Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/26923)

The rewrite relation of the following TRS is considered.

0(1(2(0(3(4(x1)))))) 0(1(0(2(3(4(x1)))))) (1)
1(2(3(1(4(4(1(x1))))))) 1(2(1(3(4(4(1(x1))))))) (2)
2(2(5(3(2(4(5(3(x1)))))))) 2(2(5(2(3(4(5(3(x1)))))))) (3)
4(2(3(5(4(2(2(0(x1)))))))) 4(2(5(3(4(2(2(0(x1)))))))) (4)
5(3(5(1(5(3(5(3(x1)))))))) 5(3(5(1(3(5(5(3(x1)))))))) (5)
3(2(3(1(3(5(4(1(1(x1))))))))) 3(1(3(3(4(2(1(5(1(x1))))))))) (6)
3(4(2(5(3(2(1(2(1(x1))))))))) 3(4(2(5(3(1(2(2(1(x1))))))))) (7)
3(1(3(0(4(5(2(4(5(0(x1)))))))))) 3(1(3(4(0(5(2(4(5(0(x1)))))))))) (8)
4(5(5(1(2(1(3(2(3(2(x1)))))))))) 5(5(0(0(5(0(3(3(2(1(x1)))))))))) (9)
5(0(1(4(1(2(5(1(5(2(x1)))))))))) 4(5(5(0(2(0(5(4(5(4(x1)))))))))) (10)
5(0(5(4(0(3(1(5(4(2(x1)))))))))) 5(0(5(4(0(1(3(5(4(2(x1)))))))))) (11)
0(3(0(2(4(1(2(2(4(0(5(x1))))))))))) 2(4(4(3(0(2(5(1(4(4(x1)))))))))) (12)
0(5(0(3(0(1(2(4(0(4(3(x1))))))))))) 5(0(4(5(1(0(1(4(2(3(x1)))))))))) (13)
0(5(2(4(1(5(0(2(1(5(1(x1))))))))))) 5(0(1(1(1(4(2(0(3(0(x1)))))))))) (14)
0(5(3(5(0(5(3(3(2(1(3(x1))))))))))) 0(3(4(0(5(1(4(4(5(2(x1)))))))))) (15)
0(5(4(4(1(0(5(1(4(5(5(x1))))))))))) 0(1(3(4(4(1(4(3(5(0(x1)))))))))) (16)
1(3(2(2(5(5(1(5(1(2(1(x1))))))))))) 4(4(0(0(1(2(4(1(5(4(x1)))))))))) (17)
1(4(4(5(2(2(1(0(0(0(2(x1))))))))))) 3(5(1(1(4(2(4(0(2(0(x1)))))))))) (18)
1(5(0(4(0(1(2(5(5(5(3(x1))))))))))) 4(3(2(3(2(4(1(0(3(3(x1)))))))))) (19)
1(5(1(2(2(1(1(0(5(5(0(x1))))))))))) 0(3(0(1(1(2(2(0(1(4(x1)))))))))) (20)
2(0(0(4(5(0(5(3(0(4(2(x1))))))))))) 2(0(0(4(5(5(0(3(4(0(2(x1))))))))))) (21)
2(1(5(3(3(1(4(1(0(4(0(x1))))))))))) 4(4(3(5(0(0(5(3(2(4(x1)))))))))) (22)
2(2(4(1(3(2(4(0(2(5(4(x1))))))))))) 0(4(2(3(1(2(3(0(1(4(x1)))))))))) (23)
2(3(2(4(0(0(3(4(3(2(0(x1))))))))))) 5(2(1(5(0(5(1(1(5(3(x1)))))))))) (24)
2(3(4(1(2(3(2(2(0(5(0(x1))))))))))) 2(2(3(0(5(1(4(1(1(2(x1)))))))))) (25)
2(5(0(1(0(5(3(1(2(1(0(x1))))))))))) 4(1(3(0(5(0(0(0(2(3(x1)))))))))) (26)
