Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/27006)

The rewrite relation of the following TRS is considered.

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

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 +
14/3
[4(x1)] = x1 +
14/3
[3(x1)] = x1 +
4
[2(x1)] = x1 +
11/3
[1(x1)] = x1 +
11/3
[0(x1)] = x1 +
4
all of the following rules can be deleted.
3(3(3(1(0(3(0(5(1(4(x1)))))))))) 2(4(1(2(1(3(1(1(1(4(x1)))))))))) (8)
4(3(1(3(0(4(3(3(4(4(x1)))))))))) 1(5(2(2(4(1(5(3(3(1(x1)))))))))) (9)
5(2(5(3(4(0(4(5(5(5(x1)))))))))) 5(2(4(0(1(1(2(1(3(5(x1)))))))))) (10)
0(0(3(1(5(2(5(1(0(0(2(x1))))))))))) 1(5(3(2(1(5(5(3(4(5(x1)))))))))) (11)
0(0(4(1(2(3(0(2(3(2(0(x1))))))))))) 3(2(1(4(4(3(5(1(4(5(x1)))))))))) (12)
0(2(3(5(5(1(4(0(3(4(2(x1))))))))))) 1(3(2(3(4(4(1(3(3(5(x1)))))))))) (13)
0(3(1(0(4(4(2(2(1(2(4(x1))))))))))) 3(0(5(0(2(2(0(1(3(3(x1)))))))))) (14)
0(4(2(2(1(5(0(3(0(5(1(x1))))))))))) 0(3(3(4(2(2(0(3(5(5(x1)))))))))) (15)
0(5(0(2(4(1(1(2(5(4(1(x1))))))))))) 1(1(1(1(4(4(0(3(5(0(x1)))))))))) (16)
0(5(2(1(2(5(4(2(3(0(0(x1))))))))))) 3(0(3(5(0(1(1(4(0(2(x1)))))))))) (17)
1(0(3(2(4(5(3(0(2(1(5(x1))))))))))) 0(3(0(3(5(1(3(3(2(0(x1)))))))))) (18)
1(1(1(0(5(0(5(2(0(0(4(x1))))))))))) 5(2(1(5(2(3(0(4(5(4(x1)))))))))) (19)
1(1(4(0(2(5(5(3(4(5(2(x1))))))))))) 3(4(1(4(1(1(5(1(1(4(x1)))))))))) (20)
1(1(4(1(2(1(4(0(2(1(4(x1))))))))))) 1(0(2(3(2(2(4(1(0(5(x1)))))))))) (21)
1(2(3(0(2(0(4(0(1(4(2(x1))))))))))) 4(2(2(0(2(1(0(1(5(1(x1)))))))))) (22)
1(3(0(1(4(0(4(4(4(1(3(x1))))))))))) 3(0(5(4(4(4(0(3(1(0(x1)))))))))) (23)
1(3(3(1(1(1(4(0(0(5(3(x1))))))))))) 0(3(4(1(2(0(2(3(1(5(x1)))))))))) (24)
1(5(2(1(1(5(1(0(0(5(4(x1))))))))))) 1(0(5(0(1(2(5(4(3(2(x1)))))))))) (25)
2(0(4(3(4(2(3(1(4(0(3(x1))))))))))) 4(0(0(3(2(0(5(1(0(3(x1)))))))))) (26)
2(1(1(4(1(0(0(4(5(1(0(x1))))))))))) 1(2(3(4(2(4(3(4(5(2(x1)))))))))) (27)
2(4(1(5(3(0(2(4(4(0(3(x1))))))))))) 1(1(5(2(4(4(2(2(4(5(x1)))))))))) (28)
2(4(5(0(1(0(1(4(5(0(1(x1))))))))))) 0(5(5(4(4(4(1(0(1(4(x1)))))))))) (29)
2(5(0(2(5(0(0(3(4(4(3(x1))))))))))) 1(1(1(5(0(5(3(0(3(2(x1)))))))))) (30)
2(5(2(0(2(0(2(3(2(5(3(x1))))))))))) 3(0(4(3(4(5(2(1(1(3(x1)))))))))) (31)
2(5(2(1(5(3(3(4(4(0(2(x1))))))))))) 3(5(1(1(2(1(4(0(3(0(x1)))))))))) (32)
2(5(5(0(2(1(0(5(1(4(2(x1))))))))))) 2(0(1(5(0(1(0(1(3(2(x1)))))))))) (33)
3(2(0(4(2(1(0(4(4(5(5(x1))))))))))) 1(3(2(4(0(2(0(4(2(4(x1)))))))))) (34)
3(3(4(0(4(5(5(5(5(2(5(x1))))))))))) 5(2(4(5(2(4(3(4(3(0(x1)))))))))) (35)
3(4(2(0(5(1(2(2(0(1(4(x1))))))))))) 5(2(2(2(5(3(1(1(2(4(x1)))))))))) (36)
3(5(0(2(5(1(1(1(5(2(0(x1))))))))))) 3(5(2(0(0(2(3(3(1(3(x1)))))))))) (37)
3(5(0(5(1(1(4(0(2(0(5(x1))))))))))) 2(5(1(2(0(2(4(0(2(3(x1)))))))))) (38)
3(5(4(1(2(5(0(1(2(0(3(x1))))))))))) 4(2(2(1(0(1(3(3(4(5(x1)))))))))) (39)
3(5(5(4(3(0(1(4(0(4(1(x1))))))))))) 2(4(2(5(2(4(3(0(1(4(x1)))))))))) (40)
4(0(0(2(5(0(5(0(1(3(3(x1))))))))))) 1(4(0(1(4(4(2(2(4(3(x1)))))))))) (41)
4(0(2(4(0(5(1(3(3(2(1(x1))))))))))) 0(1(1(3(4(0(3(3(1(4(x1)))))))))) (42)
4(1(1(0(2(2(4(0(1(3(5(x1))))))))))) 3(0(5(4(5(2(1(2(2(5(x1)))))))))) (43)
4(1(3(0(3(3(4(5(2(4(4(x1))))))))))) 1(1(4(5(4(5(2(0(1(5(x1)))))))))) (44)
4(2(4(4(1(3(1(2(5(1(4(x1))))))))))) 0(5(5(5(1(1(3(1(0(1(x1)))))))))) (45)
4(3(0(2(4(3(4(1(4(1(2(x1))))))))))) 0(1(2(4(3(1(0(5(1(4(x1)))))))))) (46)
4(4(3(1(3(2(5(2(5(2(4(x1))))))))))) 0(3(0(1(3(5(2(2(3(0(x1)))))))))) (47)
4(4(3(5(0(2(5(4(3(1(0(x1))))))))))) 2(2(5(3(1(4(5(5(4(5(x1)))))))))) (48)
5(5(3(0(0(2(4(4(0(3(1(x1))))))))))) 2(3(3(3(0(2(0(5(5(0(x1)))))))))) (49)
0(5(2(1(2(5(0(2(1(4(5(0(x1)))))))))))) 4(3(3(0(1(5(5(1(5(4(x1)))))))))) (50)
0(5(3(0(1(4(5(1(4(1(5(1(x1)))))))))))) 0(0(4(0(0(1(3(5(1(5(x1)))))))))) (51)
1(0(2(3(2(3(1(0(3(1(0(5(x1)))))))))))) 5(0(4(5(1(3(2(4(0(3(x1)))))))))) (52)
2(1(3(2(1(0(5(4(3(1(5(2(x1)))))))))))) 1(3(2(0(5(0(5(2(0(1(x1)))))))))) (53)
2(2(1(4(1(1(0(1(3(5(0(4(x1)))))))))))) 1(4(5(5(0(4(3(2(4(5(x1)))))))))) (54)
2(4(2(1(1(2(4(3(1(5(0(2(x1)))))))))))) 3(4(5(5(5(0(3(1(5(0(x1)))))))))) (56)
3(3(4(3(4(1(1(1(3(5(2(2(x1)))))))))))) 4(2(1(5(5(5(1(5(2(1(x1)))))))))) (57)
5(2(1(1(4(4(4(0(1(4(1(2(x1)))))))))))) 4(0(2(5(1(5(4(4(2(0(x1)))))))))) (59)

