Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/85345)

The rewrite relation of the following TRS is considered.

0(0(1(2(x1)))) → 2(1(2(x1))) (1)
0(3(4(2(2(1(x1)))))) → 2(2(5(5(3(x1))))) (2)
2(2(2(5(0(2(1(x1))))))) → 4(2(4(5(4(4(5(x1))))))) (3)
2(2(5(1(1(3(4(x1))))))) → 2(5(0(3(3(4(x1)))))) (4)
3(2(2(4(0(3(4(x1))))))) → 3(2(1(0(4(0(x1)))))) (5)
4(5(1(4(2(4(1(x1))))))) → 0(0(4(4(0(1(x1)))))) (6)
0(3(4(3(0(0(0(3(x1)))))))) → 2(2(1(1(4(2(3(x1))))))) (7)
0(4(4(3(4(1(0(4(x1)))))))) → 2(5(2(0(3(1(x1)))))) (8)
4(4(1(5(1(3(1(1(3(x1))))))))) → 3(2(2(1(3(0(0(x1))))))) (9)
1(4(5(3(3(1(1(1(4(3(x1)))))))))) → 3(0(4(0(5(0(0(3(x1)))))))) (10)
1(0(3(2(4(5(3(3(2(3(5(x1))))))))))) → 4(0(3(0(1(2(3(2(3(5(x1)))))))))) (11)
1(0(3(5(5(2(0(0(4(2(5(x1))))))))))) → 3(0(2(0(1(1(1(3(3(3(5(3(x1)))))))))))) (12)
3(1(5(1(2(4(5(1(4(5(2(x1))))))))))) → 3(2(2(2(5(0(0(3(2(x1))))))))) (13)
3(3(4(4(0(4(0(1(2(0(4(x1))))))))))) → 3(0(5(1(5(0(3(1(5(3(1(x1))))))))))) (14)
1(4(5(0(1(4(1(0(3(3(1(3(x1)))))))))))) → 0(2(2(4(5(4(3(5(0(0(x1)))))))))) (15)
1(5(0(5(4(3(1(3(4(4(1(4(x1)))))))))))) → 4(5(0(3(4(5(0(5(3(2(x1)))))))))) (16)
2(0(5(2(2(4(4(1(5(2(2(4(x1)))))))))))) → 2(3(5(3(1(0(1(0(3(4(2(4(1(x1))))))))))))) (17)
0(3(3(0(5(1(3(0(4(3(3(2(3(x1))))))))))))) → 0(5(0(1(2(3(0(2(0(2(3(x1))))))))))) (18)
3(3(5(4(5(5(4(2(3(1(1(0(1(x1))))))))))))) → 4(1(3(3(3(4(5(2(4(3(4(1(5(x1))))))))))))) (19)
3(5(3(0(0(1(4(5(4(2(5(1(0(x1))))))))))))) → 3(0(2(0(0(1(2(5(3(0(0(x1))))))))))) (20)
3(5(3(0(2(5(0(1(5(2(1(1(3(x1))))))))))))) → 3(3(3(3(5(1(3(5(0(4(0(1(5(5(x1)))))))))))))) (21)
0(4(0(4(1(3(4(0(2(4(0(5(0(3(x1)))))))))))))) → 2(3(3(3(1(0(0(2(1(5(5(5(3(x1))))))))))))) (22)
1(1(0(4(1(3(0(3(2(4(2(3(2(2(x1)))))))))))))) → 3(0(1(4(1(5(0(5(2(0(4(4(5(1(x1)))))))))))))) (23)
5(0(1(4(5(5(2(2(2(0(3(4(4(3(4(x1))))))))))))))) → 5(4(1(4(1(1(3(1(2(2(2(1(3(1(2(x1))))))))))))))) (24)
0(4(4(3(4(2(3(5(0(4(2(4(3(1(1(0(x1)))))))))))))))) → 2(5(2(1(0(5(4(0(4(3(3(2(1(3(x1)))))))))))))) (25)
1(3(3(0(1(2(4(1(1(4(5(5(2(3(0(0(x1)))))))))))))))) → 0(3(5(0(5(0(0(4(4(4(2(1(4(0(2(2(x1)))))))))))))))) (26)
