Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/133827)

The rewrite relation of the following TRS is considered.

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4#(5(5(5(2(x1))))) (97)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4#(5(5(4(5(2(4(4(5(5(5(2(x1)))))))))))) (98)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4#(5(2(4(4(5(5(5(2(x1))))))))) (99)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4#(5(1(0(1(3(4(5(5(4(5(2(4(4(5(5(5(2(x1)))))))))))))))))) (100)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4#(4(5(5(5(2(x1)))))) (101)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 3#(4(5(5(4(5(2(4(4(5(5(5(2(x1))))))))))))) (102)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 0#(1(3(4(5(5(4(5(2(4(4(5(5(5(2(x1))))))))))))))) (103)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 4#(5(3(3(x1)))) (104)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 4#(4(0(2(5(5(0(5(3(2(3(4(5(3(3(x1))))))))))))))) (105)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 4#(0(2(5(5(0(5(3(2(3(4(5(3(3(x1)))))))))))))) (106)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3#(x1) (107)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3#(4(5(3(3(x1))))) (108)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3#(4(4(0(2(5(5(0(5(3(2(3(4(5(3(3(x1)))))))))))))))) (109)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3#(3(x1)) (110)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3#(2(3(4(5(3(3(x1))))))) (111)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 0#(5(3(2(3(4(5(3(3(x1))))))))) (112)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 0#(2(5(5(0(5(3(2(3(4(5(3(3(x1))))))))))))) (113)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 4#(4(2(3(0(1(1(0(3(5(3(1(0(2(x1)))))))))))))) (114)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 4#(2(3(0(1(1(0(3(5(3(1(0(2(x1))))))))))))) (115)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 3#(5(3(1(0(2(x1)))))) (116)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 3#(1(0(2(x1)))) (117)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 3#(0(1(1(0(3(5(3(1(0(2(x1))))))))))) (118)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 0#(3(5(3(1(0(2(x1))))))) (119)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 0#(1(1(0(3(5(3(1(0(2(x1)))))))))) (120)

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 +
1
[3(x1)] = x1 +
2/3
[2(x1)] = x1 +
1
[1(x1)] = x1 +
4/3
[0(x1)] = x1 +
2/3
[4#(x1)] = x1 +
0
[3#(x1)] = x1 +
1
[0#(x1)] = x1 +
1
together with the usable rules
0(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 4(4(2(3(0(1(1(0(3(5(3(1(0(2(x1)))))))))))))) (12)
3(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3(4(4(0(2(5(5(0(5(3(2(3(4(5(3(3(x1)))))))))))))))) (57)
4(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4(5(1(0(1(3(4(5(5(4(5(2(4(4(5(5(5(2(x1)))))))))))))))))) (84)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4#(5(5(5(2(x1))))) (97)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4#(5(5(4(5(2(4(4(5(5(5(2(x1)))))))))))) (98)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4#(5(2(4(4(5(5(5(2(x1))))))))) (99)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 4#(4(5(5(5(2(x1)))))) (101)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 3#(4(5(5(4(5(2(4(4(5(5(5(2(x1))))))))))))) (102)
4#(2(4(0(3(1(3(1(3(2(2(3(x1)))))))))))) → 0#(1(3(4(5(5(4(5(2(4(4(5(5(5(2(x1))))))))))))))) (103)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 4#(5(3(3(x1)))) (104)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 4#(4(0(2(5(5(0(5(3(2(3(4(5(3(3(x1))))))))))))))) (105)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 4#(0(2(5(5(0(5(3(2(3(4(5(3(3(x1)))))))))))))) (106)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3#(x1) (107)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3#(4(5(3(3(x1))))) (108)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3#(3(x1)) (110)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 3#(2(3(4(5(3(3(x1))))))) (111)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 0#(5(3(2(3(4(5(3(3(x1))))))))) (112)
3#(2(0(4(2(3(0(3(3(0(4(4(x1)))))))))))) → 0#(2(5(5(0(5(3(2(3(4(5(3(3(x1))))))))))))) (113)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 4#(4(2(3(0(1(1(0(3(5(3(1(0(2(x1)))))))))))))) (114)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 4#(2(3(0(1(1(0(3(5(3(1(0(2(x1))))))))))))) (115)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 3#(5(3(1(0(2(x1)))))) (116)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 3#(1(0(2(x1)))) (117)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 3#(0(1(1(0(3(5(3(1(0(2(x1))))))))))) (118)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 0#(3(5(3(1(0(2(x1))))))) (119)
0#(4(1(3(1(4(4(4(4(1(0(2(x1)))))))))))) → 0#(1(1(0(3(5(3(1(0(2(x1)))))))))) (120)
and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.