Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/98362)

The rewrite relation of the following TRS is considered.

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

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
3(2(1(0(x1)))) → 4(2(3(x1))) (36)
4(0(3(3(2(x1))))) → 5(1(2(4(x1)))) (37)
4(1(1(3(3(0(x1)))))) → 3(1(0(3(2(3(x1)))))) (38)
0(4(0(3(1(5(x1)))))) → 2(3(3(5(3(x1))))) (39)
1(0(4(1(2(3(0(x1))))))) → 1(0(3(1(5(0(4(x1))))))) (40)
0(2(5(2(0(2(2(x1))))))) → 3(1(5(2(1(2(2(x1))))))) (41)
2(2(4(4(2(3(5(4(x1)))))))) → 2(2(0(0(0(5(0(4(x1)))))))) (42)
1(1(0(3(2(5(2(0(1(x1))))))))) → 1(4(4(2(0(2(1(5(1(x1))))))))) (43)
4(4(2(1(5(3(0(4(4(x1))))))))) → 0(0(5(3(0(2(5(1(x1)))))))) (44)
4(2(2(0(5(0(2(1(4(0(x1)))))))))) → 4(3(5(2(1(0(4(5(2(x1))))))))) (45)
5(1(0(1(4(2(1(4(5(4(x1)))))))))) → 5(2(5(1(3(1(0(1(1(x1))))))))) (46)
3(4(5(4(3(2(3(0(1(5(x1)))))))))) → 3(5(2(4(3(5(0(2(5(x1))))))))) (47)
3(5(2(3(5(0(3(4(0(3(1(x1))))))))))) → 3(5(3(3(1(3(0(0(0(3(1(x1))))))))))) (48)
5(4(5(0(5(4(4(0(0(5(4(x1))))))))))) → 5(4(1(3(4(2(2(2(4(2(x1)))))))))) (49)
4(1(0(4(3(1(4(5(0(0(3(2(x1)))))))))))) → 4(2(4(1(1(5(4(3(3(0(4(x1))))))))))) (50)
0(4(5(5(2(1(5(0(3(4(1(5(x1)))))))))))) → 5(1(5(2(4(1(5(0(0(5(2(x1))))))))))) (51)
1(5(2(2(5(5(5(0(2(5(1(5(x1)))))))))))) → 2(0(1(5(0(3(5(0(0(4(2(2(x1)))))))))))) (52)
5(5(4(4(5(2(0(2(1(5(4(5(4(x1))))))))))))) → 3(5(5(2(4(0(3(3(4(3(1(1(x1)))))))))))) (53)
5(0(5(4(0(1(1(2(0(4(0(5(5(x1))))))))))))) → 5(2(0(5(1(1(2(4(3(0(1(1(x1)))))))))))) (54)
1(5(1(3(1(1(3(1(3(0(1(5(0(5(x1)))))))))))))) → 2(2(2(3(0(1(0(5(5(2(1(4(4(5(x1)))))))))))))) (55)
1(2(4(1(5(1(2(0(4(4(0(3(5(5(x1)))))))))))))) → 0(5(4(2(5(3(3(4(2(5(1(4(5(x1))))))))))))) (56)
1(3(2(5(1(3(0(1(2(2(3(2(4(1(0(x1))))))))))))))) → 2(4(0(1(4(3(4(2(5(2(4(2(5(1(x1)))))))))))))) (57)
0(2(3(3(4(4(2(4(2(2(0(0(2(3(0(x1))))))))))))))) → 0(1(3(3(4(0(1(2(4(4(0(1(2(3(x1)))))))))))))) (58)
5(0(0(1(4(3(0(4(1(0(1(0(3(4(4(2(x1)))))))))))))))) → 5(4(1(2(3(3(5(2(2(2(2(1(3(5(2(x1))))))))))))))) (59)
1(5(4(0(3(3(0(2(1(1(5(1(1(0(3(0(5(x1))))))))))))))))) → 2(5(1(3(1(4(5(1(0(2(1(0(2(5(4(3(x1)))))))))))))))) (60)
2(4(4(3(3(1(3(2(2(5(0(5(4(5(2(5(5(0(x1)))))))))))))))))) → 2(5(3(4(1(1(4(4(1(5(2(2(3(4(4(2(0(5(x1)))))))))))))))))) (61)
5(4(0(2(1(4(1(5(4(4(2(0(5(0(1(0(2(1(x1)))))))))))))))))) → 5(3(2(4(1(3(5(3(0(4(1(5(2(5(1(2(1(x1))))))))))))))))) (62)
5(1(0(3(4(4(4(1(2(5(3(0(0(3(3(4(4(1(x1)))))))))))))))))) → 5(3(4(3(2(1(3(4(4(5(1(5(2(0(3(3(5(3(x1)))))))))))))))))) (63)
3(0(1(4(5(0(4(4(2(3(1(4(2(5(4(1(5(3(x1)))))))))))))))))) → 3(2(1(3(3(5(5(0(0(3(4(1(5(4(5(5(0(3(x1)))))))))))))))))) (64)
1(5(3(2(2(0(3(3(1(0(5(4(3(0(4(0(5(4(x1)))))))))))))))))) → 1(0(1(0(2(0(0(0(5(3(2(1(1(0(3(5(1(x1))))))))))))))))) (65)
5(0(1(5(1(4(3(3(2(2(5(5(1(3(3(5(3(5(x1)))))))))))))))))) → 5(1(5(1(4(2(0(0(2(5(0(4(3(4(4(0(5(3(x1)))))))))))))))))) (66)
0(3(3(5(1(2(3(0(4(0(4(2(1(5(1(0(4(5(x1)))))))))))))))))) → 1(2(4(5(5(0(1(1(0(5(1(0(3(0(1(2(4(x1))))))))))))))))) (67)
0(1(5(2(4(0(3(4(0(0(1(0(0(3(0(4(2(0(5(3(x1)))))))))))))))))))) → 0(0(0(2(0(4(4(2(5(4(3(5(2(3(1(1(4(1(1(x1))))))))))))))))))) (68)
0(3(4(3(2(2(5(4(2(1(2(3(5(5(3(0(2(0(1(4(x1)))))))))))))))))))) → 0(3(5(3(5(0(5(4(4(0(0(1(2(2(1(4(5(5(0(0(x1)))))))))))))))))))) (69)
4(1(4(1(5(3(2(2(0(5(3(4(1(5(4(1(2(0(0(3(4(x1))))))))))))))))))))) → 3(1(3(2(1(4(2(1(1(2(5(1(2(4(0(1(1(4(3(4(x1)))))))))))))))))))) (70)

