Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/26882)

The rewrite relation of the following TRS is considered.

0(1(0(1(2(3(4(5(x1)))))))) 1(1(0(0(2(3(4(5(x1)))))))) (1)
4(1(2(0(1(4(5(4(3(x1))))))))) 4(5(1(1(0(2(3(4(4(x1))))))))) (2)
4(4(5(5(0(0(0(3(1(x1))))))))) 4(4(5(0(5(0(0(3(1(x1))))))))) (3)
0(3(0(5(1(3(2(4(5(5(x1)))))))))) 1(2(3(3(0(4(1(5(1(3(x1)))))))))) (4)
3(3(1(4(3(1(0(5(5(4(x1)))))))))) 5(4(1(1(4(5(2(3(3(2(x1)))))))))) (5)
5(0(1(5(2(5(2(0(5(4(x1)))))))))) 3(5(1(2(4(1(0(0(1(3(x1)))))))))) (6)
5(0(5(4(5(0(0(2(1(3(x1)))))))))) 3(5(3(0(2(4(3(2(0(4(x1)))))))))) (7)
0(2(5(4(4(2(0(3(5(1(5(x1))))))))))) 0(0(2(4(4(4(1(0(5(3(x1)))))))))) (8)
0(3(1(5(4(1(5(5(5(3(2(x1))))))))))) 1(0(4(4(1(4(2(0(5(3(x1)))))))))) (9)
1(0(4(5(4(0(3(0(3(2(5(x1))))))))))) 0(5(5(5(0(1(1(1(2(4(x1)))))))))) (10)
1(2(4(1(0(3(2(5(5(3(1(x1))))))))))) 3(1(1(3(5(2(5(5(2(1(x1)))))))))) (11)
1(4(0(1(2(1(4(5(4(4(0(x1))))))))))) 4(3(3(4(1(0(3(1(2(4(x1)))))))))) (12)
1(4(2(1(4(1(2(5(2(3(3(x1))))))))))) 3(0(0(2(3(2(4(1(4(3(x1)))))))))) (13)
1(5(2(4(1(5(0(1(0(2(0(x1))))))))))) 1(1(3(4(5(2(5(4(1(1(x1)))))))))) (14)
2(0(0(1(2(1(1(4(5(5(1(x1))))))))))) 3(1(3(4(1(0(0(0(2(2(x1)))))))))) (15)
2(1(0(2(3(0(5(3(5(0(3(x1))))))))))) 0(1(1(0(3(2(0(5(4(2(x1)))))))))) (16)
2(1(0(4(3(3(1(3(0(5(0(x1))))))))))) 5(1(5(0(4(4(2(5(0(2(x1)))))))))) (17)
2(1(2(3(4(0(4(2(2(3(5(x1))))))))))) 3(1(2(2(4(3(3(3(2(4(x1)))))))))) (18)
2(1(3(1(3(2(3(4(5(5(0(x1))))))))))) 2(1(3(1(3(2(4(3(5(5(0(x1))))))))))) (19)
2(1(4(5(5(1(2(5(4(3(5(x1))))))))))) 4(5(5(4(2(2(2(3(2(0(x1)))))))))) (20)
2(3(0(3(2(4(3(5(3(3(1(x1))))))))))) 1(1(1(3(1(0(3(1(5(5(x1)))))))))) (21)
2(3(3(1(3(2(3(3(1(1(2(x1))))))))))) 2(1(4(1(4(5(4(2(2(1(x1)))))))))) (22)
2(4(0(0(4(1(0(2(1(0(4(x1))))))))))) 1(3(5(4(4(4(0(3(4(2(x1)))))))))) (23)
2(4(0(2(0(0(0(1(0(5(3(x1))))))))))) 5(1(4(4(4(0(1(4(5(1(x1)))))))))) (24)
2(5(2(1(1(0(3(1(3(4(3(x1))))))))))) 4(4(4(5(3(3(0(3(1(3(x1)))))))))) (25)
3(0(5(1(2(4(4(5(1(0(3(x1))))))))))) 3(0(2(4(4(1(5(1(0(5(3(x1))))))))))) (26)
3(2(0(0(5(2(3(2(1(0(2(x1))))))))))) 4(1(2(4(0(2(5(3(4(2(x1)))))))))) (27)
3(2(3(5(3(5(2(5(3(5(5(x1))))))))))) 3(2(3(5(3(5(2(3(5(5(5(x1))))))))))) (28)
3(3(2(4(5(0(3(2(2(4(3(x1))))))))))) 0(3(0(3(0(3(1(0(1(4(x1)))))))))) (29)
