Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/133486)

The relative rewrite relation R/S is considered where R is the following TRS

0(0(5(4(1(1(3(2(3(4(0(2(x1)))))))))))) 4(5(0(5(5(4(3(1(5(1(1(2(0(5(3(0(x1)))))))))))))))) (1)
0(1(3(1(3(4(2(2(2(2(2(5(x1)))))))))))) 5(5(3(3(1(3(1(1(4(5(5(1(0(2(5(x1))))))))))))))) (2)
0(1(5(0(2(0(2(5(0(3(2(0(x1)))))))))))) 2(5(4(4(5(4(5(4(4(4(3(2(4(4(3(4(x1)))))))))))))))) (3)
0(1(5(1(1(5(3(5(4(1(1(3(x1)))))))))))) 2(0(5(4(4(4(5(5(0(4(0(0(3(1(1(x1))))))))))))))) (4)
0(1(5(2(1(4(0(2(2(5(0(4(x1)))))))))))) 4(4(5(5(5(4(1(1(1(4(5(4(4(4(4(1(0(4(x1)))))))))))))))))) (5)
0(2(2(1(4(0(3(2(4(1(2(4(x1)))))))))))) 2(0(0(1(0(5(3(4(5(3(0(1(5(5(x1)))))))))))))) (6)
0(3(0(3(2(3(0(1(1(3(0(3(x1)))))))))))) 0(5(5(0(5(4(3(2(4(5(4(3(5(0(2(0(3(x1))))))))))))))))) (7)
0(3(1(4(3(3(2(4(1(4(0(4(x1)))))))))))) 0(3(0(4(5(5(1(3(1(4(4(1(0(4(1(2(x1)))))))))))))))) (8)
0(3(2(4(5(2(2(3(2(0(1(5(x1)))))))))))) 5(4(5(5(0(2(5(1(2(4(1(0(5(3(3(5(x1)))))))))))))))) (9)
0(3(3(2(0(4(0(2(3(5(0(4(x1)))))))))))) 5(4(3(0(4(3(4(5(3(5(5(5(5(1(3(x1))))))))))))))) (10)
0(4(2(5(1(1(1(1(1(3(2(4(x1)))))))))))) 3(5(0(5(0(2(4(3(2(4(3(1(4(1(0(x1))))))))))))))) (11)
0(4(3(3(4(1(5(0(3(2(0(4(x1)))))))))))) 5(1(2(4(4(1(5(5(4(1(4(2(5(4(0(0(x1)))))))))))))))) (12)
0(4(4(1(1(4(0(3(0(4(0(0(x1)))))))))))) 0(4(1(0(1(0(4(5(4(4(4(5(4(2(4(0(2(x1))))))))))))))))) (13)
1(0(1(1(3(4(3(2(0(4(1(4(x1)))))))))))) 0(0(5(5(4(1(2(5(5(0(1(4(4(2(x1)))))))))))))) (14)
1(0(4(2(2(3(1(0(0(3(3(4(x1)))))))))))) 1(1(0(4(4(4(0(0(5(5(0(5(1(5(1(x1))))))))))))))) (15)
1(0(5(5(2(3(3(3(2(0(0(0(x1)))))))))))) 4(1(3(1(4(3(5(1(5(4(4(5(5(3(4(3(1(x1))))))))))))))))) (16)
1(1(1(0(4(3(0(0(4(2(3(0(x1)))))))))))) 4(5(0(4(4(3(4(0(5(3(4(0(0(5(5(x1))))))))))))))) (17)
1(1(1(1(2(3(1(3(0(2(3(4(x1)))))))))))) 1(0(5(1(3(5(5(4(5(5(4(4(4(1(2(5(5(2(x1)))))))))))))))))) (18)
1(1(1(3(1(2(3(1(5(0(1(5(x1)))))))))))) 5(5(5(1(0(5(3(1(3(5(3(3(1(5(x1)))))))))))))) (19)
1(1(1(3(2(1(1(1(5(4(3(2(x1)))))))))))) 3(1(4(4(3(5(2(0(5(2(4(3(0(2(0(x1))))))))))))))) (20)