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[3(x1)] = 1 · x1 + 1
[2(x1)] = 1 · x1 + 1
[1(x1)] = 1 · x1 + 1
[0(x1)] = 1 · x1 + 1
[4(x1)] = 1 · x1 + 1
[5(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
3(2(1(0(x1)))) → 4(2(3(x1))) (36)
4(0(3(3(2(x1))))) → 5(1(2(4(x1)))) (37)
0(4(0(3(1(5(x1)))))) → 2(3(3(5(3(x1))))) (39)
4(4(2(1(5(3(0(4(4(x1))))))))) → 0(0(5(3(0(2(5(1(x1)))))))) (44)
4(2(2(0(5(0(2(1(4(0(x1)))))))))) → 4(3(5(2(1(0(4(5(2(x1))))))))) (45)
5(1(0(1(4(2(1(4(5(4(x1)))))))))) → 5(2(5(1(3(1(0(1(1(x1))))))))) (46)
3(4(5(4(3(2(3(0(1(5(x1)))))))))) → 3(5(2(4(3(5(0(2(5(x1))))))))) (47)
5(4(5(0(5(4(4(0(0(5(4(x1))))))))))) → 5(4(1(3(4(2(2(2(4(2(x1)))))))))) (49)
4(1(0(4(3(1(4(5(0(0(3(2(x1)))))))))))) → 4(2(4(1(1(5(4(3(3(0(4(x1))))))))))) (50)
0(4(5(5(2(1(5(0(3(4(1(5(x1)))))))))))) → 5(1(5(2(4(1(5(0(0(5(2(x1))))))))))) (51)
5(5(4(4(5(2(0(2(1(5(4(5(4(x1))))))))))))) → 3(5(5(2(4(0(3(3(4(3(1(1(x1)))))))))))) (53)
5(0(5(4(0(1(1(2(0(4(0(5(5(x1))))))))))))) → 5(2(0(5(1(1(2(4(3(0(1(1(x1)))))))))))) (54)
1(2(4(1(5(1(2(0(4(4(0(3(5(5(x1)))))))))))))) → 0(5(4(2(5(3(3(4(2(5(1(4(5(x1))))))))))))) (56)
1(3(2(5(1(3(0(1(2(2(3(2(4(1(0(x1))))))))))))))) → 2(4(0(1(4(3(4(2(5(2(4(2(5(1(x1)))))))))))))) (57)
0(2(3(3(4(4(2(4(2(2(0(0(2(3(0(x1))))))))))))))) → 0(1(3(3(4(0(1(2(4(4(0(1(2(3(x1)))))))))))))) (58)
5(0(0(1(4(3(0(4(1(0(1(0(3(4(4(2(x1)))))))))))))))) → 5(4(1(2(3(3(5(2(2(2(2(1(3(5(2(x1))))))))))))))) (59)
1(5(4(0(3(3(0(2(1(1(5(1(1(0(3(0(5(x1))))))))))))))))) → 2(5(1(3(1(4(5(1(0(2(1(0(2(5(4(3(x1)))))))))))))))) (60)
5(4(0(2(1(4(1(5(4(4(2(0(5(0(1(0(2(1(x1)))))))))))))))))) → 5(3(2(4(1(3(5(3(0(4(1(5(2(5(1(2(1(x1))))))))))))))))) (62)
1(5(3(2(2(0(3(3(1(0(5(4(3(0(4(0(5(4(x1)))))))))))))))))) → 1(0(1(0(2(0(0(0(5(3(2(1(1(0(3(5(1(x1))))))))))))))))) (65)
0(3(3(5(1(2(3(0(4(0(4(2(1(5(1(0(4(5(x1)))))))))))))))))) → 1(2(4(5(5(0(1(1(0(5(1(0(3(0(1(2(4(x1))))))))))))))))) (67)
0(1(5(2(4(0(3(4(0(0(1(0(0(3(0(4(2(0(5(3(x1)))))))))))))))))))) → 0(0(0(2(0(4(4(2(5(4(3(5(2(3(1(1(4(1(1(x1))))))))))))))))))) (68)
4(1(4(1(5(3(2(2(0(5(3(4(1(5(4(1(2(0(0(3(4(x1))))))))))))))))))))) → 3(1(3(2(1(4(2(1(1(2(5(1(2(4(0(1(1(4(3(4(x1)))))))))))))))))))) (70)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 157 ruless (increase limit for explicit display).

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.