3(4(0(3(5(3(4(1(2(4(3(x1))))))))))) 5(1(0(5(3(3(1(4(2(0(x1)))))))))) (30)
3(5(0(3(4(4(4(3(4(4(3(x1))))))))))) 3(1(1(2(3(4(1(5(4(1(x1)))))))))) (31)
3(5(2(5(0(0(1(4(1(4(5(x1))))))))))) 2(2(2(4(3(3(0(3(5(2(x1)))))))))) (32)
4(0(4(1(3(4(5(2(2(1(5(x1))))))))))) 5(0(3(0(4(2(2(3(4(5(x1)))))))))) (33)
4(2(1(5(0(4(3(5(3(3(0(x1))))))))))) 1(3(4(4(3(4(4(0(1(2(x1)))))))))) (34)
4(2(4(3(1(3(0(0(2(1(0(x1))))))))))) 4(0(1(4(2(0(0(0(4(4(x1)))))))))) (35)
4(4(5(3(1(0(5(4(1(3(3(x1))))))))))) 0(5(4(4(4(3(2(3(2(2(x1)))))))))) (36)
4(5(3(2(5(4(5(0(4(2(3(x1))))))))))) 0(0(1(5(0(0(2(4(2(2(x1)))))))))) (37)
4(5(5(3(5(4(3(4(5(4(0(x1))))))))))) 0(2(5(0(2(4(3(4(5(1(x1)))))))))) (38)
5(0(1(1(5(0(3(0(0(2(3(x1))))))))))) 3(3(4(5(5(0(3(1(1(1(x1)))))))))) (39)
5(0(1(3(0(0(0(4(3(1(4(x1))))))))))) 1(4(4(5(5(0(4(2(5(2(x1)))))))))) (40)
5(0(2(5(0(4(3(0(1(3(5(x1))))))))))) 2(5(3(0(0(0(3(3(2(2(x1)))))))))) (41)
5(1(5(4(2(0(5(4(3(4(0(x1))))))))))) 4(0(2(4(4(4(4(0(4(2(x1)))))))))) (42)
5(5(4(2(1(5(5(3(3(4(4(x1))))))))))) 5(4(4(3(4(0(0(2(2(3(x1)))))))))) (43)
5(5(4(2(2(5(0(1(2(2(3(x1))))))))))) 5(5(4(2(2(5(1(0(2(2(3(x1))))))))))) (44)
0(1(0(2(2(1(2(3(4(1(1(5(x1)))))))))))) 0(5(5(0(2(0(3(5(4(1(x1)))))))))) (45)
0(3(1(1(4(4(3(1(2(2(0(1(x1)))))))))))) 3(2(3(3(5(5(0(1(5(3(x1)))))))))) (46)
1(2(5(4(0(2(4(2(2(1(2(5(x1)))))))))))) 5(4(0(3(1(2(0(2(1(2(x1)))))))))) (47)
2(5(2(2(4(3(2(4(1(1(4(1(x1)))))))))))) 0(2(4(2(2(3(2(0(1(3(x1)))))))))) (48)
3(2(2(5(1(0(1(2(5(3(1(3(x1)))))))))))) 4(0(5(4(5(5(5(4(4(2(x1)))))))))) (49)
3(2(5(4(2(4(1(3(1(1(2(5(x1)))))))))))) 3(3(1(5(3(0(4(0(0(1(x1)))))))))) (50)
3(4(3(4(4(1(3(4(5(0(4(1(x1)))))))))))) 4(2(0(0(0(2(4(1(3(3(x1)))))))))) (51)
3(5(3(4(1(3(4(0(1(2(1(1(x1)))))))))))) 2(5(4(5(5(2(1(0(1(4(x1)))))))))) (52)
4(0(0(4(4(2(1(4(4(1(2(0(x1)))))))))))) 4(4(5(2(1(0(1(0(2(4(x1)))))))))) (53)
4(3(1(2(4(3(2(5(2(4(5(5(x1)))))))))))) 5(3(0(1(0(5(5(5(0(3(x1)))))))))) (54)
4(4(1(3(0(4(1(2(3(0(0(4(x1)))))))))))) 3(4(0(1(5(1(4(3(1(0(x1)))))))))) (55)
5(2(3(5(0(1(4(2(4(5(0(2(x1)))))))))))) 4(5(1(3(1(5(5(5(1(0(x1)))))))))) (56)
5(5(4(1(2(0(3(1(4(4(1(1(x1)))))))))))) 4(4(5(5(0(2(2(3(3(2(x1)))))))))) (57)
0(1(0(2(0(1(5(1(4(2(5(4(1(x1))))))))))))) 0(1(0(2(0(1(1(5(4(2(5(4(1(x1))))))))))))) (58)
