Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/98362)

The rewrite relation of the following TRS is considered.

0(1(2(3(x1)))) 3(2(4(x1))) (1)
2(3(3(0(4(x1))))) 4(2(1(5(x1)))) (2)
0(3(3(1(1(4(x1)))))) 3(2(3(0(1(3(x1)))))) (3)
5(1(3(0(4(0(x1)))))) 3(5(3(3(2(x1))))) (4)
0(3(2(1(4(0(1(x1))))))) 4(0(5(1(3(0(1(x1))))))) (5)
2(2(0(2(5(2(0(x1))))))) 2(2(1(2(5(1(3(x1))))))) (6)
4(5(3(2(4(4(2(2(x1)))))))) 4(0(5(0(0(0(2(2(x1)))))))) (7)
1(0(2(5(2(3(0(1(1(x1))))))))) 1(5(1(2(0(2(4(4(1(x1))))))))) (8)
4(4(0(3(5(1(2(4(4(x1))))))))) 1(5(2(0(3(5(0(0(x1)))))))) (9)
0(4(1(2(0(5(0(2(2(4(x1)))))))))) 2(5(4(0(1(2(5(3(4(x1))))))))) (10)
4(5(4(1(2(4(1(0(1(5(x1)))))))))) 1(1(0(1(3(1(5(2(5(x1))))))))) (11)
5(1(0(3(2(3(4(5(4(3(x1)))))))))) 5(2(0(5(3(4(2(5(3(x1))))))))) (12)
1(3(0(4(3(0(5(3(2(5(3(x1))))))))))) 1(3(0(0(0(3(1(3(3(5(3(x1))))))))))) (13)
4(5(0(0(4(4(5(0(5(4(5(x1))))))))))) 2(4(2(2(2(4(3(1(4(5(x1)))))))))) (14)
2(3(0(0(5(4(1(3(4(0(1(4(x1)))))))))))) 4(0(3(3(4(5(1(1(4(2(4(x1))))))))))) (15)
5(1(4(3(0(5(1(2(5(5(4(0(x1)))))))))))) 2(5(0(0(5(1(4(2(5(1(5(x1))))))))))) (16)
5(1(5(2(0(5(5(5(2(2(5(1(x1)))))))))))) 2(2(4(0(0(5(3(0(5(1(0(2(x1)))))))))))) (17)
4(5(4(5(1(2(0(2(5(4(4(5(5(x1))))))))))))) 1(1(3(4(3(3(0(4(2(5(5(3(x1)))))))))))) (18)
5(5(0(4(0(2(1(1(0(4(5(0(5(x1))))))))))))) 1(1(0(3(4(2(1(1(5(0(2(5(x1)))))))))))) (19)
5(0(5(1(0(3(1(3(1(1(3(1(5(1(x1)))))))))))))) 5(4(4(1(2(5(5(0(1(0(3(2(2(2(x1)))))))))))))) (20)
5(5(3(0(4(4(0(2(1(5(1(4(2(1(x1)))))))))))))) 5(4(1(5(2(4(3(3(5(2(4(5(0(x1))))))))))))) (21)
0(1(4(2(3(2(2(1(0(3(1(5(2(3(1(x1))))))))))))))) 1(5(2(4(2(5(2(4(3(4(1(0(4(2(x1)))))))))))))) (22)
0(3(2(0(0(2(2(4(2(4(4(3(3(2(0(x1))))))))))))))) 3(2(1(0(4(4(2(1(0(4(3(3(1(0(x1)))))))))))))) (23)
2(4(4(3(0(1(0(1(4(0(3(4(1(0(0(5(x1)))))))))))))))) 2(5(3(1(2(2(2(2(5(3(3(2(1(4(5(x1))))))))))))))) (24)
5(0(3(0(1(1(5(1(1(2(0(3(3(0(4(5(1(x1))))))))))))))))) 3(4(5(2(0(1(2(0(1(5(4(1(3(1(5(2(x1)))))))))))))))) (25)
0(5(5(2(5(4(5(0(5(2(2(3(1(3(3(4(4(2(x1)))))))))))))))))) 5(0(2(4(4(3(2(2(5(1(4(4(1(1(4(3(5(2(x1)))))))))))))))))) (26)
1(2(0(1(0(5(0(2(4(4(5(1(4(1(2(0(4(5(x1)))))))))))))))))) 1(2(1(5(2(5(1(4(0(3(5(3(1(4(2(3(5(x1))))))))))))))))) (27)
1(4(4(3(3(0(0(3(5(2(1(4(4(4(3(0(1(5(x1)))))))))))))))))) 3(5(3(3(0(2(5(1(5(4(4(3(1(2(3(4(3(5(x1)))))))))))))))))) (28)
3(5(1(4(5(2(4(1(3(2(4(4(0(5(4(1(0(3(x1)))))))))))))))))) 3(0(5(5(4(5(1(4(3(0(0(5(5(3(3(1(2(3(x1)))))))))))))))))) (29)
4(5(0(4(0(3(4(5(0(1(3(3(0(2(2(3(5(1(x1)))))))))))))))))) 1(5(3(0(1(1(2(3(5(0(0(0(2(0(1(0(1(x1))))))))))))))))) (30)
5(3(5(3(3(1(5(5(2(2(3(3(4(1(5(1(0(5(x1)))))))))))))))))) 3(5(0(4(4(3(4(0(5(2(0(0(2(4(1(5(1(5(x1)))))))))))))))))) (31)
5(4(0(1(5(1(2(4(0(4(0(3(2(1(5(3(3(0(x1)))))))))))))))))) 4(2(1(0(3(0(1(5(0(1(1(0(5(5(4(2(1(x1))))))))))))))))) (32)
3(5(0(2(4(0(3(0(0(1(0(0(4(3(0(4(2(5(1(0(x1)))))))))))))))))))) 1(1(4(1(1(3(2(5(3(4(5(2(4(4(0(2(0(0(0(x1))))))))))))))))))) (33)
4(1(0(2(0(3(5(5(3(2(1(2(4(5(2(2(3(4(3(0(x1)))))))))))))))))))) 0(0(5(5(4(1(2(2(1(0(0(4(4(5(0(5(3(5(3(0(x1)))))))))))))))))))) (34)
4(3(0(0(2(1(4(5(1(4(3(5(0(2(2(3(5(1(4(1(4(x1))))))))))))))))))))) 4(3(4(1(1(0(4(2(1(5(2(1(1(2(4(1(2(3(1(3(x1)))))))))))))))))))) (35)

