Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/135328)

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

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