Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/138142)

The rewrite relation of the following TRS is considered.

0(0(0(3(0(4(3(1(4(5(2(0(x1)))))))))))) → 1(0(2(3(5(5(1(0(3(2(2(3(3(4(2(4(x1)))))))))))))))) (1)
0(0(0(3(1(5(0(5(1(0(3(1(x1)))))))))))) → 3(3(1(2(5(5(1(2(4(4(4(1(2(3(3(x1))))))))))))))) (2)
0(0(1(1(4(3(4(0(5(1(4(3(x1)))))))))))) → 3(4(1(1(2(5(5(2(2(2(2(3(5(0(4(2(x1)))))))))))))))) (3)
0(0(1(3(4(3(4(4(4(0(3(0(x1)))))))))))) → 2(5(5(0(2(5(2(5(4(4(2(1(1(2(4(3(5(1(x1)))))))))))))))))) (4)
0(0(2(1(0(3(0(0(1(0(1(0(x1)))))))))))) → 2(1(2(4(3(4(2(5(5(3(5(1(0(0(3(x1))))))))))))))) (5)
0(0(4(3(0(0(1(4(0(1(3(5(x1)))))))))))) → 5(1(5(1(0(2(2(2(3(3(4(5(3(4(2(2(2(0(x1)))))))))))))))))) (6)
0(0(5(4(0(0(3(1(3(0(1(5(x1)))))))))))) → 3(1(0(5(5(5(5(4(1(1(2(2(2(4(5(2(1(2(x1)))))))))))))))))) (7)
0(0(5(4(5(0(5(0(0(3(0(3(x1)))))))))))) → 2(5(5(3(3(0(4(1(4(2(1(1(2(4(4(1(0(x1))))))))))))))))) (8)
0(1(0(0(1(3(1(0(3(2(5(0(x1)))))))))))) → 5(1(1(4(0(3(2(2(4(2(2(1(2(5(1(x1))))))))))))))) (9)
0(1(0(0(1(4(4(4(4(2(1(0(x1)))))))))))) → 5(5(2(4(1(5(3(1(2(2(2(4(2(3(2(1(3(4(x1)))))))))))))))))) (10)
0(1(1(3(4(4(0(2(0(1(1(1(x1)))))))))))) → 0(5(3(2(2(2(5(1(4(1(3(2(4(2(2(1(2(1(x1)))))))))))))))))) (11)
0(1(4(5(0(0(5(1(3(0(4(5(x1)))))))))))) → 2(3(3(1(2(5(4(1(2(5(3(4(1(2(5(5(3(5(x1)))))))))))))))))) (12)
0(2(0(4(4(1(2(1(0(4(3(3(x1)))))))))))) → 2(1(2(4(3(5(2(5(2(5(3(4(4(1(x1)))))))))))))) (13)
0(2(4(3(4(1(2(0(1(4(4(5(x1)))))))))))) → 4(2(5(1(2(1(2(3(0(2(2(2(2(2(3(3(x1)))))))))))))))) (14)
0(3(0(4(4(0(0(4(4(0(1(3(x1)))))))))))) → 5(2(1(2(4(2(2(4(0(4(3(1(0(2(3(2(5(3(x1)))))))))))))))))) (15)
0(3(4(1(3(4(1(0(1(4(3(1(x1)))))))))))) → 2(2(2(3(2(0(2(5(2(2(2(1(5(2(4(1(x1)))))))))))))))) (16)
0(3(4(3(5(0(0(4(4(3(1(1(x1)))))))))))) → 4(2(1(1(2(2(5(1(2(0(4(5(4(2(1(2(5(x1))))))))))))))))) (17)
0(3(5(4(0(1(1(4(0(5(5(0(x1)))))))))))) → 3(2(4(1(0(5(1(1(2(3(5(3(2(1(2(2(x1)))))))))))))))) (18)
0(4(0(3(1(4(0(4(1(4(5(5(x1)))))))))))) → 2(5(1(2(2(4(2(5(2(5(4(1(0(4(x1)))))))))))))) (19)
0(4(3(1(3(0(4(5(1(5(0(0(x1)))))))))))) → 3(5(1(2(1(3(2(2(2(2(0(3(2(2(2(2(4(5(x1)))))))))))))))))) (20)
0(4(3(5(3(1(4(4(0(0(0(4(x1)))))))))))) → 5(2(3(3(2(4(3(5(2(3(4(2(2(3(3(2(0(x1))))))))))))))))) (21)
0(4(5(0(0(2(0(3(3(5(5(5(x1)))))))))))) → 2(4(2(2(2(5(1(0(2(4(4(2(1(2(2(x1))))))))))))))) (22)
0(5(0(4(1(0(1(0(5(4(0(5(x1)))))))))))) → 5(3(1(5(3(2(5(3(1(0(2(3(5(2(0(4(x1)))))))))))))))) (23)