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(1(2(3(x1)))) 3(2(4(x1))) (1)
2(3(3(0(4(x1))))) 4(2(1(5(x1)))) (2)
5(1(3(0(4(0(x1)))))) 3(5(3(3(2(x1))))) (4)
4(4(0(3(5(1(2(4(4(x1))))))))) 1(5(2(0(3(5(0(0(x1)))))))) (9)
0(4(1(2(0(5(0(2(2(4(x1)))))))))) 2(5(4(0(1(2(5(3(4(x1))))))))) (10)
4(5(4(1(2(4(1(0(1(5(x1)))))))))) 1(1(0(1(3(1(5(2(5(x1))))))))) (11)
5(1(0(3(2(3(4(5(4(3(x1)))))))))) 5(2(0(5(3(4(2(5(3(x1))))))))) (12)
4(5(0(0(4(4(5(0(5(4(5(x1))))))))))) 2(4(2(2(2(4(3(1(4(5(x1)))))))))) (14)
2(3(0(0(5(4(1(3(4(0(1(4(x1)))))))))))) 4(0(3(3(4(5(1(1(4(2(4(x1))))))))))) (15)
5(1(4(3(0(5(1(2(5(5(4(0(x1)))))))))))) 2(5(0(0(5(1(4(2(5(1(5(x1))))))))))) (16)
4(5(4(5(1(2(0(2(5(4(4(5(5(x1))))))))))))) 1(1(3(4(3(3(0(4(2(5(5(3(x1)))))))))))) (18)
5(5(0(4(0(2(1(1(0(4(5(0(5(x1))))))))))))) 1(1(0(3(4(2(1(1(5(0(2(5(x1)))))))))))) (19)
5(5(3(0(4(4(0(2(1(5(1(4(2(1(x1)))))))))))))) 5(4(1(5(2(4(3(3(5(2(4(5(0(x1))))))))))))) (21)
0(1(4(2(3(2(2(1(0(3(1(5(2(3(1(x1))))))))))))))) 1(5(2(4(2(5(2(4(3(4(1(0(4(2(x1)))))))))))))) (22)
0(3(2(0(0(2(2(4(2(4(4(3(3(2(0(x1))))))))))))))) 3(2(1(0(4(4(2(1(0(4(3(3(1(0(x1)))))))))))))) (23)
2(4(4(3(0(1(0(1(4(0(3(4(1(0(0(5(x1)))))))))))))))) 2(5(3(1(2(2(2(2(5(3(3(2(1(4(5(x1))))))))))))))) (24)
5(0(3(0(1(1(5(1(1(2(0(3(3(0(4(5(1(x1))))))))))))))))) 3(4(5(2(0(1(2(0(1(5(4(1(3(1(5(2(x1)))))))))))))))) (25)
1(2(0(1(0(5(0(2(4(4(5(1(4(1(2(0(4(5(x1)))))))))))))))))) 1(2(1(5(2(5(1(4(0(3(5(3(1(4(2(3(5(x1))))))))))))))))) (27)
4(5(0(4(0(3(4(5(0(1(3(3(0(2(2(3(5(1(x1)))))))))))))))))) 1(5(3(0(1(1(2(3(5(0(0(0(2(0(1(0(1(x1))))))))))))))))) (30)
5(4(0(1(5(1(2(4(0(4(0(3(2(1(5(3(3(0(x1)))))))))))))))))) 4(2(1(0(3(0(1(5(0(1(1(0(5(5(4(2(1(x1))))))))))))))))) (32)
3(5(0(2(4(0(3(0(0(1(0(0(4(3(0(4(2(5(1(0(x1)))))))))))))))))))) 1(1(4(1(1(3(2(5(3(4(5(2(4(4(0(2(0(0(0(x1))))))))))))))))))) (33)
4(3(0(0(2(1(4(5(1(4(3(5(0(2(2(3(5(1(4(1(4(x1))))))))))))))))))))) 4(3(4(1(1(0(4(2(1(5(2(1(1(2(4(1(2(3(1(3(x1)))))))))))))))))))) (35)

