Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/84977)

The rewrite relation of the following TRS is considered.

0(0(1(2(x1)))) → 3(3(0(x1))) (1)
0(3(1(3(x1)))) → 2(4(3(x1))) (2)
5(1(0(0(x1)))) → 5(3(3(x1))) (3)
0(3(2(4(5(x1))))) → 2(5(1(5(x1)))) (4)
3(0(0(0(4(x1))))) → 5(5(1(4(x1)))) (5)
5(4(2(4(3(x1))))) → 0(5(3(1(3(x1))))) (6)
5(3(0(0(2(0(x1)))))) → 5(3(2(4(3(x1))))) (7)
4(3(5(0(3(0(5(3(x1)))))))) → 4(1(0(3(3(0(5(0(x1)))))))) (8)
0(0(3(3(3(4(3(3(1(x1))))))))) → 5(5(2(4(0(0(1(0(3(x1))))))))) (9)
1(0(2(1(5(1(1(5(4(x1))))))))) → 1(0(0(0(1(4(3(5(4(x1))))))))) (10)
3(3(1(5(4(0(4(0(4(x1))))))))) → 4(0(1(3(4(3(4(4(x1)))))))) (11)
1(3(3(2(4(1(2(2(2(1(x1)))))))))) → 1(4(3(5(3(3(1(2(1(1(x1)))))))))) (12)
4(1(4(5(2(4(0(1(5(4(x1)))))))))) → 4(0(3(4(3(5(2(2(4(x1))))))))) (13)
5(4(1(4(3(2(2(2(1(2(x1)))))))))) → 0(5(2(0(1(5(3(2(0(2(x1)))))))))) (14)
1(2(1(2(5(2(4(3(3(4(5(x1))))))))))) → 1(4(3(4(0(4(2(5(5(5(x1)))))))))) (15)
3(2(0(2(2(3(3(5(4(1(5(x1))))))))))) → 2(1(2(0(2(3(4(4(5(4(0(x1))))))))))) (16)
4(5(4(0(1(5(5(4(5(2(4(x1))))))))))) → 3(1(2(1(0(4(5(2(0(2(4(x1))))))))))) (17)
0(1(4(1(0(0(3(4(1(5(3(1(x1)))))))))))) → 0(4(4(1(2(3(0(0(0(1(3(1(x1)))))))))))) (18)
1(3(4(5(1(0(0(4(5(5(3(1(x1)))))))))))) → 1(1(2(0(2(1(2(3(0(4(3(1(x1)))))))))))) (19)
5(1(2(3(2(2(4(2(1(5(4(0(x1)))))))))))) → 5(5(4(0(5(2(2(5(5(2(5(x1))))))))))) (20)
0(3(0(4(4(4(1(1(4(5(4(3(3(x1))))))))))))) → 0(2(3(2(3(3(4(1(1(4(2(3(x1)))))))))))) (21)
4(1(1(0(1(3(4(5(2(3(5(5(1(x1))))))))))))) → 4(1(1(0(3(4(5(2(4(3(3(0(1(x1))))))))))))) (22)
5(2(5(4(4(3(4(1(0(1(0(1(2(x1))))))))))))) → 5(4(2(0(1(3(2(4(5(4(3(4(x1)))))))))))) (23)
3(3(3(1(3(0(5(2(5(4(4(3(2(1(1(x1))))))))))))))) → 3(4(5(5(1(2(5(2(2(1(4(1(0(1(1(x1))))))))))))))) (24)
3(5(0(2(1(2(2(3(5(3(1(3(1(1(4(x1))))))))))))))) → 4(3(1(0(2(5(3(3(2(4(1(5(5(4(x1)))))))))))))) (25)