3(0(0(4(5(2(5(3(5(4(1(x1))))))))))) 3(4(2(2(3(3(3(4(5(0(x1)))))))))) (27)
3(0(3(5(4(0(4(3(5(2(5(x1))))))))))) 2(4(0(3(4(1(2(0(1(5(x1)))))))))) (28)
3(1(4(0(5(0(4(1(1(0(5(x1))))))))))) 0(4(4(5(0(0(0(3(1(0(x1)))))))))) (29)
3(2(0(3(0(3(1(3(5(5(5(x1))))))))))) 0(3(2(4(1(5(1(0(5(1(x1)))))))))) (30)
3(2(2(2(0(5(1(2(3(5(0(x1))))))))))) 3(2(2(2(5(0(1(2(3(5(0(x1))))))))))) (31)
3(2(3(0(0(0(2(4(2(1(1(x1))))))))))) 2(4(0(0(4(3(0(2(5(3(x1)))))))))) (32)
3(5(0(5(4(0(0(4(4(2(4(x1))))))))))) 2(0(3(5(5(0(2(1(4(3(x1)))))))))) (33)
3(5(4(2(3(0(4(0(5(1(2(x1))))))))))) 1(3(4(2(3(5(5(1(1(2(x1)))))))))) (34)
4(1(3(0(4(5(4(2(4(2(5(x1))))))))))) 0(5(2(1(0(2(3(3(5(3(x1)))))))))) (35)
4(2(2(0(4(0(1(4(5(3(3(x1))))))))))) 1(0(0(0(0(5(1(0(5(3(x1)))))))))) (36)
4(3(3(4(3(1(0(2(5(4(1(x1))))))))))) 0(5(5(0(5(1(4(2(1(5(x1)))))))))) (37)
4(3(4(1(1(5(5(5(3(4(2(x1))))))))))) 4(5(5(0(1(2(3(1(3(5(x1)))))))))) (38)
4(3(4(4(2(4(4(2(0(4(4(x1))))))))))) 5(3(1(2(0(4(3(5(3(2(x1)))))))))) (39)
4(4(3(2(1(5(3(4(0(0(4(x1))))))))))) 5(0(2(4(0(1(4(1(4(4(x1)))))))))) (40)
4(4(5(4(1(4(1(1(5(3(5(x1))))))))))) 2(5(5(2(3(2(5(4(3(0(x1)))))))))) (41)
4(5(1(5(4(4(3(5(1(2(1(x1))))))))))) 4(3(0(4(3(0(3(3(5(4(x1)))))))))) (42)
5(1(4(3(3(2(0(5(1(4(2(x1))))))))))) 4(3(5(1(5(5(5(1(2(2(x1)))))))))) (43)
5(3(4(4(0(0(1(1(5(1(4(x1))))))))))) 3(2(4(2(0(0(4(1(3(0(x1)))))))))) (44)
5(3(5(2(3(2(2(1(0(3(5(x1))))))))))) 0(3(2(1(0(0(0(0(2(4(x1)))))))))) (45)
5(4(0(1(2(1(1(4(5(2(0(x1))))))))))) 5(4(5(4(3(4(1(1(3(1(x1)))))))))) (46)
5(5(2(4(3(5(3(4(0(3(4(x1))))))))))) 2(0(0(3(2(4(0(1(3(5(x1)))))))))) (47)
5(5(5(0(0(3(5(4(4(2(1(x1))))))))))) 1(0(3(2(1(5(5(1(0(0(x1)))))))))) (48)
0(0(1(0(4(1(4(5(4(3(1(0(x1)))))))))))) 0(0(1(0(1(4(4(5(4(3(1(0(x1)))))))))))) (49)
0(1(3(3(5(0(3(1(5(4(3(5(x1)))))))))))) 5(3(3(2(1(0(1(1(3(1(x1)))))))))) (50)
1(1(0(5(1(1(3(1(4(2(4(0(x1)))))))))))) 4(5(1(5(2(1(3(2(0(4(x1)))))))))) (51)
