Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/124269)

The relative rewrite relation R/S is considered where R is the following TRS

0(0(3(0(1(4(2(4(4(3(0(1(x1)))))))))))) 3(1(5(5(4(2(1(2(5(2(2(2(1(4(5(0(1(x1))))))))))))))))) (1)
0(0(4(1(3(2(1(3(4(3(4(1(x1)))))))))))) 0(0(5(2(3(0(4(0(0(5(2(5(2(0(2(2(1(x1))))))))))))))))) (2)
0(1(2(4(3(5(0(3(2(4(3(4(x1)))))))))))) 2(2(2(2(1(2(2(1(4(2(4(2(2(5(5(0(x1)))))))))))))))) (3)
0(1(4(4(0(3(0(4(0(4(0(0(x1)))))))))))) 0(4(0(4(3(2(0(0(0(0(0(1(5(5(5(1(x1)))))))))))))))) (4)
0(1(4(4(3(5(1(4(4(5(0(0(x1)))))))))))) 5(5(5(3(4(3(0(0(3(2(2(5(2(5(5(5(4(2(x1)))))))))))))))))) (5)
0(2(0(4(4(0(3(3(5(4(4(4(x1)))))))))))) 5(4(3(5(5(2(5(1(1(5(2(4(0(2(x1)))))))))))))) (6)
0(2(1(3(5(1(4(0(3(4(3(0(x1)))))))))))) 2(0(3(5(1(5(5(5(1(5(0(1(5(4(0(5(2(2(x1)))))))))))))))))) (7)
0(2(2(2(5(2(4(4(5(4(0(0(x1)))))))))))) 5(5(5(0(1(5(5(3(2(5(0(5(2(4(x1)))))))))))))) (8)
0(2(3(5(4(3(4(0(0(4(5(2(x1)))))))))))) 0(1(2(5(2(1(5(5(3(1(2(0(0(5(3(0(x1)))))))))))))))) (9)
0(2(4(3(0(1(0(3(2(2(0(3(x1)))))))))))) 4(1(3(0(0(5(5(5(2(0(0(0(5(5(5(5(2(x1))))))))))))))))) (10)
0(2(4(4(3(5(0(0(3(3(0(3(x1)))))))))))) 0(3(0(5(0(4(5(2(5(0(0(5(2(5(0(2(x1)))))))))))))))) (11)
0(3(1(2(4(4(3(1(5(4(0(1(x1)))))))))))) 2(1(0(1(2(0(0(5(5(0(0(2(2(2(0(2(3(x1))))))))))))))))) (12)
0(3(1(3(3(1(2(4(1(4(3(5(x1)))))))))))) 2(1(1(5(0(2(5(5(5(5(0(1(3(1(1(5(5(1(x1)))))))))))))))))) (13)
0(3(1(5(3(2(1(3(2(1(3(3(x1)))))))))))) 1(2(2(2(5(1(0(3(2(2(0(2(0(2(x1)))))))))))))) (14)
0(3(5(1(1(1(4(5(3(3(1(1(x1)))))))))))) 2(5(5(1(5(5(2(5(1(4(0(5(1(5(0(3(5(x1))))))))))))))))) (15)
0(4(0(2(0(3(4(0(3(0(4(1(x1)))))))))))) 0(5(2(1(0(4(2(2(0(1(0(5(1(5(0(2(x1)))))))))))))))) (16)
0(4(0(2(3(2(0(1(1(0(5(4(x1)))))))))))) 2(4(5(2(1(2(0(1(2(0(0(2(0(4(x1)))))))))))))) (17)
0(4(0(3(1(3(0(3(3(4(3(4(x1)))))))))))) 0(2(5(4(1(5(0(1(5(5(3(1(2(3(x1)))))))))))))) (18)
0(4(1(0(3(5(3(0(5(1(1(4(x1)))))))))))) 2(1(0(1(1(2(2(2(0(5(5(5(1(2(5(2(0(x1))))))))))))))))) (19)
0(4(1(2(4(4(0(1(4(5(1(4(x1)))))))))))) 0(5(5(5(2(0(4(3(2(5(5(4(1(2(0(0(1(x1))))))))))))))))) (20)
0(4(1(5(5(4(1(3(5(1(1(3(x1)))))))))))) 3(3(5(5(5(2(1(0(2(2(0(2(0(5(2(x1))))))))))))))) (21)
0(4(4(3(0(1(3(5(4(4(3(1(x1)))))))))))) 5(5(2(2(0(3(0(5(5(2(1(4(2(1(0(1(1(2(x1)))))))))))))))))) (22)
0(4(4(5(4(5(4(4(2(3(3(4(x1)))))))))))) 5(1(2(5(0(3(5(0(3(1(5(1(5(2(x1)))))))))))))) (23)
