Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/54097)

The rewrite relation of the following TRS is considered.

0(0(1(1(1(1(2(2(3(4(1(3(0(3(4(2(1(0(3(0(1(4(5(3(3(4(1(5(1(0(1(1(0(5(0(5(3(4(1(1(3(2(3(3(3(0(3(2(1(0(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 2(5(4(2(4(1(2(0(1(3(0(3(4(0(3(0(1(0(0(5(0(2(2(5(0(2(1(0(2(2(2(3(0(1(3(0(1(4(3(0(4(2(4(4(5(3(2(2(1(3(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (1)
0(0(1(1(2(4(3(5(1(3(5(3(4(3(5(3(2(4(0(4(0(4(3(2(1(4(3(2(1(4(1(1(1(1(0(1(4(3(2(4(0(2(5(4(4(0(0(4(4(0(4(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 1(2(5(1(2(1(2(5(5(1(0(5(1(2(5(4(3(0(0(5(2(0(5(5(1(5(5(3(2(4(1(2(1(4(3(3(0(4(5(2(1(4(5(0(3(2(4(2(3(0(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (2)
0(1(4(0(4(3(1(5(4(0(4(1(2(5(4(1(4(5(1(5(2(0(4(3(3(1(0(5(4(1(0(4(1(1(4(4(2(1(5(3(0(2(4(0(2(2(1(3(0(4(5(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 2(1(0(1(3(5(3(3(1(2(5(2(0(1(2(1(1(3(5(4(4(0(1(0(2(1(1(2(2(5(5(4(1(1(3(0(2(3(0(5(1(3(1(5(1(3(2(1(5(5(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (3)
0(2(3(5(1(5(4(5(1(0(3(1(3(3(2(0(0(1(5(3(3(2(2(1(2(0(4(0(3(4(1(3(2(4(0(5(2(4(3(3(1(4(2(1(3(1(0(0(5(2(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 5(5(1(5(5(5(0(0(2(4(2(5(1(5(4(3(0(2(5(2(5(2(5(2(1(1(3(2(5(5(5(5(4(2(2(4(0(5(3(3(2(2(0(5(4(3(2(3(5(4(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (4)
0(3(2(1(4(3(0(3(1(2(1(0(3(1(2(3(3(1(2(1(2(2(1(5(1(2(0(5(0(5(2(4(2(1(2(2(5(1(2(2(3(3(4(0(0(1(5(0(1(5(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 0(5(1(3(0(0(3(4(4(2(1(3(5(0(2(0(3(1(3(5(5(3(4(0(2(0(3(1(4(5(5(1(4(5(2(2(0(1(2(0(0(5(2(5(5(0(3(1(3(1(4(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (5)
0(4(2(1(3(1(1(1(0(3(3(3(3(2(0(4(4(5(4(5(0(4(1(3(5(1(1(3(5(3(4(1(5(5(2(2(3(5(1(3(2(1(0(4(1(2(3(3(3(0(3(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 3(1(5(4(2(2(0(5(1(3(5(2(3(3(2(4(2(5(0(2(5(4(5(3(5(2(3(0(2(5(3(5(1(5(5(1(4(1(2(1(2(5(5(3(0(5(0(4(3(0(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (6)
1(0(0(5(1(3(4(5(2(5(3(4(3(2(0(5(3(4(3(2(1(1(2(2(1(4(1(1(4(3(0(4(2(3(4(1(3(0(0(0(4(5(5(2(0(0(1(2(1(1(3(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 4(3(1(3(5(1(2(5(4(2(3(0(3(1(2(3(5(0(0(5(4(5(0(4(2(0(0(0(5(4(4(0(1(2(5(4(3(2(3(2(5(4(2(3(5(1(3(4(5(3(4(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (7)
1(1(3(3(3(3(2(4(4(4(2(3(1(4(2(2(4(1(1(0(5(0(2(1(1(1(4(2(2(2(2(4(3(4(4(1(4(5(5(1(3(3(3(3(5(5(2(3(3(3(4(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 0(4(2(4(1(0(5(0(4(5(5(4(0(0(4(4(2(4(2(2(0(1(5(5(3(0(5(1(5(4(3(0(0(1(5(0(2(5(2(4(4(2(0(3(0(4(4(1(3(3(5(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (8)
2(3(0(4(0(2(4(5(4(0(4(4(4(0(2(2(3(1(1(1(3(1(4(4(1(2(1(1(3(0(0(5(2(2(4(5(4(2(0(4(5(0(2(5(0(4(4(4(2(0(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 5(4(3(0(4(4(0(5(4(0(0(0(1(2(5(5(4(4(1(0(4(2(3(1(3(5(3(5(5(0(1(3(2(0(5(1(1(3(0(5(1(5(4(3(4(2(3(3(1(5(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (9)
2(3(4(0(4(2(1(4(4(0(2(1(1(0(4(2(4(3(2(0(1(0(5(4(1(4(5(0(0(1(5(1(4(1(2(2(2(0(2(3(0(1(2(5(3(3(2(3(1(5(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 2(3(0(1(5(5(4(5(2(4(2(0(1(4(0(5(4(3(3(5(5(1(4(0(1(2(5(1(0(2(5(2(2(4(0(1(5(5(2(2(0(5(2(1(1(0(0(0(0(3(5(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (10)
