Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/4840)

The rewrite relation of the following TRS is considered.

1(2(0(x1))) → 4(0(3(3(5(4(5(1(4(3(x1)))))))))) (1)
1(0(0(4(5(x1))))) → 1(4(3(1(3(1(4(5(2(3(x1)))))))))) (2)
2(0(3(0(2(x1))))) → 3(3(1(2(2(4(5(0(4(3(x1)))))))))) (3)
2(1(0(1(0(x1))))) → 3(5(4(5(4(3(3(1(1(2(x1)))))))))) (4)
3(4(2(0(2(x1))))) → 3(5(3(0(3(3(2(5(3(2(x1)))))))))) (5)
0(3(5(2(4(0(x1)))))) → 4(4(0(2(3(2(2(5(3(2(x1)))))))))) (6)
1(1(2(0(4(5(x1)))))) → 3(0(5(4(2(1(0(2(3(3(x1)))))))))) (7)
2(1(1(0(1(2(x1)))))) → 3(4(4(1(3(2(2(2(5(5(x1)))))))))) (8)
2(2(0(1(1(1(x1)))))) → 2(3(4(1(5(2(2(2(5(4(x1)))))))))) (9)
2(4(1(0(4(2(x1)))))) → 1(5(1(3(2(3(4(4(4(0(x1)))))))))) (10)
2(4(2(1(1(1(x1)))))) → 1(3(5(4(3(4(3(1(4(4(x1)))))))))) (11)
3(0(1(0(0(2(x1)))))) → 2(4(2(5(3(5(0(3(3(2(x1)))))))))) (12)
3(0(1(1(1(1(x1)))))) → 3(2(2(4(4(5(2(4(5(1(x1)))))))))) (13)
4(1(1(2(0(2(x1)))))) → 4(0(3(4(4(4(2(3(2(3(x1)))))))))) (14)
0(2(1(1(1(1(0(x1))))))) → 0(1(5(5(3(5(2(5(5(5(x1)))))))))) (15)
0(2(4(1(1(1(5(x1))))))) → 4(4(3(4(3(2(3(0(2(2(x1)))))))))) (16)
0(4(2(0(0(4(1(x1))))))) → 4(2(5(4(1(0(4(3(3(1(x1)))))))))) (17)
0(4(3(0(5(4(1(x1))))))) → 0(3(1(5(3(1(2(5(4(1(x1)))))))))) (18)
1(0(5(2(2(0(0(x1))))))) → 1(5(4(4(3(4(5(4(5(2(x1)))))))))) (19)
1(1(3(4(5(0(0(x1))))))) → 1(3(1(5(3(4(1(4(5(3(x1)))))))))) (20)
1(4(3(1(5(0(5(x1))))))) → 5(0(3(3(2(4(1(3(3(2(x1)))))))))) (21)
1(5(0(2(0(5(5(x1))))))) → 2(5(2(5(4(2(0(0(5(5(x1)))))))))) (22)
2(0(1(5(2(0(5(x1))))))) → 4(3(3(5(5(3(1(3(5(5(x1)))))))))) (23)
2(4(0(5(4(1(4(x1))))))) → 3(4(5(5(1(5(3(5(1(4(x1)))))))))) (24)
3(4(1(4(0(4(5(x1))))))) → 3(2(2(1(3(4(3(3(0(3(x1)))))))))) (25)
4(1(0(4(2(0(0(x1))))))) → 4(2(2(3(1(0(0(3(4(0(x1)))))))))) (26)
4(1(0(4(2(0(3(x1))))))) → 0(4(3(0(0(1(5(4(3(2(x1)))))))))) (27)
4(1(1(1(0(1(2(x1))))))) → 3(3(2(3(3(0(1(5(5(2(x1)))))))))) (28)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
0(2(1(x1))) → 3(4(1(5(4(5(3(3(0(4(x1)))))))))) (29)
5(4(0(0(1(x1))))) → 3(2(5(4(1(3(1(3(4(1(x1)))))))))) (30)
2(0(3(0(2(x1))))) → 3(4(0(5(4(2(2(1(3(3(x1)))))))))) (31)
0(1(0(1(2(x1))))) → 2(1(1(3(3(4(5(4(5(3(x1)))))))))) (32)
2(0(2(4(3(x1))))) → 2(3(5(2(3(3(0(3(5(3(x1)))))))))) (33)
0(4(2(5(3(0(x1)))))) → 2(3(5(2(2(3(2(0(4(4(x1)))))))))) (34)
5(4(0(2(1(1(x1)))))) → 3(3(2(0(1(2(4(5(0(3(x1)))))))))) (35)
2(1(0(1(1(2(x1)))))) → 5(5(2(2(2(3(1(4(4(3(x1)))))))))) (36)
1(1(1(0(2(2(x1)))))) → 4(5(2(2(2(5(1(4(3(2(x1)))))))))) (37)
2(4(0(1(4(2(x1)))))) → 0(4(4(4(3(2(3(1(5(1(x1)))))))))) (38)
1(1(1(2(4(2(x1)))))) → 4(4(1(3(4(3(4(5(3(1(x1)))))))))) (39)
2(0(0(1(0(3(x1)))))) → 2(3(3(0(5(3(5(2(4(2(x1)))))))))) (40)
1(1(1(1(0(3(x1)))))) → 1(5(4(2(5(4(4(2(2(3(x1)))))))))) (41)
2(0(2(1(1(4(x1)))))) → 3(2(3(2(4(4(4(3(0(4(x1)))))))))) (42)
0(1(1(1(1(2(0(x1))))))) → 5(5(5(2(5(3(5(5(1(0(x1)))))))))) (43)
5(1(1(1(4(2(0(x1))))))) → 2(2(0(3(2(3(4(3(4(4(x1)))))))))) (44)
1(4(0(0(2(4(0(x1))))))) → 1(3(3(4(0(1(4(5(2(4(x1)))))))))) (45)
1(4(5(0(3(4(0(x1))))))) → 1(4(5(2(1(3(5(1(3(0(x1)))))))))) (46)
0(0(2(2(5(0(1(x1))))))) → 2(5(4(5(4(3(4(4(5(1(x1)))))))))) (47)
0(0(5(4(3(1(1(x1))))))) → 3(5(4(1(4(3(5(1(3(1(x1)))))))))) (48)
5(0(5(1(3(4(1(x1))))))) → 2(3(3(1(4(2(3(3(0(5(x1)))))))))) (49)
5(5(0(2(0(5(1(x1))))))) → 5(5(0(0(2(4(5(2(5(2(x1)))))))))) (50)
5(0(2(5(1(0(2(x1))))))) → 5(5(3(1(3(5(5(3(3(4(x1)))))))))) (51)
4(1(4(5(0(4(2(x1))))))) → 4(1(5(3(5(1(5(5(4(3(x1)))))))))) (52)
5(4(0(4(1(4(3(x1))))))) → 3(0(3(3(4(3(1(2(2(3(x1)))))))))) (53)
0(0(2(4(0(1(4(x1))))))) → 0(4(3(0(0(1(3(2(2(4(x1)))))))))) (54)
3(0(2(4(0(1(4(x1))))))) → 2(3(4(5(1(0(0(3(4(0(x1)))))))))) (55)
2(1(0(1(1(1(4(x1))))))) → 2(5(5(1(0(3(3(2(3(3(x1)))))))))) (56)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 269 ruless (increase limit for explicit display).

