Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/96029)

The rewrite relation of the following TRS is considered.

0(1(2(3(x1)))) → 2(3(3(x1))) (1)
2(3(4(1(5(x1))))) → 2(5(0(3(2(x1))))) (2)
3(2(3(2(1(5(x1)))))) → 3(2(0(0(3(0(x1)))))) (3)
0(0(4(1(4(3(2(x1))))))) → 5(5(3(3(0(1(x1)))))) (4)
5(1(2(1(0(0(3(2(x1)))))))) → 5(1(0(2(1(0(3(2(x1)))))))) (5)
4(3(2(4(0(2(0(5(2(x1))))))))) → 0(5(1(1(2(2(1(2(x1)))))))) (6)
2(0(3(1(4(2(5(2(4(5(x1)))))))))) → 5(5(4(5(5(5(2(5(0(x1))))))))) (7)
2(4(4(0(5(0(1(0(4(1(2(x1))))))))))) → 2(4(1(3(4(0(2(0(2(5(2(x1))))))))))) (8)
3(5(3(1(4(4(5(5(4(2(1(x1))))))))))) → 3(3(5(2(3(4(5(4(3(0(1(x1))))))))))) (9)
5(0(0(5(2(5(4(1(0(2(1(x1))))))))))) → 5(4(5(4(5(1(1(1(3(2(x1)))))))))) (10)
4(1(4(3(3(2(3(0(1(3(5(1(x1)))))))))))) → 2(1(5(3(5(5(5(2(2(3(2(1(x1)))))))))))) (11)
5(4(0(4(0(4(2(2(5(5(4(5(x1)))))))))))) → 5(4(5(2(5(3(2(2(5(1(4(5(x1)))))))))))) (12)
5(4(2(5(0(3(1(4(3(0(1(4(1(x1))))))))))))) → 5(4(4(2(5(0(2(5(5(3(2(2(1(x1))))))))))))) (13)
3(4(4(1(0(2(3(3(1(4(5(5(5(3(x1)))))))))))))) → 1(3(4(5(5(3(0(2(2(1(3(2(3(3(x1)))))))))))))) (14)
5(2(5(1(5(0(4(3(2(2(2(3(0(4(0(x1))))))))))))))) → 5(4(3(4(0(3(2(2(1(3(4(4(5(4(4(x1))))))))))))))) (15)
0(5(2(1(1(1(3(4(1(0(4(5(5(4(2(5(x1)))))))))))))))) → 0(2(2(3(3(1(3(2(1(4(1(0(0(0(2(5(x1)))))))))))))))) (16)
5(2(4(5(2(1(5(2(4(1(0(3(3(0(0(2(x1)))))))))))))))) → 2(2(2(0(3(0(5(0(3(2(4(4(4(0(1(3(x1)))))))))))))))) (17)
0(1(0(2(3(5(3(4(0(2(4(1(1(2(5(1(3(5(x1)))))))))))))))))) → 0(1(3(5(2(0(5(1(2(3(5(1(2(3(3(1(4(x1))))))))))))))))) (18)
1(0(3(4(2(4(5(2(2(4(3(3(3(3(3(2(3(5(x1)))))))))))))))))) → 1(0(0(2(5(3(5(2(3(4(2(0(0(3(2(3(2(4(1(x1))))))))))))))))))) (19)
2(2(0(3(5(5(1(3(3(0(0(3(0(3(4(3(5(4(x1)))))))))))))))))) → 2(2(5(2(4(4(1(0(0(2(3(5(0(1(2(4(1(4(x1)))))))))))))))))) (20)
3(1(2(2(4(3(3(3(5(3(4(3(1(3(0(5(2(3(x1)))))))))))))))))) → 3(1(5(3(4(3(2(5(5(5(3(2(5(5(1(5(2(3(x1)))))))))))))))))) (21)
5(0(2(1(2(2(1(5(1(4(2(0(3(1(5(4(4(5(x1)))))))))))))))))) → 5(4(1(4(1(5(4(3(5(3(2(4(0(0(0(2(5(2(x1)))))))))))))))))) (22)
4(3(1(4(5(1(3(5(1(2(0(4(2(4(3(1(5(1(0(1(x1)))))))))))))))))))) → 5(1(3(5(3(4(4(2(2(5(1(0(0(2(4(4(2(3(0(0(1(x1))))))))))))))))))))) (23)
1(1(2(4(5(1(4(4(1(3(1(3(1(4(4(4(3(3(4(1(2(x1))))))))))))))))))))) → 0(4(2(2(2(2(5(1(2(1(4(4(5(1(1(0(4(5(2(3(4(x1))))))))))))))))))))) (24)
1(4(0(2(0(0(5(1(4(4(0(3(4(2(1(4(1(5(5(1(5(x1))))))))))))))))))))) → 1(1(3(3(2(2(2(0(5(3(4(5(3(2(1(0(1(2(4(3(4(x1))))))))))))))))))))) (25)

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 150 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 900 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 +
10
[51(x1)] = x1 +
10
[52(x1)] = x1 +
11
[53(x1)] = x1 +
12876/13
[54(x1)] = x1 +
8381/13
[55(x1)] = x1 +
1277/13
[40(x1)] = x1 +
10
[41(x1)] = x1 +
11
[42(x1)] = x1 +
22774/13
[43(x1)] = x1 +
22644/13
[44(x1)] = x1 +
22657/13
[45(x1)] = x1 +
11058/13
[30(x1)] = x1 +
12358/13
[31(x1)] = x1 +
19087/13
[32(x1)] = x1 +
10915/13
[33(x1)] = x1 +
1306
[34(x1)] = x1 +
22644/13
[35(x1)] = x1 +
16114/13
[20(x1)] = x1 +
1
[21(x1)] = x1 +
22657/13
[22(x1)] = x1 +
10
[23(x1)] = x1 +
0
[24(x1)] = x1 +
22644/13
[25(x1)] = x1 +
10
[10(x1)] = x1 +
22774/13
[11(x1)] = x1 +
22644/13
[12(x1)] = x1 +
370/13
[13(x1)] = x1 +
22774/13
[14(x1)] = x1 +
11526/13
[15(x1)] = x1 +
11045/13
[00(x1)] = x1 +
10
[01(x1)] = x1 +
22774/13
[02(x1)] = x1 +
22644/13
[03(x1)] = x1 +
12580/13
[04(x1)] = x1 +
10
[05(x1)] = x1 +
864/13
all of the following rules can be deleted.

There are 900 ruless (increase limit for explicit display).

1.1.1.1 R is empty

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