Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/85477)

The rewrite relation of the following TRS is considered.

0(0(1(2(2(x1))))) 2(3(3(2(2(x1))))) (1)
0(4(1(0(4(5(x1)))))) 5(2(5(3(4(5(x1)))))) (2)
0(5(2(4(2(0(0(5(2(x1))))))))) 1(5(2(0(2(3(5(2(x1)))))))) (3)
3(4(2(2(1(5(3(3(2(x1))))))))) 3(1(0(3(5(3(3(0(2(x1))))))))) (4)
0(5(5(3(1(3(3(0(2(0(x1)))))))))) 3(3(3(2(5(0(4(0(5(0(x1)))))))))) (5)
5(0(3(5(1(4(2(1(0(2(x1)))))))))) 5(2(0(1(3(1(1(0(2(x1))))))))) (6)
2(4(5(2(4(3(4(3(1(2(3(x1))))))))))) 5(3(1(3(3(3(2(2(4(1(x1)))))))))) (7)
0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) (8)
3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) (9)
3(3(3(2(3(3(1(2(2(4(3(3(x1)))))))))))) 1(2(2(1(3(2(3(5(2(5(5(x1))))))))))) (10)
5(5(4(0(3(0(2(3(3(2(3(3(x1)))))))))))) 5(2(2(3(0(1(1(3(4(4(5(x1))))))))))) (11)
0(0(3(1(0(2(2(3(2(5(3(0(1(x1))))))))))))) 0(1(3(4(3(1(0(0(3(3(4(0(x1)))))))))))) (12)
2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) (13)
2(4(2(3(3(5(0(4(5(5(2(5(5(x1))))))))))))) 3(5(4(3(3(2(0(1(1(4(3(5(x1)))))))))))) (14)
5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) (15)
2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) (16)
2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) (17)
2(5(3(5(2(4(2(1(3(0(5(0(3(1(3(x1))))))))))))))) 5(3(2(3(5(1(5(3(5(3(5(1(5(x1))))))))))))) (18)
0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) (19)
0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) (20)
3(3(2(0(1(5(4(5(4(2(4(2(3(4(4(3(x1)))))))))))))))) 5(2(5(5(5(5(2(3(3(1(2(5(0(4(0(x1))))))))))))))) (21)
3(0(0(2(0(3(5(3(0(2(5(3(5(5(2(3(4(x1))))))))))))))))) 5(3(4(3(0(4(2(2(1(0(4(1(0(0(3(4(x1)))))))))))))))) (22)
0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) (23)
1(0(3(2(0(2(4(1(3(2(5(0(0(4(5(0(3(4(3(x1))))))))))))))))))) 4(2(2(0(0(0(0(4(1(4(0(5(3(5(0(2(4(3(x1)))))))))))))))))) (24)
1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) (25)
3(3(5(4(1(1(3(2(1(4(4(0(1(1(0(4(3(1(0(x1))))))))))))))))))) 3(4(0(2(3(1(3(0(4(4(3(4(4(4(3(2(1(0(x1)))))))))))))))))) (26)
2(5(0(4(0(3(4(3(4(0(0(2(4(2(4(1(0(1(2(3(x1)))))))))))))))))))) 0(1(2(0(5(4(3(2(0(2(3(3(3(0(1(5(5(5(1(x1))))))))))))))))))) (27)
0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) (28)
2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) (29)
4(3(1(4(3(3(3(3(4(4(2(5(1(4(5(1(4(3(2(3(3(x1))))))))))))))))))))) 2(5(5(1(1(1(4(5(3(2(3(3(0(4(2(3(1(5(4(5(x1)))))))))))))))))))) (30)