1(2(4(1(4(4(2(1(5(4(0(0(x1)))))))))))) 1(1(3(4(5(1(2(4(3(2(x1)))))))))) (52)
1(3(0(0(4(5(3(4(3(0(0(1(x1)))))))))))) 3(5(4(5(0(3(4(4(2(5(x1)))))))))) (53)
2(2(3(4(2(0(5(4(3(0(0(0(x1)))))))))))) 0(1(2(2(3(2(4(0(1(3(x1)))))))))) (54)
2(2(4(2(3(0(5(3(0(5(5(3(x1)))))))))))) 0(4(4(3(5(4(4(1(5(1(x1)))))))))) (55)
3(0(3(1(5(0(3(1(1(0(2(3(x1)))))))))))) 5(1(4(4(5(4(1(1(1(1(x1)))))))))) (56)
3(1(0(4(0(2(5(2(4(3(0(0(x1)))))))))))) 2(1(2(2(3(1(0(4(1(4(x1)))))))))) (57)
5(1(3(3(3(4(0(5(3(4(5(5(x1)))))))))))) 3(0(1(4(4(5(2(5(5(5(x1)))))))))) (58)
5(3(0(3(5(3(0(1(5(3(4(0(x1)))))))))))) 4(3(2(1(0(3(2(4(0(0(x1)))))))))) (59)
5(5(4(4(0(4(4(3(4(4(5(5(x1)))))))))))) 5(0(4(4(4(4(1(5(1(4(x1)))))))))) (60)
1(0(2(0(1(1(0(4(5(3(0(3(5(x1))))))))))))) 3(0(2(0(1(5(4(4(2(4(x1)))))))))) (61)
3(0(4(0(4(0(4(3(5(0(0(1(3(x1))))))))))))) 3(0(4(0(4(0(4(5(3(0(0(1(3(x1))))))))))))) (62)
4(3(4(1(0(3(5(3(2(2(0(2(3(x1))))))))))))) 4(3(4(1(0(3(5(2(3(2(0(2(3(x1))))))))))))) (63)
4(4(3(4(3(1(4(5(4(2(2(0(5(4(x1)))))))))))))) 4(4(3(4(3(1(4(5(2(2(0(4(5(4(x1)))))))))))))) (64)
5(0(0(1(0(3(4(3(5(5(5(3(0(0(x1)))))))))))))) 5(2(4(2(1(1(3(1(1(2(x1)))))))))) (65)
2(0(5(1(1(0(0(2(4(0(0(1(4(1(0(x1))))))))))))))) 2(0(5(1(1(0(0(2(4(0(1(0(4(1(0(x1))))))))))))))) (66)
4(0(5(4(2(5(4(0(2(5(2(2(1(3(2(x1))))))))))))))) 4(2(4(4(5(5(2(3(5(2(2(0(1(0(2(x1))))))))))))))) (67)
1(1(4(4(4(2(3(1(4(4(0(5(0(4(4(4(x1)))))))))))))))) 1(1(4(4(4(2(3(1(4(0(4(5(0(4(4(4(x1)))))))))))))))) (68)
5(3(5(0(2(0(1(4(5(0(1(0(4(3(5(4(x1)))))))))))))))) 5(3(0(5(2(0(1(4(5(0(1(0(4(3(5(4(x1)))))))))))))))) (69)
2(5(3(4(0(3(4(5(5(4(5(3(4(4(5(3(4(x1))))))))))))))))) 2(5(3(4(0(3(4(5(5(5(4(3(4(4(5(3(4(x1))))))))))))))))) (70)
4(3(2(0(5(1(4(1(4(0(0(0(3(1(5(5(5(x1))))))))))))))))) 4(3(2(0(1(5(4(1(4(0(0(0(3(1(5(5(5(x1))))))))))))))))) (71)
4(5(2(5(4(0(0(5(5(3(2(3(5(0(5(4(4(x1))))))))))))))))) 4(5(5(2(4(0(0(5(5(3(3(2(0(5(4(5(4(x1))))))))))))))))) (72)