1(1(3(0(3(2(1(5(4(5(1(0(x1)))))))))))) 1(1(4(0(5(4(5(5(4(5(1(4(1(1(4(x1))))))))))))))) (21)
1(1(3(3(2(1(4(4(5(1(1(4(x1)))))))))))) 0(0(3(5(1(5(4(1(1(5(1(0(2(4(5(x1))))))))))))))) (22)
1(1(3(4(2(0(2(2(4(2(0(3(x1)))))))))))) 2(3(5(5(0(2(4(5(3(2(2(5(4(4(1(x1))))))))))))))) (23)
1(1(4(3(1(2(2(3(3(5(3(2(x1)))))))))))) 1(5(5(3(2(2(3(2(4(5(5(4(3(3(x1)))))))))))))) (24)
1(2(0(3(3(2(4(3(5(2(1(0(x1)))))))))))) 3(2(2(5(4(0(4(3(5(4(0(0(3(0(x1)))))))))))))) (25)
1(2(0(3(3(3(5(0(0(3(2(1(x1)))))))))))) 5(2(2(1(0(5(4(4(5(1(0(5(5(5(5(3(x1)))))))))))))))) (26)
1(2(0(4(4(1(3(3(0(2(1(1(x1)))))))))))) 3(4(4(1(2(5(4(5(5(3(3(1(2(4(4(x1))))))))))))))) (27)
1(2(2(0(1(2(3(0(1(5(0(5(x1)))))))))))) 3(5(5(5(5(4(5(5(0(5(5(0(0(2(1(x1))))))))))))))) (28)
1(2(2(3(3(3(2(4(5(4(1(2(x1)))))))))))) 1(5(3(5(5(1(4(3(0(4(4(4(1(2(4(2(x1)))))))))))))))) (29)
1(2(3(2(1(0(3(0(2(1(2(1(x1)))))))))))) 3(1(4(1(5(5(4(5(1(4(1(4(5(0(2(0(1(x1))))))))))))))))) (30)
1(2(3(3(1(3(2(1(3(0(4(0(x1)))))))))))) 4(5(4(4(1(2(3(5(5(1(3(5(1(3(1(4(3(0(x1)))))))))))))))))) (31)
1(2(4(1(4(0(1(3(5(5(2(1(x1)))))))))))) 3(0(5(4(4(0(0(3(1(5(0(2(2(1(x1)))))))))))))) (32)
1(3(2(0(0(3(2(5(2(2(2(2(x1)))))))))))) 0(2(2(1(3(3(0(4(5(1(3(5(1(4(2(x1))))))))))))))) (33)
1(3(2(0(2(3(0(0(0(1(0(5(x1)))))))))))) 4(5(1(0(4(4(3(5(4(5(2(4(5(0(x1)))))))))))))) (34)
1(3(2(4(2(3(2(2(1(2(2(5(x1)))))))))))) 4(4(4(0(3(5(1(1(2(0(1(4(4(4(5(5(5(x1))))))))))))))))) (35)
1(5(4(0(3(3(1(2(2(2(2(0(x1)))))))))))) 0(0(4(2(4(2(5(4(5(4(0(0(5(4(4(1(2(x1))))))))))))))))) (36)
2(0(1(1(5(0(3(2(5(5(1(0(x1)))))))))))) 4(4(5(2(5(1(3(2(5(1(4(5(5(5(4(4(2(x1))))))))))))))))) (37)
2(0(1(2(2(2(3(2(1(5(0(2(x1)))))))))))) 4(2(4(0(5(2(2(4(2(0(4(5(3(4(4(4(x1)))))))))))))))) (38)
2(0(2(1(5(3(2(2(5(2(0(3(x1)))))))))))) 3(5(3(5(5(1(4(5(0(5(5(4(1(2(4(4(1(x1))))))))))))))))) (39)
2(0(2(2(3(1(2(2(3(2(1(2(x1)))))))))))) 2(2(2(5(0(4(5(0(0(3(4(4(0(2(5(5(4(x1))))))))))))))))) (40)
2(0(2(4(5(3(1(2(2(2(1(5(x1)))))))))))) 1(2(1(0(4(4(4(0(1(2(5(0(4(5(x1)))))))))))))) (41)
2(1(0(2(3(5(2(0(2(1(3(1(x1)))))))))))) 0(4(4(5(1(0(0(3(5(5(4(4(4(4(2(2(5(x1))))))))))))))))) (42)
