Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/137623)

The rewrite relation of the following TRS is considered.

0(0(2(5(3(2(2(5(3(5(3(2(x1)))))))))))) → 2(0(0(0(3(0(4(0(0(3(0(2(5(0(5(2(0(2(x1)))))))))))))))))) (1)
0(1(5(5(4(4(2(5(4(2(5(1(x1)))))))))))) → 0(0(0(5(0(5(3(4(1(0(4(2(2(4(x1)))))))))))))) (2)
0(2(2(4(5(0(1(3(3(2(5(5(x1)))))))))))) → 1(0(4(0(0(4(3(4(2(1(1(4(4(0(2(3(x1)))))))))))))))) (3)
0(2(5(3(5(1(5(1(5(5(5(5(x1)))))))))))) → 2(0(4(0(4(1(5(1(0(4(5(1(1(1(1(0(3(2(x1)))))))))))))))))) (4)
0(3(0(2(5(2(5(3(3(3(5(4(x1)))))))))))) → 2(4(4(2(0(4(4(2(4(4(2(3(1(2(1(1(1(5(x1)))))))))))))))))) (5)
0(3(1(2(1(2(3(1(5(3(3(5(x1)))))))))))) → 0(1(4(4(2(2(0(0(5(4(2(3(1(0(2(1(0(2(x1)))))))))))))))))) (6)
0(5(3(2(2(1(2(3(3(2(5(0(x1)))))))))))) → 3(0(5(5(4(3(1(1(5(4(0(1(4(0(x1)))))))))))))) (7)
0(5(5(3(2(4(0(5(4(4(0(4(x1)))))))))))) → 2(1(0(4(4(3(0(1(0(4(0(2(2(4(4(4(x1)))))))))))))))) (8)
1(0(3(3(5(5(0(5(4(5(1(0(x1)))))))))))) → 2(3(4(0(3(0(1(4(0(3(4(1(2(4(2(x1))))))))))))))) (9)
1(1(2(5(0(1(2(3(5(5(3(4(x1)))))))))))) → 2(4(2(4(4(3(3(2(2(3(2(1(4(0(x1)))))))))))))) (10)
1(2(2(5(3(1(4(5(1(3(4(1(x1)))))))))))) → 2(3(4(0(4(0(4(5(1(2(0(0(4(5(0(5(x1)))))))))))))))) (11)
1(2(4(1(5(3(1(1(0(5(4(2(x1)))))))))))) → 2(0(1(2(0(4(2(2(1(1(4(4(1(0(0(4(4(0(x1)))))))))))))))))) (12)
1(3(0(5(5(0(1(2(5(1(5(4(x1)))))))))))) → 4(0(4(3(0(3(0(4(0(1(0(4(5(4(x1)))))))))))))) (13)
1(3(3(1(5(2(3(3(5(5(5(5(x1)))))))))))) → 0(2(4(0(1(4(0(4(4(4(5(2(4(3(1(5(2(3(x1)))))))))))))))))) (14)
1(3(5(5(1(3(2(4(2(3(2(5(x1)))))))))))) → 1(2(0(3(0(4(4(1(4(0(1(5(1(1(x1)))))))))))))) (15)
1(5(0(5(1(4(5(4(3(5(0(2(x1)))))))))))) → 0(1(5(4(0(1(5(0(4(4(4(4(2(0(0(2(x1)))))))))))))))) (16)
1(5(0(5(5(1(5(5(3(5(3(0(x1)))))))))))) → 1(0(5(5(2(0(4(0(5(4(1(1(0(4(0(5(0(4(x1)))))))))))))))))) (17)
1(5(1(3(3(2(2(1(5(1(5(5(x1)))))))))))) → 0(2(1(1(1(4(1(2(4(5(2(2(0(1(2(4(x1)))))))))))))))) (18)
1(5(2(0(3(3(5(3(1(5(2(3(x1)))))))))))) → 5(2(4(0(5(4(5(0(5(3(4(2(0(3(x1)))))))))))))) (19)
1(5(3(1(1(2(3(3(2(5(1(1(x1)))))))))))) → 3(1(1(3(1(2(4(0(0(2(5(1(2(0(0(4(x1)))))))))))))))) (20)
1(5(3(5(3(3(3(2(2(0(1(1(x1)))))))))))) → 2(4(0(2(2(0(0(4(1(1(5(3(5(1(3(4(1(x1))))))))))))))))) (21)
1(5(5(0(5(1(5(3(5(0(3(4(x1)))))))))))) → 4(1(2(0(0(1(3(1(1(0(5(4(2(3(2(x1))))))))))))))) (22)
2(0(0(1(5(1(2(5(3(2(1(4(x1)))))))))))) → 2(5(4(0(0(2(2(1(4(1(0(4(4(1(3(0(x1)))))))))))))))) (23)
2(0(5(4(2(2(3(3(2(5(2(5(x1)))))))))))) → 4(5(3(2(0(4(4(0(1(2(2(1(0(2(0(0(4(x1))))))))))))))))) (24)