1.1 Closure Under Flat Contexts

Using the flat contexts

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

We obtain the transformed TRS

There are 252 ruless (increase limit for explicit display).

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 1512 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 +
5
[52(x1)] = x1 +
5
[53(x1)] = x1 +
3
[54(x1)] = x1 +
3
[55(x1)] = x1 +
12
[40(x1)] = x1 +
1
[41(x1)] = x1 +
3
[42(x1)] = x1 +
0
[43(x1)] = x1 +
1
[44(x1)] = x1 +
0
[45(x1)] = x1 +
0
[30(x1)] = x1 +
0
[31(x1)] = x1 +
12
[32(x1)] = x1 +
0
[33(x1)] = x1 +
5
[34(x1)] = x1 +
0
[35(x1)] = x1 +
5
[20(x1)] = x1 +
5
[21(x1)] = x1 +
0
[22(x1)] = x1 +
0
[23(x1)] = x1 +
0
[24(x1)] = x1 +
0
[25(x1)] = x1 +
0
[10(x1)] = x1 +
0
[11(x1)] = x1 +
0
[12(x1)] = x1 +
5
[13(x1)] = x1 +
12
[14(x1)] = x1 +
1
[15(x1)] = x1 +
0
[00(x1)] = x1 +
3
[01(x1)] = x1 +
5
[02(x1)] = x1 +
5
[03(x1)] = x1 +
0
[04(x1)] = x1 +
12
[05(x1)] = x1 +
1
all of the following rules can be deleted.

There are 1512 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.