5(3(4(5(1(2(5(0(0(0(5(4(1(4(4(x1))))))))))))))) → 5(4(5(5(5(1(3(3(4(1(5(5(1(4(x1)))))))))))))) (26)
1(5(2(4(4(3(4(2(3(3(1(1(4(2(5(2(x1)))))))))))))))) → 1(5(1(3(0(0(0(2(4(4(5(0(4(0(5(2(5(x1))))))))))))))))) (27)
3(1(2(3(5(2(0(0(4(4(3(3(3(1(4(0(4(x1))))))))))))))))) → 1(2(3(0(1(0(2(1(0(2(3(5(3(4(1(0(4(x1))))))))))))))))) (28)
2(2(0(3(4(0(1(4(2(0(3(0(2(2(4(2(4(4(x1)))))))))))))))))) → 5(5(3(0(2(1(2(1(4(4(5(5(1(1(0(3(1(4(x1)))))))))))))))))) (29)
0(2(3(0(4(0(1(0(1(4(5(3(5(2(4(1(0(3(3(x1))))))))))))))))))) → 0(3(0(5(3(5(4(3(1(4(0(3(4(5(3(1(1(1(x1)))))))))))))))))) (30)
0(4(1(2(3(5(3(1(4(0(1(0(5(5(4(0(3(3(0(x1))))))))))))))))))) → 1(5(5(2(5(0(4(4(4(4(5(3(3(4(2(3(4(x1))))))))))))))))) (31)
3(0(4(3(0(2(3(4(4(1(2(3(0(0(2(0(4(1(1(x1))))))))))))))))))) → 3(0(3(0(5(0(0(0(3(3(2(1(1(2(3(0(1(1(1(0(x1)))))))))))))))))))) (32)
3(2(3(1(4(4(1(2(5(1(5(0(4(5(1(0(1(0(2(x1))))))))))))))))))) → 0(5(5(4(5(0(1(1(3(5(1(5(2(0(4(0(2(0(2(x1))))))))))))))))))) (33)
5(0(0(3(0(4(0(3(5(3(1(3(1(4(1(2(0(1(3(x1))))))))))))))))))) → 2(2(0(3(4(5(1(3(2(0(4(4(0(5(4(0(2(3(x1)))))))))))))))))) (34)
1(0(4(1(0(4(0(1(3(1(1(0(2(5(4(1(0(3(3(2(x1)))))))))))))))))))) → 1(5(5(0(3(5(0(5(4(3(4(3(5(1(3(5(4(1(x1)))))))))))))))))) (35)
2(0(5(5(5(0(3(5(0(0(4(2(1(5(5(0(2(4(1(1(x1)))))))))))))))))))) → 2(0(0(5(1(1(4(2(5(5(3(2(5(1(4(3(4(5(2(x1))))))))))))))))))) (36)
3(1(1(3(4(3(1(5(5(4(1(1(3(5(0(1(2(3(0(0(x1)))))))))))))))))))) → 1(1(1(3(3(4(1(4(5(1(4(0(5(1(2(2(1(4(3(2(x1)))))))))))))))))))) (37)
0(0(1(3(4(3(1(1(3(3(2(0(4(4(3(0(0(5(5(2(0(x1))))))))))))))))))))) → 2(4(5(5(2(1(5(2(4(2(1(0(2(4(4(3(5(2(4(3(x1)))))))))))))))))))) (38)
0(5(1(4(0(5(1(3(4(3(3(1(1(4(5(1(4(1(4(0(0(x1))))))))))))))))))))) → 5(4(1(2(5(2(0(0(2(2(2(0(3(0(4(4(0(0(5(3(x1)))))))))))))))))))) (39)
4(2(3(1(3(1(1(5(3(4(2(0(5(5(3(3(2(0(5(4(1(x1))))))))))))))))))))) → 4(3(0(4(5(3(2(4(3(0(0(5(1(5(5(4(5(3(0(0(4(x1))))))))))))))))))))) (40)