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 +
1
[4(x1)] = x1 +
1
[3(x1)] = x1 +
1
[2(x1)] = x1 +
1
[1(x1)] = x1 +
1
[0(x1)] = x1 +
1
all of the following rules can be deleted.
0(5(2(4(2(0(0(5(2(x1))))))))) 1(5(2(0(2(3(5(2(x1)))))))) (3)
5(0(3(5(1(4(2(1(0(2(x1)))))))))) 5(2(0(1(3(1(1(0(2(x1))))))))) (6)
2(4(5(2(4(3(4(3(1(2(3(x1))))))))))) 5(3(1(3(3(3(2(2(4(1(x1)))))))))) (7)
3(3(3(2(3(3(1(2(2(4(3(3(x1)))))))))))) 1(2(2(1(3(2(3(5(2(5(5(x1))))))))))) (10)
5(5(4(0(3(0(2(3(3(2(3(3(x1)))))))))))) 5(2(2(3(0(1(1(3(4(4(5(x1))))))))))) (11)
0(0(3(1(0(2(2(3(2(5(3(0(1(x1))))))))))))) 0(1(3(4(3(1(0(0(3(3(4(0(x1)))))))))))) (12)
2(4(2(3(3(5(0(4(5(5(2(5(5(x1))))))))))))) 3(5(4(3(3(2(0(1(1(4(3(5(x1)))))))))))) (14)
2(5(3(5(2(4(2(1(3(0(5(0(3(1(3(x1))))))))))))))) 5(3(2(3(5(1(5(3(5(3(5(1(5(x1))))))))))))) (18)
3(3(2(0(1(5(4(5(4(2(4(2(3(4(4(3(x1)))))))))))))))) 5(2(5(5(5(5(2(3(3(1(2(5(0(4(0(x1))))))))))))))) (21)
3(0(0(2(0(3(5(3(0(2(5(3(5(5(2(3(4(x1))))))))))))))))) 5(3(4(3(0(4(2(2(1(0(4(1(0(0(3(4(x1)))))))))))))))) (22)
1(0(3(2(0(2(4(1(3(2(5(0(0(4(5(0(3(4(3(x1))))))))))))))))))) 4(2(2(0(0(0(0(4(1(4(0(5(3(5(0(2(4(3(x1)))))))))))))))))) (24)
3(3(5(4(1(1(3(2(1(4(4(0(1(1(0(4(3(1(0(x1))))))))))))))))))) 3(4(0(2(3(1(3(0(4(4(3(4(4(4(3(2(1(0(x1)))))))))))))))))) (26)
2(5(0(4(0(3(4(3(4(0(0(2(4(2(4(1(0(1(2(3(x1)))))))))))))))))))) 0(1(2(0(5(4(3(2(0(2(3(3(3(0(1(5(5(5(1(x1))))))))))))))))))) (27)
4(3(1(4(3(3(3(3(4(4(2(5(1(4(5(1(4(3(2(3(3(x1))))))))))))))))))))) 2(5(5(1(1(1(4(5(3(2(3(3(0(4(2(3(1(5(4(5(x1)))))))))))))))))))) (30)