1.1 Closure Under Flat Contexts

Using the flat contexts

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

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

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 468 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 +
1
[53(x1)] = x1 +
13
[54(x1)] = x1 +
0
[55(x1)] = x1 +
0
[40(x1)] = x1 +
13
[41(x1)] = x1 +
0
[42(x1)] = x1 +
1
[43(x1)] = x1 +
5
[44(x1)] = x1 +
0
[45(x1)] = x1 +
0
[30(x1)] = x1 +
0
[31(x1)] = x1 +
0
[32(x1)] = x1 +
0
[33(x1)] = x1 +
1
[34(x1)] = x1 +
0
[35(x1)] = x1 +
1
[20(x1)] = x1 +
1
[21(x1)] = x1 +
0
[22(x1)] = x1 +
0
[23(x1)] = x1 +
0
[24(x1)] = x1 +
0
[25(x1)] = x1 +
0
[10(x1)] = x1 +
5
[11(x1)] = x1 +
5
[12(x1)] = x1 +
0
[13(x1)] = x1 +
0
[14(x1)] = x1 +
5
[15(x1)] = x1 +
5
[00(x1)] = x1 +
1
[01(x1)] = x1 +
1
[02(x1)] = x1 +
0
[03(x1)] = x1 +
0
[04(x1)] = x1 +
0
[05(x1)] = x1 +
0
all of the following rules can be deleted.

There are 468 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.