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 240 ruless (increase limit for explicit display).

1.1 Closure Under Flat Contexts

Using the flat contexts

{5(☐), 4(☐), 3(☐), 2(☐), 1(☐), 0(☐)}

We obtain the transformed TRS

There are 1440 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

The following interpretations form a model of the rules.

As carrier we take the set {0,...,35}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 36):

[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 51840 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[50(x1)] = x1 +
0
[56(x1)] = x1 +
34974
[512(x1)] = x1 +
6365
[518(x1)] = x1 +
0
[524(x1)] = x1 +
0
[530(x1)] = x1 +
6365
[51(x1)] = x1 +
0
[57(x1)] = x1 +
34974
[513(x1)] = x1 +
1
[519(x1)] = x1 +
34974
[525(x1)] = x1 +
34974
[531(x1)] = x1 +
34974
[52(x1)] = x1 +
8
[58(x1)] = x1 +
34974
[514(x1)] = x1 +
0
[520(x1)] = x1 +
1
[526(x1)] = x1 +
34975
[532(x1)] = x1 +
25059
[53(x1)] = x1 +
0
[59(x1)] = x1 +
8
[515(x1)] = x1 +
34974
[521(x1)] = x1 +
0
[527(x1)] = x1 +
0
[533(x1)] = x1 +
0
[54(x1)] = x1 +
0
[510(x1)] = x1 +
25393
[516(x1)] = x1 +
34974
[522(x1)] = x1 +
0
[528(x1)] = x1 +
34982
[534(x1)] = x1 +
34975
[55(x1)] = x1 +
0
[511(x1)] = x1 +
0
[517(x1)] = x1 +
34975
[523(x1)] = x1 +
30485
[529(x1)] = x1 +
34974
[535(x1)] = x1 +
34974
[40(x1)] = x1 +
0
[46(x1)] = x1 +
28617
[412(x1)] = x1 +
8
[418(x1)] = x1 +
29287
[424(x1)] = x1 +
34982
[430(x1)] = x1 +
0
[41(x1)] = x1 +
0
[47(x1)] = x1 +
0
[413(x1)] = x1 +
34974
[419(x1)] = x1 +
0
[425(x1)] = x1 +
5293
[431(x1)] = x1 +
8
[42(x1)] = x1 +
34640
[48(x1)] = x1 +
30418
[414(x1)] = x1 +
34974
[420(x1)] = x1 +
9924
[426(x1)] = x1 +
34974
[432(x1)] = x1 +
34974
[43(x1)] = x1 +
0
[49(x1)] = x1 +
34982
[415(x1)] = x1 +
0
[421(x1)] = x1 +
0
[427(x1)] = x1 +
0
[433(x1)] = x1 +
34974
[44(x1)] = x1 +
34975
[410(x1)] = x1 +
34974
[416(x1)] = x1 +
34982
[422(x1)] = x1 +
4690
[428(x1)] = x1 +
34974
[434(x1)] = x1 +
0
[45(x1)] = x1 +
0
[411(x1)] = x1 +
32696
[417(x1)] = x1 +
34974
[423(x1)] = x1 +
0
[429(x1)] = x1 +
34974
[435(x1)] = x1 +
29823
[30(x1)] = x1 +
0
[36(x1)] = x1 +
0
[312(x1)] = x1 +
34974
[318(x1)] = x1 +
0
[324(x1)] = x1 +
9917
[330(x1)] = x1 +
34974
[31(x1)] = x1 +
0
[37(x1)] = x1 +
0
[313(x1)] = x1 +
34974
[319(x1)] = x1 +
8
[325(x1)] = x1 +
1
[331(x1)] = x1 +
0
[32(x1)] = x1 +
34974
[38(x1)] = x1 +
0
[314(x1)] = x1 +
34974
[320(x1)] = x1 +
34974
[326(x1)] = x1 +
34974
[332(x1)] = x1 +
0
[33(x1)] = x1 +
0
[39(x1)] = x1 +
9849
[315(x1)] = x1 +
9916
[321(x1)] = x1 +
34974
[327(x1)] = x1 +
1
[333(x1)] = x1 +
34974
[34(x1)] = x1 +
34982
[310(x1)] = x1 +
34974
[316(x1)] = x1 +
34974
[322(x1)] = x1 +