2(1(1(1(3(1(1(3(2(0(3(2(x1)))))))))))) → 2(2(1(0(0(0(2(1(2(2(2(1(4(4(x1)))))))))))))) (25)
2(2(0(1(2(5(0(3(2(5(1(2(x1)))))))))))) → 3(2(1(4(4(4(2(1(0(1(2(0(0(0(4(0(4(5(x1)))))))))))))))))) (26)
2(2(0(1(2(5(5(2(1(1(1(3(x1)))))))))))) → 2(4(4(0(4(1(3(0(0(5(2(0(4(0(4(4(4(0(x1)))))))))))))))))) (27)
2(2(0(3(2(1(0(3(5(5(5(5(x1)))))))))))) → 4(1(2(2(3(4(0(2(4(1(0(4(2(4(0(0(4(3(x1)))))))))))))))))) (28)
2(2(1(5(5(3(4(3(2(5(3(2(x1)))))))))))) → 2(2(2(1(3(3(4(0(0(4(3(3(1(3(x1)))))))))))))) (29)
2(2(4(1(5(5(3(3(5(2(5(5(x1)))))))))))) → 5(0(5(0(4(4(1(2(1(3(3(4(2(4(4(x1))))))))))))))) (30)
2(3(3(0(2(5(5(2(2(4(5(4(x1)))))))))))) → 4(0(0(0(1(0(0(5(4(1(4(1(4(2(1(x1))))))))))))))) (31)
2(3(4(3(2(5(3(5(1(0(1(4(x1)))))))))))) → 2(0(0(4(4(5(0(2(3(5(1(2(2(2(4(x1))))))))))))))) (32)
2(4(3(5(2(1(5(5(2(1(5(2(x1)))))))))))) → 4(2(1(4(1(0(4(5(0(4(2(1(2(3(5(x1))))))))))))))) (33)
2(5(0(5(5(2(4(3(2(5(1(5(x1)))))))))))) → 3(4(1(4(0(1(1(0(4(1(1(0(0(2(4(4(5(2(x1)))))))))))))))))) (34)
2(5(1(5(5(1(1(0(5(5(5(5(x1)))))))))))) → 3(3(2(4(4(4(0(2(5(1(0(2(5(1(3(x1))))))))))))))) (35)
2(5(1(5(5(5(5(3(4(0(2(4(x1)))))))))))) → 4(2(4(4(2(2(4(5(0(4(2(2(1(0(x1)))))))))))))) (36)
2(5(2(0(5(3(4(1(5(2(5(5(x1)))))))))))) → 1(2(2(0(1(4(1(2(1(2(5(1(3(0(0(4(4(1(x1)))))))))))))))))) (37)
2(5(3(3(3(3(5(2(1(4(2(0(x1)))))))))))) → 0(1(3(4(0(5(5(3(0(2(2(4(3(2(3(x1))))))))))))))) (38)
2(5(4(5(1(5(3(1(5(2(4(1(x1)))))))))))) → 2(4(1(0(4(1(5(2(1(1(1(4(0(5(2(3(4(0(x1)))))))))))))))))) (39)
3(0(5(5(5(2(0(3(1(2(5(0(x1)))))))))))) → 0(4(0(4(2(2(2(0(1(3(4(4(1(0(5(x1))))))))))))))) (40)
3(1(1(5(0(3(2(3(3(3(3(4(x1)))))))))))) → 2(0(2(4(4(5(0(5(4(3(1(0(0(3(x1)))))))))))))) (41)
3(1(3(5(3(5(5(2(5(3(0(1(x1)))))))))))) → 2(3(1(2(0(3(2(3(5(4(0(0(0(5(0(0(x1)))))))))))))))) (42)
3(2(1(1(5(4(1(5(0(2(3(5(x1)))))))))))) → 2(5(4(2(3(0(0(4(5(4(4(2(4(0(0(0(2(x1))))))))))))))))) (43)
3(2(3(1(5(2(0(3(3(4(0(5(x1)))))))))))) → 3(0(0(0(4(0(3(0(2(3(3(2(1(4(1(x1))))))))))))))) (44)
3(2(5(3(2(5(4(0(3(1(2(2(x1)))))))))))) → 0(2(3(0(4(5(0(5(4(4(4(4(1(4(2(3(0(1(x1)))))))))))))))))) (45)
3(2(5(4(3(5(2(5(1(5(2(2(x1)))))))))))) → 4(2(3(3(0(5(3(5(1(0(0(2(1(5(0(4(x1)))))))))))))))) (46)
3(2(5(5(5(4(4(3(2(3(1(0(x1)))))))))))) → 0(0(2(2(4(5(4(0(4(5(5(4(0(5(x1)))))))))))))) (47)
3(3(1(1(4(5(3(4(3(5(1(5(x1)))))))))))) → 2(4(1(2(5(0(1(3(0(5(0(4(2(0(x1)))))))))))))) (48)