5(5(2(4(2(2(1(4(2(1(4(1(2(2(4(0(4(x1))))))))))))))))) 5(2(2(2(4(4(5(2(1(1(4(0(1(2(2(4(4(x1))))))))))))))))) (73)
4(4(0(2(5(1(2(1(0(3(4(4(4(5(4(2(0(0(4(x1))))))))))))))))))) 4(4(0(2(5(1(2(1(0(3(4(4(4(5(2(4(0(0(4(x1))))))))))))))))))) (74)
0(3(4(5(3(2(0(0(4(4(4(5(0(1(3(3(4(3(1(1(x1)))))))))))))))))))) 0(3(4(3(5(2(0(0(4(4(4(5(0(1(3(3(4(3(1(1(x1)))))))))))))))))))) (75)
1(4(0(4(1(5(4(3(4(3(0(5(5(1(1(1(2(2(1(4(x1)))))))))))))))))))) 1(4(4(0(1(4(3(5(4(3(0(5(1(5(1(2(1(2(1(4(x1)))))))))))))))))))) (76)
3(5(1(3(1(4(4(2(3(1(0(4(3(2(0(4(3(2(5(1(x1)))))))))))))))))))) 3(5(1(3(1(4(4(2(3(1(0(4(3(0(2(4(3(2(5(1(x1)))))))))))))))))))) (77)
3(0(3(2(3(2(3(2(4(3(3(0(1(2(4(2(3(5(2(3(3(x1))))))))))))))))))))) 3(0(3(2(3(2(3(4(2(3(3(0(1(2(4(2(3(5(2(3(3(x1))))))))))))))))))))) (78)
1(3(0(2(0(4(1(2(2(2(5(5(4(3(5(2(4(1(2(3(3(3(x1)))))))))))))))))))))) 1(3(0(2(0(4(1(2(2(2(5(5(3(4(5(2(4(1(2(3(3(3(x1)))))))))))))))))))))) (79)
3(4(2(2(2(3(4(0(3(0(1(1(4(1(3(0(0(0(1(1(5(3(x1)))))))))))))))))))))) 3(4(0(2(2(4(3(2(1(0(4(5(1(3(1(1(0(0(3(1(0(3(x1)))))))))))))))))))))) (80)
4(1(5(4(5(4(5(3(1(3(1(1(1(3(4(1(0(1(3(2(0(3(x1)))))))))))))))))))))) 4(1(5(5(4(4(5(3(1(3(1(1(1(3(4(1(0(1(3(2(0(3(x1)))))))))))))))))))))) (81)
0(5(4(5(3(0(2(1(5(2(0(0(4(5(4(1(0(1(2(0(1(5(0(x1))))))))))))))))))))))) 0(5(4(3(5(2(0(5(5(1(2(4(0(0(0(0(1(1(2(4(1(5(0(x1))))))))))))))))))))))) (82)
0(3(1(4(5(3(0(4(5(4(2(5(3(4(3(0(5(5(0(2(3(3(0(3(x1)))))))))))))))))))))))) 0(3(1(4(5(3(4(0(5(4(2(5(3(4(3(0(5(5(0(2(3(3(0(3(x1)))))))))))))))))))))))) (83)
1(4(3(0(1(0(2(2(5(5(1(0(1(2(2(5(4(3(2(5(2(2(0(2(0(x1))))))))))))))))))))))))) 1(4(3(0(1(0(2(2(5(5(1(0(1(2(5(2(4(3(2(5(2(2(0(2(0(x1))))))))))))))))))))))))) (84)
2(3(3(4(0(0(2(0(4(5(4(3(5(4(3(4(3(1(5(3(2(3(5(3(3(x1))))))))))))))))))))))))) 2(3(3(4(0(0(2(4(0(5(4(3(5(4(3(4(3(1(5(3(2(3(5(3(3(x1))))))))))))))))))))))))) (85)