2(4(2(4(5(0(3(2(4(0(3(5(0(x1))))))))))))) 2(4(2(4(5(0(2(3(4(0(3(5(0(x1))))))))))))) (59)
3(0(0(0(2(1(0(3(4(3(0(5(2(3(3(x1))))))))))))))) 3(0(0(0(2(1(0(3(4(3(0(2(5(3(3(x1))))))))))))))) (60)
3(4(1(0(4(0(1(0(1(5(1(3(0(5(0(x1))))))))))))))) 3(4(0(1(4(0(1(0(1(5(1(3(0(5(0(x1))))))))))))))) (61)
1(0(1(4(1(4(0(1(1(1(4(5(2(0(5(5(3(x1))))))))))))))))) 1(0(1(4(1(4(0(1(1(1(4(2(5(0(5(5(3(x1))))))))))))))))) (62)
5(2(0(2(2(1(0(3(2(5(1(1(2(1(0(1(1(x1))))))))))))))))) 5(2(0(2(2(1(0(3(2(5(2(1(1(0(1(1(1(x1))))))))))))))))) (63)
0(3(1(0(3(4(1(2(0(2(1(3(5(1(2(4(5(2(x1)))))))))))))))))) 0(3(1(3(0(4(1(2(0(2(1(3(5(1(2(4(5(2(x1)))))))))))))))))) (64)
1(2(0(4(2(4(5(4(1(2(2(2(5(0(5(2(3(1(x1)))))))))))))))))) 1(2(0(4(4(2(5(4(1(2(2(2(5(0(5(2(3(1(x1)))))))))))))))))) (65)
4(5(2(3(5(2(2(4(2(4(2(1(4(3(4(3(3(2(0(1(x1)))))))))))))))))))) 4(5(2(3(5(2(2(4(4(2(2(1(4(3(4(3(3(2(0(1(x1)))))))))))))))))))) (66)
0(0(0(3(1(2(2(0(1(1(4(5(5(4(2(3(5(3(0(3(1(3(x1)))))))))))))))))))))) 0(0(3(0(1(2(2(1(0(1(4(5(5(4(2(3(5(3(0(3(1(3(x1)))))))))))))))))))))) (67)
2(0(3(3(4(3(0(3(4(2(3(3(2(4(5(2(5(0(0(0(3(2(4(x1))))))))))))))))))))))) 2(0(3(3(4(3(0(3(4(2(3(3(2(4(2(5(5(0(0(0(3(2(4(x1))))))))))))))))))))))) (68)
2(4(3(3(3(4(1(1(5(3(2(0(0(4(2(2(2(0(5(3(0(1(0(x1))))))))))))))))))))))) 2(4(3(3(3(4(1(1(5(3(2(0(0(2(4(2(2(0(5(3(0(1(0(x1))))))))))))))))))))))) (69)
3(5(5(0(0(2(3(5(2(3(5(1(0(2(3(0(3(4(1(4(1(4(3(x1))))))))))))))))))))))) 3(5(5(0(0(2(5(3(2(3(5(1(0(2(3(0(3(4(1(4(1(4(3(x1))))))))))))))))))))))) (70)
4(2(5(3(3(2(4(4(5(5(3(5(3(3(2(3(0(2(2(5(1(5(1(x1))))))))))))))))))))))) 4(2(5(3(3(2(4(4(5(5(5(3(3(3(2(3(0(2(2(5(1(5(1(x1))))))))))))))))))))))) (71)
1(0(2(3(1(1(2(3(4(3(4(2(4(0(1(0(3(1(0(0(2(2(5(1(x1)))))))))))))))))))))))) 1(0(2(3(1(1(2(4(3(3(4(2(4(0(1(0(3(1(0(0(2(2(5(1(x1)))))))))))))))))))))))) (72)
1(1(5(5(5(4(4(2(3(0(3(5(0(4(0(3(5(3(5(5(2(4(1(4(x1)))))))))))))))))))))))) 1(1(5(5(5(4(4(2(3(0(3(5(0(4(0(3(3(5(5(5(2(4(1(4(x1)))))))))))))))))))))))) (73)
1(2(1(1(4(3(3(0(0(5(5(3(2(1(5(5(5(5(5(5(2(5(2(2(x1)))))))))))))))))))))))) 1(2(1(1(4(3(3(0(0(5(5(2(3(1(5(5(5(5(5(5(2(5(2(2(x1)))))))))))))))))))))))) (74)
5(1(2(3(3(1(0(0(5(5(5(0(3(5(4(2(0(4(1(1(4(2(2(4(x1)))))))))))))))))))))))) 5(1(2(3(3(1(0(0(5(5(5(0(3(5(4(2(0(1(4(1(4(2(2(4(x1)))))))))))))))))))))))) (75)
