Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/48262)

The rewrite relation of the following TRS is considered.

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

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 540 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 3240 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 +
162
[51(x1)] = x1 +
3
[52(x1)] = x1 +
162
[53(x1)] = x1 +
162
[54(x1)] = x1 +
3
[55(x1)] = x1 +
126
[40(x1)] = x1 +
18
[41(x1)] = x1 +
0
[42(x1)] = x1 +
90
[43(x1)] = x1 +
0
[44(x1)] = x1 +
0
[45(x1)] = x1 +
0
[30(x1)] = x1 +
1
[31(x1)] = x1 +
90
[32(x1)] = x1 +
72
[33(x1)] = x1 +
36
[34(x1)] = x1 +
162
[35(x1)] = x1 +
18
[20(x1)] = x1 +
144
[21(x1)] = x1 +
0
[22(x1)] = x1 +
171
[23(x1)] = x1 +
90
[24(x1)] = x1 +
1
[25(x1)] = x1 +
165
[10(x1)] = x1 +
108
[11(x1)] = x1 +
162
[12(x1)] = x1 +
36
[13(x1)] = x1 +
0
[14(x1)] = x1 +
171
[15(x1)] = x1 +
55
[00(x1)] = x1 +
0
[01(x1)] = x1 +
165
[02(x1)] = x1 +
0
[03(x1)] = x1 +
36
[04(x1)] = x1 +
163
[05(x1)] = x1 +
18
all of the following rules can be deleted.

There are 3240 ruless (increase limit for explicit display).

1.1.1.1 R is empty

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