0(5(1(5(0(0(5(4(5(0(4(1(x1)))))))))))) → 5(5(2(4(2(5(5(0(4(0(1(1(2(5(5(5(5(x1))))))))))))))))) (24)
0(5(3(1(1(1(4(4(3(0(1(4(x1)))))))))))) → 2(0(2(4(5(5(2(2(3(3(5(0(5(2(3(2(x1)))))))))))))))) (25)
0(5(3(1(2(5(1(3(4(3(5(0(x1)))))))))))) → 5(2(2(5(2(0(5(5(2(5(5(5(3(5(4(2(x1)))))))))))))))) (26)
0(5(4(2(1(0(4(3(0(2(3(3(x1)))))))))))) → 3(3(3(2(3(2(0(5(4(2(3(4(1(2(3(x1))))))))))))))) (27)
0(5(4(5(0(3(3(0(4(0(5(4(x1)))))))))))) → 5(4(2(5(4(1(2(1(3(3(4(2(2(2(1(2(5(0(x1)))))))))))))))))) (28)
1(0(0(0(1(0(5(0(1(1(3(0(x1)))))))))))) → 3(5(5(2(1(2(5(5(5(0(2(3(3(1(0(2(x1)))))))))))))))) (29)
1(0(1(3(2(5(2(2(3(4(1(2(x1)))))))))))) → 3(2(1(2(0(2(3(5(4(1(2(2(2(2(x1)))))))))))))) (30)
1(0(4(3(0(0(5(4(4(1(0(5(x1)))))))))))) → 1(3(2(2(3(0(0(2(4(2(1(2(5(3(x1)))))))))))))) (31)
1(1(4(5(0(5(4(4(0(5(0(0(x1)))))))))))) → 5(4(4(1(1(4(1(2(2(5(0(2(1(2(0(5(x1)))))))))))))))) (32)
1(2(5(4(4(0(0(4(3(1(1(2(x1)))))))))))) → 5(2(2(2(3(0(4(1(5(4(2(3(2(2(x1)))))))))))))) (33)
1(3(1(3(3(4(4(4(3(0(4(4(x1)))))))))))) → 2(5(5(5(3(3(2(3(0(5(0(4(2(1(x1)))))))))))))) (34)
1(3(3(5(4(0(4(4(5(2(0(0(x1)))))))))))) → 4(2(2(1(2(5(3(3(3(5(4(5(1(5(5(x1))))))))))))))) (35)
1(4(1(0(4(3(1(0(0(5(4(0(x1)))))))))))) → 2(5(1(5(1(3(3(1(4(1(0(2(0(0(2(x1))))))))))))))) (36)
1(4(3(5(3(3(0(4(0(0(4(0(x1)))))))))))) → 1(5(3(2(2(2(2(1(5(2(5(1(0(1(5(0(x1)))))))))))))))) (37)
2(0(0(3(1(2(4(3(0(1(4(3(x1)))))))))))) → 1(1(2(2(5(0(2(1(1(5(0(5(5(2(2(3(x1)))))))))))))))) (38)
2(0(3(1(4(5(2(0(2(5(1(4(x1)))))))))))) → 1(2(5(3(2(0(1(2(4(1(3(2(5(3(5(x1))))))))))))))) (39)
2(1(0(1(5(3(1(3(1(0(3(5(x1)))))))))))) → 5(3(5(2(4(4(0(3(5(2(2(1(2(2(1(3(x1)))))))))))))))) (40)
2(2(4(0(4(0(0(5(5(4(3(5(x1)))))))))))) → 2(2(5(4(2(3(0(1(2(2(1(2(2(3(5(3(x1)))))))))))))))) (41)
2(4(0(4(4(4(0(4(1(3(2(0(x1)))))))))))) → 3(0(5(5(2(2(0(2(2(5(2(2(5(3(x1)))))))))))))) (42)
2(4(0(4(5(4(0(3(4(4(1(5(x1)))))))))))) → 2(4(1(1(5(2(3(2(1(4(0(2(2(5(5(2(3(3(x1)))))))))))))))))) (43)
2(5(1(4(0(0(5(4(0(0(4(1(x1)))))))))))) → 5(5(4(1(3(3(1(1(2(2(3(3(2(2(2(3(5(5(x1)))))))))))))))))) (44)
3(1(0(3(4(4(4(3(1(3(4(1(x1)))))))))))) → 1(2(2(4(4(1(2(0(5(2(4(1(5(4(x1)))))))))))))) (45)
3(3(4(0(4(2(2(3(4(3(0(0(x1)))))))))))) → 1(2(3(0(4(1(5(3(0(2(2(5(2(4(1(x1))))))))))))))) (46)
3(4(0(4(4(4(3(0(3(4(4(2(x1)))))))))))) → 1(2(4(5(3(2(2(2(1(2(4(0(5(3(2(0(0(1(x1)))))))))))))))))) (47)
3(4(4(4(0(4(0(0(2(3(1(4(x1)))))))))))) → 2(5(2(2(2(2(3(5(2(1(3(2(1(3(4(5(5(2(x1)))))))))))))))))) (48)
4(0(0(3(2(2(4(5(0(4(3(3(x1)))))))))))) → 5(2(2(5(2(1(3(2(2(3(3(5(5(2(2(2(4(3(x1)))))))))))))))))) (49)