2(4(0(4(1(4(3(2(0(3(2(0(3(4(4(4(2(1(3(3(0(4(5(3(0(4(5(4(0(0(2(1(3(4(4(0(0(3(4(1(4(4(2(5(1(2(4(0(1(1(3(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 3(0(4(2(4(4(5(3(5(2(5(0(3(3(5(2(5(4(2(4(0(3(3(2(4(2(4(0(0(5(5(5(4(0(0(4(2(5(3(0(0(4(5(0(3(4(0(2(4(0(5(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (11)
2(4(5(3(1(1(0(2(0(1(1(2(0(5(0(0(0(0(0(4(5(2(3(2(3(4(4(0(4(4(4(1(0(1(5(1(4(0(1(1(4(2(2(1(3(5(0(1(5(2(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 3(3(5(5(5(0(5(2(2(0(3(0(5(5(4(5(1(3(4(0(3(5(4(5(2(5(0(2(5(1(3(4(1(0(3(5(3(0(2(5(3(4(0(2(0(2(2(5(2(5(5(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (12)
3(4(2(2(1(2(1(2(1(3(2(4(4(5(4(1(1(5(0(5(4(1(4(3(4(1(4(2(4(4(4(1(2(1(1(4(1(0(0(3(0(4(1(4(4(1(1(0(1(2(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 3(0(4(0(2(5(3(4(1(3(5(5(5(3(5(4(4(3(0(2(1(0(2(4(1(2(0(2(2(0(1(3(1(1(4(1(3(1(1(4(4(5(1(5(2(1(1(3(0(1(5(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (13)
4(0(3(4(4(2(4(2(2(2(1(1(4(2(2(0(1(4(2(1(4(1(1(0(2(5(1(2(5(0(5(0(1(3(1(4(0(1(5(3(1(0(0(0(0(4(2(3(4(2(4(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 5(5(5(0(3(3(1(3(0(3(3(0(5(1(0(3(5(4(2(1(0(1(5(1(4(4(1(2(4(4(1(0(3(0(1(0(2(4(1(5(4(0(5(3(0(2(5(0(4(1(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (14)
4(4(3(2(3(4(0(2(0(2(3(3(1(0(1(1(1(1(5(1(1(0(4(2(2(3(4(3(5(2(4(4(3(2(5(2(3(4(5(3(4(4(3(0(1(0(4(4(5(4(3(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 0(0(1(5(3(2(1(4(1(4(5(1(5(0(0(2(1(0(2(0(2(1(3(3(5(5(3(3(2(4(1(3(5(5(2(5(4(5(3(2(5(4(3(5(1(0(5(1(0(5(5(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (15)
5(1(0(0(3(0(1(1(4(3(1(2(5(0(5(1(2(0(0(4(3(3(2(4(4(3(4(3(5(5(3(1(5(3(5(1(1(3(3(4(0(0(2(4(2(0(0(4(2(4(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 3(2(0(3(4(0(0(5(2(2(3(1(5(5(5(5(1(2(5(4(5(0(4(2(0(0(0(2(5(3(0(3(3(2(0(1(0(5(5(0(3(4(3(5(0(2(5(1(1(5(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (16)
5(1(5(4(4(5(0(2(0(2(3(0(5(5(3(0(0(1(3(0(0(4(5(1(1(1(3(5(1(4(2(1(2(2(4(1(4(3(3(4(2(3(2(3(1(0(4(2(5(3(3(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 0(5(5(0(5(1(1(5(5(0(1(0(0(5(0(4(1(1(5(3(4(1(3(2(4(3(0(2(0(2(2(3(1(1(3(0(3(5(1(2(4(5(0(5(3(3(4(4(0(0(5(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (17)
5(2(4(2(2(2(3(2(3(0(4(2(2(2(2(4(1(4(0(2(3(0(4(0(3(2(2(3(3(1(4(0(0(5(3(2(3(1(3(1(1(1(3(0(3(2(0(2(0(1(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 4(5(1(0(4(3(1(1(5(1(2(0(3(5(3(5(3(4(0(5(0(2(1(1(2(4(3(5(5(3(5(0(5(3(5(0(3(5(4(0(0(1(5(5(4(4(3(5(3(1(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (18)
2(2(0(1(1(0(1(1(1(0(3(3(2(5(0(1(5(3(4(5(2(1(2(0(2(2(2(3(4(4(4(1(0(4(2(4(1(0(1(1(3(4(4(4(2(2(0(4(0(0(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 1(4(4(5(4(0(2(0(0(5(0(5(2(5(1(0(3(4(5(0(3(3(4(4(0(3(1(4(2(5(5(2(0(4(3(0(0(0(0(2(0(2(5(0(4(4(4(0(1(2(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (19)
2(3(2(3(4(4(4(2(1(0(4(1(1(3(2(2(0(2(3(2(5(1(2(2(0(4(3(1(1(5(4(1(3(2(4(3(3(1(4(4(3(2(0(3(2(4(3(2(5(4(3(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) 4(2(3(3(1(1(0(0(2(0(5(5(1(0(5(4(5(2(5(3(1(0(1(3(2(0(3(3(0(0(0(5(2(3(0(3(2(2(4(1(2(5(5(2(3(2(1(5(4(0(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (20)