0(4(5(3(0(1(2(3(2(1(5(4(x1)))))))))))) 4(5(2(5(5(5(2(3(1(5(0(5(1(0(3(5(x1)))))))))))))))) (24)
0(5(0(2(4(2(2(4(5(3(4(4(x1)))))))))))) 0(5(2(1(2(5(2(0(5(5(1(2(2(2(2(x1))))))))))))))) (25)
0(5(1(4(4(4(5(2(3(4(5(3(x1)))))))))))) 1(5(5(4(3(4(2(5(2(5(4(5(5(2(5(5(x1)))))))))))))))) (26)
1(1(1(4(1(3(3(3(2(1(5(1(x1)))))))))))) 3(5(2(2(5(0(5(5(5(5(1(1(4(4(x1)))))))))))))) (27)
1(1(3(3(4(4(3(3(3(4(3(3(x1)))))))))))) 0(0(4(5(5(1(4(1(2(3(0(0(5(5(x1)))))))))))))) (28)
1(1(3(4(4(3(3(4(1(1(0(5(x1)))))))))))) 3(5(0(2(5(3(5(2(5(5(2(0(3(5(5(x1))))))))))))))) (29)
1(1(4(4(5(1(3(0(1(4(3(1(x1)))))))))))) 1(5(4(2(2(0(2(2(5(5(1(5(2(2(1(0(1(3(x1)))))))))))))))))) (30)
1(2(4(0(1(5(4(3(2(3(3(0(x1)))))))))))) 3(1(4(2(2(1(4(1(5(2(1(5(2(5(x1)))))))))))))) (31)
1(3(0(0(4(5(3(5(3(5(3(3(x1)))))))))))) 1(3(5(0(2(2(0(0(2(0(1(5(2(5(5(2(x1)))))))))))))))) (32)
1(3(2(0(2(3(2(4(5(3(1(1(x1)))))))))))) 2(0(1(5(5(2(1(2(2(1(1(5(5(2(0(3(2(x1))))))))))))))))) (33)
1(3(2(4(2(5(1(3(5(0(1(4(x1)))))))))))) 5(2(5(1(0(5(0(5(0(3(3(2(5(2(4(x1))))))))))))))) (34)
1(3(2(5(1(3(2(1(1(2(0(3(x1)))))))))))) 4(5(5(0(2(5(5(4(5(0(3(2(5(5(2(4(x1)))))))))))))))) (35)
1(4(0(3(3(5(2(1(3(5(4(2(x1)))))))))))) 4(2(5(5(2(3(3(4(0(3(3(1(2(5(1(x1))))))))))))))) (36)
1(4(2(3(5(4(1(3(3(1(1(4(x1)))))))))))) 3(5(5(1(2(4(1(2(0(5(0(4(5(4(x1)))))))))))))) (37)
1(4(4(4(1(4(5(4(3(0(3(0(x1)))))))))))) 3(2(4(0(4(4(0(2(1(1(0(2(5(5(5(2(x1)))))))))))))))) (38)
1(4(5(1(1(4(4(0(3(2(4(0(x1)))))))))))) 0(2(1(3(4(5(5(0(1(0(2(4(0(0(1(5(5(2(x1)))))))))))))))))) (39)
1(5(1(1(1(1(1(1(4(2(1(4(x1)))))))))))) 4(5(1(1(5(5(3(5(5(5(2(1(0(3(1(2(0(x1))))))))))))))))) (40)
2(0(3(3(0(1(4(3(4(1(1(2(x1)))))))))))) 2(5(4(5(5(5(0(5(1(1(2(4(2(5(x1)))))))))))))) (41)
2(0(4(0(2(1(3(3(0(1(5(2(x1)))))))))))) 5(2(5(2(2(0(5(2(2(2(2(0(5(0(1(2(2(x1))))))))))))))))) (42)
2(0(5(4(4(4(2(4(5(4(0(1(x1)))))))))))) 1(2(5(5(2(1(2(4(0(5(2(3(1(4(0(2(2(x1))))))))))))))))) (43)
2(1(4(1(1(4(0(1(1(4(3(4(x1)))))))))))) 5(1(5(1(2(4(2(0(5(5(0(3(1(1(0(x1))))))))))))))) (44)
2(1(4(4(1(1(4(5(4(3(0(0(x1)))))))))))) 5(3(5(5(5(1(0(5(0(5(2(1(1(0(0(2(5(2(x1)))))))))))))))))) (45)
2(1(4(4(1(3(3(4(2(4(5(4(x1)))))))))))) 5(2(2(0(5(3(4(2(0(1(0(3(2(4(x1)))))))))))))) (46)
2(2(3(4(0(5(4(4(3(1(0(1(x1)))))))))))) 1(2(1(5(0(5(5(5(5(5(0(5(3(5(0(x1))))))))))))))) (47)
