Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/96464)

The rewrite relation of the following TRS is considered.

0(1(1(x1))) 2(1(1(x1))) (1)
3(4(0(x1))) 4(5(5(x1))) (2)
2(2(1(4(0(x1))))) 2(2(1(3(5(x1))))) (3)
4(5(3(1(0(0(4(x1))))))) 4(5(0(1(4(5(4(x1))))))) (4)
1(3(3(2(1(4(4(0(x1)))))))) 1(1(3(0(4(4(0(2(x1)))))))) (5)
2(2(1(5(0(2(2(2(4(x1))))))))) 2(0(0(4(5(5(0(x1))))))) (6)
0(2(3(0(2(0(3(5(2(4(x1)))))))))) 2(0(4(5(4(4(5(4(0(x1))))))))) (7)
3(2(3(1(3(5(5(2(2(0(x1)))))))))) 4(0(0(0(0(2(2(2(0(x1))))))))) (8)
3(4(1(1(0(0(5(0(5(0(x1)))))))))) 3(0(1(0(1(1(3(0(4(5(x1)))))))))) (9)
4(1(3(0(0(5(1(5(0(1(x1)))))))))) 4(0(4(2(2(3(0(3(0(1(x1)))))))))) (10)
3(3(0(2(0(1(4(5(5(5(0(x1))))))))))) 1(2(3(5(4(4(0(2(3(5(5(x1))))))))))) (11)
3(0(2(0(2(1(2(5(4(3(5(4(x1)))))))))))) 3(2(2(0(3(0(5(4(5(1(5(4(x1)))))))))))) (12)
3(0(5(4(5(3(0(0(0(5(1(1(x1)))))))))))) 3(2(4(1(4(0(3(0(2(1(1(3(x1)))))))))))) (13)
3(3(0(4(4(1(1(1(3(1(2(1(x1)))))))))))) 3(3(5(3(0(1(4(4(0(0(1(x1))))))))))) (14)
3(2(0(2(1(1(4(1(3(1(4(5(2(4(x1)))))))))))))) 0(2(2(4(4(5(4(1(0(0(5(0(0(x1))))))))))))) (15)
4(1(3(3(0(5(2(3(3(3(0(5(4(3(x1)))))))))))))) 4(4(2(2(0(3(1(1(5(1(3(0(5(3(x1)))))))))))))) (16)
3(4(0(2(1(1(0(4(1(1(0(2(3(5(3(x1))))))))))))))) 2(4(5(0(0(3(0(3(0(4(4(1(1(3(x1)))))))))))))) (17)
5(5(1(4(1(3(4(2(3(4(3(1(2(0(5(x1))))))))))))))) 1(3(5(1(1(2(1(2(0(3(0(1(1(2(0(x1))))))))))))))) (18)
0(5(0(3(3(3(2(3(1(1(1(2(5(1(2(5(x1)))))))))))))))) 0(5(1(4(3(3(4(4(1(1(4(1(0(1(4(x1))))))))))))))) (19)
2(4(0(5(0(0(4(4(2(2(3(5(3(4(5(0(x1)))))))))))))))) 2(3(1(3(0(1(1(1(0(5(5(0(2(3(4(0(x1)))))))))))))))) (20)
3(1(3(5(4(1(0(4(1(2(4(2(3(3(1(3(1(3(x1)))))))))))))))))) 0(3(3(0(0(3(3(4(4(1(4(1(5(2(2(4(3(x1))))))))))))))))) (21)
2(5(0(5(2(1(3(1(4(5(5(0(0(2(3(0(3(2(4(3(x1)))))))))))))))))))) 0(2(0(0(3(1(4(2(5(2(0(5(4(4(1(1(0(4(3(x1))))))))))))))))))) (22)
2(5(2(1(1(0(4(3(3(0(4(5(4(3(0(4(1(0(4(0(x1)))))))))))))))))))) 5(1(1(0(1(3(5(5(0(5(3(4(5(1(0(0(2(3(2(0(x1)))))))))))))))))))) (23)
1(3(0(5(3(1(1(1(0(3(3(0(1(3(3(4(3(5(3(0(5(x1))))))))))))))))))))) 1(5(3(3(1(3(3(4(0(5(5(0(3(2(4(1(2(3(3(4(5(x1))))))))))))))))))))) (24)
3(3(0(5(3(5(1(2(5(1(2(1(2(1(4(0(1(2(3(1(5(x1))))))))))))))))))))) 0(2(0(5(5(4(5(2(5(0(0(4(2(0(5(5(5(3(1(5(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.