2(1(2(2(3(1(1(4(5(1(0(2(x1)))))))))))) 4(1(0(4(3(5(1(3(5(5(4(1(0(5(4(4(0(x1))))))))))))))))) (43)
2(1(5(3(2(0(5(0(5(1(5(1(x1)))))))))))) 5(4(4(0(4(4(5(5(3(3(1(4(5(1(4(x1))))))))))))))) (44)
2(2(0(3(2(1(3(2(0(3(5(2(x1)))))))))))) 0(3(0(3(5(4(1(3(1(0(4(1(4(4(4(1(0(1(x1)))))))))))))))))) (45)
2(2(0(5(5(5(1(3(4(2(1(5(x1)))))))))))) 4(4(0(0(5(3(2(0(4(5(3(5(1(3(x1)))))))))))))) (46)
2(2(1(1(3(0(0(0(3(1(4(1(x1)))))))))))) 1(4(5(0(0(4(4(4(4(0(5(4(4(4(4(1(2(3(x1)))))))))))))))))) (47)
2(2(1(5(3(1(0(1(1(0(2(3(x1)))))))))))) 1(4(4(3(4(3(1(3(1(5(5(0(0(0(2(x1))))))))))))))) (48)
2(2(3(1(3(2(2(2(0(1(5(0(x1)))))))))))) 2(1(2(4(4(0(4(5(2(3(0(3(5(1(x1)))))))))))))) (49)
2(2(3(2(0(5(0(1(1(4(3(2(x1)))))))))))) 1(5(1(2(4(4(4(4(3(5(4(1(3(4(0(5(4(x1))))))))))))))))) (50)
2(2(3(2(3(3(3(4(3(2(1(4(x1)))))))))))) 0(1(0(5(4(4(1(2(3(0(4(4(3(2(2(2(0(x1))))))))))))))))) (51)
2(2(3(3(2(1(4(1(1(3(0(4(x1)))))))))))) 3(3(0(4(5(5(4(4(5(3(3(5(1(4(4(2(1(5(x1)))))))))))))))))) (52)
2(2(5(0(5(0(4(1(1(2(2(0(x1)))))))))))) 5(5(3(4(4(4(5(4(4(5(0(2(4(1(3(2(1(x1))))))))))))))))) (53)
2(3(2(1(1(4(5(4(3(2(1(1(x1)))))))))))) 3(0(2(1(5(5(5(1(2(4(4(5(4(3(0(4(0(x1))))))))))))))))) (54)
2(3(4(1(4(3(0(4(2(0(2(0(x1)))))))))))) 5(3(1(1(5(4(2(5(4(5(4(4(5(4(3(5(4(x1))))))))))))))))) (55)
2(3(4(3(5(3(3(0(4(2(4(0(x1)))))))))))) 0(5(5(4(0(5(4(4(1(5(1(2(1(2(1(x1))))))))))))))) (56)
2(4(4(2(4(0(3(3(3(3(5(2(x1)))))))))))) 0(5(4(4(1(0(0(4(4(5(2(0(4(2(x1)))))))))))))) (57)
2(5(0(0(5(2(1(1(1(3(1(0(x1)))))))))))) 4(5(5(4(3(2(2(1(0(1(5(4(5(5(5(0(x1)))))))))))))))) (58)
2(5(2(1(4(0(1(4(2(1(1(2(x1)))))))))))) 5(4(4(2(3(5(0(4(4(1(2(1(5(1(x1)))))))))))))) (59)
2(5(3(2(5(1(2(3(2(5(1(2(x1)))))))))))) 0(4(4(0(4(2(5(2(5(3(5(5(5(5(1(x1))))))))))))))) (60)
3(0(0(2(2(3(1(0(3(5(3(0(x1)))))))))))) 3(0(5(4(4(0(5(0(0(5(1(0(0(0(4(x1))))))))))))))) (61)
3(0(0(2(4(2(1(3(2(5(1(2(x1)))))))))))) 1(1(4(5(0(5(4(0(0(5(4(4(2(4(4(4(4(1(x1)))))))))))))))))) (62)
3(0(1(4(1(5(5(0(1(5(1(2(x1)))))))))))) 1(0(1(5(0(0(4(5(0(0(5(3(4(4(5(x1))))))))))))))) (63)
3(0(3(3(1(2(0(3(0(1(1(5(x1)))))))))))) 5(5(2(4(2(3(4(5(1(5(1(1(5(4(1(0(x1)))))))))))))))) (64)
