Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/96370)

The rewrite relation of the following TRS is considered.

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

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.
5(0(1(3(2(0(x1)))))) 5(5(1(0(1(x1))))) (3)
4(5(0(4(2(3(2(0(x1)))))))) 1(1(4(2(3(2(0(x1))))))) (7)
0(1(2(5(0(2(0(5(0(0(1(x1))))))))))) 0(1(4(1(2(0(3(1(2(1(x1)))))))))) (11)
2(1(4(1(5(2(4(4(2(5(0(1(0(x1))))))))))))) 0(2(1(0(1(4(4(3(1(0(3(0(x1)))))))))))) (13)
2(0(0(4(2(5(5(1(4(3(2(3(0(1(5(x1))))))))))))))) 2(1(0(1(3(2(4(5(2(4(4(5(2(4(x1)))))))))))))) (16)
2(2(5(1(0(2(3(3(5(4(5(5(3(1(2(3(x1)))))))))))))))) 2(4(2(1(1(1(3(5(1(0(4(3(5(3(3(x1))))))))))))))) (17)
5(5(5(2(3(4(0(4(2(2(2(4(1(4(5(5(5(4(x1)))))))))))))))))) 3(0(1(1(2(3(3(2(2(5(0(2(1(4(3(4(4(x1))))))))))))))))) (20)
3(2(2(4(2(5(0(4(2(3(4(0(3(0(0(4(0(5(3(5(0(x1))))))))))))))))))))) 1(1(4(1(5(4(1(3(2(1(5(5(1(3(5(2(0(4(4(x1))))))))))))))))))) (23)
5(1(0(4(5(2(3(0(2(3(2(1(4(3(1(3(3(1(3(4(2(x1))))))))))))))))))))) 0(1(1(3(5(0(4(5(2(2(3(0(4(4(3(1(0(0(1(3(x1)))))))))))))))))))) (25)

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(0(0(2(x1))))))) 5(0(0(2(3(4(4(x1))))))) (26)
5(3(2(4(4(1(3(x1))))))) 5(5(5(3(2(1(3(x1))))))) (27)
5(1(5(1(0(2(1(1(x1)))))))) 5(1(3(2(1(1(0(4(x1)))))))) (28)
5(0(1(0(0(5(5(5(0(x1))))))))) 5(0(2(2(1(0(4(0(0(x1))))))))) (29)
5(1(2(5(1(2(0(1(3(x1))))))))) 5(1(4(2(5(4(3(5(3(x1))))))))) (30)
5(0(2(4(0(1(5(4(2(3(x1)))))))))) 5(2(2(0(3(4(0(0(3(3(x1)))))))))) (31)
5(1(5(1(3(0(5(2(0(0(x1)))))))))) 5(1(1(3(2(5(0(2(2(5(x1)))))))))) (32)
5(5(1(0(2(4(1(4(2(1(0(x1))))))))))) 5(1(3(5(1(2(4(0(3(2(3(x1))))))))))) (33)
5(1(3(3(2(3(1(1(5(1(4(1(2(0(x1)))))))))))))) 5(1(3(5(5(4(4(4(5(5(3(2(4(3(x1)))))))))))))) (34)
5(2(2(1(0(0(0(5(2(2(4(1(1(1(x1)))))))))))))) 5(1(1(4(0(2(5(2(0(2(2(4(2(5(x1)))))))))))))) (35)
5(0(3(5(5(5(3(1(0(0(5(1(4(2(1(1(x1)))))))))))))))) 5(0(3(5(2(1(0(1(2(1(3(2(4(5(2(1(x1)))))))))))))))) (36)
5(3(1(2(0(2(4(1(4(4(4(2(3(3(2(4(0(0(x1)))))))))))))))))) 5(2(5(1(2(1(4(4(0(2(1(5(4(1(3(2(0(0(x1)))))))))))))))))) (37)
5(0(5(0(0(4(2(0(0(3(4(4(0(5(0(4(1(2(0(x1))))))))))))))))))) 5(0(3(5(5(3(0(4(2(2(3(2(5(1(2(4(0(2(4(x1))))))))))))))))))) (38)