2(2(1(5(0(2(2(2(4(x1))))))))) 2(0(0(4(5(5(0(x1))))))) (6)
0(2(3(0(2(0(3(5(2(4(x1)))))))))) 2(0(4(5(4(4(5(4(0(x1))))))))) (7)
3(2(3(1(3(5(5(2(2(0(x1)))))))))) 4(0(0(0(0(2(2(2(0(x1))))))))) (8)
3(3(0(4(4(1(1(1(3(1(2(1(x1)))))))))))) 3(3(5(3(0(1(4(4(0(0(1(x1))))))))))) (14)
3(2(0(2(1(1(4(1(3(1(4(5(2(4(x1)))))))))))))) 0(2(2(4(4(5(4(1(0(0(5(0(0(x1))))))))))))) (15)
3(4(0(2(1(1(0(4(1(1(0(2(3(5(3(x1))))))))))))))) 2(4(5(0(0(3(0(3(0(4(4(1(1(3(x1)))))))))))))) (17)
0(5(0(3(3(3(2(3(1(1(1(2(5(1(2(5(x1)))))))))))))))) 0(5(1(4(3(3(4(4(1(1(4(1(0(1(4(x1))))))))))))))) (19)
3(1(3(5(4(1(0(4(1(2(4(2(3(3(1(3(1(3(x1)))))))))))))))))) 0(3(3(0(0(3(3(4(4(1(4(1(5(2(2(4(3(x1))))))))))))))))) (21)
2(5(0(5(2(1(3(1(4(5(5(0(0(2(3(0(3(2(4(3(x1)))))))))))))))))))) 0(2(0(0(3(1(4(2(5(2(0(5(4(4(1(1(0(4(3(x1))))))))))))))))))) (22)
3(3(0(5(3(5(1(2(5(1(2(1(2(1(4(0(1(2(3(1(5(x1))))))))))))))))))))) 0(2(0(5(5(4(5(2(5(0(0(4(2(0(5(5(5(3(1(5(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(1(1(x1)))) 5(2(1(1(x1)))) (26)
5(3(4(0(x1)))) 5(4(5(5(x1)))) (27)
5(2(2(1(4(0(x1)))))) 5(2(2(1(3(5(x1)))))) (28)
5(4(5(3(1(0(0(4(x1)))))))) 5(4(5(0(1(4(5(4(x1)))))))) (29)
5(1(3(3(2(1(4(4(0(x1))))))))) 5(1(1(3(0(4(4(0(2(x1))))))))) (30)
5(3(4(1(1(0(0(5(0(5(0(x1))))))))))) 5(3(0(1(0(1(1(3(0(4(5(x1))))))))))) (31)
5(4(1(3(0(0(5(1(5(0(1(x1))))))))))) 5(4(0(4(2(2(3(0(3(0(1(x1))))))))))) (32)
5(3(3(0(2(0(1(4(5(5(5(0(x1)))))))))))) 5(1(2(3(5(4(4(0(2(3(5(5(x1)))))))))))) (33)
5(3(0(2(0(2(1(2(5(4(3(5(4(x1))))))))))))) 5(3(2(2(0(3(0(5(4(5(1(5(4(x1))))))))))))) (34)
5(3(0(5(4(5(3(0(0(0(5(1(1(x1))))))))))))) 5(3(2(4(1(4(0(3(0(2(1(1(3(x1))))))))))))) (35)
5(4(1(3(3(0(5(2(3(3(3(0(5(4(3(x1))))))))))))))) 5(4(4(2(2(0(3(1(1(5(1(3(0(5(3(x1))))))))))))))) (36)
5(5(5(1(4(1(3(4(2(3(4(3(1(2(0(5(x1)))))))))))))))) 5(1(3(5(1(1(2(1(2(0(3(0(1(1(2(0(x1)))))))))))))))) (37)