2(5(1(5(4(0(4(3(0(4(0(1(2(4(0(5(0(5(4(3(1(5(2(3(0(x1))))))))))))))))))))))))) 2(5(1(5(4(0(4(3(0(4(0(1(2(4(0(5(0(5(3(4(1(5(2(3(0(x1))))))))))))))))))))))))) (76)
3(1(5(5(5(3(5(5(1(0(4(3(5(5(4(1(1(2(4(2(4(2(0(4(3(x1))))))))))))))))))))))))) 3(1(5(5(5(3(5(5(1(0(3(4(5(5(4(1(1(2(4(2(4(2(0(4(3(x1))))))))))))))))))))))))) (77)
2(0(1(2(3(2(5(5(0(0(5(5(1(2(1(3(4(5(4(5(0(3(0(4(3(4(x1)))))))))))))))))))))))))) 2(0(1(3(5(0(4(5(2(2(2(0(5(3(1(0(5(4(3(5(4(0(5(1(3(4(x1)))))))))))))))))))))))))) (78)
2(3(1(2(0(0(4(5(5(3(2(4(3(3(4(5(4(1(3(4(4(2(3(5(0(3(2(x1))))))))))))))))))))))))))) 2(3(1(2(0(0(4(5(5(3(2(4(3(3(4(5(4(1(3(4(2(4(3(5(0(3(2(x1))))))))))))))))))))))))))) (79)
3(1(0(4(1(1(0(3(1(3(3(1(0(4(0(1(4(0(4(3(1(2(4(5(3(2(3(x1))))))))))))))))))))))))))) 3(0(1(4(1(1(3(0(3(1(1(0(4(3(1(4(0(0(4(3(1(2(3(5(4(2(3(x1))))))))))))))))))))))))))) (80)
1(2(1(3(0(4(5(2(3(1(3(3(5(0(2(0(3(5(2(0(4(5(1(3(0(1(0(2(x1)))))))))))))))))))))))))))) 1(2(1(3(0(5(4(2(3(1(3(3(5(0(2(0(3(5(2(0(4(5(1(3(0(1(0(2(x1)))))))))))))))))))))))))))) (81)
2(0(0(2(1(5(3(4(5(2(3(4(1(4(0(4(5(0(4(4(0(5(3(0(0(4(5(5(x1)))))))))))))))))))))))))))) 2(0(0(2(1(5(3(4(2(5(3(4(1(4(0(4(5(0(4(4(0(5(3(0(0(4(5(5(x1)))))))))))))))))))))))))))) (82)
3(3(5(5(0(1(0(4(3(5(5(2(5(0(2(0(4(1(4(3(4(0(1(4(2(1(0(2(x1)))))))))))))))))))))))))))) 3(3(5(5(0(0(1(4(3(5(5(2(5(0(2(0(4(1(4(3(4(0(1(4(2(1(0(2(x1)))))))))))))))))))))))))))) (83)
0(1(0(4(4(5(3(0(0(3(5(4(3(1(2(2(3(0(4(0(1(4(0(0(5(3(0(4(0(x1))))))))))))))))))))))))))))) 0(1(0(4(4(5(0(3(0(3(5(4(3(1(2(2(3(0(4(0(1(4(0(0(5(3(0(4(0(x1))))))))))))))))))))))))))))) (84)
1(0(4(0(4(5(3(1(5(2(2(1(5(2(3(2(3(3(3(2(1(5(5(1(3(5(4(3(4(x1))))))))))))))))))))))))))))) 0(0(1(4(3(1(5(5(4(5(1(3(2(2(2(2(1(3(5(3(2(3(3(1(4(5(5(3(4(x1))))))))))))))))))))))))))))) (85)
4(3(1(4(5(2(5(1(4(4(5(2(0(2(4(2(1(4(2(4(0(0(4(0(3(0(0(2(4(x1))))))))))))))))))))))))))))) 4(3(1(4(5(2(5(1(4(4(5(0(2(2(4(2(1(4(2(4(0(0(4(0(3(0(0(2(4(x1))))))))))))))))))))))))))))) (86)
0(1(5(5(2(4(5(2(2(3(2(3(2(3(0(4(1(5(3(5(0(3(0(5(4(2(4(5(5(0(x1)))))))))))))))))))))))))))))) 0(1(5(5(2(4(5(2(2(3(2(3(2(3(0(4(1(5(3(5(0(3(5(0(4(2(4(5(5(0(x1)))))))))))))))))))))))))))))) (87)