1.1.1 Split

We split (P,R) into the relative DP-problem (PD,P-PD,RD,R-RD) and (P-PD,R-RD) where the pairs PD

There are 245 ruless (increase limit for explicit display).

and the rules RD

There are no rules.

are deleted.

1.1.1.1 Semantic Labeling Processor

The following interpretations form a model of the rules.

As carrier we take the set {0,1}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 2):

[5(x1)] = 0
[3#(x1)] = 0
[0#(x1)] = 0
[4#(x1)] = 0
[1#(x1)] = 0
[5#(x1)] = 0
[2#(x1)] = 0
[0(x1)] = 0
[1(x1)] = 1
[2(x1)] = 0
[3(x1)] = 0
[4(x1)] = 1

We obtain the set of labeled pairs

There are 538 ruless (increase limit for explicit display).

and the set of labeled rules:
00(21(10(x1))) → 31(41(10(51(40(50(30(30(01(40(x1)))))))))) (864)
00(21(11(x1))) → 31(41(10(51(40(50(30(30(01(41(x1)))))))))) (865)
51(40(00(01(10(x1))))) → 30(20(51(41(10(31(10(31(41(10(x1)))))))))) (866)
51(40(00(01(11(x1))))) → 30(20(51(41(10(31(10(31(41(11(x1)))))))))) (867)
20(00(30(00(20(x1))))) → 31(40(00(51(40(20(21(10(30(30(x1)))))))))) (868)
20(00(30(00(21(x1))))) → 31(40(00(51(40(20(21(10(30(31(x1)))))))))) (869)
01(10(01(10(20(x1))))) → 21(11(10(30(31(40(51(40(50(30(x1)))))))))) (870)
01(10(01(10(21(x1))))) → 21(11(10(30(31(40(51(40(50(31(x1)))))))))) (871)
20(00(21(40(30(x1))))) → 20(30(50(20(30(30(00(30(50(30(x1)))))))))) (872)
20(00(21(40(31(x1))))) → 20(30(50(20(30(30(00(30(50(31(x1)))))))))) (873)
01(40(20(50(30(00(x1)))))) → 20(30(50(20(20(30(20(01(41(40(x1)))))))))) (874)
01(40(20(50(30(01(x1)))))) → 20(30(50(20(20(30(20(01(41(41(x1)))))))))) (875)
51(40(00(21(11(10(x1)))))) → 30(30(20(01(10(21(40(50(00(30(x1)))))))))) (876)
51(40(00(21(11(11(x1)))))) → 30(30(20(01(10(21(40(50(00(31(x1)))))))))) (877)
21(10(01(11(10(20(x1)))))) → 50(50(20(20(20(31(11(41(40(30(x1)))))))))) (878)
21(10(01(11(10(21(x1)))))) → 50(50(20(20(20(31(11(41(40(31(x1)))))))))) (879)
11(11(10(00(20(20(x1)))))) → 40(50(20(20(20(51(11(40(30(20(x1)))))))))) (880)
11(11(10(00(20(21(x1)))))) → 40(50(20(20(20(51(11(40(30(21(x1)))))))))) (881)
21(40(01(11(40(20(x1)))))) → 01(41(41(40(30(20(31(10(51(10(x1)))))))))) (882)
21(40(01(11(40(21(x1)))))) → 01(41(41(40(30(20(31(10(51(11(x1)))))))))) (883)
11(11(10(21(40(20(x1)))))) → 41(41(10(31(40(31(40(50(31(10(x1)))))))))) (884)
11(11(10(21(40(21(x1)))))) → 41(41(10(31(40(31(40(50(31(11(x1)))))))))) (885)
20(00(01(10(00(30(x1)))))) → 20(30(30(00(50(30(50(21(40(20(x1)))))))))) (886)
20(00(01(10(00(31(x1)))))) → 20(30(30(00(50(30(50(21(40(21(x1)))))))))) (887)
11(11(11(10(00(30(x1)))))) → 10(51(40(20(51(41(40(20(20(30(x1)))))))))) (888)