1.1 Closure Under Flat Contexts

Using the flat contexts

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

We obtain the transformed TRS
5(0(0(1(2(2(x1)))))) 5(2(3(3(2(2(x1)))))) (31)
5(0(4(1(0(4(5(x1))))))) 5(5(2(5(3(4(5(x1))))))) (32)
5(3(4(2(2(1(5(3(3(2(x1)))))))))) 5(3(1(0(3(5(3(3(0(2(x1)))))))))) (33)
5(0(5(5(3(1(3(3(0(2(0(x1))))))))))) 5(3(3(3(2(5(0(4(0(5(0(x1))))))))))) (34)
5(0(2(1(5(3(5(3(0(5(5(5(0(x1))))))))))))) 5(2(2(5(4(4(2(3(3(0(3(5(4(x1))))))))))))) (35)
5(3(2(0(0(4(0(0(0(2(5(1(0(x1))))))))))))) 5(1(5(2(4(0(4(2(0(0(5(0(2(x1))))))))))))) (36)
5(2(1(3(3(4(3(0(2(0(4(0(3(1(x1)))))))))))))) 5(2(4(1(3(0(1(3(3(2(1(3(4(4(x1)))))))))))))) (37)
5(5(2(3(1(2(1(5(4(5(1(1(0(0(x1)))))))))))))) 5(5(2(0(5(4(3(4(3(3(0(5(0(0(x1)))))))))))))) (38)
5(2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1)))))))))))))))) 5(5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1)))))))))))))))) (39)
5(2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1)))))))))))))))) 5(0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1)))))))))))))))) (40)
5(0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1))))))))))))))))) 5(0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1))))))))))))))))) (41)
5(0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1))))))))))))))))) 5(0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1))))))))))))))))) (42)
5(0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1)))))))))))))))))))) 5(2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1)))))))))))))))))))) (43)
5(1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1)))))))))))))))))))) 5(1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))))))))))) (44)
5(0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1)))))))))))))))))))))) 5(2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))))))))))))) (45)
5(2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1)))))))))))))))))))))) 5(2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))))))))))) (46)
4(0(0(1(2(2(x1)))))) 4(2(3(3(2(2(x1)))))) (47)
4(0(4(1(0(4(5(x1))))))) 4(5(2(5(3(4(5(x1))))))) (48)
4(3(4(2(2(1(5(3(3(2(x1)))))))))) 4(3(1(0(3(5(3(3(0(2(x1)))))))))) (49)
4(0(5(5(3(1(3(3(0(2(0(x1))))))))))) 4(3(3(3(2(5(0(4(0(5(0(x1))))))))))) (50)
4(0(2(1(5(3(5(3(0(5(5(5(0(x1))))))))))))) 4(2(2(5(4(4(2(3(3(0(3(5(4(x1))))))))))))) (51)
4(3(2(0(0(4(0(0(0(2(5(1(0(x1))))))))))))) 4(1(5(2(4(0(4(2(0(0(5(0(2(x1))))))))))))) (52)
4(2(1(3(3(4(3(0(2(0(4(0(3(1(x1)))))))))))))) 4(2(4(1(3(0(1(3(3(2(1(3(4(4(x1)))))))))))))) (53)
4(5(2(3(1(2(1(5(4(5(1(1(0(0(x1)))))))))))))) 4(5(2(0(5(4(3(4(3(3(0(5(0(0(x1)))))))))))))) (54)
4(2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1)))))))))))))))) 4(5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1)))))))))))))))) (55)
4(2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1)))))))))))))))) 4(0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1)))))))))))))))) (56)
4(0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1))))))))))))))))) 4(0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1))))))))))))))))) (57)
4(0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1))))))))))))))))) 4(0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1))))))))))))))))) (58)
4(0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1)))))))))))))))))))) 4(2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1)))))))))))))))))))) (59)
4(1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1)))))))))))))))))))) 4(1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))))))))))) (60)
4(0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1)))))))))))))))))))))) 4(2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))))))))))))) (61)
4(2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1)))))))))))))))))))))) 4(2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))))))))))) (62)
3(0(0(1(2(2(x1)))))) 3(2(3(3(2(2(x1)))))) (63)
3(0(4(1(0(4(5(x1))))))) 3(5(2(5(3(4(5(x1))))))) (64)
3(3(4(2(2(1(5(3(3(2(x1)))))))))) 3(3(1(0(3(5(3(3(0(2(x1)))))))))) (65)
3(0(5(5(3(1(3(3(0(2(0(x1))))))))))) 3(3(3(3(2(5(0(4(0(5(0(x1))))))))))) (66)
3(0(2(1(5(3(5(3(0(5(5(5(0(x1))))))))))))) 3(2(2(5(4(4(2(3(3(0(3(5(4(x1))))))))))))) (67)
3(3(2(0(0(4(0(0(0(2(5(1(0(x1))))))))))))) 3(1(5(2(4(0(4(2(0(0(5(0(2(x1))))))))))))) (68)
3(2(1(3(3(4(3(0(2(0(4(0(3(1(x1)))))))))))))) 3(2(4(1(3(0(1(3(3(2(1(3(4(4(x1)))))))))))))) (69)
3(5(2(3(1(2(1(5(4(5(1(1(0(0(x1)))))))))))))) 3(5(2(0(5(4(3(4(3(3(0(5(0(0(x1)))))))))))))) (70)
3(2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1)))))))))))))))) 3(5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1)))))))))))))))) (71)
3(2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1)))))))))))))))) 3(0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1)))))))))))))))) (72)
3(0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1))))))))))))))))) 3(0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1))))))))))))))))) (73)
3(0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1))))))))))))))))) 3(0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1))))))))))))))))) (74)
3(0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1)))))))))))))))))))) 3(2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1)))))))))))))))))))) (75)
3(1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1)))))))))))))))))))) 3(1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))))))))))) (76)
3(0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1)))))))))))))))))))))) 3(2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))))))))))))) (77)
3(2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1)))))))))))))))))))))) 3(2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))))))))))) (78)