2(3(0(5(1(3(5(4(4(2(3(3(x1)))))))))))) 1(2(2(2(2(5(0(0(1(3(1(5(4(5(2(x1))))))))))))))) (48)
2(3(3(4(3(4(1(0(4(4(3(3(x1)))))))))))) 3(1(0(0(5(0(4(1(5(2(3(2(3(1(2(5(x1)))))))))))))))) (49)
2(3(4(4(1(1(3(0(0(3(2(1(x1)))))))))))) 0(2(2(5(2(0(5(5(2(3(2(2(5(5(2(0(3(x1))))))))))))))))) (50)
2(4(0(1(4(5(3(5(3(3(3(5(x1)))))))))))) 2(3(1(2(2(5(5(5(2(0(2(1(3(1(2(5(2(x1))))))))))))))))) (51)
2(4(2(3(2(3(3(3(0(2(3(0(x1)))))))))))) 5(0(0(5(5(5(0(5(4(4(2(5(5(0(0(5(0(5(x1)))))))))))))))))) (52)
2(4(3(4(2(4(2(4(4(2(2(5(x1)))))))))))) 1(2(2(0(5(2(2(5(0(2(1(2(2(1(2(0(0(x1))))))))))))))))) (53)
2(5(1(2(1(5(3(3(3(3(1(4(x1)))))))))))) 1(2(5(5(3(2(2(0(5(2(2(3(1(3(x1)))))))))))))) (54)
3(0(4(1(3(1(2(4(2(1(1(2(x1)))))))))))) 0(5(5(4(5(2(5(2(1(0(2(0(4(1(2(2(x1)))))))))))))))) (55)
3(0(4(4(4(5(1(5(4(5(3(0(x1)))))))))))) 1(4(5(5(0(1(3(3(0(5(0(0(5(3(5(x1))))))))))))))) (56)
3(1(0(3(2(4(4(1(0(3(3(2(x1)))))))))))) 3(0(5(5(2(0(5(5(4(2(5(5(2(0(x1)))))))))))))) (57)
3(1(2(3(4(2(3(2(1(4(0(5(x1)))))))))))) 5(5(0(2(3(1(0(1(3(1(2(1(2(2(x1)))))))))))))) (58)
3(1(3(3(3(4(4(2(1(0(4(2(x1)))))))))))) 2(1(2(1(2(2(2(5(5(5(0(2(4(0(5(4(x1)))))))))))))))) (59)
3(1(3(4(5(0(0(3(4(0(1(4(x1)))))))))))) 3(2(5(4(2(0(5(2(2(5(2(1(2(3(x1)))))))))))))) (60)
3(1(4(0(1(3(2(3(4(1(0(0(x1)))))))))))) 3(0(5(5(5(3(1(2(5(4(5(1(2(0(4(1(x1)))))))))))))))) (61)
3(1(5(4(0(3(4(5(3(3(3(4(x1)))))))))))) 2(5(1(5(5(5(5(3(2(4(3(5(2(5(0(x1))))))))))))))) (62)
3(2(2(1(5(1(4(4(3(2(2(3(x1)))))))))))) 2(2(0(5(0(0(5(5(5(0(2(2(1(1(2(5(x1)))))))))))))))) (63)
3(2(4(1(3(4(4(4(0(1(4(4(x1)))))))))))) 0(1(2(5(4(0(5(1(2(0(0(5(1(0(5(1(x1)))))))))))))))) (64)
3(3(1(1(4(4(2(3(2(1(4(0(x1)))))))))))) 2(5(1(3(1(2(2(5(0(4(2(5(4(2(2(2(x1)))))))))))))))) (65)
3(3(1(2(4(2(0(2(1(4(3(0(x1)))))))))))) 2(1(5(0(1(0(2(0(5(0(0(5(2(5(2(3(x1)))))))))))))))) (66)
3(3(3(0(0(3(1(4(4(4(4(5(x1)))))))))))) 2(2(0(5(0(5(1(2(0(0(2(2(0(2(2(0(4(5(x1)))))))))))))))))) (67)
3(3(3(3(3(5(5(1(3(4(1(0(x1)))))))))))) 4(2(2(0(2(2(3(5(5(5(5(1(2(1(2(2(2(x1))))))))))))))))) (68)
3(3(4(4(0(4(5(4(1(3(5(1(x1)))))))))))) 4(1(0(0(1(1(2(1(2(2(0(0(3(4(1(x1))))))))))))))) (69)
3(3(5(3(4(3(5(0(1(4(3(0(x1)))))))))))) 2(0(0(2(2(0(1(3(2(3(5(0(2(2(x1)))))))))))))) (70)
3(4(4(0(2(4(2(0(4(5(3(4(x1)))))))))))) 3(4(2(5(4(5(5(5(5(5(5(0(2(5(x1)))))))))))))) (71)
3(5(1(4(4(2(1(5(4(4(4(2(x1)))))))))))) 2(3(3(5(0(1(5(2(5(2(1(5(2(3(2(5(5(x1))))))))))))))))) (72)