3(1(2(1(1(1(4(4(1(0(2(5(2(1(3(3(x1)))))))))))))))) → 1(3(4(2(0(3(0(5(3(1(0(1(4(4(5(3(x1)))))))))))))))) (27)
3(3(2(5(2(3(0(5(3(1(0(2(0(1(0(4(x1)))))))))))))))) → 0(2(0(3(5(1(2(1(5(2(4(2(3(0(4(5(x1)))))))))))))))) (28)
3(3(3(5(5(4(0(2(1(0(1(0(4(3(4(3(x1)))))))))))))))) → 3(2(4(4(3(5(4(0(4(4(0(4(1(4(3(3(x1)))))))))))))))) (29)
0(5(0(2(3(3(5(1(4(5(2(5(2(0(3(4(1(2(x1)))))))))))))))))) → 0(3(0(2(1(4(5(3(0(5(1(1(3(5(2(0(4(2(x1)))))))))))))))))) (30)
1(5(5(2(3(3(1(3(4(0(4(5(1(0(2(5(2(0(x1)))))))))))))))))) → 2(0(3(5(3(1(0(4(5(1(1(2(1(3(4(4(1(2(5(x1))))))))))))))))))) (31)
3(0(3(5(1(3(4(0(1(2(3(4(5(4(4(1(1(0(x1)))))))))))))))))) → 0(2(0(1(1(5(0(0(4(2(3(4(5(3(3(0(x1)))))))))))))))) (32)
5(1(1(0(0(2(4(2(1(5(0(5(4(1(3(5(4(0(x1)))))))))))))))))) → 5(0(3(1(4(5(5(2(5(1(5(1(4(3(5(5(4(x1))))))))))))))))) (33)
1(0(1(0(4(4(1(1(2(1(2(1(4(0(4(2(1(2(5(x1))))))))))))))))))) → 0(2(4(4(5(4(1(1(1(3(5(4(0(4(5(2(1(3(3(3(x1)))))))))))))))))))) (34)
1(5(5(4(1(5(2(3(0(3(3(2(4(5(3(4(1(1(4(x1))))))))))))))))))) → 4(4(0(4(4(0(0(5(5(0(5(4(0(3(3(0(1(1(4(x1))))))))))))))))))) (35)
2(2(3(5(1(5(4(0(0(1(4(0(1(3(5(0(3(5(1(x1))))))))))))))))))) → 2(1(5(0(2(1(4(4(3(1(3(5(0(4(3(1(2(x1))))))))))))))))) (36)
3(5(4(3(1(5(0(1(4(5(0(5(2(1(5(2(3(2(2(x1))))))))))))))))))) → 3(4(2(4(1(4(3(4(0(1(2(2(0(0(2(2(5(2(x1)))))))))))))))))) (37)
4(2(2(4(0(5(5(0(5(3(2(5(3(1(3(2(3(4(4(x1))))))))))))))))))) → 2(4(5(1(4(2(1(3(2(0(1(5(4(4(4(5(2(4(x1)))))))))))))))))) (38)
1(4(1(1(5(0(1(4(4(1(2(0(5(1(0(0(4(5(1(0(1(x1))))))))))))))))))))) → 0(4(3(5(3(4(2(4(2(4(1(0(2(4(4(1(3(4(4(4(1(x1))))))))))))))))))))) (39)
4(0(3(0(3(4(2(2(1(3(5(4(4(5(3(1(2(0(1(4(1(x1))))))))))))))))))))) → 2(1(0(2(5(0(0(2(0(3(3(0(5(0(2(1(3(1(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 +
374
[512(x1)] = x1 +
4
[518(x1)] = x1 +
374
[524(x1)] = x1 +
374
[530(x1)] = x1 +
374
[51(x1)] = x1 +
374
[57(x1)] = x1 +
0
[513(x1)] = x1 +
340
[519(x1)] = x1 +
0
[525(x1)] = x1 +
4
[531(x1)] = x1 +
4
[52(x1)] = x1 +
4
[58(x1)] = x1 +
0
[514(x1)] = x1 +
375