5(4(1(5(4(3(2(5(4(5(2(0(2(2(2(4(1(0(2(1(0(x1))))))))))))))))))))) 5(4(5(5(2(3(2(0(4(2(0(1(4(0(4(3(1(3(2(0(0(x1))))))))))))))))))))) (39)
5(2(0(4(1(2(2(3(5(3(5(3(4(4(1(5(3(0(5(1(0(2(x1)))))))))))))))))))))) 5(2(2(2(4(4(2(0(0(1(4(5(3(3(3(5(3(0(4(0(3(2(x1)))))))))))))))))))))) (40)
5(5(1(0(1(1(5(2(1(5(5(4(3(5(2(5(2(4(3(1(5(3(x1)))))))))))))))))))))) 5(1(1(0(3(4(5(4(3(2(2(5(4(0(1(4(3(0(2(5(0(3(x1)))))))))))))))))))))) (41)
4(0(0(1(0(0(2(x1))))))) 4(0(0(2(3(4(4(x1))))))) (42)
4(3(2(4(4(1(3(x1))))))) 4(5(5(3(2(1(3(x1))))))) (43)
4(1(5(1(0(2(1(1(x1)))))))) 4(1(3(2(1(1(0(4(x1)))))))) (44)
4(0(1(0(0(5(5(5(0(x1))))))))) 4(0(2(2(1(0(4(0(0(x1))))))))) (45)
4(1(2(5(1(2(0(1(3(x1))))))))) 4(1(4(2(5(4(3(5(3(x1))))))))) (46)
4(0(2(4(0(1(5(4(2(3(x1)))))))))) 4(2(2(0(3(4(0(0(3(3(x1)))))))))) (47)
4(1(5(1(3(0(5(2(0(0(x1)))))))))) 4(1(1(3(2(5(0(2(2(5(x1)))))))))) (48)
4(5(1(0(2(4(1(4(2(1(0(x1))))))))))) 4(1(3(5(1(2(4(0(3(2(3(x1))))))))))) (49)
4(1(3(3(2(3(1(1(5(1(4(1(2(0(x1)))))))))))))) 4(1(3(5(5(4(4(4(5(5(3(2(4(3(x1)))))))))))))) (50)
4(2(2(1(0(0(0(5(2(2(4(1(1(1(x1)))))))))))))) 4(1(1(4(0(2(5(2(0(2(2(4(2(5(x1)))))))))))))) (51)
4(0(3(5(5(5(3(1(0(0(5(1(4(2(1(1(x1)))))))))))))))) 4(0(3(5(2(1(0(1(2(1(3(2(4(5(2(1(x1)))))))))))))))) (52)
4(3(1(2(0(2(4(1(4(4(4(2(3(3(2(4(0(0(x1)))))))))))))))))) 4(2(5(1(2(1(4(4(0(2(1(5(4(1(3(2(0(0(x1)))))))))))))))))) (53)
4(0(5(0(0(4(2(0(0(3(4(4(0(5(0(4(1(2(0(x1))))))))))))))))))) 4(0(3(5(5(3(0(4(2(2(3(2(5(1(2(4(0(2(4(x1))))))))))))))))))) (54)
4(4(1(5(4(3(2(5(4(5(2(0(2(2(2(4(1(0(2(1(0(x1))))))))))))))))))))) 4(4(5(5(2(3(2(0(4(2(0(1(4(0(4(3(1(3(2(0(0(x1))))))))))))))))))))) (55)
4(2(0(4(1(2(2(3(5(3(5(3(4(4(1(5(3(0(5(1(0(2(x1)))))))))))))))))))))) 4(2(2(2(4(4(2(0(0(1(4(5(3(3(3(5(3(0(4(0(3(2(x1)))))))))))))))))))))) (56)
4(5(1(0(1(1(5(2(1(5(5(4(3(5(2(5(2(4(3(1(5(3(x1)))))))))))))))))))))) 4(1(1(0(3(4(5(4(3(2(2(5(4(0(1(4(3(0(2(5(0(3(x1)))))))))))))))))))))) (57)
3(0(0(1(0(0(2(x1))))))) 3(0(0(2(3(4(4(x1))))))) (58)
3(3(2(4(4(1(3(x1))))))) 3(5(5(3(2(1(3(x1))))))) (59)
3(1(5(1(0(2(1(1(x1)))))))) 3(1(3(2(1(1(0(4(x1)))))))) (60)
3(0(1(0(0(5(5(5(0(x1))))))))) 3(0(2(2(1(0(4(0(0(x1))))))))) (61)