0(5(5(5(5(0(1(1(5(5(1(0(0(4(4(1(3(3(1(1(4(1(3(1(3(1(1(2(5(4(x1)))))))))))))))))))))))))))))) 0(5(5(5(5(0(1(1(5(5(1(0(0(4(4(1(3(3(1(1(4(3(1(1(3(1(1(2(5(4(x1)))))))))))))))))))))))))))))) (88)
2(2(5(2(4(4(0(1(2(2(2(0(3(2(5(4(5(0(2(5(0(3(5(1(4(5(1(1(5(1(x1)))))))))))))))))))))))))))))) 2(2(5(2(4(4(0(1(2(2(2(3(0(2(5(4(5(0(2(5(0(3(5(1(4(5(1(1(5(1(x1)))))))))))))))))))))))))))))) (89)
3(1(3(3(1(5(1(0(0(0(3(4(1(2(0(5(3(4(5(2(4(3(2(4(5(5(0(1(4(5(x1)))))))))))))))))))))))))))))) 3(1(3(3(1(5(1(0(0(0(3(4(1(2(0(5(3(4(5(2(4(3(4(2(5(5(0(1(4(5(x1)))))))))))))))))))))))))))))) (90)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[5(x1)] = x1 +
6
[4(x1)] = x1 +
5
[3(x1)] = x1 +
5
[2(x1)] = x1 +
5
[1(x1)] = x1 +
5
[0(x1)] = x1 +
6
all of the following rules can be deleted.
0(3(0(5(1(3(2(4(5(5(x1)))))))))) 1(2(3(3(0(4(1(5(1(3(x1)))))))))) (4)
3(3(1(4(3(1(0(5(5(4(x1)))))))))) 5(4(1(1(4(5(2(3(3(2(x1)))))))))) (5)
5(0(1(5(2(5(2(0(5(4(x1)))))))))) 3(5(1(2(4(1(0(0(1(3(x1)))))))))) (6)
5(0(5(4(5(0(0(2(1(3(x1)))))))))) 3(5(3(0(2(4(3(2(0(4(x1)))))))))) (7)
0(2(5(4(4(2(0(3(5(1(5(x1))))))))))) 0(0(2(4(4(4(1(0(5(3(x1)))))))))) (8)
0(3(1(5(4(1(5(5(5(3(2(x1))))))))))) 1(0(4(4(1(4(2(0(5(3(x1)))))))))) (9)
1(0(4(5(4(0(3(0(3(2(5(x1))))))))))) 0(5(5(5(0(1(1(1(2(4(x1)))))))))) (10)
1(2(4(1(0(3(2(5(5(3(1(x1))))))))))) 3(1(1(3(5(2(5(5(2(1(x1)))))))))) (11)
1(4(0(1(2(1(4(5(4(4(0(x1))))))))))) 4(3(3(4(1(0(3(1(2(4(x1)))))))))) (12)
1(4(2(1(4(1(2(5(2(3(3(x1))))))))))) 3(0(0(2(3(2(4(1(4(3(x1)))))))))) (13)
1(5(2(4(1(5(0(1(0(2(0(x1))))))))))) 1(1(3(4(5(2(5(4(1(1(x1)))))))))) (14)
2(0(0(1(2(1(1(4(5(5(1(x1))))))))))) 3(1(3(4(1(0(0(0(2(2(x1)))))))))) (15)
2(1(0(2(3(0(5(3(5(0(3(x1))))))))))) 0(1(1(0(3(2(0(5(4(2(x1)))))))))) (16)
2(1(0(4(3(3(1(3(0(5(0(x1))))))))))) 5(1(5(0(4(4(2(5(0(2(x1)))))))))) (17)
2(1(2(3(4(0(4(2(2(3(5(x1))))))))))) 3(1(2(2(4(3(3(3(2(4(x1)))))))))) (18)
2(1(4(5(5(1(2(5(4(3(5(x1))))))))))) 4(5(5(4(2(2(2(3(2(0(x1)))))))))) (20)
2(3(0(3(2(4(3(5(3(3(1(x1))))))))))) 1(1(1(3(1(0(3(1(5(5(x1)))))))))) (21)
2(3(3(1(3(2(3(3(1(1(2(x1))))))))))) 2(1(4(1(4(5(4(2(2(1(x1)))))))))) (22)
2(4(0(0(4(1(0(2(1(0(4(x1))))))))))) 1(3(5(4(4(4(0(3(4(2(x1)))))))))) (23)
2(4(0(2(0(0(0(1(0(5(3(x1))))))))))) 5(1(4(4(4(0(1(4(5(1(x1)))))))))) (24)
2(5(2(1(1(0(3(1(3(4(3(x1))))))))))) 4(4(4(5(3(3(0(3(1(3(x1)))))))))) (25)