0
[328(x1)] = x1 +
34974
[334(x1)] = x1 +
34974
[35(x1)] = x1 +
0
[311(x1)] = x1 +
10251
[317(x1)] = x1 +
25125
[323(x1)] = x1 +
20569
[329(x1)] = x1 +
0
[335(x1)] = x1 +
34974
[20(x1)] = x1 +
1
[26(x1)] = x1 +
34974
[212(x1)] = x1 +
8
[218(x1)] = x1 +
24865
[224(x1)] = x1 +
1
[230(x1)] = x1 +
8
[21(x1)] = x1 +
8
[27(x1)] = x1 +
8
[213(x1)] = x1 +
29614
[219(x1)] = x1 +
8
[225(x1)] = x1 +
34982
[231(x1)] = x1 +
8
[22(x1)] = x1 +
34974
[28(x1)] = x1 +
8
[214(x1)] = x1 +
34974
[220(x1)] = x1 +
34974
[226(x1)] = x1 +
34974
[232(x1)] = x1 +
34974
[23(x1)] = x1 +
0
[29(x1)] = x1 +
34974
[215(x1)] = x1 +
34974
[221(x1)] = x1 +
9447
[227(x1)] = x1 +
8
[233(x1)] = x1 +
8
[24(x1)] = x1 +
8
[210(x1)] = x1 +
8
[216(x1)] = x1 +
34974
[222(x1)] = x1 +
4690
[228(x1)] = x1 +
0
[234(x1)] = x1 +
1
[25(x1)] = x1 +
25058
[211(x1)] = x1 +
34974
[217(x1)] = x1 +
24329
[223(x1)] = x1 +
0
[229(x1)] = x1 +
24724
[235(x1)] = x1 +
29823
[10(x1)] = x1 +
0
[16(x1)] = x1 +
34982
[112(x1)] = x1 +
34975
[118(x1)] = x1 +
0
[124(x1)] = x1 +
34974
[130(x1)] = x1 +
34974
[11(x1)] = x1 +
34974
[17(x1)] = x1 +
9179
[113(x1)] = x1 +
1
[119(x1)] = x1 +
9179
[125(x1)] = x1 +
34974
[131(x1)] = x1 +
34974
[12(x1)] = x1 +
34974
[18(x1)] = x1 +
34974
[114(x1)] = x1 +
34982
[120(x1)] = x1 +
8
[126(x1)] = x1 +
34974
[132(x1)] = x1 +
14740
[13(x1)] = x1 +
5160
[19(x1)] = x1 +
15410
[115(x1)] = x1 +
34974
[121(x1)] = x1 +
15820
[127(x1)] = x1 +
5160
[133(x1)] = x1 +
15820
[14(x1)] = x1 +
34974
[110(x1)] = x1 +
34974
[116(x1)] = x1 +
34974
[122(x1)] = x1 +
8
[128(x1)] = x1 +
0
[134(x1)] = x1 +
16415
[15(x1)] = x1 +
25058
[111(x1)] = x1 +
32629
[117(x1)] = x1 +
0
[123(x1)] = x1 +
0
[129(x1)] = x1 +
34974
[135(x1)] = x1 +
34975
[00(x1)] = x1 +
34974
[06(x1)] = x1 +
34974
[012(x1)] = x1 +
1
[018(x1)] = x1 +
0
[024(x1)] = x1 +
34974
[030(x1)] = x1 +
0
[01(x1)] = x1 +
25795
[07(x1)] = x1 +
0
[013(x1)] = x1 +
34974
[019(x1)] = x1 +
0
[025(x1)] = x1 +
34982
[031(x1)] = x1 +
34974
[02(x1)] = x1 +
0
[08(x1)] = x1 +
1
[014(x1)] = x1 +
34639
[020(x1)] = x1 +
34982
[026(x1)] = x1 +
34982
[032(x1)] = x1 +
9917
[03(x1)] = x1 +
8
[09(x1)] = x1 +
0
[015(x1)] = x1 +
1
[021(x1)] = x1 +
25059
[027(x1)] = x1 +
14405
[033(x1)] = x1 +
25059
[04(x1)] = x1 +
0
[010(x1)] = x1 +
34975
[016(x1)] = x1 +
34974
[022(x1)] = x1 +
34975
[028(x1)] = x1 +
0
[034(x1)] = x1 +
34974
[05(x1)] = x1 +
25059
[011(x1)] = x1 +
34982
[017(x1)] = x1 +
34974
[023(x1)] = x1 +
1
[029(x1)] = x1 +
34974
[035(x1)] = x1 +
14740
all of the following rules can be deleted.

There are 51840 ruless (increase limit for explicit display).

1.1.1.1.1 R is empty

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