5(2(4(0(5(0(0(4(4(2(2(3(5(3(4(5(0(x1))))))))))))))))) 5(2(3(1(3(0(1(1(1(0(5(5(0(2(3(4(0(x1))))))))))))))))) (38)
5(2(5(2(1(1(0(4(3(3(0(4(5(4(3(0(4(1(0(4(0(x1))))))))))))))))))))) 5(5(1(1(0(1(3(5(5(0(5(3(4(5(1(0(0(2(3(2(0(x1))))))))))))))))))))) (39)
5(1(3(0(5(3(1(1(1(0(3(3(0(1(3(3(4(3(5(3(0(5(x1)))))))))))))))))))))) 5(1(5(3(3(1(3(3(4(0(5(5(0(3(2(4(1(2(3(3(4(5(x1)))))))))))))))))))))) (40)
4(0(1(1(x1)))) 4(2(1(1(x1)))) (41)
4(3(4(0(x1)))) 4(4(5(5(x1)))) (42)
4(2(2(1(4(0(x1)))))) 4(2(2(1(3(5(x1)))))) (43)
4(4(5(3(1(0(0(4(x1)))))))) 4(4(5(0(1(4(5(4(x1)))))))) (44)
4(1(3(3(2(1(4(4(0(x1))))))))) 4(1(1(3(0(4(4(0(2(x1))))))))) (45)
4(3(4(1(1(0(0(5(0(5(0(x1))))))))))) 4(3(0(1(0(1(1(3(0(4(5(x1))))))))))) (46)
4(4(1(3(0(0(5(1(5(0(1(x1))))))))))) 4(4(0(4(2(2(3(0(3(0(1(x1))))))))))) (47)
4(3(3(0(2(0(1(4(5(5(5(0(x1)))))))))))) 4(1(2(3(5(4(4(0(2(3(5(5(x1)))))))))))) (48)
4(3(0(2(0(2(1(2(5(4(3(5(4(x1))))))))))))) 4(3(2(2(0(3(0(5(4(5(1(5(4(x1))))))))))))) (49)
4(3(0(5(4(5(3(0(0(0(5(1(1(x1))))))))))))) 4(3(2(4(1(4(0(3(0(2(1(1(3(x1))))))))))))) (50)
4(4(1(3(3(0(5(2(3(3(3(0(5(4(3(x1))))))))))))))) 4(4(4(2(2(0(3(1(1(5(1(3(0(5(3(x1))))))))))))))) (51)
4(5(5(1(4(1(3(4(2(3(4(3(1(2(0(5(x1)))))))))))))))) 4(1(3(5(1(1(2(1(2(0(3(0(1(1(2(0(x1)))))))))))))))) (52)
4(2(4(0(5(0(0(4(4(2(2(3(5(3(4(5(0(x1))))))))))))))))) 4(2(3(1(3(0(1(1(1(0(5(5(0(2(3(4(0(x1))))))))))))))))) (53)
4(2(5(2(1(1(0(4(3(3(0(4(5(4(3(0(4(1(0(4(0(x1))))))))))))))))))))) 4(5(1(1(0(1(3(5(5(0(5(3(4(5(1(0(0(2(3(2(0(x1))))))))))))))))))))) (54)
4(1(3(0(5(3(1(1(1(0(3(3(0(1(3(3(4(3(5(3(0(5(x1)))))))))))))))))))))) 4(1(5(3(3(1(3(3(4(0(5(5(0(3(2(4(1(2(3(3(4(5(x1)))))))))))))))))))))) (55)
3(0(1(1(x1)))) 3(2(1(1(x1)))) (56)
3(3(4(0(x1)))) 3(4(5(5(x1)))) (57)
3(2(2(1(4(0(x1)))))) 3(2(2(1(3(5(x1)))))) (58)
3(4(5(3(1(0(0(4(x1)))))))) 3(4(5(0(1(4(5(4(x1)))))))) (59)
3(1(3(3(2(1(4(4(0(x1))))))))) 3(1(1(3(0(4(4(0(2(x1))))))))) (60)