3(3(3(1(0(2(4(3(3(4(4(5(x1)))))))))))) → 0(2(0(4(4(4(0(5(5(0(1(3(3(5(x1)))))))))))))) (49)
3(4(0(5(3(4(0(0(4(5(2(4(x1)))))))))))) → 0(0(0(5(4(5(4(4(0(4(2(4(4(0(0(x1))))))))))))))) (50)
3(4(1(5(3(1(2(3(1(0(1(3(x1)))))))))))) → 1(3(4(3(3(5(4(0(4(0(4(4(5(0(1(x1))))))))))))))) (51)
3(4(3(3(4(5(2(5(4(2(4(5(x1)))))))))))) → 4(0(0(3(5(1(0(0(4(2(4(4(3(5(0(2(2(x1))))))))))))))))) (52)
3(4(5(3(5(3(1(5(3(1(1(1(x1)))))))))))) → 1(2(3(0(4(4(4(4(4(1(4(4(4(0(5(x1))))))))))))))) (53)
3(5(1(5(0(1(4(4(3(5(5(1(x1)))))))))))) → 4(4(2(5(0(4(0(2(2(4(0(0(4(2(0(1(2(4(x1)))))))))))))))))) (54)
3(5(1(5(3(3(1(3(3(2(4(5(x1)))))))))))) → 2(3(4(5(1(0(0(2(4(2(3(1(3(1(4(4(x1)))))))))))))))) (55)
3(5(3(2(1(3(2(1(5(5(3(5(x1)))))))))))) → 3(3(2(1(4(2(1(2(1(5(0(1(0(4(x1)))))))))))))) (56)
3(5(3(2(3(3(5(3(2(5(3(4(x1)))))))))))) → 4(1(1(3(0(4(2(1(2(1(0(4(4(1(1(1(2(x1))))))))))))))))) (57)
4(1(1(5(5(5(3(5(5(2(3(5(x1)))))))))))) → 4(0(1(5(1(0(2(4(1(2(0(4(2(2(3(4(0(0(x1)))))))))))))))))) (58)
4(1(2(4(5(4(1(5(5(5(2(1(x1)))))))))))) → 4(4(3(0(4(1(1(2(1(1(0(4(1(3(0(4(0(0(x1)))))))))))))))))) (59)
4(2(5(3(3(3(3(2(1(5(4(5(x1)))))))))))) → 3(0(3(4(3(0(1(1(4(3(4(2(0(4(1(3(1(2(x1)))))))))))))))))) (60)
4(2(5(5(3(4(5(4(2(5(2(0(x1)))))))))))) → 0(2(2(2(5(1(4(0(4(0(2(1(5(0(x1)))))))))))))) (61)
4(3(0(0(1(5(5(3(0(3(3(2(x1)))))))))))) → 3(1(3(5(0(0(0(3(1(4(5(0(4(0(x1)))))))))))))) (62)
4(3(5(2(5(3(5(1(3(0(5(2(x1)))))))))))) → 2(5(0(4(4(1(4(4(0(4(0(0(4(2(4(1(3(x1))))))))))))))))) (63)
4(5(0(5(0(1(1(2(1(2(4(0(x1)))))))))))) → 4(2(0(0(0(5(4(2(3(0(4(4(0(1(x1)))))))))))))) (64)
4(5(5(2(3(2(5(0(2(0(3(1(x1)))))))))))) → 1(0(0(3(4(1(5(1(0(4(4(0(3(4(0(0(x1)))))))))))))))) (65)
4(5(5(3(2(1(0(1(1(0(5(5(x1)))))))))))) → 0(5(0(4(0(3(0(1(2(2(3(2(3(4(x1)))))))))))))) (66)
5(0(5(5(3(1(5(5(3(3(5(0(x1)))))))))))) → 0(4(4(5(4(0(3(5(3(4(3(5(0(4(1(3(1(2(x1)))))))))))))))))) (67)
5(1(0(3(1(5(5(3(0(3(5(1(x1)))))))))))) → 5(4(1(1(0(4(0(3(3(1(3(0(4(0(1(4(4(4(x1)))))))))))))))))) (68)
5(1(1(0(3(1(5(3(1(1(0(5(x1)))))))))))) → 4(1(0(5(4(0(4(5(2(1(5(1(4(0(3(x1))))))))))))))) (69)
5(1(1(4(3(1(2(5(5(4(2(2(x1)))))))))))) → 0(1(0(0(2(4(0(2(5(0(0(5(0(0(x1)))))))))))))) (70)
5(1(2(2(5(2(4(1(3(2(3(1(x1)))))))))))) → 2(1(4(4(2(4(4(0(0(4(4(4(3(1(4(0(3(1(x1)))))))))))))))))) (71)
5(1(2(3(3(1(1(2(3(1(5(1(x1)))))))))))) → 5(2(0(3(3(5(2(4(4(1(4(4(3(4(1(x1))))))))))))))) (72)
5(1(5(1(3(2(4(5(3(5(4(0(x1)))))))))))) → 0(1(4(0(0(4(0(5(2(1(4(4(4(3(3(3(x1)))))))))))))))) (73)