4(0(1(0(1(0(0(3(3(4(0(4(x1)))))))))))) → 4(4(5(2(1(3(2(5(1(2(0(2(5(1(3(2(2(x1))))))))))))))))) (50)
4(0(1(1(4(1(3(4(5(4(3(5(x1)))))))))))) → 5(2(5(2(3(2(4(1(3(2(2(0(5(1(4(x1))))))))))))))) (51)
4(0(1(4(3(0(5(4(5(1(0(4(x1)))))))))))) → 2(1(1(5(2(1(5(5(0(3(2(4(3(2(5(x1))))))))))))))) (52)
4(0(3(1(5(1(3(1(4(0(1(2(x1)))))))))))) → 2(2(5(2(3(2(3(5(5(3(4(4(1(5(3(5(2(2(x1)))))))))))))))))) (53)
4(0(3(4(4(3(0(2(5(4(1(1(x1)))))))))))) → 0(5(3(5(5(2(2(1(2(2(3(2(5(5(4(3(x1)))))))))))))))) (54)
4(0(4(3(1(3(3(4(3(4(0(0(x1)))))))))))) → 5(3(2(2(2(5(2(2(1(0(4(2(2(4(3(2(3(x1))))))))))))))))) (55)
4(1(0(0(0(3(5(4(3(3(0(2(x1)))))))))))) → 2(2(5(3(3(5(2(3(3(1(2(2(0(2(5(3(3(x1))))))))))))))))) (56)
4(1(0(4(3(0(5(0(3(2(3(0(x1)))))))))))) → 2(3(3(2(1(4(4(5(2(0(1(5(2(5(1(1(x1)))))))))))))))) (57)
4(1(1(3(1(3(2(4(3(1(4(4(x1)))))))))))) → 4(0(1(2(3(2(2(2(3(2(3(1(4(1(2(5(3(2(x1)))))))))))))))))) (58)
4(1(1(3(2(3(1(5(0(4(0(0(x1)))))))))))) → 5(0(5(5(3(4(2(2(3(2(2(2(5(1(0(5(x1)))))))))))))))) (59)
4(1(5(2(3(0(1(5(4(3(3(0(x1)))))))))))) → 5(5(4(2(2(0(5(3(2(3(2(1(3(2(1(3(5(1(x1)))))))))))))))))) (60)
4(2(0(0(4(3(0(3(0(4(4(5(x1)))))))))))) → 2(0(5(3(2(2(3(3(5(3(5(0(1(1(2(3(0(2(x1)))))))))))))))))) (61)
4(2(2(0(4(1(4(4(0(0(3(4(x1)))))))))))) → 3(5(1(2(3(4(1(2(5(2(0(5(3(3(2(x1))))))))))))))) (62)
4(3(2(0(0(5(4(4(5(4(4(0(x1)))))))))))) → 4(1(2(3(1(2(2(0(3(5(2(0(5(2(5(1(5(0(x1)))))))))))))))))) (63)
4(3(2(1(4(4(5(0(3(0(0(4(x1)))))))))))) → 2(1(2(5(3(3(5(2(0(5(3(4(2(5(2(3(5(3(x1)))))))))))))))))) (64)
4(3(2(5(4(4(4(1(4(5(1(0(x1)))))))))))) → 2(1(5(5(2(3(3(1(2(3(2(2(3(0(1(5(x1)))))))))))))))) (65)
4(3(5(4(5(0(5(4(3(4(3(5(x1)))))))))))) → 5(3(4(2(2(0(2(2(5(5(4(3(4(0(1(5(5(x1))))))))))))))))) (66)
4(4(0(4(3(1(0(5(4(4(0(0(x1)))))))))))) → 0(4(5(5(4(3(4(5(5(3(5(1(1(3(2(5(x1)))))))))))))))) (67)
4(4(3(1(2(1(4(3(0(4(5(4(x1)))))))))))) → 3(3(5(4(4(1(2(3(0(0(1(1(2(1(2(x1))))))))))))))) (68)
4(4(3(4(3(1(4(3(4(5(0(4(x1)))))))))))) → 0(2(4(2(5(1(3(2(0(2(0(0(1(2(x1)))))))))))))) (69)
4(4(4(0(3(5(5(0(5(2(3(1(x1)))))))))))) → 5(2(2(2(2(0(2(4(1(3(0(0(1(5(3(x1))))))))))))))) (70)
4(4(4(4(4(0(3(5(1(0(2(0(x1)))))))))))) → 1(2(5(5(3(3(3(1(2(3(0(2(5(3(3(3(0(2(x1)))))))))))))))))) (71)
4(4(5(0(3(5(4(1(1(5(1(0(x1)))))))))))) → 2(2(5(5(2(3(2(2(5(3(1(2(0(2(x1)))))))))))))) (72)
4(5(0(0(1(0(2(5(4(4(0(4(x1)))))))))))) → 3(5(5(0(2(4(2(2(4(0(1(1(2(2(1(x1))))))))))))))) (73)