3(1(2(5(1(2(0(1(3(x1))))))))) 3(1(4(2(5(4(3(5(3(x1))))))))) (62)
3(0(2(4(0(1(5(4(2(3(x1)))))))))) 3(2(2(0(3(4(0(0(3(3(x1)))))))))) (63)
3(1(5(1(3(0(5(2(0(0(x1)))))))))) 3(1(1(3(2(5(0(2(2(5(x1)))))))))) (64)
3(5(1(0(2(4(1(4(2(1(0(x1))))))))))) 3(1(3(5(1(2(4(0(3(2(3(x1))))))))))) (65)
3(1(3(3(2(3(1(1(5(1(4(1(2(0(x1)))))))))))))) 3(1(3(5(5(4(4(4(5(5(3(2(4(3(x1)))))))))))))) (66)
3(2(2(1(0(0(0(5(2(2(4(1(1(1(x1)))))))))))))) 3(1(1(4(0(2(5(2(0(2(2(4(2(5(x1)))))))))))))) (67)
3(0(3(5(5(5(3(1(0(0(5(1(4(2(1(1(x1)))))))))))))))) 3(0(3(5(2(1(0(1(2(1(3(2(4(5(2(1(x1)))))))))))))))) (68)
3(3(1(2(0(2(4(1(4(4(4(2(3(3(2(4(0(0(x1)))))))))))))))))) 3(2(5(1(2(1(4(4(0(2(1(5(4(1(3(2(0(0(x1)))))))))))))))))) (69)
3(0(5(0(0(4(2(0(0(3(4(4(0(5(0(4(1(2(0(x1))))))))))))))))))) 3(0(3(5(5(3(0(4(2(2(3(2(5(1(2(4(0(2(4(x1))))))))))))))))))) (70)
3(4(1(5(4(3(2(5(4(5(2(0(2(2(2(4(1(0(2(1(0(x1))))))))))))))))))))) 3(4(5(5(2(3(2(0(4(2(0(1(4(0(4(3(1(3(2(0(0(x1))))))))))))))))))))) (71)
3(2(0(4(1(2(2(3(5(3(5(3(4(4(1(5(3(0(5(1(0(2(x1)))))))))))))))))))))) 3(2(2(2(4(4(2(0(0(1(4(5(3(3(3(5(3(0(4(0(3(2(x1)))))))))))))))))))))) (72)
3(5(1(0(1(1(5(2(1(5(5(4(3(5(2(5(2(4(3(1(5(3(x1)))))))))))))))))))))) 3(1(1(0(3(4(5(4(3(2(2(5(4(0(1(4(3(0(2(5(0(3(x1)))))))))))))))))))))) (73)
2(0(0(1(0(0(2(x1))))))) 2(0(0(2(3(4(4(x1))))))) (74)
2(3(2(4(4(1(3(x1))))))) 2(5(5(3(2(1(3(x1))))))) (75)
2(1(5(1(0(2(1(1(x1)))))))) 2(1(3(2(1(1(0(4(x1)))))))) (76)
2(0(1(0(0(5(5(5(0(x1))))))))) 2(0(2(2(1(0(4(0(0(x1))))))))) (77)
2(1(2(5(1(2(0(1(3(x1))))))))) 2(1(4(2(5(4(3(5(3(x1))))))))) (78)
2(0(2(4(0(1(5(4(2(3(x1)))))))))) 2(2(2(0(3(4(0(0(3(3(x1)))))))))) (79)
2(1(5(1(3(0(5(2(0(0(x1)))))))))) 2(1(1(3(2(5(0(2(2(5(x1)))))))))) (80)
2(5(1(0(2(4(1(4(2(1(0(x1))))))))))) 2(1(3(5(1(2(4(0(3(2(3(x1))))))))))) (81)
2(1(3(3(2(3(1(1(5(1(4(1(2(0(x1)))))))))))))) 2(1(3(5(5(4(4(4(5(5(3(2(4(3(x1)))))))))))))) (82)
2(2(2(1(0(0(0(5(2(2(4(1(1(1(x1)))))))))))))) 2(1(1(4(0(2(5(2(0(2(2(4(2(5(x1)))))))))))))) (83)
2(0(3(5(5(5(3(1(0(0(5(1(4(2(1(1(x1)))))))))))))))) 2(0(3(5(2(1(0(1(2(1(3(2(4(5(2(1(x1)))))))))))))))) (84)
2(3(1(2(0(2(4(1(4(4(4(2(3(3(2(4(0(0(x1)))))))))))))))))) 2(2(5(1(2(1(4(4(0(2(1(5(4(1(3(2(0(0(x1)))))))))))))))))) (85)