5(2(1(4(3(5(3(1(5(3(5(1(x1)))))))))))) → 0(1(1(1(1(2(0(5(0(0(0(2(3(0(0(1(x1)))))))))))))))) (74)
5(2(1(5(5(0(3(5(5(5(3(0(x1)))))))))))) → 2(3(4(1(1(0(2(3(0(4(1(0(2(0(0(0(3(3(x1)))))))))))))))))) (75)
5(2(3(5(3(0(3(3(1(4(0(2(x1)))))))))))) → 4(5(4(0(0(2(1(1(0(3(4(4(4(3(x1)))))))))))))) (76)
5(2(4(3(0(2(5(2(1(1(5(0(x1)))))))))))) → 0(1(1(0(0(0(0(0(4(0(0(5(4(5(x1)))))))))))))) (77)
5(2(5(4(4(2(3(5(0(5(2(5(x1)))))))))))) → 4(1(4(2(3(4(3(4(4(4(2(1(4(0(0(0(1(x1))))))))))))))))) (78)
5(3(2(0(2(0(4(2(5(1(5(2(x1)))))))))))) → 4(3(0(4(5(0(3(4(0(3(5(1(1(1(4(x1))))))))))))))) (79)
5(3(3(2(1(1(3(3(2(5(5(1(x1)))))))))))) → 3(1(4(0(1(4(2(0(3(0(0(3(1(2(4(x1))))))))))))))) (80)
5(3(3(3(1(5(2(0(1(5(4(0(x1)))))))))))) → 3(5(0(3(5(0(0(4(2(3(4(0(1(2(2(3(0(0(x1)))))))))))))))))) (81)
5(3(5(2(5(5(2(4(2(1(2(2(x1)))))))))))) → 0(1(3(4(4(4(0(1(5(0(1(1(2(1(x1)))))))))))))) (82)
5(3(5(4(4(5(1(3(4(5(5(5(x1)))))))))))) → 5(3(4(4(0(4(4(0(1(5(0(5(2(2(3(4(0(4(x1)))))))))))))))))) (83)
5(4(3(3(0(3(2(5(4(0(5(5(x1)))))))))))) → 1(1(4(1(3(2(2(0(1(2(2(0(0(1(1(4(3(x1))))))))))))))))) (84)
5(4(3(3(3(2(5(2(4(1(4(2(x1)))))))))))) → 3(0(4(4(4(0(4(4(1(4(4(1(1(2(2(4(5(4(x1)))))))))))))))))) (85)
5(5(0(0(3(4(3(2(3(4(5(5(x1)))))))))))) → 0(2(1(0(1(2(4(0(0(5(3(2(2(2(0(x1))))))))))))))) (86)
5(5(1(2(5(2(5(3(3(3(4(5(x1)))))))))))) → 3(5(1(3(1(5(2(4(0(2(3(3(4(4(2(x1))))))))))))))) (87)
5(5(1(3(3(2(0(5(5(1(0(0(x1)))))))))))) → 5(0(4(0(3(4(5(1(2(4(4(4(3(0(0(0(1(x1))))))))))))))))) (88)
5(5(1(5(1(5(3(4(2(5(4(5(x1)))))))))))) → 0(5(4(0(4(0(4(1(2(2(1(1(3(4(4(0(3(3(x1)))))))))))))))))) (89)
5(5(1(5(2(5(1(5(1(1(1(4(x1)))))))))))) → 1(4(1(3(4(4(5(4(0(5(4(5(0(4(3(5(2(x1))))))))))))))))) (90)
5(5(2(5(3(3(1(0(1(4(5(5(x1)))))))))))) → 2(4(3(0(5(2(3(0(5(4(4(0(3(2(3(2(5(x1))))))))))))))))) (91)
5(5(3(1(0(3(3(0(2(3(5(3(x1)))))))))))) → 2(4(5(2(2(2(0(0(4(4(3(0(4(0(0(2(4(4(x1)))))))))))))))))) (92)
5(5(3(3(5(3(3(0(4(5(3(0(x1)))))))))))) → 4(4(0(2(4(0(3(4(1(4(1(3(3(4(4(4(4(3(x1)))))))))))))))))) (93)
5(5(3(4(0(2(5(5(3(2(4(1(x1)))))))))))) → 2(4(3(0(4(0(1(3(2(4(1(1(0(1(1(1(0(1(x1)))))))))))))))))) (94)
5(5(4(5(5(2(5(3(1(2(1(1(x1)))))))))))) → 0(3(0(2(2(4(4(3(1(4(4(2(5(0(0(0(x1)))))))))))))))) (95)
5(5(5(5(5(3(2(5(2(0(5(5(x1)))))))))))) → 5(3(5(3(3(1(5(5(4(1(3(4(0(1(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 +
351
[4(x1)] = x1 +
0
[3(x1)] = x1 +
235
[2(x1)] = x1 +
50
[1(x1)] = x1 +
40
[0(x1)] = x1 +
0
all of the following rules can be deleted.