4(5(2(0(5(3(1(2(4(0(2(1(x1)))))))))))) → 4(2(5(3(5(5(3(5(2(2(4(3(3(2(4(x1))))))))))))))) (74)
4(5(2(1(4(2(0(3(0(0(4(0(x1)))))))))))) → 3(4(5(3(5(2(5(2(1(4(0(2(2(2(0(x1))))))))))))))) (75)
4(5(3(0(0(5(0(4(5(2(4(0(x1)))))))))))) → 5(2(2(4(3(2(2(3(3(2(3(4(1(1(2(2(0(x1))))))))))))))))) (76)
4(5(4(5(4(4(5(2(1(0(0(5(x1)))))))))))) → 5(3(5(5(5(3(3(1(2(3(4(5(2(2(3(3(5(x1))))))))))))))))) (77)
5(0(0(4(4(5(0(3(3(4(3(1(x1)))))))))))) → 5(1(1(0(4(0(2(2(0(2(5(2(2(5(5(1(1(2(x1)))))))))))))))))) (78)
5(0(1(4(0(5(2(0(0(0(0(0(x1)))))))))))) → 1(0(1(1(1(2(3(3(2(2(5(5(3(0(4(5(x1)))))))))))))))) (79)
5(0(3(5(3(4(2(1(0(3(0(4(x1)))))))))))) → 2(1(2(5(3(2(0(2(1(2(5(4(3(5(2(1(x1)))))))))))))))) (80)
5(0(5(0(4(0(0(5(2(0(0(0(x1)))))))))))) → 0(3(2(2(3(5(4(2(0(2(5(2(2(1(3(4(x1)))))))))))))))) (81)
5(1(0(0(4(2(1(1(2(1(4(2(x1)))))))))))) → 2(2(2(2(3(3(2(3(3(2(5(5(2(2(1(0(2(x1))))))))))))))))) (82)
5(1(1(1(4(4(0(3(1(4(5(0(x1)))))))))))) → 4(1(2(3(1(5(3(5(2(0(4(2(2(4(2(2(x1)))))))))))))))) (83)
5(1(3(0(3(0(3(4(1(0(5(5(x1)))))))))))) → 4(3(2(5(5(2(5(5(4(1(1(5(5(2(x1)))))))))))))) (84)
5(1(5(0(0(3(5(2(1(4(4(5(x1)))))))))))) → 3(4(2(3(2(5(2(2(3(2(2(2(1(2(3(2(5(3(x1)))))))))))))))))) (85)
5(1(5(1(1(4(5(2(4(2(1(4(x1)))))))))))) → 5(1(2(2(1(2(3(1(1(4(1(4(2(2(2(2(2(5(x1)))))))))))))))))) (86)
5(3(0(3(5(1(0(0(0(0(5(4(x1)))))))))))) → 2(3(3(1(2(2(3(2(1(5(4(1(5(1(2(1(2(x1))))))))))))))))) (87)
5(4(0(3(4(0(0(2(3(0(0(2(x1)))))))))))) → 2(1(3(5(1(1(1(2(5(4(3(3(2(1(3(x1))))))))))))))) (88)
5(4(0(4(0(4(5(3(5(4(4(4(x1)))))))))))) → 2(4(1(5(5(2(0(2(0(3(5(4(2(3(1(x1))))))))))))))) (89)
5(4(3(4(4(3(4(2(4(3(2(0(x1)))))))))))) → 2(1(2(3(0(4(2(4(2(5(3(4(1(5(x1)))))))))))))) (90)
5(4(4(0(5(2(0(3(0(0(4(2(x1)))))))))))) → 3(1(2(4(0(1(5(1(4(4(1(2(5(4(x1)))))))))))))) (91)
5(4(4(4(0(2(0(4(5(4(3(4(x1)))))))))))) → 5(2(0(2(1(1(1(1(1(0(4(2(1(1(4(x1))))))))))))))) (92)
5(4(5(1(4(3(1(3(0(0(1(0(x1)))))))))))) → 5(5(3(0(1(3(5(2(0(2(1(1(1(2(3(4(x1)))))))))))))))) (93)
5(4(5(5(5(0(3(4(3(4(4(1(x1)))))))))))) → 1(0(5(5(1(2(2(1(3(5(2(0(5(3(5(4(1(x1))))))))))))))))) (94)
5(5(0(5(0(5(0(0(3(0(3(4(x1)))))))))))) → 3(4(2(2(1(4(0(1(5(1(2(1(2(5(4(5(3(1(x1)))))))))))))))))) (95)
5(5(2(2(1(0(4(0(0(5(5(0(x1)))))))))))) → 5(3(2(1(3(1(1(2(2(2(0(5(3(3(2(5(5(x1))))))))))))))))) (96)

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 +
0
[4(x1)] = x1 +
41/10
[3(x1)] = x1 +
13/5
[2(x1)] = x1 +
0
[1(x1)] = x1 +
3/2
[0(x1)] = x1 +
26/5
all of the following rules can be deleted.