2(0(5(0(0(4(2(0(0(3(4(4(0(5(0(4(1(2(0(x1))))))))))))))))))) 2(0(3(5(5(3(0(4(2(2(3(2(5(1(2(4(0(2(4(x1))))))))))))))))))) (86)
2(4(1(5(4(3(2(5(4(5(2(0(2(2(2(4(1(0(2(1(0(x1))))))))))))))))))))) 2(4(5(5(2(3(2(0(4(2(0(1(4(0(4(3(1(3(2(0(0(x1))))))))))))))))))))) (87)
2(2(0(4(1(2(2(3(5(3(5(3(4(4(1(5(3(0(5(1(0(2(x1)))))))))))))))))))))) 2(2(2(2(4(4(2(0(0(1(4(5(3(3(3(5(3(0(4(0(3(2(x1)))))))))))))))))))))) (88)
2(5(1(0(1(1(5(2(1(5(5(4(3(5(2(5(2(4(3(1(5(3(x1)))))))))))))))))))))) 2(1(1(0(3(4(5(4(3(2(2(5(4(0(1(4(3(0(2(5(0(3(x1)))))))))))))))))))))) (89)
1(0(0(1(0(0(2(x1))))))) 1(0(0(2(3(4(4(x1))))))) (90)
1(3(2(4(4(1(3(x1))))))) 1(5(5(3(2(1(3(x1))))))) (91)
1(1(5(1(0(2(1(1(x1)))))))) 1(1(3(2(1(1(0(4(x1)))))))) (92)
1(0(1(0(0(5(5(5(0(x1))))))))) 1(0(2(2(1(0(4(0(0(x1))))))))) (93)
1(1(2(5(1(2(0(1(3(x1))))))))) 1(1(4(2(5(4(3(5(3(x1))))))))) (94)
1(0(2(4(0(1(5(4(2(3(x1)))))))))) 1(2(2(0(3(4(0(0(3(3(x1)))))))))) (95)
1(1(5(1(3(0(5(2(0(0(x1)))))))))) 1(1(1(3(2(5(0(2(2(5(x1)))))))))) (96)
1(5(1(0(2(4(1(4(2(1(0(x1))))))))))) 1(1(3(5(1(2(4(0(3(2(3(x1))))))))))) (97)
1(1(3(3(2(3(1(1(5(1(4(1(2(0(x1)))))))))))))) 1(1(3(5(5(4(4(4(5(5(3(2(4(3(x1)))))))))))))) (98)
1(2(2(1(0(0(0(5(2(2(4(1(1(1(x1)))))))))))))) 1(1(1(4(0(2(5(2(0(2(2(4(2(5(x1)))))))))))))) (99)
1(0(3(5(5(5(3(1(0(0(5(1(4(2(1(1(x1)))))))))))))))) 1(0(3(5(2(1(0(1(2(1(3(2(4(5(2(1(x1)))))))))))))))) (100)
1(3(1(2(0(2(4(1(4(4(4(2(3(3(2(4(0(0(x1)))))))))))))))))) 1(2(5(1(2(1(4(4(0(2(1(5(4(1(3(2(0(0(x1)))))))))))))))))) (101)
1(0(5(0(0(4(2(0(0(3(4(4(0(5(0(4(1(2(0(x1))))))))))))))))))) 1(0(3(5(5(3(0(4(2(2(3(2(5(1(2(4(0(2(4(x1))))))))))))))))))) (102)
1(4(1(5(4(3(2(5(4(5(2(0(2(2(2(4(1(0(2(1(0(x1))))))))))))))))))))) 1(4(5(5(2(3(2(0(4(2(0(1(4(0(4(3(1(3(2(0(0(x1))))))))))))))))))))) (103)
1(2(0(4(1(2(2(3(5(3(5(3(4(4(1(5(3(0(5(1(0(2(x1)))))))))))))))))))))) 1(2(2(2(4(4(2(0(0(1(4(5(3(3(3(5(3(0(4(0(3(2(x1)))))))))))))))))))))) (104)
1(5(1(0(1(1(5(2(1(5(5(4(3(5(2(5(2(4(3(1(5(3(x1)))))))))))))))))))))) 1(1(1(0(3(4(5(4(3(2(2(5(4(0(1(4(3(0(2(5(0(3(x1)))))))))))))))))))))) (105)
0(0(0(1(0(0(2(x1))))))) 0(0(0(2(3(4(4(x1))))))) (106)