3(0(3(5(3(1(3(5(2(3(0(5(x1)))))))))))) 0(0(1(5(3(5(0(5(3(5(5(1(4(0(2(x1))))))))))))))) (65)
3(0(4(2(2(1(5(1(5(5(0(3(x1)))))))))))) 3(4(5(5(5(0(5(3(4(4(0(2(5(3(x1)))))))))))))) (66)
3(2(0(2(1(3(4(3(5(3(4(3(x1)))))))))))) 1(3(1(4(4(0(2(1(0(4(1(1(0(4(3(5(x1)))))))))))))))) (67)
3(3(3(3(2(1(1(3(3(0(3(5(x1)))))))))))) 3(4(4(1(4(5(5(0(5(3(4(5(0(2(0(0(4(0(x1)))))))))))))))))) (68)
3(3(3(5(2(2(0(1(0(0(0(4(x1)))))))))))) 3(5(4(3(4(2(4(5(3(1(4(4(4(5(1(3(0(4(x1)))))))))))))))))) (69)
3(3(4(3(4(3(2(5(3(3(0(4(x1)))))))))))) 3(5(0(1(3(4(5(4(5(4(3(0(3(0(x1)))))))))))))) (70)
3(3(5(3(4(4(0(3(3(3(2(5(x1)))))))))))) 3(5(1(4(5(1(4(3(5(4(3(3(4(4(4(4(3(x1))))))))))))))))) (71)
3(4(3(1(5(1(4(0(5(4(2(2(x1)))))))))))) 1(0(5(3(0(2(5(5(4(5(2(5(5(2(x1)))))))))))))) (72)
3(5(2(5(2(3(2(4(2(3(2(5(x1)))))))))))) 4(4(4(2(0(2(4(2(1(0(4(2(0(2(x1)))))))))))))) (73)
4(0(0(3(3(2(3(2(3(4(4(3(x1)))))))))))) 2(5(4(1(4(4(5(5(2(5(4(4(5(0(2(5(1(x1))))))))))))))))) (74)
4(1(2(2(1(0(1(4(2(0(3(1(x1)))))))))))) 0(0(4(5(1(5(4(0(2(1(5(5(0(4(0(x1))))))))))))))) (75)
4(1(2(5(2(2(1(0(2(1(2(1(x1)))))))))))) 1(2(4(5(5(5(0(0(0(4(5(2(3(0(1(x1))))))))))))))) (76)
4(1(3(1(0(3(1(2(2(1(1(4(x1)))))))))))) 3(5(4(5(1(4(5(1(4(3(3(4(0(1(5(x1))))))))))))))) (77)
4(1(3(3(3(0(1(5(5(3(0(1(x1)))))))))))) 4(5(1(5(0(0(4(4(5(3(5(1(4(1(4(3(5(5(x1)))))))))))))))))) (78)
4(2(0(1(1(3(1(5(1(0(5(1(x1)))))))))))) 0(5(4(4(3(4(5(5(4(4(2(4(0(0(1(1(x1)))))))))))))))) (79)
4(2(1(3(3(1(1(1(3(1(3(2(x1)))))))))))) 1(2(5(1(2(5(4(3(3(0(5(0(0(4(2(4(5(5(x1)))))))))))))))))) (80)
4(2(1(4(2(3(2(0(1(2(1(0(x1)))))))))))) 3(4(5(5(3(0(1(5(4(2(3(1(3(5(x1)))))))))))))) (81)
4(2(4(1(1(1(2(3(2(2(1(0(x1)))))))))))) 4(2(4(3(4(3(1(5(2(3(4(4(4(3(x1)))))))))))))) (82)
4(2(4(3(3(3(3(3(4(2(2(0(x1)))))))))))) 5(0(2(5(5(5(4(1(0(4(3(3(5(2(4(3(x1)))))))))))))))) (83)
4(3(1(1(1(3(4(0(0(2(2(1(x1)))))))))))) 5(1(4(0(2(4(5(4(4(2(1(5(1(5(4(x1))))))))))))))) (84)
4(3(2(3(2(5(3(3(3(5(3(0(x1)))))))))))) 3(1(1(1(2(4(0(0(5(5(1(0(4(3(x1)))))))))))))) (85)
4(3(2(3(3(3(1(1(1(1(0(2(x1)))))))))))) 5(4(5(1(5(1(2(2(1(4(4(2(4(0(4(3(x1)))))))))))))))) (86)