3(5(2(4(0(1(1(4(4(3(2(3(x1)))))))))))) 4(2(1(5(1(0(5(2(5(2(3(5(4(3(x1)))))))))))))) (73)
4(1(0(4(4(4(2(4(3(5(4(2(x1)))))))))))) 1(1(5(0(1(2(1(4(3(1(0(2(2(5(0(1(x1)))))))))))))))) (74)
4(1(2(1(4(0(4(5(5(4(1(4(x1)))))))))))) 5(5(5(1(5(1(0(4(5(3(3(2(1(3(x1)))))))))))))) (75)
4(1(3(0(3(0(4(1(3(1(1(4(x1)))))))))))) 1(2(0(0(0(2(1(0(1(0(3(1(3(5(x1)))))))))))))) (76)
4(1(4(2(4(5(0(4(1(3(3(0(x1)))))))))))) 0(2(2(5(2(0(1(2(3(5(5(5(3(1(2(2(0(x1))))))))))))))))) (77)
4(3(0(4(4(3(2(4(0(4(4(2(x1)))))))))))) 2(1(0(1(3(4(5(0(3(3(3(2(0(5(5(2(1(x1))))))))))))))))) (78)
4(3(1(4(4(1(1(1(4(5(4(0(x1)))))))))))) 4(1(2(5(1(4(3(0(1(2(0(0(5(2(2(5(5(5(x1)))))))))))))))))) (79)
4(3(3(2(4(1(4(4(0(4(1(0(x1)))))))))))) 1(1(2(5(2(4(5(1(0(2(5(4(5(5(5(1(x1)))))))))))))))) (80)
4(3(4(1(2(3(5(1(0(2(3(5(x1)))))))))))) 2(5(5(2(5(0(3(1(5(5(2(0(5(3(0(5(0(2(x1)))))))))))))))))) (81)
4(3(4(4(4(1(4(1(3(0(1(3(x1)))))))))))) 1(1(0(5(5(5(5(2(1(4(1(2(5(1(x1)))))))))))))) (82)
4(3(5(3(0(4(1(5(1(1(1(3(x1)))))))))))) 2(2(4(2(2(1(5(5(2(1(2(2(0(0(2(x1))))))))))))))) (83)
4(4(0(4(2(4(0(3(5(0(3(1(x1)))))))))))) 4(5(3(4(0(0(1(0(3(0(5(5(5(5(2(2(2(2(x1)))))))))))))))))) (84)
4(4(2(1(4(5(5(4(1(5(0(1(x1)))))))))))) 4(5(2(0(3(4(0(5(5(5(2(5(2(0(5(5(2(x1))))))))))))))))) (85)
4(4(2(3(5(1(2(1(3(1(4(4(x1)))))))))))) 2(2(5(2(0(5(2(1(2(1(5(3(5(2(2(1(5(x1))))))))))))))))) (86)
4(4(4(4(5(1(3(2(3(5(3(3(x1)))))))))))) 5(5(5(0(2(3(1(4(1(5(0(0(5(0(2(0(5(x1))))))))))))))))) (87)
4(4(4(5(4(0(4(0(4(1(1(4(x1)))))))))))) 1(1(5(0(0(1(2(3(1(1(1(0(0(5(0(5(x1)))))))))))))))) (88)
4(4(5(3(2(0(5(3(2(4(1(5(x1)))))))))))) 0(1(2(0(1(0(3(4(1(0(2(4(2(5(x1)))))))))))))) (89)
4(4(5(4(4(4(0(1(2(4(0(0(x1)))))))))))) 0(3(3(1(2(2(2(2(2(0(2(0(5(1(1(2(5(2(x1)))))))))))))))))) (90)
4(5(1(1(0(3(4(5(3(0(1(0(x1)))))))))))) 1(5(5(2(1(5(3(3(5(4(5(0(2(5(5(5(0(x1))))))))))))))))) (91)
4(5(4(1(4(4(5(4(3(0(1(4(x1)))))))))))) 3(5(1(5(3(2(3(0(0(5(5(0(0(0(5(5(x1)))))))))))))))) (92)
5(1(1(5(4(5(1(5(4(4(3(4(x1)))))))))))) 5(0(1(2(5(5(5(5(0(0(3(1(5(0(2(0(0(1(x1)))))))))))))))))) (93)
5(3(1(0(4(1(4(4(2(4(2(3(x1)))))))))))) 5(2(1(2(3(5(2(0(3(2(0(0(5(5(0(x1))))))))))))))) (94)
5(3(3(5(4(4(2(1(5(0(0(1(x1)))))))))))) 3(3(1(0(1(5(3(0(5(5(5(5(5(5(1(5(x1)))))))))))))))) (95)
5(5(0(2(4(0(4(3(4(4(3(0(x1)))))))))))) 2(2(2(1(2(2(2(3(1(3(0(0(5(0(2(x1))))))))))))))) (96)