0(3(2(4(4(1(3(x1))))))) 0(5(5(3(2(1(3(x1))))))) (107)
0(1(5(1(0(2(1(1(x1)))))))) 0(1(3(2(1(1(0(4(x1)))))))) (108)
0(0(1(0(0(5(5(5(0(x1))))))))) 0(0(2(2(1(0(4(0(0(x1))))))))) (109)
0(1(2(5(1(2(0(1(3(x1))))))))) 0(1(4(2(5(4(3(5(3(x1))))))))) (110)
0(0(2(4(0(1(5(4(2(3(x1)))))))))) 0(2(2(0(3(4(0(0(3(3(x1)))))))))) (111)
0(1(5(1(3(0(5(2(0(0(x1)))))))))) 0(1(1(3(2(5(0(2(2(5(x1)))))))))) (112)
0(5(1(0(2(4(1(4(2(1(0(x1))))))))))) 0(1(3(5(1(2(4(0(3(2(3(x1))))))))))) (113)
0(1(3(3(2(3(1(1(5(1(4(1(2(0(x1)))))))))))))) 0(1(3(5(5(4(4(4(5(5(3(2(4(3(x1)))))))))))))) (114)
0(2(2(1(0(0(0(5(2(2(4(1(1(1(x1)))))))))))))) 0(1(1(4(0(2(5(2(0(2(2(4(2(5(x1)))))))))))))) (115)
0(0(3(5(5(5(3(1(0(0(5(1(4(2(1(1(x1)))))))))))))))) 0(0(3(5(2(1(0(1(2(1(3(2(4(5(2(1(x1)))))))))))))))) (116)
0(3(1(2(0(2(4(1(4(4(4(2(3(3(2(4(0(0(x1)))))))))))))))))) 0(2(5(1(2(1(4(4(0(2(1(5(4(1(3(2(0(0(x1)))))))))))))))))) (117)
0(0(5(0(0(4(2(0(0(3(4(4(0(5(0(4(1(2(0(x1))))))))))))))))))) 0(0(3(5(5(3(0(4(2(2(3(2(5(1(2(4(0(2(4(x1))))))))))))))))))) (118)
0(4(1(5(4(3(2(5(4(5(2(0(2(2(2(4(1(0(2(1(0(x1))))))))))))))))))))) 0(4(5(5(2(3(2(0(4(2(0(1(4(0(4(3(1(3(2(0(0(x1))))))))))))))))))))) (119)
0(2(0(4(1(2(2(3(5(3(5(3(4(4(1(5(3(0(5(1(0(2(x1)))))))))))))))))))))) 0(2(2(2(4(4(2(0(0(1(4(5(3(3(3(5(3(0(4(0(3(2(x1)))))))))))))))))))))) (120)
0(5(1(0(1(1(5(2(1(5(5(4(3(5(2(5(2(4(3(1(5(3(x1)))))))))))))))))))))) 0(1(1(0(3(4(5(4(3(2(2(5(4(0(1(4(3(0(2(5(0(3(x1)))))))))))))))))))))) (121)

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 +
0
[53(x1)] = x1 +
6
[54(x1)] = x1 +
1
[55(x1)] = x1 +
0
[40(x1)] = x1 +
0
[41(x1)] = x1 +
0
[42(x1)] = x1 +
0
[43(x1)] = x1 +
0
[44(x1)] = x1 +
0
[45(x1)] = x1 +
0
[30(x1)] = x1 +
0
[31(x1)] = x1 +
0
[32(x1)] = x1 +
6
[33(x1)] = x1 +
0
[34(x1)] = x1 +
53
[35(x1)] = x1 +
0
[20(x1)] = x1 +
0
[21(x1)] = x1 +
1
[22(x1)] = x1 +
1
[23(x1)] = x1 +
0
[24(x1)] = x1 +
0
[25(x1)] = x1 +
6
[10(x1)] = x1 +
0
[11(x1)] = x1 +
1
[12(x1)] = x1 +
0
[13(x1)] = x1 +
24
[14(x1)] = x1 +
1
[15(x1)] = x1 +
107
[00(x1)] = x1 +
0
[01(x1)] = x1 +
0
[02(x1)] = x1 +
1
[03(x1)] = x1 +
1
[04(x1)] = x1 +
24
[05(x1)] = x1 +
6
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.