0(0(2(5(3(2(2(5(3(5(3(2(x1)))))))))))) → 2(0(0(0(3(0(4(0(0(3(0(2(5(0(5(2(0(2(x1)))))))))))))))))) (1)
0(1(5(5(4(4(2(5(4(2(5(1(x1)))))))))))) → 0(0(0(5(0(5(3(4(1(0(4(2(2(4(x1)))))))))))))) (2)
0(2(2(4(5(0(1(3(3(2(5(5(x1)))))))))))) → 1(0(4(0(0(4(3(4(2(1(1(4(4(0(2(3(x1)))))))))))))))) (3)
0(2(5(3(5(1(5(1(5(5(5(5(x1)))))))))))) → 2(0(4(0(4(1(5(1(0(4(5(1(1(1(1(0(3(2(x1)))))))))))))))))) (4)
0(3(0(2(5(2(5(3(3(3(5(4(x1)))))))))))) → 2(4(4(2(0(4(4(2(4(4(2(3(1(2(1(1(1(5(x1)))))))))))))))))) (5)
0(3(1(2(1(2(3(1(5(3(3(5(x1)))))))))))) → 0(1(4(4(2(2(0(0(5(4(2(3(1(0(2(1(0(2(x1)))))))))))))))))) (6)
0(5(3(2(2(1(2(3(3(2(5(0(x1)))))))))))) → 3(0(5(5(4(3(1(1(5(4(0(1(4(0(x1)))))))))))))) (7)
0(5(5(3(2(4(0(5(4(4(0(4(x1)))))))))))) → 2(1(0(4(4(3(0(1(0(4(0(2(2(4(4(4(x1)))))))))))))))) (8)
1(0(3(3(5(5(0(5(4(5(1(0(x1)))))))))))) → 2(3(4(0(3(0(1(4(0(3(4(1(2(4(2(x1))))))))))))))) (9)
1(1(2(5(0(1(2(3(5(5(3(4(x1)))))))))))) → 2(4(2(4(4(3(3(2(2(3(2(1(4(0(x1)))))))))))))) (10)
1(2(2(5(3(1(4(5(1(3(4(1(x1)))))))))))) → 2(3(4(0(4(0(4(5(1(2(0(0(4(5(0(5(x1)))))))))))))))) (11)
1(2(4(1(5(3(1(1(0(5(4(2(x1)))))))))))) → 2(0(1(2(0(4(2(2(1(1(4(4(1(0(0(4(4(0(x1)))))))))))))))))) (12)
1(3(0(5(5(0(1(2(5(1(5(4(x1)))))))))))) → 4(0(4(3(0(3(0(4(0(1(0(4(5(4(x1)))))))))))))) (13)
1(3(3(1(5(2(3(3(5(5(5(5(x1)))))))))))) → 0(2(4(0(1(4(0(4(4(4(5(2(4(3(1(5(2(3(x1)))))))))))))))))) (14)
1(3(5(5(1(3(2(4(2(3(2(5(x1)))))))))))) → 1(2(0(3(0(4(4(1(4(0(1(5(1(1(x1)))))))))))))) (15)
1(5(0(5(1(4(5(4(3(5(0(2(x1)))))))))))) → 0(1(5(4(0(1(5(0(4(4(4(4(2(0(0(2(x1)))))))))))))))) (16)
1(5(0(5(5(1(5(5(3(5(3(0(x1)))))))))))) → 1(0(5(5(2(0(4(0(5(4(1(1(0(4(0(5(0(4(x1)))))))))))))))))) (17)
1(5(1(3(3(2(2(1(5(1(5(5(x1)))))))))))) → 0(2(1(1(1(4(1(2(4(5(2(2(0(1(2(4(x1)))))))))))))))) (18)
1(5(2(0(3(3(5(3(1(5(2(3(x1)))))))))))) → 5(2(4(0(5(4(5(0(5(3(4(2(0(3(x1)))))))))))))) (19)
1(5(3(1(1(2(3(3(2(5(1(1(x1)))))))))))) → 3(1(1(3(1(2(4(0(0(2(5(1(2(0(0(4(x1)))))))))))))))) (20)
1(5(3(5(3(3(3(2(2(0(1(1(x1)))))))))))) → 2(4(0(2(2(0(0(4(1(1(5(3(5(1(3(4(1(x1))))))))))))))))) (21)
1(5(5(0(5(1(5(3(5(0(3(4(x1)))))))))))) → 4(1(2(0(0(1(3(1(1(0(5(4(2(3(2(x1))))))))))))))) (22)
2(0(0(1(5(1(2(5(3(2(1(4(x1)))))))))))) → 2(5(4(0(0(2(2(1(4(1(0(4(4(1(3(0(x1)))))))))))))))) (23)
2(0(5(4(2(2(3(3(2(5(2(5(x1)))))))))))) → 4(5(3(2(0(4(4(0(1(2(2(1(0(2(0(0(4(x1))))))))))))))))) (24)
2(1(1(1(3(1(1(3(2(0(3(2(x1)))))))))))) → 2(2(1(0(0(0(2(1(2(2(2(1(4(4(x1)))))))))))))) (25)
2(2(0(1(2(5(0(3(2(5(1(2(x1)))))))))))) → 3(2(1(4(4(4(2(1(0(1(2(0(0(0(4(0(4(5(x1)))))))))))))))))) (26)