and S is the following TRS.

0(1(2(3(4(5(x1)))))) 0(1(2(3(4(5(x1)))))) (97)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

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

We obtain the transformed TRS

There are 576 ruless (increase limit for explicit display).

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 3456 ruless (increase limit for explicit display).

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 +
42
[52(x1)] = x1 +
21
[53(x1)] = x1 +
0
[54(x1)] = x1 +
0
[55(x1)] = x1 +
0
[40(x1)] = x1 +
49
[41(x1)] = x1 +
56
[42(x1)] = x1 +
56
[43(x1)] = x1 +
56
[44(x1)] = x1 +
56
[45(x1)] = x1 +
56
[30(x1)] = x1 +
57
[31(x1)] = x1 +
56
[32(x1)] = x1 +
56
[33(x1)] = x1 +
56
[34(x1)] = x1 +
56
[35(x1)] = x1 +
56
[20(x1)] = x1 +
0
[21(x1)] = x1 +
42
[22(x1)] = x1 +
0
[23(x1)] = x1 +
0
[24(x1)] = x1 +
1
[25(x1)] = x1 +
0
[10(x1)] = x1 +
0
[11(x1)] = x1 +
57
[12(x1)] = x1 +
57
[13(x1)] = x1 +
0
[14(x1)] = x1 +
56
[15(x1)] = x1 +
1
[00(x1)] = x1 +
0
[01(x1)] = x1 +
56
[02(x1)] = x1 +
56
[03(x1)] = x1 +
0
[04(x1)] = x1 +
15
[05(x1)] = x1 +
1
all of the following rules can be deleted.

There are 3456 ruless (increase limit for explicit display).

1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.