2(0(0(1(2(2(x1)))))) 2(2(3(3(2(2(x1)))))) (79)
2(0(4(1(0(4(5(x1))))))) 2(5(2(5(3(4(5(x1))))))) (80)
2(3(4(2(2(1(5(3(3(2(x1)))))))))) 2(3(1(0(3(5(3(3(0(2(x1)))))))))) (81)
2(0(5(5(3(1(3(3(0(2(0(x1))))))))))) 2(3(3(3(2(5(0(4(0(5(0(x1))))))))))) (82)
2(0(2(1(5(3(5(3(0(5(5(5(0(x1))))))))))))) 2(2(2(5(4(4(2(3(3(0(3(5(4(x1))))))))))))) (83)
2(3(2(0(0(4(0(0(0(2(5(1(0(x1))))))))))))) 2(1(5(2(4(0(4(2(0(0(5(0(2(x1))))))))))))) (84)
2(2(1(3(3(4(3(0(2(0(4(0(3(1(x1)))))))))))))) 2(2(4(1(3(0(1(3(3(2(1(3(4(4(x1)))))))))))))) (85)
2(5(2(3(1(2(1(5(4(5(1(1(0(0(x1)))))))))))))) 2(5(2(0(5(4(3(4(3(3(0(5(0(0(x1)))))))))))))) (86)
2(2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1)))))))))))))))) 2(5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1)))))))))))))))) (87)
2(2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1)))))))))))))))) 2(0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1)))))))))))))))) (88)
2(0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1))))))))))))))))) 2(0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1))))))))))))))))) (89)
2(0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1))))))))))))))))) 2(0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1))))))))))))))))) (90)
2(0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1)))))))))))))))))))) 2(2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1)))))))))))))))))))) (91)
2(1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1)))))))))))))))))))) 2(1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))))))))))) (92)
2(0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1)))))))))))))))))))))) 2(2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))))))))))))) (93)
2(2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1)))))))))))))))))))))) 2(2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))))))))))) (94)
1(0(0(1(2(2(x1)))))) 1(2(3(3(2(2(x1)))))) (95)
1(0(4(1(0(4(5(x1))))))) 1(5(2(5(3(4(5(x1))))))) (96)
1(3(4(2(2(1(5(3(3(2(x1)))))))))) 1(3(1(0(3(5(3(3(0(2(x1)))))))))) (97)
1(0(5(5(3(1(3(3(0(2(0(x1))))))))))) 1(3(3(3(2(5(0(4(0(5(0(x1))))))))))) (98)
1(0(2(1(5(3(5(3(0(5(5(5(0(x1))))))))))))) 1(2(2(5(4(4(2(3(3(0(3(5(4(x1))))))))))))) (99)
1(3(2(0(0(4(0(0(0(2(5(1(0(x1))))))))))))) 1(1(5(2(4(0(4(2(0(0(5(0(2(x1))))))))))))) (100)
1(2(1(3(3(4(3(0(2(0(4(0(3(1(x1)))))))))))))) 1(2(4(1(3(0(1(3(3(2(1(3(4(4(x1)))))))))))))) (101)
1(5(2(3(1(2(1(5(4(5(1(1(0(0(x1)))))))))))))) 1(5(2(0(5(4(3(4(3(3(0(5(0(0(x1)))))))))))))) (102)
1(2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1)))))))))))))))) 1(5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1)))))))))))))))) (103)
1(2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1)))))))))))))))) 1(0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1)))))))))))))))) (104)
1(0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1))))))))))))))))) 1(0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1))))))))))))))))) (105)
1(0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1))))))))))))))))) 1(0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1))))))))))))))))) (106)
1(0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1)))))))))))))))))))) 1(2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1)))))))))))))))))))) (107)
1(1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1)))))))))))))))))))) 1(1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))))))))))) (108)
1(0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1)))))))))))))))))))))) 1(2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))))))))))))) (109)
1(2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1)))))))))))))))))))))) 1(2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))))))))))) (110)
0(0(0(1(2(2(x1)))))) 0(2(3(3(2(2(x1)))))) (111)
0(0(4(1(0(4(5(x1))))))) 0(5(2(5(3(4(5(x1))))))) (112)
0(3(4(2(2(1(5(3(3(2(x1)))))))))) 0(3(1(0(3(5(3(3(0(2(x1)))))))))) (113)
0(0(5(5(3(1(3(3(0(2(0(x1))))))))))) 0(3(3(3(2(5(0(4(0(5(0(x1))))))))))) (114)
0(0(2(1(5(3(5(3(0(5(5(5(0(x1))))))))))))) 0(2(2(5(4(4(2(3(3(0(3(5(4(x1))))))))))))) (115)
0(3(2(0(0(4(0(0(0(2(5(1(0(x1))))))))))))) 0(1(5(2(4(0(4(2(0(0(5(0(2(x1))))))))))))) (116)
0(2(1(3(3(4(3(0(2(0(4(0(3(1(x1)))))))))))))) 0(2(4(1(3(0(1(3(3(2(1(3(4(4(x1)))))))))))))) (117)
0(5(2(3(1(2(1(5(4(5(1(1(0(0(x1)))))))))))))) 0(5(2(0(5(4(3(4(3(3(0(5(0(0(x1)))))))))))))) (118)
0(2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1)))))))))))))))) 0(5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1)))))))))))))))) (119)
0(2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1)))))))))))))))) 0(0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1)))))))))))))))) (120)
0(0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1))))))))))))))))) 0(0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1))))))))))))))))) (121)
0(0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1))))))))))))))))) 0(0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1))))))))))))))))) (122)
0(0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1)))))))))))))))))))) 0(2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1)))))))))))))))))))) (123)
0(1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1)))))))))))))))))))) 0(1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))))))))))) (124)
0(0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1)))))))))))))))))))))) 0(2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))))))))))))) (125)
0(2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1)))))))))))))))))))))) 0(2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))))))))))) (126)

