0(2(1(2(x1))))))))))))) 2(2(2(2(2(2(0(2(2(2(2(2(0(0(2(2(1(x1))))))))))))))))) (137)
0(2(2(1(2(0(0(1(0(1(2(1(2(x1))))))))))))) 1(0(0(0(1(2(1(0(0(0(0(0(0(1(1(2(2(x1))))))))))))))))) (138)
0(1(2(1(2(2(2(2(1(2(2(1(2(x1))))))))))))) 0(0(0(0(0(0(0(0(1(0(0(0(0(2(0(0(2(x1))))))))))))))))) (139)
0(0(1(2(2(0(1(2(1(0(0(2(2(x1))))))))))))) 0(2(2(0(2(2(0(2(2(0(2(2(2(2(2(2(2(x1))))))))))))))))) (140)
0(2(1(0(2(1(2(1(2(0(0(2(2(x1))))))))))))) 1(2(0(0(1(0(0(0(0(1(0(2(2(2(2(2(2(x1))))))))))))))))) (141)
0(1(0(1(2(1(1(0(2(2(0(2(2(x1))))))))))))) 1(0(0(2(0(0(1(2(2(2(2(2(2(2(2(2(2(x1))))))))))))))))) (142)
0(2(0(2(2(1(2(1(2(2(0(2(2(x1))))))))))))) 1(0(0(0(1(0(2(2(0(2(2(2(0(2(2(2(2(x1))))))))))))))))) (143)
1(0(1(0(2(2(1(1(2(0(1(2(2(x1))))))))))))) 2(1(0(0(0(0(0(1(0(0(0(1(0(2(2(2(2(x1))))))))))))))))) (144)
2(0(0(2(1(1(0(1(2(1(1(2(2(x1))))))))))))) 2(2(2(2(2(0(0(0(2(1(0(0(0(0(2(2(2(x1))))))))))))))))) (145)
0(1(2(0(0(2(0(2(1(2(2(1(2(x1))))))))))))) 0(2(1(2(2(2(2(2(2(0(2(2(2(0(2(1(1(x1))))))))))))))))) (146)

1.1 Closure Under Flat Contexts

Using the flat contexts

{0(), 2(), 1()}

We obtain the transformed TRS

There are 131 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 393 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

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

There are 377 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[20(x1)] = 1 · x1
[01(x1)] = 1 + 1 · x1
[12(x1)] = 1 + 1 · x1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1
[10(x1)] = 1 · x1
[11(x1)] = 1 + 1 · x1
all of the following rules can be deleted.
00(02(20(01(12(22(20(00(02(22(20(00(00(x1))))))))))))) 00(02(22(21(10(00(00(00(00(00(00(00(00(01(10(02(20(x1))))))))))))))))) (243)
00(02(20(01(12(22(20(00(02(22(20(00(01(x1))))))))))))) 00(02(22(21(10(00(00(00(00(00(00(00(00(01(10(02(21(x1))))))))))))))))) (245)
10(00(02(22(22(20(01(12(20(00(02(21(12(x1))))))))))))) 10(01(12(22(20(02(20(02(22(22(21(10(00(00(00(00(02(x1))))))))))))))))) (325)
20(01(10(00(02(21(11(10(00(00(00(02(22(x1))))))))))))) 22(22(22(20(02(20(02(22(22(22(21(10(00(00(01(10(02(x1))))))))))))))))) (334)
00(01(11(10(02(22(20(01(12(22(20(00(00(00(x1)))))))))))))) 01(11(12(20(00(00(00(00(00(00(00(00(00(00(02(21(10(00(x1)))))))))))))))))) (372)
00(01(11(10(02(22(20(01(12(22(20(00(00(02(x1)))))))))))))) 01(11(12(20(00(00(00(00(00(00(00(00(00(00(02(21(10(02(x1)))))))))))))))))) (373)
00(01(11(10(02(22(20(01(12(22(20(00(00(01(x1)))))))))))))) 01(11(12(20(00(00(00(00(00(00(00(00(00(00(02(21(10(01(x1)))))))))))))))))) (374)
20(01(11(10(02(22(20(01(12(22(20(00(00(00(x1)))))))))))))) 21(11(12(20(00(00(00(00(00(00(00(00(00(00(02(21(10(00(x1)))))))))))))))))) (375)
20(01(11(10(02(22(20(01(12(22(20(00(00(02(x1)))))))))))))) 21(11(12(20(00(00(00(00(00(00(00(00(00(00(02(21(10(02(x1)))))))))))))))))) (376)
20(01(11(10(02(22(20(01(12(22(20(00(00(01(x1)))))))))))))) 21(11(12(20(00(00(00(00(00(00(00(00(00(00(02(21(10(01(x1)))))))))))))))))) (377)
10(01(11(10(02(22(20(01(12(22(20(00(00(00(x1)))))))))))))) 11(11(12(20(00(00(00(00(00(00(00(00(00(00(02(21(10(00(x1)))))))))))))))))) (378)
10(01(11(10(02(22(20(01(12(22(20(00(00(02(x1)))))))))))))) 11(11(12(20(00(00(00(00(00(00(00(00(00(00(02(21(10(02(x1)))))))))))))))))) (379)
10(01(11(10(02(22(20(01(12(22(20(00(00(01(x1)))))))))))))) 11(11(12(20(00(00(00(00(00(00(00(00(00(00(02(21(10(01(x1)))))))))))))))))) (380)
01(12(20(00(02(20(02(21(12(22(21(12(22(x1))))))))))))) 02(21(12(22(22(22(22(22(20(02(22(22(20(02(21(11(12(x1))))))))))))))))) (625)