3(3(4(1(1(0(0(5(0(5(0(x1))))))))))) 3(3(0(1(0(1(1(3(0(4(5(x1))))))))))) (61)
3(4(1(3(0(0(5(1(5(0(1(x1))))))))))) 3(4(0(4(2(2(3(0(3(0(1(x1))))))))))) (62)
3(3(3(0(2(0(1(4(5(5(5(0(x1)))))))))))) 3(1(2(3(5(4(4(0(2(3(5(5(x1)))))))))))) (63)
3(3(0(2(0(2(1(2(5(4(3(5(4(x1))))))))))))) 3(3(2(2(0(3(0(5(4(5(1(5(4(x1))))))))))))) (64)
3(3(0(5(4(5(3(0(0(0(5(1(1(x1))))))))))))) 3(3(2(4(1(4(0(3(0(2(1(1(3(x1))))))))))))) (65)
3(4(1(3(3(0(5(2(3(3(3(0(5(4(3(x1))))))))))))))) 3(4(4(2(2(0(3(1(1(5(1(3(0(5(3(x1))))))))))))))) (66)
3(5(5(1(4(1(3(4(2(3(4(3(1(2(0(5(x1)))))))))))))))) 3(1(3(5(1(1(2(1(2(0(3(0(1(1(2(0(x1)))))))))))))))) (67)
3(2(4(0(5(0(0(4(4(2(2(3(5(3(4(5(0(x1))))))))))))))))) 3(2(3(1(3(0(1(1(1(0(5(5(0(2(3(4(0(x1))))))))))))))))) (68)
3(2(5(2(1(1(0(4(3(3(0(4(5(4(3(0(4(1(0(4(0(x1))))))))))))))))))))) 3(5(1(1(0(1(3(5(5(0(5(3(4(5(1(0(0(2(3(2(0(x1))))))))))))))))))))) (69)
3(1(3(0(5(3(1(1(1(0(3(3(0(1(3(3(4(3(5(3(0(5(x1)))))))))))))))))))))) 3(1(5(3(3(1(3(3(4(0(5(5(0(3(2(4(1(2(3(3(4(5(x1)))))))))))))))))))))) (70)
2(0(1(1(x1)))) 2(2(1(1(x1)))) (71)
2(3(4(0(x1)))) 2(4(5(5(x1)))) (72)
2(2(2(1(4(0(x1)))))) 2(2(2(1(3(5(x1)))))) (73)
2(4(5(3(1(0(0(4(x1)))))))) 2(4(5(0(1(4(5(4(x1)))))))) (74)
2(1(3(3(2(1(4(4(0(x1))))))))) 2(1(1(3(0(4(4(0(2(x1))))))))) (75)
2(3(4(1(1(0(0(5(0(5(0(x1))))))))))) 2(3(0(1(0(1(1(3(0(4(5(x1))))))))))) (76)
2(4(1(3(0(0(5(1(5(0(1(x1))))))))))) 2(4(0(4(2(2(3(0(3(0(1(x1))))))))))) (77)
2(3(3(0(2(0(1(4(5(5(5(0(x1)))))))))))) 2(1(2(3(5(4(4(0(2(3(5(5(x1)))))))))))) (78)
2(3(0(2(0(2(1(2(5(4(3(5(4(x1))))))))))))) 2(3(2(2(0(3(0(5(4(5(1(5(4(x1))))))))))))) (79)
2(3(0(5(4(5(3(0(0(0(5(1(1(x1))))))))))))) 2(3(2(4(1(4(0(3(0(2(1(1(3(x1))))))))))))) (80)
2(4(1(3(3(0(5(2(3(3(3(0(5(4(3(x1))))))))))))))) 2(4(4(2(2(0(3(1(1(5(1(3(0(5(3(x1))))))))))))))) (81)
2(5(5(1(4(1(3(4(2(3(4(3(1(2(0(5(x1)))))))))))))))) 2(1(3(5(1(1(2(1(2(0(3(0(1(1(2(0(x1)))))))))))))))) (82)