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 576 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 +
6
[53(x1)] = x1 +
0
[54(x1)] = x1 +
7
[55(x1)] = x1 +
6
[40(x1)] = x1 +
0
[41(x1)] = x1 +
0
[42(x1)] = x1 +
1
[43(x1)] = x1 +
0
[44(x1)] = x1 +
6
[45(x1)] = x1 +
0
[30(x1)] = x1 +
0
[31(x1)] = x1 +
7
[32(x1)] = x1 +
0
[33(x1)] = x1 +
1
[34(x1)] = x1 +
0
[35(x1)] = x1 +
0
[20(x1)] = x1 +
1
[21(x1)] = x1 +
0
[22(x1)] = x1 +
1
[23(x1)] = x1 +
1
[24(x1)] = x1 +
6
[25(x1)] = x1 +
6
[10(x1)] = x1 +
6
[11(x1)] = x1 +
0
[12(x1)] = x1 +
6
[13(x1)] = x1 +
6
[14(x1)] = x1 +
6
[15(x1)] = x1 +
6
[00(x1)] = x1 +
0
[01(x1)] = x1 +
6
[02(x1)] = x1 +
6
[03(x1)] = x1 +
6
[04(x1)] = x1 +
1
[05(x1)] = x1 +
7
all of the following rules can be deleted.

There are 576 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.