3(2(0(5(2(1(2(1(1(0(5(0(3(4(2(2(3(0(2(4(3(4(0(0(1(x1))))))))))))))))))))))))) 3(2(0(5(2(1(2(1(1(0(5(0(3(4(2(2(3(0(4(2(3(4(0(0(1(x1))))))))))))))))))))))))) (86)
3(4(1(5(5(1(1(2(0(5(3(3(0(5(2(5(0(2(0(3(5(5(0(5(3(x1))))))))))))))))))))))))) 3(4(1(5(5(1(1(2(0(5(3(3(0(5(5(2(0(2(0(3(5(5(0(5(3(x1))))))))))))))))))))))))) (87)
4(5(2(0(0(1(2(1(5(1(2(2(5(0(2(0(2(0(0(2(2(5(3(1(1(x1))))))))))))))))))))))))) 4(5(2(0(0(1(2(1(5(1(2(2(0(5(2(0(2(0(0(2(2(5(3(1(1(x1))))))))))))))))))))))))) (88)
1(3(2(3(3(1(0(5(5(3(4(0(4(2(4(3(2(5(2(2(1(5(2(4(1(5(x1)))))))))))))))))))))))))) 1(3(3(2(1(0(3(5(5(4(3(4(0(2(4(3(2(5(2(2(5(1(2(4(1(5(x1)))))))))))))))))))))))))) (89)
0(4(4(2(1(1(2(1(1(0(4(3(4(1(1(0(0(3(4(3(5(4(4(2(0(3(5(x1))))))))))))))))))))))))))) 0(4(4(2(1(1(2(1(1(0(4(3(4(1(1(0(0(3(4(3(5(4(4(0(2(3(5(x1))))))))))))))))))))))))))) (90)
5(2(0(4(2(2(5(5(4(5(5(2(4(3(5(5(3(3(2(5(0(4(3(0(1(2(2(x1))))))))))))))))))))))))))) 5(2(0(4(2(2(5(5(4(5(5(2(4(3(5(5(3(3(2(5(0(3(4(0(1(2(2(x1))))))))))))))))))))))))))) (91)
5(3(5(4(5(5(4(1(5(5(3(4(1(2(1(1(2(1(0(1(3(0(4(2(5(1(2(x1))))))))))))))))))))))))))) 5(5(5(4(3(5(5(1(4(4(5(3(1(1(1(0(1(2(2(1(0(4(3(5(2(1(2(x1))))))))))))))))))))))))))) (92)
2(0(1(5(2(1(5(0(5(2(4(2(0(0(5(2(0(1(0(2(2(5(0(3(5(4(2(5(2(x1))))))))))))))))))))))))))))) 2(0(1(5(2(1(5(0(5(2(4(2(0(0(5(2(0(1(0(2(2(5(3(0(5(4(2(5(2(x1))))))))))))))))))))))))))))) (93)
5(1(4(3(1(0(4(3(5(5(4(3(3(3(3(2(3(5(0(1(5(0(3(3(5(1(2(3(0(x1))))))))))))))))))))))))))))) 5(1(4(3(1(0(4(3(5(5(4(3(3(3(3(2(3(5(0(5(1(0(3(3(5(1(2(3(0(x1))))))))))))))))))))))))))))) (94)
0(3(1(0(1(5(1(0(2(3(2(3(3(0(1(3(5(2(4(5(2(0(2(4(1(4(4(5(1(4(x1)))))))))))))))))))))))))))))) 0(3(1(0(1(5(1(0(2(3(2(3(3(0(1(3(5(2(4(5(2(0(4(2(1(4(4(5(1(4(x1)))))))))))))))))))))))))))))) (95)

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 +
7
[4(x1)] = x1 +
7
[3(x1)] = x1 +
7
[2(x1)] = x1 +
8
[1(x1)] = x1 +
8
[0(x1)] = x1 +
7
all of the following rules can be deleted.