0(0(0(3(0(4(3(1(4(5(2(0(x1)))))))))))) → 1(0(2(3(5(5(1(0(3(2(2(3(3(4(2(4(x1)))))))))))))))) (1)
0(0(0(3(1(5(0(5(1(0(3(1(x1)))))))))))) → 3(3(1(2(5(5(1(2(4(4(4(1(2(3(3(x1))))))))))))))) (2)
0(0(1(1(4(3(4(0(5(1(4(3(x1)))))))))))) → 3(4(1(1(2(5(5(2(2(2(2(3(5(0(4(2(x1)))))))))))))))) (3)
0(0(1(3(4(3(4(4(4(0(3(0(x1)))))))))))) → 2(5(5(0(2(5(2(5(4(4(2(1(1(2(4(3(5(1(x1)))))))))))))))))) (4)
0(0(2(1(0(3(0(0(1(0(1(0(x1)))))))))))) → 2(1(2(4(3(4(2(5(5(3(5(1(0(0(3(x1))))))))))))))) (5)
0(0(4(3(0(0(1(4(0(1(3(5(x1)))))))))))) → 5(1(5(1(0(2(2(2(3(3(4(5(3(4(2(2(2(0(x1)))))))))))))))))) (6)
0(0(5(4(0(0(3(1(3(0(1(5(x1)))))))))))) → 3(1(0(5(5(5(5(4(1(1(2(2(2(4(5(2(1(2(x1)))))))))))))))))) (7)
0(0(5(4(5(0(5(0(0(3(0(3(x1)))))))))))) → 2(5(5(3(3(0(4(1(4(2(1(1(2(4(4(1(0(x1))))))))))))))))) (8)
0(1(0(0(1(3(1(0(3(2(5(0(x1)))))))))))) → 5(1(1(4(0(3(2(2(4(2(2(1(2(5(1(x1))))))))))))))) (9)
0(1(0(0(1(4(4(4(4(2(1(0(x1)))))))))))) → 5(5(2(4(1(5(3(1(2(2(2(4(2(3(2(1(3(4(x1)))))))))))))))))) (10)
0(1(1(3(4(4(0(2(0(1(1(1(x1)))))))))))) → 0(5(3(2(2(2(5(1(4(1(3(2(4(2(2(1(2(1(x1)))))))))))))))))) (11)
0(1(4(5(0(0(5(1(3(0(4(5(x1)))))))))))) → 2(3(3(1(2(5(4(1(2(5(3(4(1(2(5(5(3(5(x1)))))))))))))))))) (12)
0(2(0(4(4(1(2(1(0(4(3(3(x1)))))))))))) → 2(1(2(4(3(5(2(5(2(5(3(4(4(1(x1)))))))))))))) (13)
0(2(4(3(4(1(2(0(1(4(4(5(x1)))))))))))) → 4(2(5(1(2(1(2(3(0(2(2(2(2(2(3(3(x1)))))))))))))))) (14)
0(3(0(4(4(0(0(4(4(0(1(3(x1)))))))))))) → 5(2(1(2(4(2(2(4(0(4(3(1(0(2(3(2(5(3(x1)))))))))))))))))) (15)
0(3(4(1(3(4(1(0(1(4(3(1(x1)))))))))))) → 2(2(2(3(2(0(2(5(2(2(2(1(5(2(4(1(x1)))))))))))))))) (16)
0(3(4(3(5(0(0(4(4(3(1(1(x1)))))))))))) → 4(2(1(1(2(2(5(1(2(0(4(5(4(2(1(2(5(x1))))))))))))))))) (17)
0(3(5(4(0(1(1(4(0(5(5(0(x1)))))))))))) → 3(2(4(1(0(5(1(1(2(3(5(3(2(1(2(2(x1)))))))))))))))) (18)
0(4(0(3(1(4(0(4(1(4(5(5(x1)))))))))))) → 2(5(1(2(2(4(2(5(2(5(4(1(0(4(x1)))))))))))))) (19)
0(4(3(1(3(0(4(5(1(5(0(0(x1)))))))))))) → 3(5(1(2(1(3(2(2(2(2(0(3(2(2(2(2(4(5(x1)))))))))))))))))) (20)
0(4(3(5(3(1(4(4(0(0(0(4(x1)))))))))))) → 5(2(3(3(2(4(3(5(2(3(4(2(2(3(3(2(0(x1))))))))))))))))) (21)
0(4(5(0(0(2(0(3(3(5(5(5(x1)))))))))))) → 2(4(2(2(2(5(1(0(2(4(4(2(1(2(2(x1))))))))))))))) (22)
0(5(0(4(1(0(1(0(5(4(0(5(x1)))))))))))) → 5(3(1(5(3(2(5(3(1(0(2(3(5(2(0(4(x1)))))))))))))))) (23)
0(5(1(5(0(0(5(4(5(0(4(1(x1)))))))))))) → 5(5(2(4(2(5(5(0(4(0(1(1(2(5(5(5(5(x1))))))))))))))))) (24)
0(5(3(1(1(1(4(4(3(0(1(4(x1)))))))))))) → 2(0(2(4(5(5(2(2(3(3(5(0(5(2(3(2(x1)))))))))))))))) (25)
0(5(3(1(2(5(1(3(4(3(5(0(x1)))))))))))) → 5(2(2(5(2(0(5(5(2(5(5(5(3(5(4(2(x1)))))))))))))))) (26)
0(5(4(2(1(0(4(3(0(2(3(3(x1)))))))))))) → 3(3(3(2(3(2(0(5(4(2(3(4(1(2(3(x1))))))))))))))) (27)
0(5(4(5(0(3(3(0(4(0(5(4(x1)))))))))))) → 5(4(2(5(4(1(2(1(3(3(4(2(2(2(1(2(5(0(x1)))))))))))))))))) (28)
1(0(0(0(1(0(5(0(1(1(3(0(x1)))))))))))) → 3(5(5(2(1(2(5(5(5(0(2(3(3(1(0(2(x1)))))))))))))))) (29)
1(0(1(3(2(5(2(2(3(4(1(2(x1)))))))))))) → 3(2(1(2(0(2(3(5(4(1(2(2(2(2(x1)))))))))))))) (30)
1(0(4(3(0(0(5(4(4(1(0(5(x1)))))))))))) → 1(3(2(2(3(0(0(2(4(2(1(2(5(3(x1)))))))))))))) (31)
1(1(4(5(0(5(4(4(0(5(0(0(x1)))))))))))) → 5(4(4(1(1(4(1(2(2(5(0(2(1(2(0(5(x1)))))))))))))))) (32)
