Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/131633)

The rewrite relation of the following TRS is considered.

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

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 +
8
[51(x1)] = x1 +
0
[52(x1)] = x1 +
0
[53(x1)] = x1 +
0
[54(x1)] = x1 +
0
[55(x1)] = x1 +
9
[40(x1)] = x1 +
0
[41(x1)] = x1 +
8
[42(x1)] = x1 +
1
[43(x1)] = x1 +
8
[44(x1)] = x1 +
1
[45(x1)] = x1 +
8
[30(x1)] = x1 +
0
[31(x1)] = x1 +
1
[32(x1)] = x1 +
0
[33(x1)] = x1 +
0
[34(x1)] = x1 +
0
[35(x1)] = x1 +
0
[20(x1)] = x1 +
8
[21(x1)] = x1 +
9
[22(x1)] = x1 +
0
[23(x1)] = x1 +
8
[24(x1)] = x1 +
1
[25(x1)] = x1 +
8
[10(x1)] = x1 +
0
[11(x1)] = x1 +
0
[12(x1)] = x1 +
0
[13(x1)] = x1 +
1
[14(x1)] = x1 +
0
[15(x1)] = x1 +
0
[00(x1)] = x1 +
9
[01(x1)] = x1 +
8
[02(x1)] = x1 +
1
[03(x1)] = x1 +
8
[04(x1)] = x1 +
0
[05(x1)] = x1 +
9
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.