Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/85650)

The rewrite relation of the following TRS is considered.

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

1.1 Closure Under Flat Contexts

Using the flat contexts

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

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

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 540 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 +
1
[51(x1)] = x1 +
0
[52(x1)] = x1 +
1
[53(x1)] = x1 +
10
[54(x1)] = x1 +
1
[55(x1)] = x1 +
10
[40(x1)] = x1 +
0
[41(x1)] = x1 +
0
[42(x1)] = x1 +
0
[43(x1)] = x1 +
10
[44(x1)] = x1 +
11
[45(x1)] = x1 +
1
[30(x1)] = x1 +
11
[31(x1)] = x1 +
0
[32(x1)] = x1 +
0
[33(x1)] = x1 +
11
[34(x1)] = x1 +
0
[35(x1)] = x1 +
0
[20(x1)] = x1 +
1
[21(x1)] = x1 +
11
[22(x1)] = x1 +
0
[23(x1)] = x1 +
0
[24(x1)] = x1 +
10
[25(x1)] = x1 +
10
[10(x1)] = x1 +
10
[11(x1)] = x1 +
11
[12(x1)] = x1 +
11
[13(x1)] = x1 +
0
[14(x1)] = x1 +
10
[15(x1)] = x1 +
0
[00(x1)] = x1 +
0
[01(x1)] = x1 +
10
[02(x1)] = x1 +
10
[03(x1)] = x1 +
0
[04(x1)] = x1 +
10
[05(x1)] = x1 +
5
all of the following rules can be deleted.

There are 540 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.