3(2(0(0(5(2(3(2(1(0(2(x1))))))))))) 4(1(2(4(0(2(5(3(4(2(x1)))))))))) (27)
3(3(2(4(5(0(3(2(2(4(3(x1))))))))))) 0(3(0(3(0(3(1(0(1(4(x1)))))))))) (29)
3(4(0(3(5(3(4(1(2(4(3(x1))))))))))) 5(1(0(5(3(3(1(4(2(0(x1)))))))))) (30)
3(5(0(3(4(4(4(3(4(4(3(x1))))))))))) 3(1(1(2(3(4(1(5(4(1(x1)))))))))) (31)
3(5(2(5(0(0(1(4(1(4(5(x1))))))))))) 2(2(2(4(3(3(0(3(5(2(x1)))))))))) (32)
4(0(4(1(3(4(5(2(2(1(5(x1))))))))))) 5(0(3(0(4(2(2(3(4(5(x1)))))))))) (33)
4(2(1(5(0(4(3(5(3(3(0(x1))))))))))) 1(3(4(4(3(4(4(0(1(2(x1)))))))))) (34)
4(2(4(3(1(3(0(0(2(1(0(x1))))))))))) 4(0(1(4(2(0(0(0(4(4(x1)))))))))) (35)
4(4(5(3(1(0(5(4(1(3(3(x1))))))))))) 0(5(4(4(4(3(2(3(2(2(x1)))))))))) (36)
4(5(3(2(5(4(5(0(4(2(3(x1))))))))))) 0(0(1(5(0(0(2(4(2(2(x1)))))))))) (37)
4(5(5(3(5(4(3(4(5(4(0(x1))))))))))) 0(2(5(0(2(4(3(4(5(1(x1)))))))))) (38)
5(0(1(1(5(0(3(0(0(2(3(x1))))))))))) 3(3(4(5(5(0(3(1(1(1(x1)))))))))) (39)
5(0(1(3(0(0(0(4(3(1(4(x1))))))))))) 1(4(4(5(5(0(4(2(5(2(x1)))))))))) (40)
5(0(2(5(0(4(3(0(1(3(5(x1))))))))))) 2(5(3(0(0(0(3(3(2(2(x1)))))))))) (41)
5(1(5(4(2(0(5(4(3(4(0(x1))))))))))) 4(0(2(4(4(4(4(0(4(2(x1)))))))))) (42)
5(5(4(2(1(5(5(3(3(4(4(x1))))))))))) 5(4(4(3(4(0(0(2(2(3(x1)))))))))) (43)
0(1(0(2(2(1(2(3(4(1(1(5(x1)))))))))))) 0(5(5(0(2(0(3(5(4(1(x1)))))))))) (45)
0(3(1(1(4(4(3(1(2(2(0(1(x1)))))))))))) 3(2(3(3(5(5(0(1(5(3(x1)))))))))) (46)
1(2(5(4(0(2(4(2(2(1(2(5(x1)))))))))))) 5(4(0(3(1(2(0(2(1(2(x1)))))))))) (47)
2(5(2(2(4(3(2(4(1(1(4(1(x1)))))))))))) 0(2(4(2(2(3(2(0(1(3(x1)))))))))) (48)
3(2(2(5(1(0(1(2(5(3(1(3(x1)))))))))))) 4(0(5(4(5(5(5(4(4(2(x1)))))))))) (49)
3(2(5(4(2(4(1(3(1(1(2(5(x1)))))))))))) 3(3(1(5(3(0(4(0(0(1(x1)))))))))) (50)
3(4(3(4(4(1(3(4(5(0(4(1(x1)))))))))))) 4(2(0(0(0(2(4(1(3(3(x1)))))))))) (51)
3(5(3(4(1(3(4(0(1(2(1(1(x1)))))))))))) 2(5(4(5(5(2(1(0(1(4(x1)))))))))) (52)
4(0(0(4(4(2(1(4(4(1(2(0(x1)))))))))))) 4(4(5(2(1(0(1(0(2(4(x1)))))))))) (53)
4(3(1(2(4(3(2(5(2(4(5(5(x1)))))))))))) 5(3(0(1(0(5(5(5(0(3(x1)))))))))) (54)
4(4(1(3(0(4(1(2(3(0(0(4(x1)))))))))))) 3(4(0(1(5(1(4(3(1(0(x1)))))))))) (55)
5(2(3(5(0(1(4(2(4(5(0(2(x1)))))))))))) 4(5(1(3(1(5(5(5(1(0(x1)))))))))) (56)
5(5(4(1(2(0(3(1(4(4(1(1(x1)))))))))))) 4(4(5(5(0(2(2(3(3(2(x1)))))))))) (57)