2(2(0(1(2(5(5(2(1(1(1(3(x1)))))))))))) → 2(4(4(0(4(1(3(0(0(5(2(0(4(0(4(4(4(0(x1)))))))))))))))))) (27)
2(2(0(3(2(1(0(3(5(5(5(5(x1)))))))))))) → 4(1(2(2(3(4(0(2(4(1(0(4(2(4(0(0(4(3(x1)))))))))))))))))) (28)
2(2(1(5(5(3(4(3(2(5(3(2(x1)))))))))))) → 2(2(2(1(3(3(4(0(0(4(3(3(1(3(x1)))))))))))))) (29)
2(2(4(1(5(5(3(3(5(2(5(5(x1)))))))))))) → 5(0(5(0(4(4(1(2(1(3(3(4(2(4(4(x1))))))))))))))) (30)
2(3(3(0(2(5(5(2(2(4(5(4(x1)))))))))))) → 4(0(0(0(1(0(0(5(4(1(4(1(4(2(1(x1))))))))))))))) (31)
2(3(4(3(2(5(3(5(1(0(1(4(x1)))))))))))) → 2(0(0(4(4(5(0(2(3(5(1(2(2(2(4(x1))))))))))))))) (32)
2(4(3(5(2(1(5(5(2(1(5(2(x1)))))))))))) → 4(2(1(4(1(0(4(5(0(4(2(1(2(3(5(x1))))))))))))))) (33)
2(5(0(5(5(2(4(3(2(5(1(5(x1)))))))))))) → 3(4(1(4(0(1(1(0(4(1(1(0(0(2(4(4(5(2(x1)))))))))))))))))) (34)
2(5(1(5(5(1(1(0(5(5(5(5(x1)))))))))))) → 3(3(2(4(4(4(0(2(5(1(0(2(5(1(3(x1))))))))))))))) (35)
2(5(1(5(5(5(5(3(4(0(2(4(x1)))))))))))) → 4(2(4(4(2(2(4(5(0(4(2(2(1(0(x1)))))))))))))) (36)
2(5(2(0(5(3(4(1(5(2(5(5(x1)))))))))))) → 1(2(2(0(1(4(1(2(1(2(5(1(3(0(0(4(4(1(x1)))))))))))))))))) (37)
2(5(4(5(1(5(3(1(5(2(4(1(x1)))))))))))) → 2(4(1(0(4(1(5(2(1(1(1(4(0(5(2(3(4(0(x1)))))))))))))))))) (39)
3(0(5(5(5(2(0(3(1(2(5(0(x1)))))))))))) → 0(4(0(4(2(2(2(0(1(3(4(4(1(0(5(x1))))))))))))))) (40)
3(1(1(5(0(3(2(3(3(3(3(4(x1)))))))))))) → 2(0(2(4(4(5(0(5(4(3(1(0(0(3(x1)))))))))))))) (41)
3(1(3(5(3(5(5(2(5(3(0(1(x1)))))))))))) → 2(3(1(2(0(3(2(3(5(4(0(0(0(5(0(0(x1)))))))))))))))) (42)
3(2(1(1(5(4(1(5(0(2(3(5(x1)))))))))))) → 2(5(4(2(3(0(0(4(5(4(4(2(4(0(0(0(2(x1))))))))))))))))) (43)
3(2(3(1(5(2(0(3(3(4(0(5(x1)))))))))))) → 3(0(0(0(4(0(3(0(2(3(3(2(1(4(1(x1))))))))))))))) (44)
3(2(5(3(2(5(4(0(3(1(2(2(x1)))))))))))) → 0(2(3(0(4(5(0(5(4(4(4(4(1(4(2(3(0(1(x1)))))))))))))))))) (45)
3(2(5(4(3(5(2(5(1(5(2(2(x1)))))))))))) → 4(2(3(3(0(5(3(5(1(0(0(2(1(5(0(4(x1)))))))))))))))) (46)
3(2(5(5(5(4(4(3(2(3(1(0(x1)))))))))))) → 0(0(2(2(4(5(4(0(4(5(5(4(0(5(x1)))))))))))))) (47)
3(3(1(1(4(5(3(4(3(5(1(5(x1)))))))))))) → 2(4(1(2(5(0(1(3(0(5(0(4(2(0(x1)))))))))))))) (48)
3(3(3(1(0(2(4(3(3(4(4(5(x1)))))))))))) → 0(2(0(4(4(4(0(5(5(0(1(3(3(5(x1)))))))))))))) (49)
3(4(0(5(3(4(0(0(4(5(2(4(x1)))))))))))) → 0(0(0(5(4(5(4(4(0(4(2(4(4(0(0(x1))))))))))))))) (50)
3(4(1(5(3(1(2(3(1(0(1(3(x1)))))))))))) → 1(3(4(3(3(5(4(0(4(0(4(4(5(0(1(x1))))))))))))))) (51)
3(4(3(3(4(5(2(5(4(2(4(5(x1)))))))))))) → 4(0(0(3(5(1(0(0(4(2(4(4(3(5(0(2(2(x1))))))))))))))))) (52)
3(4(5(3(5(3(1(5(3(1(1(1(x1)))))))))))) → 1(2(3(0(4(4(4(4(4(1(4(4(4(0(5(x1))))))))))))))) (53)