[520(x1)] = x1 +
0
[526(x1)] = x1 +
0
[532(x1)] = x1 +
4
[53(x1)] = x1 +
0
[59(x1)] = x1 +
0
[515(x1)] = x1 +
374
[521(x1)] = x1 +
374
[527(x1)] = x1 +
4
[533(x1)] = x1 +
374
[54(x1)] = x1 +
0
[510(x1)] = x1 +
374
[516(x1)] = x1 +
374
[522(x1)] = x1 +
0
[528(x1)] = x1 +
378
[534(x1)] = x1 +
378
[55(x1)] = x1 +
4
[511(x1)] = x1 +
0
[517(x1)] = x1 +
0
[523(x1)] = x1 +
0
[529(x1)] = x1 +
374
[535(x1)] = x1 +
0
[40(x1)] = x1 +
374
[46(x1)] = x1 +
0
[412(x1)] = x1 +
1
[418(x1)] = x1 +
0
[424(x1)] = x1 +
374
[430(x1)] = x1 +
4
[41(x1)] = x1 +
0
[47(x1)] = x1 +
0
[413(x1)] = x1 +
374
[419(x1)] = x1 +
0
[425(x1)] = x1 +
378
[431(x1)] = x1 +
0
[42(x1)] = x1 +
0
[48(x1)] = x1 +
374
[414(x1)] = x1 +
0
[420(x1)] = x1 +
204
[426(x1)] = x1 +
374
[432(x1)] = x1 +
374
[43(x1)] = x1 +
374
[49(x1)] = x1 +
0
[415(x1)] = x1 +
374
[421(x1)] = x1 +
374
[427(x1)] = x1 +
378
[433(x1)] = x1 +
374
[44(x1)] = x1 +
310
[410(x1)] = x1 +
374
[416(x1)] = x1 +
378
[422(x1)] = x1 +
242
[428(x1)] = x1 +
374
[434(x1)] = x1 +
378
[45(x1)] = x1 +
374
[411(x1)] = x1 +
238
[417(x1)] = x1 +
340
[423(x1)] = x1 +
4
[429(x1)] = x1 +
0
[435(x1)] = x1 +
374
[30(x1)] = x1 +
374
[36(x1)] = x1 +
374
[312(x1)] = x1 +
0
[318(x1)] = x1 +
0
[324(x1)] = x1 +
0
[330(x1)] = x1 +
0
[31(x1)] = x1 +
4
[37(x1)] = x1 +
378
[313(x1)] = x1 +
378
[319(x1)] = x1 +
340
[325(x1)] = x1 +
374
[331(x1)] = x1 +
374
[32(x1)] = x1 +
0
[38(x1)] = x1 +
0
[314(x1)] = x1 +
0
[320(x1)] = x1 +
205
[326(x1)] = x1 +
374
[332(x1)] = x1 +
306
[33(x1)] = x1 +
374
[39(x1)] = x1 +
375
[315(x1)] = x1 +
0
[321(x1)] = x1 +
0
[327(x1)] = x1 +
1
[333(x1)] = x1 +
4
[34(x1)] = x1 +
272
[310(x1)] = x1 +
0
[316(x1)] = x1 +
374
[322(x1)] = x1 +
0
[328(x1)] = x1 +
374
[334(x1)] = x1 +
1
[35(x1)] = x1 +
34
[311(x1)] = x1 +
0
[317(x1)] = x1 +
378
[323(x1)] = x1 +
0
[329(x1)] = x1 +
0
[335(x1)] = x1 +
374
[20(x1)] = x1 +
0
[26(x1)] = x1 +
374
[212(x1)] = x1 +
374
[218(x1)] = x1 +
374
[224(x1)] = x1 +
374
[230(x1)] = x1 +
374
[21(x1)] = x1 +
1
[27(x1)] = x1 +
204
[213(x1)] = x1 +
0