1(2(5(4(4(0(0(4(3(1(1(2(x1)))))))))))) → 5(2(2(2(3(0(4(1(5(4(2(3(2(2(x1)))))))))))))) (33)
1(3(1(3(3(4(4(4(3(0(4(4(x1)))))))))))) → 2(5(5(5(3(3(2(3(0(5(0(4(2(1(x1)))))))))))))) (34)
1(3(3(5(4(0(4(4(5(2(0(0(x1)))))))))))) → 4(2(2(1(2(5(3(3(3(5(4(5(1(5(5(x1))))))))))))))) (35)
1(4(1(0(4(3(1(0(0(5(4(0(x1)))))))))))) → 2(5(1(5(1(3(3(1(4(1(0(2(0(0(2(x1))))))))))))))) (36)
1(4(3(5(3(3(0(4(0(0(4(0(x1)))))))))))) → 1(5(3(2(2(2(2(1(5(2(5(1(0(1(5(0(x1)))))))))))))))) (37)
2(0(0(3(1(2(4(3(0(1(4(3(x1)))))))))))) → 1(1(2(2(5(0(2(1(1(5(0(5(5(2(2(3(x1)))))))))))))))) (38)
2(0(3(1(4(5(2(0(2(5(1(4(x1)))))))))))) → 1(2(5(3(2(0(1(2(4(1(3(2(5(3(5(x1))))))))))))))) (39)
2(2(4(0(4(0(0(5(5(4(3(5(x1)))))))))))) → 2(2(5(4(2(3(0(1(2(2(1(2(2(3(5(3(x1)))))))))))))))) (41)
2(4(0(4(4(4(0(4(1(3(2(0(x1)))))))))))) → 3(0(5(5(2(2(0(2(2(5(2(2(5(3(x1)))))))))))))) (42)
2(4(0(4(5(4(0(3(4(4(1(5(x1)))))))))))) → 2(4(1(1(5(2(3(2(1(4(0(2(2(5(5(2(3(3(x1)))))))))))))))))) (43)
2(5(1(4(0(0(5(4(0(0(4(1(x1)))))))))))) → 5(5(4(1(3(3(1(1(2(2(3(3(2(2(2(3(5(5(x1)))))))))))))))))) (44)
3(1(0(3(4(4(4(3(1(3(4(1(x1)))))))))))) → 1(2(2(4(4(1(2(0(5(2(4(1(5(4(x1)))))))))))))) (45)
3(3(4(0(4(2(2(3(4(3(0(0(x1)))))))))))) → 1(2(3(0(4(1(5(3(0(2(2(5(2(4(1(x1))))))))))))))) (46)
3(4(0(4(4(4(3(0(3(4(4(2(x1)))))))))))) → 1(2(4(5(3(2(2(2(1(2(4(0(5(3(2(0(0(1(x1)))))))))))))))))) (47)
3(4(4(4(0(4(0(0(2(3(1(4(x1)))))))))))) → 2(5(2(2(2(2(3(5(2(1(3(2(1(3(4(5(5(2(x1)))))))))))))))))) (48)
4(0(0(3(2(2(4(5(0(4(3(3(x1)))))))))))) → 5(2(2(5(2(1(3(2(2(3(3(5(5(2(2(2(4(3(x1)))))))))))))))))) (49)
4(0(1(0(1(0(0(3(3(4(0(4(x1)))))))))))) → 4(4(5(2(1(3(2(5(1(2(0(2(5(1(3(2(2(x1))))))))))))))))) (50)
4(0(1(1(4(1(3(4(5(4(3(5(x1)))))))))))) → 5(2(5(2(3(2(4(1(3(2(2(0(5(1(4(x1))))))))))))))) (51)
4(0(1(4(3(0(5(4(5(1(0(4(x1)))))))))))) → 2(1(1(5(2(1(5(5(0(3(2(4(3(2(5(x1))))))))))))))) (52)
4(0(3(1(5(1(3(1(4(0(1(2(x1)))))))))))) → 2(2(5(2(3(2(3(5(5(3(4(4(1(5(3(5(2(2(x1)))))))))))))))))) (53)
4(0(3(4(4(3(0(2(5(4(1(1(x1)))))))))))) → 0(5(3(5(5(2(2(1(2(2(3(2(5(5(4(3(x1)))))))))))))))) (54)
4(0(4(3(1(3(3(4(3(4(0(0(x1)))))))))))) → 5(3(2(2(2(5(2(2(1(0(4(2(2(4(3(2(3(x1))))))))))))))))) (55)
4(1(0(0(0(3(5(4(3(3(0(2(x1)))))))))))) → 2(2(5(3(3(5(2(3(3(1(2(2(0(2(5(3(3(x1))))))))))))))))) (56)
4(1(0(4(3(0(5(0(3(2(3(0(x1)))))))))))) → 2(3(3(2(1(4(4(5(2(0(1(5(2(5(1(1(x1)))))))))))))))) (57)
4(1(1(3(1(3(2(4(3(1(4(4(x1)))))))))))) → 4(0(1(2(3(2(2(2(3(2(3(1(4(1(2(5(3(2(x1)))))))))))))))))) (58)
4(1(1(3(2(3(1(5(0(4(0(0(x1)))))))))))) → 5(0(5(5(3(4(2(2(3(2(2(2(5(1(0(5(x1)))))))))))))))) (59)
4(1(5(2(3(0(1(5(4(3(3(0(x1)))))))))))) → 5(5(4(2(2(0(5(3(2(3(2(1(3(2(1(3(5(1(x1)))))))))))))))))) (60)
4(2(0(0(4(3(0(3(0(4(4(5(x1)))))))))))) → 2(0(5(3(2(2(3(3(5(3(5(0(1(1(2(3(0(2(x1)))))))))))))))))) (61)
4(2(2(0(4(1(4(4(0(0(3(4(x1)))))))))))) → 3(5(1(2(3(4(1(2(5(2(0(5(3(3(2(x1))))))))))))))) (62)
4(3(2(0(0(5(4(4(5(4(4(0(x1)))))))))))) → 4(1(2(3(1(2(2(0(3(5(2(0(5(2(5(1(5(0(x1)))))))))))))))))) (63)
4(3(2(1(4(4(5(0(3(0(0(4(x1)))))))))))) → 2(1(2(5(3(3(5(2(0(5(3(4(2(5(2(3(5(3(x1)))))))))))))))))) (64)
4(3(2(5(4(4(4(1(4(5(1(0(x1)))))))))))) → 2(1(5(5(2(3(3(1(2(3(2(2(3(0(1(5(x1)))))))))))))))) (65)