2(2(4(0(5(0(0(4(4(2(2(3(5(3(4(5(0(x1))))))))))))))))) 2(2(3(1(3(0(1(1(1(0(5(5(0(2(3(4(0(x1))))))))))))))))) (83)
2(2(5(2(1(1(0(4(3(3(0(4(5(4(3(0(4(1(0(4(0(x1))))))))))))))))))))) 2(5(1(1(0(1(3(5(5(0(5(3(4(5(1(0(0(2(3(2(0(x1))))))))))))))))))))) (84)
2(1(3(0(5(3(1(1(1(0(3(3(0(1(3(3(4(3(5(3(0(5(x1)))))))))))))))))))))) 2(1(5(3(3(1(3(3(4(0(5(5(0(3(2(4(1(2(3(3(4(5(x1)))))))))))))))))))))) (85)
1(0(1(1(x1)))) 1(2(1(1(x1)))) (86)
1(3(4(0(x1)))) 1(4(5(5(x1)))) (87)
1(2(2(1(4(0(x1)))))) 1(2(2(1(3(5(x1)))))) (88)
1(4(5(3(1(0(0(4(x1)))))))) 1(4(5(0(1(4(5(4(x1)))))))) (89)
1(1(3(3(2(1(4(4(0(x1))))))))) 1(1(1(3(0(4(4(0(2(x1))))))))) (90)
1(3(4(1(1(0(0(5(0(5(0(x1))))))))))) 1(3(0(1(0(1(1(3(0(4(5(x1))))))))))) (91)
1(4(1(3(0(0(5(1(5(0(1(x1))))))))))) 1(4(0(4(2(2(3(0(3(0(1(x1))))))))))) (92)
1(3(3(0(2(0(1(4(5(5(5(0(x1)))))))))))) 1(1(2(3(5(4(4(0(2(3(5(5(x1)))))))))))) (93)
1(3(0(2(0(2(1(2(5(4(3(5(4(x1))))))))))))) 1(3(2(2(0(3(0(5(4(5(1(5(4(x1))))))))))))) (94)
1(3(0(5(4(5(3(0(0(0(5(1(1(x1))))))))))))) 1(3(2(4(1(4(0(3(0(2(1(1(3(x1))))))))))))) (95)
1(4(1(3(3(0(5(2(3(3(3(0(5(4(3(x1))))))))))))))) 1(4(4(2(2(0(3(1(1(5(1(3(0(5(3(x1))))))))))))))) (96)
1(5(5(1(4(1(3(4(2(3(4(3(1(2(0(5(x1)))))))))))))))) 1(1(3(5(1(1(2(1(2(0(3(0(1(1(2(0(x1)))))))))))))))) (97)
1(2(4(0(5(0(0(4(4(2(2(3(5(3(4(5(0(x1))))))))))))))))) 1(2(3(1(3(0(1(1(1(0(5(5(0(2(3(4(0(x1))))))))))))))))) (98)
1(2(5(2(1(1(0(4(3(3(0(4(5(4(3(0(4(1(0(4(0(x1))))))))))))))))))))) 1(5(1(1(0(1(3(5(5(0(5(3(4(5(1(0(0(2(3(2(0(x1))))))))))))))))))))) (99)
1(1(3(0(5(3(1(1(1(0(3(3(0(1(3(3(4(3(5(3(0(5(x1)))))))))))))))))))))) 1(1(5(3(3(1(3(3(4(0(5(5(0(3(2(4(1(2(3(3(4(5(x1)))))))))))))))))))))) (100)
0(0(1(1(x1)))) 0(2(1(1(x1)))) (101)
0(3(4(0(x1)))) 0(4(5(5(x1)))) (102)
0(2(2(1(4(0(x1)))))) 0(2(2(1(3(5(x1)))))) (103)
0(4(5(3(1(0(0(4(x1)))))))) 0(4(5(0(1(4(5(4(x1)))))))) (104)