[219(x1)] = x1 +
374
[225(x1)] = x1 +
374
[231(x1)] = x1 +
374
[22(x1)] = x1 +
374
[28(x1)] = x1 +
374
[214(x1)] = x1 +
374
[220(x1)] = x1 +
374
[226(x1)] = x1 +
374
[232(x1)] = x1 +
374
[23(x1)] = x1 +
0
[29(x1)] = x1 +
374
[215(x1)] = x1 +
374
[221(x1)] = x1 +
374
[227(x1)] = x1 +
1
[233(x1)] = x1 +
374
[24(x1)] = x1 +
0
[210(x1)] = x1 +
0
[216(x1)] = x1 +
4
[222(x1)] = x1 +
374
[228(x1)] = x1 +
374
[234(x1)] = x1 +
137
[25(x1)] = x1 +
374
[211(x1)] = x1 +
4
[217(x1)] = x1 +
0
[223(x1)] = x1 +
0
[229(x1)] = x1 +
374
[235(x1)] = x1 +
374
[10(x1)] = x1 +
374
[16(x1)] = x1 +
238
[112(x1)] = x1 +
0
[118(x1)] = x1 +
238
[124(x1)] = x1 +
374
[130(x1)] = x1 +
238
[11(x1)] = x1 +
374
[17(x1)] = x1 +
0
[113(x1)] = x1 +
0
[119(x1)] = x1 +
136
[125(x1)] = x1 +
374
[131(x1)] = x1 +
374
[12(x1)] = x1 +
0
[18(x1)] = x1 +
374
[114(x1)] = x1 +
4
[120(x1)] = x1 +
204
[126(x1)] = x1 +
374
[132(x1)] = x1 +
208
[13(x1)] = x1 +
0
[19(x1)] = x1 +
378
[115(x1)] = x1 +
0
[121(x1)] = x1 +
0
[127(x1)] = x1 +
374
[133(x1)] = x1 +
374
[14(x1)] = x1 +
374
[110(x1)] = x1 +
374
[116(x1)] = x1 +
4
[122(x1)] = x1 +
4
[128(x1)] = x1 +
4
[134(x1)] = x1 +
374
[15(x1)] = x1 +
4
[111(x1)] = x1 +
374
[117(x1)] = x1 +
375
[123(x1)] = x1 +
374
[129(x1)] = x1 +
374
[135(x1)] = x1 +
378
[00(x1)] = x1 +
374
[06(x1)] = x1 +
374
[012(x1)] = x1 +
0
[018(x1)] = x1 +
0
[024(x1)] = x1 +
0
[030(x1)] = x1 +
0
[01(x1)] = x1 +
4
[07(x1)] = x1 +
374
[013(x1)] = x1 +
0
[019(x1)] = x1 +
374
[025(x1)] = x1 +
374
[031(x1)] = x1 +
68
[02(x1)] = x1 +
0
[08(x1)] = x1 +
238
[014(x1)] = x1 +
374
[020(x1)] = x1 +
375
[026(x1)] = x1 +
1
[032(x1)] = x1 +
0
[03(x1)] = x1 +
374
[09(x1)] = x1 +
140
[015(x1)] = x1 +
0
[021(x1)] = x1 +
0
[027(x1)] = x1 +
374
[033(x1)] = x1 +
0
[04(x1)] = x1 +
374
[010(x1)] = x1 +
374
[016(x1)] = x1 +
374
[022(x1)] = x1 +
374
[028(x1)] = x1 +
0
[034(x1)] = x1 +
374
[05(x1)] = x1 +
0
[011(x1)] = x1 +
0
[017(x1)] = x1 +
272
[023(x1)] = x1 +
0
[029(x1)] = x1 +
378
[035(x1)] = x1 +
374
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.