1.1.1.1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1010(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0010(x1))))))))))))))))) (627)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1000(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0000(x1))))))))))))))))) (628)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1002(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0002(x1))))))))))))))))) (629)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1022(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0022(x1))))))))))))))))) (630)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1020(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0020(x1))))))))))))))))) (631)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1001(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0001(x1))))))))))))))))) (632)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1012(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0012(x1))))))))))))))))) (633)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1021(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0021(x1))))))))))))))))) (634)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1011(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0011(x1))))))))))))))))) (635)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1110(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0110(x1))))))))))))))))) (636)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1100(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0100(x1))))))))))))))))) (637)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1102(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0102(x1))))))))))))))))) (638)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1122(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0122(x1))))))))))))))))) (639)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1120(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0120(x1))))))))))))))))) (640)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1101(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0101(x1))))))))))))))))) (641)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1112(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0112(x1))))))))))))))))) (642)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1121(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0121(x1))))))))))))))))) (643)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1111(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0111(x1))))))))))))))))) (644)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[1000(x1)] = 1 · x1
[0002(x1)] = 1 · x1
[0222(x1)] = 1 · x1
[2222(x1)] = 1 · x1
[2220(x1)] = 1 · x1
[2001(x1)] = 1 · x1 + 1
[0112(x1)] = 1 · x1
[1220(x1)] = 1 · x1
[2000(x1)] = 1 · x1
[0221(x1)] = 1 · x1
[2110(x1)] = 1 · x1
[1010(x1)] = 1 · x1
[1001(x1)] = 1 · x1
[1222(x1)] = 1 · x1
[2002(x1)] = 1 · x1
[0220(x1)] = 1 · x1
[2221(x1)] = 1 · x1
[0000(x1)] = 1 · x1
[0010(x1)] = 1 · x1
[1002(x1)] = 1 · x1
[1022(x1)] = 1 · x1
[0022(x1)] = 1 · x1
[1020(x1)] = 1 · x1
[0020(x1)] = 1 · x1
[0001(x1)] = 1 · x1
[1012(x1)] = 1 · x1
[0012(x1)] = 1 · x1
[1021(x1)] = 1 · x1
[0021(x1)] = 1 · x1
[1011(x1)] = 1 · x1
[0011(x1)] = 1 · x1
[2111(x1)] = 1 · x1
[1110(x1)] = 1 · x1
[0110(x1)] = 1 · x1
[1100(x1)] = 1 · x1
[0100(x1)] = 1 · x1
[1102(x1)] = 1 · x1
[0102(x1)] = 1 · x1
[1122(x1)] = 1 · x1
[0122(x1)] = 1 · x1
[1120(x1)] = 1 · x1
[0120(x1)] = 1 · x1
[1101(x1)] = 1 · x1
[0101(x1)] = 1 · x1
[1112(x1)] = 1 · x1
[1121(x1)] = 1 · x1
[0121(x1)] = 1 · x1
[1111(x1)] = 1 · x1
[0111(x1)] = 1 · x1
all of the following rules can be deleted.
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1010(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0010(x1))))))))))))))))) (627)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1000(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0000(x1))))))))))))))))) (628)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1002(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0002(x1))))))))))))))))) (629)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1022(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0022(x1))))))))))))))))) (630)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1020(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0020(x1))))))))))))))))) (631)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1001(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0001(x1))))))))))))))))) (632)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1012(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0012(x1))))))))))))))))) (633)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1021(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0021(x1))))))))))))))))) (634)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2110(1011(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0000(0011(x1))))))))))))))))) (635)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1110(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0110(x1))))))))))))))))) (636)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1100(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0100(x1))))))))))))))))) (637)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1102(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0102(x1))))))))))))))))) (638)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1122(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0122(x1))))))))))))))))) (639)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1120(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0120(x1))))))))))))))))) (640)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1101(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0101(x1))))))))))))))))) (641)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1112(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0112(x1))))))))))))))))) (642)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1121(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0121(x1))))))))))))))))) (643)
1000(0002(0222(2222(2220(2001(0112(1220(2000(0002(0221(2111(1111(x1))))))))))))) 1001(0112(1222(2220(2002(0220(2002(0222(2222(2221(2110(1000(0000(0000(0000(0001(0111(x1))))))))))))))))) (644)

1.1.1.1.1.1.1.1 R is empty

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