Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/139025)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

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

We obtain the transformed TRS

There are 320 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,...,3}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 4):

[3(x1)] = 4x1 + 0
[2(x1)] = 4x1 + 1
[1(x1)] = 4x1 + 2
[0(x1)] = 4x1 + 3

We obtain the labeled TRS

There are 1280 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
[30(x1)] = x1 +
42
[31(x1)] = x1 +
43
[32(x1)] = x1 +
0
[33(x1)] = x1 +
42
[20(x1)] = x1 +
28
[21(x1)] = x1 +
0
[22(x1)] = x1 +
7
[23(x1)] = x1 +
36
[10(x1)] = x1 +
21
[11(x1)] = x1 +
42
[12(x1)] = x1 +
0
[13(x1)] = x1 +
1
[00(x1)] = x1 +
1
[01(x1)] = x1 +
0
[02(x1)] = x1 +
42
[03(x1)] = x1 +
0
all of the following rules can be deleted.

There are 1280 ruless (increase limit for explicit display).

1.1.1.1 R is empty

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