4(5(5(1(2(1(3(2(3(2(x1)))))))))) 5(5(0(0(5(0(3(3(2(1(x1)))))))))) (9)
5(0(1(4(1(2(5(1(5(2(x1)))))))))) 4(5(5(0(2(0(5(4(5(4(x1)))))))))) (10)
0(3(0(2(4(1(2(2(4(0(5(x1))))))))))) 2(4(4(3(0(2(5(1(4(4(x1)))))))))) (12)
0(5(0(3(0(1(2(4(0(4(3(x1))))))))))) 5(0(4(5(1(0(1(4(2(3(x1)))))))))) (13)
0(5(2(4(1(5(0(2(1(5(1(x1))))))))))) 5(0(1(1(1(4(2(0(3(0(x1)))))))))) (14)
0(5(3(5(0(5(3(3(2(1(3(x1))))))))))) 0(3(4(0(5(1(4(4(5(2(x1)))))))))) (15)
0(5(4(4(1(0(5(1(4(5(5(x1))))))))))) 0(1(3(4(4(1(4(3(5(0(x1)))))))))) (16)
1(3(2(2(5(5(1(5(1(2(1(x1))))))))))) 4(4(0(0(1(2(4(1(5(4(x1)))))))))) (17)
1(4(4(5(2(2(1(0(0(0(2(x1))))))))))) 3(5(1(1(4(2(4(0(2(0(x1)))))))))) (18)
1(5(0(4(0(1(2(5(5(5(3(x1))))))))))) 4(3(2(3(2(4(1(0(3(3(x1)))))))))) (19)
1(5(1(2(2(1(1(0(5(5(0(x1))))))))))) 0(3(0(1(1(2(2(0(1(4(x1)))))))))) (20)
2(1(5(3(3(1(4(1(0(4(0(x1))))))))))) 4(4(3(5(0(0(5(3(2(4(x1)))))))))) (22)
2(2(4(1(3(2(4(0(2(5(4(x1))))))))))) 0(4(2(3(1(2(3(0(1(4(x1)))))))))) (23)
2(3(2(4(0(0(3(4(3(2(0(x1))))))))))) 5(2(1(5(0(5(1(1(5(3(x1)))))))))) (24)
2(3(4(1(2(3(2(2(0(5(0(x1))))))))))) 2(2(3(0(5(1(4(1(1(2(x1)))))))))) (25)
2(5(0(1(0(5(3(1(2(1(0(x1))))))))))) 4(1(3(0(5(0(0(0(2(3(x1)))))))))) (26)
3(0(0(4(5(2(5(3(5(4(1(x1))))))))))) 3(4(2(2(3(3(3(4(5(0(x1)))))))))) (27)
3(0(3(5(4(0(4(3(5(2(5(x1))))))))))) 2(4(0(3(4(1(2(0(1(5(x1)))))))))) (28)
3(1(4(0(5(0(4(1(1(0(5(x1))))))))))) 0(4(4(5(0(0(0(3(1(0(x1)))))))))) (29)
3(2(0(3(0(3(1(3(5(5(5(x1))))))))))) 0(3(2(4(1(5(1(0(5(1(x1)))))))))) (30)
3(2(3(0(0(0(2(4(2(1(1(x1))))))))))) 2(4(0(0(4(3(0(2(5(3(x1)))))))))) (32)
3(5(0(5(4(0(0(4(4(2(4(x1))))))))))) 2(0(3(5(5(0(2(1(4(3(x1)))))))))) (33)
3(5(4(2(3(0(4(0(5(1(2(x1))))))))))) 1(3(4(2(3(5(5(1(1(2(x1)))))))))) (34)
4(1(3(0(4(5(4(2(4(2(5(x1))))))))))) 0(5(2(1(0(2(3(3(5(3(x1)))))))))) (35)
4(2(2(0(4(0(1(4(5(3(3(x1))))))))))) 1(0(0(0(0(5(1(0(5(3(x1)))))))))) (36)
4(3(3(4(3(1(0(2(5(4(1(x1))))))))))) 0(5(5(0(5(1(4(2(1(5(x1)))))))))) (37)
4(3(4(1(1(5(5(5(3(4(2(x1))))))))))) 4(5(5(0(1(2(3(1(3(5(x1)))))))))) (38)
4(3(4(4(2(4(4(2(0(4(4(x1))))))))))) 5(3(1(2(0(4(3(5(3(2(x1)))))))))) (39)
4(4(3(2(1(5(3(4(0(0(4(x1))))))))))) 5(0(2(4(0(1(4(1(4(4(x1)))))))))) (40)
4(4(5(4(1(4(1(1(5(3(5(x1))))))))))) 2(5(5(2(3(2(5(4(3(0(x1)))))))))) (41)
4(5(1(5(4(4(3(5(1(2(1(x1))))))))))) 4(3(0(4(3(0(3(3(5(4(x1)))))))))) (42)
5(1(4(3(3(2(0(5(1(4(2(x1))))))))))) 4(3(5(1(5(5(5(1(2(2(x1)))))))))) (43)
