Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/3995)

The rewrite relation of the following TRS is considered.

0(1(2(x1))) → 0(0(3(4(0(3(0(3(0(2(x1)))))))))) (1)
0(5(1(4(4(x1))))) → 0(5(1(5(0(0(1(0(3(2(x1)))))))))) (2)
1(2(5(5(2(x1))))) → 1(0(5(0(3(5(4(0(3(2(x1)))))))))) (3)
2(5(5(1(3(x1))))) → 2(5(0(1(1(5(5(1(5(5(x1)))))))))) (4)
3(5(2(5(2(x1))))) → 3(5(5(3(0(0(3(0(4(3(x1)))))))))) (5)
4(3(1(2(5(x1))))) → 4(3(0(2(5(0(3(4(3(1(x1)))))))))) (6)
4(5(1(1(2(x1))))) → 1(0(1(5(5(0(5(2(0(3(x1)))))))))) (7)
5(1(1(0(2(x1))))) → 5(5(3(1(1(0(0(4(2(2(x1)))))))))) (8)
0(1(2(4(2(4(x1)))))) → 0(5(3(4(1(0(0(3(2(0(x1)))))))))) (9)
0(1(2(5(1(3(x1)))))) → 0(4(0(3(0(2(1(5(5(0(x1)))))))))) (10)
0(5(2(0(1(1(x1)))))) → 0(5(5(5(1(0(3(0(1(1(x1)))))))))) (11)
1(4(1(0(1(2(x1)))))) → 2(5(2(0(2(0(0(0(3(3(x1)))))))))) (12)
2(1(1(4(1(3(x1)))))) → 4(1(1(5(5(1(5(2(1(5(x1)))))))))) (13)
2(3(1(2(1(2(x1)))))) → 1(3(3(3(0(2(0(0(3(4(x1)))))))))) (14)
2(3(5(1(1(2(x1)))))) → 2(3(4(3(2(2(0(4(0(3(x1)))))))))) (15)
2(3(5(1(1(3(x1)))))) → 1(0(3(4(0(4(0(1(0(1(x1)))))))))) (16)
2(3(5(1(2(4(x1)))))) → 1(2(1(5(5(0(3(3(0(0(x1)))))))))) (17)
3(2(5(2(1(2(x1)))))) → 4(3(1(0(0(4(2(0(4(3(x1)))))))))) (18)
3(3(1(2(5(3(x1)))))) → 0(4(2(2(2(2(3(3(1(3(x1)))))))))) (19)
4(5(1(2(0(5(x1)))))) → 1(3(4(0(3(5(5(3(0(3(x1)))))))))) (20)
5(2(4(4(1(2(x1)))))) → 5(2(3(3(0(0(3(2(2(3(x1)))))))))) (21)
0(4(1(1(4(2(3(x1))))))) → 0(1(0(0(0(1(2(0(4(3(x1)))))))))) (22)
0(5(2(5(2(3(1(x1))))))) → 0(0(1(0(2(5(5(1(2(1(x1)))))))))) (23)
1(0(1(2(5(1(2(x1))))))) → 1(5(2(3(5(1(1(0(0(3(x1)))))))))) (24)
1(1(4(1(2(1(4(x1))))))) → 1(0(2(3(0(5(0(4(3(0(x1)))))))))) (25)
1(1(4(2(1(4(1(x1))))))) → 1(3(0(0(5(4(1(5(3(1(x1)))))))))) (26)
1(1(4(2(4(3(4(x1))))))) → 5(5(4(3(0(0(4(0(0(2(x1)))))))))) (27)
1(2(1(2(2(1(5(x1))))))) → 5(4(1(0(3(4(3(3(1(3(x1)))))))))) (28)
1(2(3(2(4(2(4(x1))))))) → 5(5(0(0(3(3(5(0(4(4(x1)))))))))) (29)
1(2(3(5(3(4(5(x1))))))) → 1(0(5(3(5(1(5(2(0(1(x1)))))))))) (30)
1(3(0(5(2(3(5(x1))))))) → 1(0(0(4(2(3(5(3(2(5(x1)))))))))) (31)
1(3(4(1(1(2(5(x1))))))) → 1(5(3(4(2(2(2(0(3(1(x1)))))))))) (32)
1(4(1(1(1(2(0(x1))))))) → 1(5(0(5(1(5(2(5(2(0(x1)))))))))) (33)
2(0(1(1(1(2(5(x1))))))) → 5(2(0(3(4(0(0(4(2(5(x1)))))))))) (34)
2(0(1(4(1(1(3(x1))))))) → 2(0(2(0(0(4(5(0(4(5(x1)))))))))) (35)
2(1(2(5(2(5(1(x1))))))) → 2(4(5(3(3(5(3(1(0(1(x1)))))))))) (36)
2(1(5(0(2(4(0(x1))))))) → 2(5(3(2(5(0(3(1(5(0(x1)))))))))) (37)
2(3(1(1(1(1(4(x1))))))) → 2(0(3(3(1(4(3(5(0(5(x1)))))))))) (38)
2(3(2(0(5(1(0(x1))))))) → 4(3(5(0(3(3(0(3(5(0(x1)))))))))) (39)
2(3(2(4(0(2(4(x1))))))) → 4(4(5(4(3(0(3(3(5(4(x1)))))))))) (40)
2(3(3(5(2(5(2(x1))))))) → 1(0(0(4(2(2(0(5(1(2(x1)))))))))) (41)
3(1(4(2(1(1(2(x1))))))) → 0(1(2(2(3(2(4(0(0(4(x1)))))))))) (42)
3(2(3(1(1(3(2(x1))))))) → 0(3(1(2(0(0(2(0(0(3(x1)))))))))) (43)
3(2(3(5(1(3(1(x1))))))) → 3(1(4(3(0(5(5(5(5(1(x1)))))))))) (44)
3(4(0(2(2(4(0(x1))))))) → 0(4(5(3(0(3(0(5(3(0(x1)))))))))) (45)
5(2(0(1(4(1(3(x1))))))) → 5(4(0(0(4(1(4(5(5(4(x1)))))))))) (46)
5(2(0(2(1(1(4(x1))))))) → 5(1(3(3(3(5(3(0(3(0(x1)))))))))) (47)
5(2(0(5(4(1(5(x1))))))) → 1(1(5(2(1(5(5(1(1(5(x1)))))))))) (48)
0(2(5(1(4(x1))))) → 0(0(0(3(1(3(4(3(0(0(x1)))))))))) (49)
1(3(2(4(1(x1))))) → 2(3(0(0(3(5(5(0(0(1(x1)))))))))) (50)
2(2(3(5(1(x1))))) → 5(0(0(3(3(4(0(0(2(1(x1)))))))))) (51)
0(0(1(1(1(3(x1)))))) → 3(1(5(1(5(5(0(4(1(3(x1)))))))))) (52)
1(1(2(2(5(1(x1)))))) → 5(4(5(3(0(3(0(1(0(1(x1)))))))))) (53)
1(3(3(1(3(3(x1)))))) → 1(0(0(0(2(0(5(3(5(2(x1)))))))))) (54)
2(0(5(4(0(5(x1)))))) → 4(3(5(3(5(4(5(0(0(5(x1)))))))))) (55)
2(3(5(0(2(1(x1)))))) → 2(3(0(3(5(5(5(0(3(1(x1)))))))))) (56)
2(1(2(5(3(3(5(x1))))))) → 5(4(0(4(2(0(0(5(0(5(x1)))))))))) (57)
2(2(0(4(5(1(3(x1))))))) → 2(1(0(4(4(5(0(0(4(2(x1)))))))))) (58)
3(2(4(1(3(3(1(x1))))))) → 0(2(0(5(0(3(0(3(3(1(x1)))))))))) (59)
3(3(1(2(4(0(2(x1))))))) → 0(4(3(3(3(5(0(4(3(2(x1)))))))))) (60)