4(5(2(4(0(1(3(3(3(2(5(3(x1)))))))))))) 2(4(4(2(5(5(4(1(2(4(2(4(5(5(1(5(4(3(x1)))))))))))))))))) (87)
5(2(3(2(4(5(0(2(0(3(0(5(x1)))))))))))) 1(3(3(4(5(5(5(4(5(4(4(4(4(5(3(0(2(x1))))))))))))))))) (88)
5(2(3(4(2(4(2(3(2(0(2(1(x1)))))))))))) 4(5(1(3(0(5(1(5(1(2(4(5(1(2(4(5(x1)))))))))))))))) (89)
5(3(2(4(2(2(3(4(1(1(3(1(x1)))))))))))) 5(0(0(3(5(4(4(4(0(0(5(5(2(1(0(5(4(4(x1)))))))))))))))))) (90)
5(3(3(0(3(4(4(2(3(5(2(5(x1)))))))))))) 0(4(5(2(5(5(3(2(2(4(3(4(4(5(5(x1))))))))))))))) (91)
5(3(4(3(2(3(5(1(0(3(5(3(x1)))))))))))) 4(0(4(4(4(0(5(4(5(3(1(5(1(0(4(1(x1)))))))))))))))) (92)
5(3(5(2(1(2(1(2(0(3(5(0(x1)))))))))))) 5(5(4(1(5(4(5(4(5(5(4(2(4(0(2(4(1(4(x1)))))))))))))))))) (93)
5(3(5(2(5(1(2(0(3(2(2(4(x1)))))))))))) 4(4(5(5(5(3(3(5(5(1(2(2(5(4(5(x1))))))))))))))) (94)
5(3(5(3(5(1(1(3(2(2(5(1(x1)))))))))))) 1(0(4(5(3(5(1(5(5(5(3(4(0(2(1(x1))))))))))))))) (95)
5(5(0(2(2(1(5(2(2(1(5(3(x1)))))))))))) 5(1(5(5(2(4(3(4(3(0(3(1(5(5(x1)))))))))))))) (96)

and S is the following TRS.

0(1(2(3(4(5(x1)))))) 0(1(2(3(4(5(x1)))))) (97)

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 576 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 3456 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 +
0
[52(x1)] = x1 +
0
[53(x1)] = x1 +
3800
[54(x1)] = x1 +
1
[55(x1)] = x1 +
0
[40(x1)] = x1 +
0
[41(x1)] = x1 +
0
[42(x1)] = x1 +
405
[43(x1)] = x1 +
3701
[44(x1)] = x1 +
1
[45(x1)] = x1 +
0
[30(x1)] = x1 +
1
[31(x1)] = x1 +
4225
[32(x1)] = x1 +
4825
[33(x1)] = x1 +
4825
[34(x1)] = x1 +
1280
[35(x1)] = x1 +
1025
[20(x1)] = x1 +
425
[21(x1)] = x1 +
26
[22(x1)] = x1 +
4830
[23(x1)] = x1 +
4825
[24(x1)] = x1 +
4826
[25(x1)] = x1 +
4825
[10(x1)] = x1 +
2100
[11(x1)] = x1 +
2826
[12(x1)] = x1 +
4825
[13(x1)] = x1 +
2226
[14(x1)] = x1 +
4830
[15(x1)] = x1 +
0
[00(x1)] = x1 +
0
[01(x1)] = x1 +
0
[02(x1)] = x1 +
4825
[03(x1)] = x1 +
525
[04(x1)] = x1 +
4425
[05(x1)] = x1 +
0
all of the following rules can be deleted.

There are 3456 ruless (increase limit for explicit display).

1.1.1.1 R is empty

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