3(5(1(5(0(1(4(4(3(5(5(1(x1)))))))))))) → 4(4(2(5(0(4(0(2(2(4(0(0(4(2(0(1(2(4(x1)))))))))))))))))) (54)
3(5(1(5(3(3(1(3(3(2(4(5(x1)))))))))))) → 2(3(4(5(1(0(0(2(4(2(3(1(3(1(4(4(x1)))))))))))))))) (55)
3(5(3(2(1(3(2(1(5(5(3(5(x1)))))))))))) → 3(3(2(1(4(2(1(2(1(5(0(1(0(4(x1)))))))))))))) (56)
3(5(3(2(3(3(5(3(2(5(3(4(x1)))))))))))) → 4(1(1(3(0(4(2(1(2(1(0(4(4(1(1(1(2(x1))))))))))))))))) (57)
4(1(1(5(5(5(3(5(5(2(3(5(x1)))))))))))) → 4(0(1(5(1(0(2(4(1(2(0(4(2(2(3(4(0(0(x1)))))))))))))))))) (58)
4(1(2(4(5(4(1(5(5(5(2(1(x1)))))))))))) → 4(4(3(0(4(1(1(2(1(1(0(4(1(3(0(4(0(0(x1)))))))))))))))))) (59)
4(2(5(3(3(3(3(2(1(5(4(5(x1)))))))))))) → 3(0(3(4(3(0(1(1(4(3(4(2(0(4(1(3(1(2(x1)))))))))))))))))) (60)
4(2(5(5(3(4(5(4(2(5(2(0(x1)))))))))))) → 0(2(2(2(5(1(4(0(4(0(2(1(5(0(x1)))))))))))))) (61)
4(3(0(0(1(5(5(3(0(3(3(2(x1)))))))))))) → 3(1(3(5(0(0(0(3(1(4(5(0(4(0(x1)))))))))))))) (62)
4(3(5(2(5(3(5(1(3(0(5(2(x1)))))))))))) → 2(5(0(4(4(1(4(4(0(4(0(0(4(2(4(1(3(x1))))))))))))))))) (63)
4(5(0(5(0(1(1(2(1(2(4(0(x1)))))))))))) → 4(2(0(0(0(5(4(2(3(0(4(4(0(1(x1)))))))))))))) (64)
4(5(5(2(3(2(5(0(2(0(3(1(x1)))))))))))) → 1(0(0(3(4(1(5(1(0(4(4(0(3(4(0(0(x1)))))))))))))))) (65)
4(5(5(3(2(1(0(1(1(0(5(5(x1)))))))))))) → 0(5(0(4(0(3(0(1(2(2(3(2(3(4(x1)))))))))))))) (66)
5(0(5(5(3(1(5(5(3(3(5(0(x1)))))))))))) → 0(4(4(5(4(0(3(5(3(4(3(5(0(4(1(3(1(2(x1)))))))))))))))))) (67)
5(1(0(3(1(5(5(3(0(3(5(1(x1)))))))))))) → 5(4(1(1(0(4(0(3(3(1(3(0(4(0(1(4(4(4(x1)))))))))))))))))) (68)
5(1(1(0(3(1(5(3(1(1(0(5(x1)))))))))))) → 4(1(0(5(4(0(4(5(2(1(5(1(4(0(3(x1))))))))))))))) (69)
5(1(1(4(3(1(2(5(5(4(2(2(x1)))))))))))) → 0(1(0(0(2(4(0(2(5(0(0(5(0(0(x1)))))))))))))) (70)
5(1(2(2(5(2(4(1(3(2(3(1(x1)))))))))))) → 2(1(4(4(2(4(4(0(0(4(4(4(3(1(4(0(3(1(x1)))))))))))))))))) (71)
5(1(2(3(3(1(1(2(3(1(5(1(x1)))))))))))) → 5(2(0(3(3(5(2(4(4(1(4(4(3(4(1(x1))))))))))))))) (72)
5(1(5(1(3(2(4(5(3(5(4(0(x1)))))))))))) → 0(1(4(0(0(4(0(5(2(1(4(4(4(3(3(3(x1)))))))))))))))) (73)
5(2(1(4(3(5(3(1(5(3(5(1(x1)))))))))))) → 0(1(1(1(1(2(0(5(0(0(0(2(3(0(0(1(x1)))))))))))))))) (74)
5(2(1(5(5(0(3(5(5(5(3(0(x1)))))))))))) → 2(3(4(1(1(0(2(3(0(4(1(0(2(0(0(0(3(3(x1)))))))))))))))))) (75)
5(2(3(5(3(0(3(3(1(4(0(2(x1)))))))))))) → 4(5(4(0(0(2(1(1(0(3(4(4(4(3(x1)))))))))))))) (76)
5(2(4(3(0(2(5(2(1(1(5(0(x1)))))))))))) → 0(1(1(0(0(0(0(0(4(0(0(5(4(5(x1)))))))))))))) (77)
5(2(5(4(4(2(3(5(0(5(2(5(x1)))))))))))) → 4(1(4(2(3(4(3(4(4(4(2(1(4(0(0(0(1(x1))))))))))))))))) (78)
5(3(2(0(2(0(4(2(5(1(5(2(x1)))))))))))) → 4(3(0(4(5(0(3(4(0(3(5(1(1(1(4(x1))))))))))))))) (79)