0(1(3(3(2(1(4(4(0(x1))))))))) 0(1(1(3(0(4(4(0(2(x1))))))))) (105)
0(3(4(1(1(0(0(5(0(5(0(x1))))))))))) 0(3(0(1(0(1(1(3(0(4(5(x1))))))))))) (106)
0(4(1(3(0(0(5(1(5(0(1(x1))))))))))) 0(4(0(4(2(2(3(0(3(0(1(x1))))))))))) (107)
0(3(3(0(2(0(1(4(5(5(5(0(x1)))))))))))) 0(1(2(3(5(4(4(0(2(3(5(5(x1)))))))))))) (108)
0(3(0(2(0(2(1(2(5(4(3(5(4(x1))))))))))))) 0(3(2(2(0(3(0(5(4(5(1(5(4(x1))))))))))))) (109)
0(3(0(5(4(5(3(0(0(0(5(1(1(x1))))))))))))) 0(3(2(4(1(4(0(3(0(2(1(1(3(x1))))))))))))) (110)
0(4(1(3(3(0(5(2(3(3(3(0(5(4(3(x1))))))))))))))) 0(4(4(2(2(0(3(1(1(5(1(3(0(5(3(x1))))))))))))))) (111)
0(5(5(1(4(1(3(4(2(3(4(3(1(2(0(5(x1)))))))))))))))) 0(1(3(5(1(1(2(1(2(0(3(0(1(1(2(0(x1)))))))))))))))) (112)
0(2(4(0(5(0(0(4(4(2(2(3(5(3(4(5(0(x1))))))))))))))))) 0(2(3(1(3(0(1(1(1(0(5(5(0(2(3(4(0(x1))))))))))))))))) (113)
0(2(5(2(1(1(0(4(3(3(0(4(5(4(3(0(4(1(0(4(0(x1))))))))))))))))))))) 0(5(1(1(0(1(3(5(5(0(5(3(4(5(1(0(0(2(3(2(0(x1))))))))))))))))))))) (114)
0(1(3(0(5(3(1(1(1(0(3(3(0(1(3(3(4(3(5(3(0(5(x1)))))))))))))))))))))) 0(1(5(3(3(1(3(3(4(0(5(5(0(3(2(4(1(2(3(3(4(5(x1)))))))))))))))))))))) (115)

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 +
0
[51(x1)] = x1 +
0
[52(x1)] = x1 +
0
[53(x1)] = x1 +
0
[54(x1)] = x1 +
0
[55(x1)] = x1 +
2
[40(x1)] = x1 +
0
[41(x1)] = x1 +
0
[42(x1)] = x1 +
27
[43(x1)] = x1 +
0
[44(x1)] = x1 +
0
[45(x1)] = x1 +
1
[30(x1)] = x1 +
0
[31(x1)] = x1 +
2
[32(x1)] = x1 +
2
[33(x1)] = x1 +
0
[34(x1)] = x1 +
0
[35(x1)] = x1 +
0
[20(x1)] = x1 +
0
[21(x1)] = x1 +
0
[22(x1)] = x1 +
0
[23(x1)] = x1 +
0
[24(x1)] = x1 +
0
[25(x1)] = x1 +
2
[10(x1)] = x1 +
8
[11(x1)] = x1 +
1
[12(x1)] = x1 +
0
[13(x1)] = x1 +
0
[14(x1)] = x1 +
0
[15(x1)] = x1 +
2
[00(x1)] = x1 +
8
[01(x1)] = x1 +
1
[02(x1)] = x1 +
2
[03(x1)] = x1 +
1
[04(x1)] = x1 +
2
[05(x1)] = x1 +
8
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.