5(3(4(4(0(0(1(1(5(1(4(x1))))))))))) 3(2(4(2(0(0(4(1(3(0(x1)))))))))) (44)
5(3(5(2(3(2(2(1(0(3(5(x1))))))))))) 0(3(2(1(0(0(0(0(2(4(x1)))))))))) (45)
5(4(0(1(2(1(1(4(5(2(0(x1))))))))))) 5(4(5(4(3(4(1(1(3(1(x1)))))))))) (46)
5(5(2(4(3(5(3(4(0(3(4(x1))))))))))) 2(0(0(3(2(4(0(1(3(5(x1)))))))))) (47)
5(5(5(0(0(3(5(4(4(2(1(x1))))))))))) 1(0(3(2(1(5(5(1(0(0(x1)))))))))) (48)
0(1(3(3(5(0(3(1(5(4(3(5(x1)))))))))))) 5(3(3(2(1(0(1(1(3(1(x1)))))))))) (50)
1(1(0(5(1(1(3(1(4(2(4(0(x1)))))))))))) 4(5(1(5(2(1(3(2(0(4(x1)))))))))) (51)
1(2(4(1(4(4(2(1(5(4(0(0(x1)))))))))))) 1(1(3(4(5(1(2(4(3(2(x1)))))))))) (52)
1(3(0(0(4(5(3(4(3(0(0(1(x1)))))))))))) 3(5(4(5(0(3(4(4(2(5(x1)))))))))) (53)
2(2(3(4(2(0(5(4(3(0(0(0(x1)))))))))))) 0(1(2(2(3(2(4(0(1(3(x1)))))))))) (54)
2(2(4(2(3(0(5(3(0(5(5(3(x1)))))))))))) 0(4(4(3(5(4(4(1(5(1(x1)))))))))) (55)
3(0(3(1(5(0(3(1(1(0(2(3(x1)))))))))))) 5(1(4(4(5(4(1(1(1(1(x1)))))))))) (56)
3(1(0(4(0(2(5(2(4(3(0(0(x1)))))))))))) 2(1(2(2(3(1(0(4(1(4(x1)))))))))) (57)
5(1(3(3(3(4(0(5(3(4(5(5(x1)))))))))))) 3(0(1(4(4(5(2(5(5(5(x1)))))))))) (58)
5(3(0(3(5(3(0(1(5(3(4(0(x1)))))))))))) 4(3(2(1(0(3(2(4(0(0(x1)))))))))) (59)
5(5(4(4(0(4(4(3(4(4(5(5(x1)))))))))))) 5(0(4(4(4(4(1(5(1(4(x1)))))))))) (60)
1(0(2(0(1(1(0(4(5(3(0(3(5(x1))))))))))))) 3(0(2(0(1(5(4(4(2(4(x1)))))))))) (61)
5(0(0(1(0(3(4(3(5(5(5(3(0(0(x1)))))))))))))) 5(2(4(2(1(1(3(1(1(2(x1)))))))))) (65)

1.1 Closure Under Flat Contexts

Using the flat contexts

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

We obtain the transformed TRS

There are 270 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 1620 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 +
2
[53(x1)] = x1 +
0
[54(x1)] = x1 +
5
[55(x1)] = x1 +
2
[40(x1)] = x1 +
1
[41(x1)] = x1 +
2
[42(x1)] = x1 +
1
[43(x1)] = x1 +
0
[44(x1)] = x1 +
42
[45(x1)] = x1 +
0
[30(x1)] = x1 +
5
[31(x1)] = x1 +
0
[32(x1)] = x1 +
0
[33(x1)] = x1 +
24
[34(x1)] = x1 +
23
[35(x1)] = x1 +
2
[20(x1)] = x1 +
2
[21(x1)] = x1 +
0
[22(x1)] = x1 +
0
[23(x1)] = x1 +
23
[24(x1)] = x1 +
23
[25(x1)] = x1 +
1
[10(x1)] = x1 +
23
[11(x1)] = x1 +
24
[12(x1)] = x1 +
0
[13(x1)] = x1 +
5
[14(x1)] = x1 +
0
[15(x1)] = x1 +
0
[00(x1)] = x1 +
0
[01(x1)] = x1 +
0
[02(x1)] = x1 +
1
[03(x1)] = x1 +
0
[04(x1)] = x1 +
0
[05(x1)] = x1 +
0
all of the following rules can be deleted.

There are 1620 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.