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 360 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 2160 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 +
943949
[52(x1)] = x1 +
310
[53(x1)] = x1 +
803249
[54(x1)] = x1 +
356
[55(x1)] = x1 +
0
[40(x1)] = x1 +
0
[41(x1)] = x1 +
0
[42(x1)] = x1 +
35
[43(x1)] = x1 +
0
[44(x1)] = x1 +
434943
[45(x1)] = x1 +
6
[30(x1)] = x1 +
3720
[31(x1)] = x1 +
26484
[32(x1)] = x1 +
721128
[33(x1)] = x1 +
35463
[34(x1)] = x1 +
0
[35(x1)] = x1 +
0
[20(x1)] = x1 +
1243188
[21(x1)] = x1 +
2096242
[22(x1)] = x1 +
2412626
[23(x1)] = x1 +
7
[24(x1)] = x1 +
425907
[25(x1)] = x1 +
906895
[10(x1)] = x1 +
0
[11(x1)] = x1 +
3108573
[12(x1)] = x1 +
1936374
[13(x1)] = x1 +
3105530
[14(x1)] = x1 +
1018904
[15(x1)] = x1 +
333866
[00(x1)] = x1 +
1197191
[01(x1)] = x1 +
0
[02(x1)] = x1 +
0
[03(x1)] = x1 +
0
[04(x1)] = x1 +
1
[05(x1)] = x1 +
1
all of the following rules can be deleted.

There are 2160 ruless (increase limit for explicit display).

1.1.1.1 R is empty

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