4(4(0(4(3(1(0(5(4(4(0(0(x1)))))))))))) → 0(4(5(5(4(3(4(5(5(3(5(1(1(3(2(5(x1)))))))))))))))) (67)
4(4(3(1(2(1(4(3(0(4(5(4(x1)))))))))))) → 3(3(5(4(4(1(2(3(0(0(1(1(2(1(2(x1))))))))))))))) (68)
4(4(3(4(3(1(4(3(4(5(0(4(x1)))))))))))) → 0(2(4(2(5(1(3(2(0(2(0(0(1(2(x1)))))))))))))) (69)
4(4(4(0(3(5(5(0(5(2(3(1(x1)))))))))))) → 5(2(2(2(2(0(2(4(1(3(0(0(1(5(3(x1))))))))))))))) (70)
4(4(4(4(4(0(3(5(1(0(2(0(x1)))))))))))) → 1(2(5(5(3(3(3(1(2(3(0(2(5(3(3(3(0(2(x1)))))))))))))))))) (71)
4(4(5(0(3(5(4(1(1(5(1(0(x1)))))))))))) → 2(2(5(5(2(3(2(2(5(3(1(2(0(2(x1)))))))))))))) (72)
4(5(0(0(1(0(2(5(4(4(0(4(x1)))))))))))) → 3(5(5(0(2(4(2(2(4(0(1(1(2(2(1(x1))))))))))))))) (73)
4(5(2(0(5(3(1(2(4(0(2(1(x1)))))))))))) → 4(2(5(3(5(5(3(5(2(2(4(3(3(2(4(x1))))))))))))))) (74)
4(5(2(1(4(2(0(3(0(0(4(0(x1)))))))))))) → 3(4(5(3(5(2(5(2(1(4(0(2(2(2(0(x1))))))))))))))) (75)
4(5(3(0(0(5(0(4(5(2(4(0(x1)))))))))))) → 5(2(2(4(3(2(2(3(3(2(3(4(1(1(2(2(0(x1))))))))))))))))) (76)
4(5(4(5(4(4(5(2(1(0(0(5(x1)))))))))))) → 5(3(5(5(5(3(3(1(2(3(4(5(2(2(3(3(5(x1))))))))))))))))) (77)
5(0(0(4(4(5(0(3(3(4(3(1(x1)))))))))))) → 5(1(1(0(4(0(2(2(0(2(5(2(2(5(5(1(1(2(x1)))))))))))))))))) (78)
5(0(1(4(0(5(2(0(0(0(0(0(x1)))))))))))) → 1(0(1(1(1(2(3(3(2(2(5(5(3(0(4(5(x1)))))))))))))))) (79)
5(0(3(5(3(4(2(1(0(3(0(4(x1)))))))))))) → 2(1(2(5(3(2(0(2(1(2(5(4(3(5(2(1(x1)))))))))))))))) (80)
5(0(5(0(4(0(0(5(2(0(0(0(x1)))))))))))) → 0(3(2(2(3(5(4(2(0(2(5(2(2(1(3(4(x1)))))))))))))))) (81)
5(1(0(0(4(2(1(1(2(1(4(2(x1)))))))))))) → 2(2(2(2(3(3(2(3(3(2(5(5(2(2(1(0(2(x1))))))))))))))))) (82)
5(1(1(1(4(4(0(3(1(4(5(0(x1)))))))))))) → 4(1(2(3(1(5(3(5(2(0(4(2(2(4(2(2(x1)))))))))))))))) (83)
5(1(3(0(3(0(3(4(1(0(5(5(x1)))))))))))) → 4(3(2(5(5(2(5(5(4(1(1(5(5(2(x1)))))))))))))) (84)
5(1(5(0(0(3(5(2(1(4(4(5(x1)))))))))))) → 3(4(2(3(2(5(2(2(3(2(2(2(1(2(3(2(5(3(x1)))))))))))))))))) (85)
5(3(0(3(5(1(0(0(0(0(5(4(x1)))))))))))) → 2(3(3(1(2(2(3(2(1(5(4(1(5(1(2(1(2(x1))))))))))))))))) (87)
5(4(0(3(4(0(0(2(3(0(0(2(x1)))))))))))) → 2(1(3(5(1(1(1(2(5(4(3(3(2(1(3(x1))))))))))))))) (88)
5(4(0(4(0(4(5(3(5(4(4(4(x1)))))))))))) → 2(4(1(5(5(2(0(2(0(3(5(4(2(3(1(x1))))))))))))))) (89)
5(4(3(4(4(3(4(2(4(3(2(0(x1)))))))))))) → 2(1(2(3(0(4(2(4(2(5(3(4(1(5(x1)))))))))))))) (90)
5(4(4(0(5(2(0(3(0(0(4(2(x1)))))))))))) → 3(1(2(4(0(1(5(1(4(4(1(2(5(4(x1)))))))))))))) (91)
5(4(4(4(0(2(0(4(5(4(3(4(x1)))))))))))) → 5(2(0(2(1(1(1(1(1(0(4(2(1(1(4(x1))))))))))))))) (92)
5(4(5(1(4(3(1(3(0(0(1(0(x1)))))))))))) → 5(5(3(0(1(3(5(2(0(2(1(1(1(2(3(4(x1)))))))))))))))) (93)
5(4(5(5(5(0(3(4(3(4(4(1(x1)))))))))))) → 1(0(5(5(1(2(2(1(3(5(2(0(5(3(5(4(1(x1))))))))))))))))) (94)
5(5(0(5(0(5(0(0(3(0(3(4(x1)))))))))))) → 3(4(2(2(1(4(0(1(5(1(2(1(2(5(4(5(3(1(x1)))))))))))))))))) (95)
5(5(2(2(1(0(4(0(0(5(5(0(x1)))))))))))) → 5(3(2(1(3(1(1(2(2(2(0(5(3(3(2(5(5(x1))))))))))))))))) (96)

1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
5(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 3(1(2(2(1(2(2(5(3(0(4(4(2(5(3(5(x1)))))))))))))))) (97)
5(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5(5(1(0(4(3(4(5(5(2(2(0(2(2(4(3(5(x1))))))))))))))))) (98)