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 120 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 720 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 +
0
[52(x1)] = x1 +
0
[53(x1)] = x1 +
0
[54(x1)] = x1 +
0
[55(x1)] = x1 +
0
[40(x1)] = x1 +
36
[41(x1)] = x1 +
37
[42(x1)] = x1 +
37
[43(x1)] = x1 +
1
[44(x1)] = x1 +
36
[45(x1)] = x1 +
1
[30(x1)] = x1 +
0
[31(x1)] = x1 +
13
[32(x1)] = x1 +
36
[33(x1)] = x1 +
36
[34(x1)] = x1 +
1
[35(x1)] = x1 +
0
[20(x1)] = x1 +
0
[21(x1)] = x1 +
37
[22(x1)] = x1 +
37
[23(x1)] = x1 +
37
[24(x1)] = x1 +
37
[25(x1)] = x1 +
1
[10(x1)] = x1 +
0
[11(x1)] = x1 +
37
[12(x1)] = x1 +
1
[13(x1)] = x1 +
36
[14(x1)] = x1 +
36
[15(x1)] = x1 +
36
[00(x1)] = x1 +
0
[01(x1)] = x1 +
37
[02(x1)] = x1 +
0
[03(x1)] = x1 +
0
[04(x1)] = x1 +
37
[05(x1)] = x1 +
1
all of the following rules can be deleted.

There are 720 ruless (increase limit for explicit display).

1.1.1.1 R is empty

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