5(3(3(2(1(1(3(3(2(5(5(1(x1)))))))))))) → 3(1(4(0(1(4(2(0(3(0(0(3(1(2(4(x1))))))))))))))) (80)
5(3(3(3(1(5(2(0(1(5(4(0(x1)))))))))))) → 3(5(0(3(5(0(0(4(2(3(4(0(1(2(2(3(0(0(x1)))))))))))))))))) (81)
5(3(5(2(5(5(2(4(2(1(2(2(x1)))))))))))) → 0(1(3(4(4(4(0(1(5(0(1(1(2(1(x1)))))))))))))) (82)
5(3(5(4(4(5(1(3(4(5(5(5(x1)))))))))))) → 5(3(4(4(0(4(4(0(1(5(0(5(2(2(3(4(0(4(x1)))))))))))))))))) (83)
5(4(3(3(0(3(2(5(4(0(5(5(x1)))))))))))) → 1(1(4(1(3(2(2(0(1(2(2(0(0(1(1(4(3(x1))))))))))))))))) (84)
5(4(3(3(3(2(5(2(4(1(4(2(x1)))))))))))) → 3(0(4(4(4(0(4(4(1(4(4(1(1(2(2(4(5(4(x1)))))))))))))))))) (85)
5(5(0(0(3(4(3(2(3(4(5(5(x1)))))))))))) → 0(2(1(0(1(2(4(0(0(5(3(2(2(2(0(x1))))))))))))))) (86)
5(5(1(2(5(2(5(3(3(3(4(5(x1)))))))))))) → 3(5(1(3(1(5(2(4(0(2(3(3(4(4(2(x1))))))))))))))) (87)
5(5(1(3(3(2(0(5(5(1(0(0(x1)))))))))))) → 5(0(4(0(3(4(5(1(2(4(4(4(3(0(0(0(1(x1))))))))))))))))) (88)
5(5(1(5(1(5(3(4(2(5(4(5(x1)))))))))))) → 0(5(4(0(4(0(4(1(2(2(1(1(3(4(4(0(3(3(x1)))))))))))))))))) (89)
5(5(1(5(2(5(1(5(1(1(1(4(x1)))))))))))) → 1(4(1(3(4(4(5(4(0(5(4(5(0(4(3(5(2(x1))))))))))))))))) (90)
5(5(2(5(3(3(1(0(1(4(5(5(x1)))))))))))) → 2(4(3(0(5(2(3(0(5(4(4(0(3(2(3(2(5(x1))))))))))))))))) (91)
5(5(3(1(0(3(3(0(2(3(5(3(x1)))))))))))) → 2(4(5(2(2(2(0(0(4(4(3(0(4(0(0(2(4(4(x1)))))))))))))))))) (92)
5(5(3(3(5(3(3(0(4(5(3(0(x1)))))))))))) → 4(4(0(2(4(0(3(4(1(4(1(3(3(4(4(4(4(3(x1)))))))))))))))))) (93)
5(5(3(4(0(2(5(5(3(2(4(1(x1)))))))))))) → 2(4(3(0(4(0(1(3(2(4(1(1(0(1(1(1(0(1(x1)))))))))))))))))) (94)
5(5(4(5(5(2(5(3(1(2(1(1(x1)))))))))))) → 0(3(0(2(2(4(4(3(1(4(4(2(5(0(0(0(x1)))))))))))))))) (95)
5(5(5(5(5(3(2(5(2(0(5(5(x1)))))))))))) → 5(3(5(3(3(1(5(5(4(1(3(4(0(1(x1)))))))))))))) (96)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
2#(5(3(3(3(3(5(2(1(4(2(0(x1)))))))))))) → 2#(4(3(2(3(x1))))) (97)
2#(5(3(3(3(3(5(2(1(4(2(0(x1)))))))))))) → 2#(3(x1)) (98)
2#(5(3(3(3(3(5(2(1(4(2(0(x1)))))))))))) → 2#(2(4(3(2(3(x1)))))) (99)

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 +
1
[4(x1)] = x1 +
0
[3(x1)] = x1 +
1
[2(x1)] = x1 +
1
[1(x1)] = x1 +
1
[0(x1)] = x1 +
0
[2#(x1)] = x1 +
0
together with the usable rule
2(5(3(3(3(3(5(2(1(4(2(0(x1)))))))))))) → 0(1(3(4(0(5(5(3(0(2(2(4(3(2(3(x1))))))))))))))) (38)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
2#(5(3(3(3(3(5(2(1(4(2(0(x1)))))))))))) → 2#(4(3(2(3(x1))))) (97)
2#(5(3(3(3(3(5(2(1(4(2(0(x1)))))))))))) → 2#(3(x1)) (98)
2#(5(3(3(3(3(5(2(1(4(2(0(x1)))))))))))) → 2#(2(4(3(2(3(x1)))))) (99)
and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.