4(1(2(4(2(5(4(1(1(5(1(5(x1)))))))))))) → 5(2(2(2(2(2(4(1(4(1(1(3(2(1(2(2(1(5(x1)))))))))))))))))) (99)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5#(x1) (100)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5#(5(2(2(0(2(2(4(3(5(x1)))))))))) (101)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5#(5(1(0(4(3(4(5(5(2(2(0(2(2(4(3(5(x1))))))))))))))))) (102)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5#(2(2(0(2(2(4(3(5(x1))))))))) (103)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5#(1(0(4(3(4(5(5(2(2(0(2(2(4(3(5(x1)))))))))))))))) (104)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 4#(5(5(2(2(0(2(2(4(3(5(x1))))))))))) (105)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 4#(3(5(x1))) (106)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 4#(3(4(5(5(2(2(0(2(2(4(3(5(x1))))))))))))) (107)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 5#(x1) (108)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 5#(3(5(x1))) (109)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 5#(3(0(4(4(2(5(3(5(x1))))))))) (110)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 4#(4(2(5(3(5(x1)))))) (111)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 4#(2(5(3(5(x1))))) (112)
4#(1(2(4(2(5(4(1(1(5(1(5(x1)))))))))))) → 5#(2(2(2(2(2(4(1(4(1(1(3(2(1(2(2(1(5(x1)))))))))))))))))) (113)
4#(1(2(4(2(5(4(1(1(5(1(5(x1)))))))))))) → 4#(1(4(1(1(3(2(1(2(2(1(5(x1)))))))))))) (114)
4#(1(2(4(2(5(4(1(1(5(1(5(x1)))))))))))) → 4#(1(1(3(2(1(2(2(1(5(x1)))))))))) (115)

1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[5(x1)] = x1 +
0
[4(x1)] = x1 +
4/3
[3(x1)] = x1 +
2/3
[2(x1)] = x1 +
0
[1(x1)] = x1 +
2/3
[0(x1)] = x1 +
4/3
[5#(x1)] = x1 +
0
[4#(x1)] = x1 +
7/3
together with the usable rules
5(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 3(1(2(2(1(2(2(5(3(0(4(4(2(5(3(5(x1)))))))))))))))) (97)
5(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5(5(1(0(4(3(4(5(5(2(2(0(2(2(4(3(5(x1))))))))))))))))) (98)
4(1(2(4(2(5(4(1(1(5(1(5(x1)))))))))))) → 5(2(2(2(2(2(4(1(4(1(1(3(2(1(2(2(1(5(x1)))))))))))))))))) (99)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5#(x1) (100)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5#(5(2(2(0(2(2(4(3(5(x1)))))))))) (101)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 5#(2(2(0(2(2(4(3(5(x1))))))))) (103)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 4#(5(5(2(2(0(2(2(4(3(5(x1))))))))))) (105)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 4#(3(5(x1))) (106)
5#(3(4(3(4(5(0(5(4(5(3(4(x1)))))))))))) → 4#(3(4(5(5(2(2(0(2(2(4(3(5(x1))))))))))))) (107)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 5#(x1) (108)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 5#(3(5(x1))) (109)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 5#(3(0(4(4(2(5(3(5(x1))))))))) (110)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 4#(4(2(5(3(5(x1)))))) (111)
5#(3(0(1(3(1(3(5(1(0(1(2(x1)))))))))))) → 4#(2(5(3(5(x1))))) (112)
4#(1(2(4(2(5(4(1(1(5(1(5(x1)))))))))))) → 5#(2(2(2(2(2(4(1(4(1(1(3(2(1(2(2(1(5(x1)))))))))))))))))) (113)
4#(1(2(4(2(5(4(1(1(5(1(5(x1)))))))))))) → 4#(1(1(3(2(1(2(2(1(5(x1)))))))))) (115)
and no rules could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.