11(11(11(10(00(31(x1)))))) → 10(51(40(20(51(41(40(20(20(31(x1)))))))))) (889)
20(00(21(11(11(40(x1)))))) → 30(20(30(21(41(41(40(30(01(40(x1)))))))))) (890)
20(00(21(11(11(41(x1)))))) → 30(20(30(21(41(41(40(30(01(41(x1)))))))))) (891)
01(11(11(11(10(20(00(x1))))))) → 50(50(50(20(50(30(50(51(10(00(x1)))))))))) (892)
01(11(11(11(10(20(01(x1))))))) → 50(50(50(20(50(30(50(51(10(01(x1)))))))))) (893)
51(11(11(11(40(20(00(x1))))))) → 20(20(00(30(20(31(40(31(41(40(x1)))))))))) (894)
51(11(11(11(40(20(01(x1))))))) → 20(20(00(30(20(31(40(31(41(41(x1)))))))))) (895)
11(40(00(00(21(40(00(x1))))))) → 10(30(31(40(01(11(40(50(21(40(x1)))))))))) (896)
11(40(00(00(21(40(01(x1))))))) → 10(30(31(40(01(11(40(50(21(41(x1)))))))))) (897)
11(40(50(00(31(40(00(x1))))))) → 11(40(50(21(10(30(51(10(30(00(x1)))))))))) (898)
11(40(50(00(31(40(01(x1))))))) → 11(40(50(21(10(30(51(10(30(01(x1)))))))))) (899)
00(00(20(20(50(01(10(x1))))))) → 20(51(40(51(40(31(41(40(51(10(x1)))))))))) (900)
00(00(20(20(50(01(11(x1))))))) → 20(51(40(51(40(31(41(40(51(11(x1)))))))))) (901)
00(00(51(40(31(11(10(x1))))))) → 30(51(41(11(40(30(51(10(31(10(x1)))))))))) (902)
00(00(51(40(31(11(11(x1))))))) → 30(51(41(11(40(30(51(10(31(11(x1)))))))))) (903)
50(00(51(10(31(41(10(x1))))))) → 20(30(31(11(40(20(30(30(00(50(x1)))))))))) (904)
50(00(51(10(31(41(11(x1))))))) → 20(30(31(11(40(20(30(30(00(51(x1)))))))))) (905)
50(50(00(20(00(51(10(x1))))))) → 50(50(00(00(21(40(50(20(50(20(x1)))))))))) (906)
50(50(00(20(00(51(11(x1))))))) → 50(50(00(00(21(40(50(20(50(21(x1)))))))))) (907)
50(00(20(51(10(00(20(x1))))))) → 50(50(31(10(30(50(50(30(31(40(x1)))))))))) (908)
50(00(20(51(10(00(21(x1))))))) → 50(50(31(10(30(50(50(30(31(41(x1)))))))))) (909)
41(11(40(50(01(40(20(x1))))))) → 41(10(50(30(51(10(50(51(40(30(x1)))))))))) (910)
41(11(40(50(01(40(21(x1))))))) → 41(10(50(30(51(10(50(51(40(31(x1)))))))))) (911)
51(40(01(41(11(40(30(x1))))))) → 30(00(30(31(40(31(10(20(20(30(x1)))))))))) (912)
51(40(01(41(11(40(31(x1))))))) → 30(00(30(31(40(31(10(20(20(31(x1)))))))))) (913)
00(00(21(40(01(11(40(x1))))))) → 01(40(30(00(01(10(30(20(21(40(x1)))))))))) (914)
00(00(21(40(01(11(41(x1))))))) → 01(40(30(00(01(10(30(20(21(41(x1)))))))))) (915)
30(00(21(40(01(11(40(x1))))))) → 20(31(40(51(10(00(00(31(40(00(x1)))))))))) (916)
30(00(21(40(01(11(41(x1))))))) → 20(31(40(51(10(00(00(31(40(01(x1)))))))))) (917)
21(10(01(11(11(11(40(x1))))))) → 20(50(51(10(00(30(30(20(30(30(x1)))))))))) (918)
21(10(01(11(11(11(41(x1))))))) → 20(50(51(10(00(30(30(20(30(31(x1)))))))))) (919)