1.1 Closure Under Flat Contexts

Using the flat contexts

{5(), 4(), 3(), 2(), 1(), 0()}

We obtain the transformed TRS

There are 240 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

The following interpretations form a model of the rules.

As carrier we take the set {0,...,5}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 6):

[5(x1)] = 6x1 + 0
[4(x1)] = 6x1 + 1
[3(x1)] = 6x1 + 2
[2(x1)] = 6x1 + 3
[1(x1)] = 6x1 + 4
[0(x1)] = 6x1 + 5

We obtain the labeled TRS

There are 1440 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[50(x1)] = x1 +
0
[51(x1)] = x1 +
0
[52(x1)] = x1 +
3
[53(x1)] = x1 +
6
[54(x1)] = x1 +
4
[55(x1)] = x1 +
0
[40(x1)] = x1 +
1
[41(x1)] = x1 +
0
[42(x1)] = x1 +
0
[43(x1)] = x1 +
1
[44(x1)] = x1 +
0
[45(x1)] = x1 +
0
[30(x1)] = x1 +
0
[31(x1)] = x1 +
1
[32(x1)] = x1 +
0
[33(x1)] = x1 +
3
[34(x1)] = x1 +
6
[35(x1)] = x1 +
6
[20(x1)] = x1 +
0
[21(x1)] = x1 +
0
[22(x1)] = x1 +
1
[23(x1)] = x1 +
0
[24(x1)] = x1 +
15
[25(x1)] = x1 +
22
[10(x1)] = x1 +
4
[11(x1)] = x1 +
0
[12(x1)] = x1 +
6
[13(x1)] = x1 +
26
[14(x1)] = x1 +
3
[15(x1)] = x1 +
15
[00(x1)] = x1 +
0
[01(x1)] = x1 +
3
[02(x1)] = x1 +
0
[03(x1)] = x1 +
0
[04(x1)] = x1 +
0
[05(x1)] = x1 +
6
all of the following rules can be deleted.

There are 1440 ruless (increase limit for explicit display).

1.1.1.1.1 R is empty

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