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

1.1.1.2 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals
[0#(x1)] = 2 + x1
[2#(x1)] = -2 + x1
[4#(x1)] = -1 + x1
[5#(x1)] = x1
[0(x1)] = 2 + x1
[3#(x1)] = -2 + x1
[2(x1)] = x1
[3(x1)] = -2 + x1
[4(x1)] = x1
[1(x1)] = 2 + x1
[5(x1)] = x1
[1#(x1)] = x1
the pairs
0#(2(1(x1))) → 0#(4(x1)) (65)
0#(2(1(x1))) → 4#(x1) (66)
0#(1(0(1(2(x1))))) → 3#(x1) (95)
0#(4(2(5(3(0(x1)))))) → 4#(4(x1)) (113)
0#(4(2(5(3(0(x1)))))) → 4#(x1) (114)
1#(1(1(0(2(2(x1)))))) → 5#(1(4(3(2(x1))))) (140)
2#(4(0(1(4(2(x1)))))) → 1#(x1) (153)
1#(1(1(2(4(2(x1)))))) → 1#(x1) (163)
5#(1(1(1(4(2(0(x1))))))) → 4#(4(x1)) (209)
5#(1(1(1(4(2(0(x1))))))) → 4#(x1) (210)
1#(4(0(0(2(4(0(x1))))))) → 0#(1(4(5(2(4(x1)))))) (215)
1#(4(0(0(2(4(0(x1))))))) → 2#(4(x1)) (219)
1#(4(0(0(2(4(0(x1))))))) → 4#(x1) (220)
1#(4(5(0(3(4(0(x1))))))) → 3#(0(x1)) (229)
0#(0(2(2(5(0(1(x1))))))) → 5#(1(x1)) (238)
4#(1(4(5(0(4(2(x1))))))) → 3#(x1) (287)
0#(0(2(4(0(1(4(x1))))))) → 0#(4(3(0(0(1(3(2(2(4(x1)))))))))) (297)
0#(0(2(4(0(1(4(x1))))))) → 3#(0(0(1(3(2(2(4(x1)))))))) (299)
0#(0(2(4(0(1(4(x1))))))) → 0#(0(1(3(2(2(4(x1))))))) (300)
0#(0(2(4(0(1(4(x1))))))) → 0#(1(3(2(2(4(x1)))))) (301)
0#(0(2(4(0(1(4(x1))))))) → 2#(4(x1)) (305)
3#(0(2(4(0(1(4(x1))))))) → 0#(x1) (315)
could be deleted.

1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.