Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/64160)

The rewrite relation of the following TRS is considered.

1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))) 1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))) (1)
2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))) 2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))) (2)
2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))) 0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))) (3)
2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))) 2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))) (4)
2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))) 0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))) (5)
2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))) 0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))) (6)
2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))) 0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))) (7)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{2(), 1(), 0()}

We obtain the transformed TRS
2(1(2(1(1(0(0(1(0(2(1(1(0(2(x1)))))))))))))) 2(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1)))))))))))))))))) (8)
2(2(0(1(2(1(2(1(2(1(0(0(1(0(x1)))))))))))))) 2(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1)))))))))))))))))) (9)
2(2(1(0(0(2(1(1(1(0(1(2(2(0(x1)))))))))))))) 2(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1)))))))))))))))))) (10)
2(2(2(0(0(1(0(2(1(1(1(0(0(1(x1)))))))))))))) 2(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1)))))))))))))))))) (11)
2(2(2(1(0(1(0(1(1(0(0(0(2(0(x1)))))))))))))) 2(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1)))))))))))))))))) (12)
2(2(2(1(1(0(2(0(0(0(2(2(0(0(x1)))))))))))))) 2(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1)))))))))))))))))) (13)
2(2(2(2(1(0(2(0(0(1(0(1(0(2(x1)))))))))))))) 2(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1)))))))))))))))))) (14)
1(1(2(1(1(0(0(1(0(2(1(1(0(2(x1)))))))))))))) 1(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1)))))))))))))))))) (15)
1(2(0(1(2(1(2(1(2(1(0(0(1(0(x1)))))))))))))) 1(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1)))))))))))))))))) (16)
1(2(1(0(0(2(1(1(1(0(1(2(2(0(x1)))))))))))))) 1(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1)))))))))))))))))) (17)
1(2(2(0(0(1(0(2(1(1(1(0(0(1(x1)))))))))))))) 1(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1)))))))))))))))))) (18)
1(2(2(1(0(1(0(1(1(0(0(0(2(0(x1)))))))))))))) 1(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1)))))))))))))))))) (19)
1(2(2(1(1(0(2(0(0(0(2(2(0(0(x1)))))))))))))) 1(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1)))))))))))))))))) (20)
1(2(2(2(1(0(2(0(0(1(0(1(0(2(x1)))))))))))))) 1(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1)))))))))))))))))) (21)
0(1(2(1(1(0(0(1(0(2(1(1(0(2(x1)))))))))))))) 0(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1)))))))))))))))))) (22)
0(2(0(1(2(1(2(1(2(1(0(0(1(0(x1)))))))))))))) 0(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1)))))))))))))))))) (23)
0(2(1(0(0(2(1(1(1(0(1(2(2(0(x1)))))))))))))) 0(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1)))))))))))))))))) (24)
0(2(2(0(0(1(0(2(1(1(1(0(0(1(x1)))))))))))))) 0(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1)))))))))))))))))) (25)
0(2(2(1(0(1(0(1(1(0(0(0(2(0(x1)))))))))))))) 0(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1)))))))))))))))))) (26)
0(2(2(1(1(0(2(0(0(0(2(2(0(0(x1)))))))))))))) 0(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1)))))))))))))))))) (27)
0(2(2(2(1(0(2(0(0(1(0(1(0(2(x1)))))))))))))) 0(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1)))))))))))))))))) (28)

1.1 Closure Under Flat Contexts

Using the flat contexts

{2(), 1(), 0()}

We obtain the transformed TRS
2(2(1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))))) 2(2(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))))) (29)
2(2(2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))))) 2(2(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))))) (30)
2(2(2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))))) 2(2(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))))) (31)
2(2(2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))))) 2(2(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))))) (32)
2(2(2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))))) 2(2(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))))) (33)
2(2(2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))))) 2(2(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))))) (34)
2(2(2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))))) 2(2(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))))) (35)
2(1(1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))))) 2(1(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))))) (36)
2(1(2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))))) 2(1(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))))) (37)
2(1(2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))))) 2(1(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))))) (38)
2(1(2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))))) 2(1(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))))) (39)
2(1(2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))))) 2(1(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))))) (40)
2(1(2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))))) 2(1(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))))) (41)
2(1(2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))))) 2(1(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))))) (42)
2(0(1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))))) 2(0(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))))) (43)
2(0(2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))))) 2(0(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))))) (44)
2(0(2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))))) 2(0(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))))) (45)
2(0(2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))))) 2(0(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))))) (46)
2(0(2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))))) 2(0(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))))) (47)
2(0(2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))))) 2(0(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))))) (48)
2(0(2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))))) 2(0(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))))) (49)
1(2(1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))))) 1(2(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))))) (50)
1(2(2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))))) 1(2(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))))) (51)
1(2(2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))))) 1(2(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))))) (52)
1(2(2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))))) 1(2(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))))) (53)
1(2(2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))))) 1(2(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))))) (54)
1(2(2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))))) 1(2(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))))) (55)
1(2(2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))))) 1(2(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))))) (56)
1(1(1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))))) 1(1(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))))) (57)
1(1(2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))))) 1(1(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))))) (58)
1(1(2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))))) 1(1(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))))) (59)
1(1(2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))))) 1(1(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))))) (60)
1(1(2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))))) 1(1(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))))) (61)
1(1(2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))))) 1(1(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))))) (62)
1(1(2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))))) 1(1(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))))) (63)
1(0(1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))))) 1(0(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))))) (64)
1(0(2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))))) 1(0(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))))) (65)
1(0(2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))))) 1(0(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))))) (66)
1(0(2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))))) 1(0(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))))) (67)
1(0(2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))))) 1(0(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))))) (68)
1(0(2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))))) 1(0(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))))) (69)
1(0(2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))))) 1(0(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))))) (70)
0(2(1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))))) 0(2(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))))) (71)
0(2(2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))))) 0(2(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))))) (72)
0(2(2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))))) 0(2(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))))) (73)
0(2(2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))))) 0(2(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))))) (74)
0(2(2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))))) 0(2(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))))) (75)
0(2(2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))))) 0(2(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))))) (76)
0(2(2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))))) 0(2(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))))) (77)
0(1(1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))))) 0(1(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))))) (78)
0(1(2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))))) 0(1(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))))) (79)
0(1(2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))))) 0(1(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))))) (80)
0(1(2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))))) 0(1(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))))) (81)
0(1(2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))))) 0(1(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))))) (82)
0(1(2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))))) 0(1(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))))) (83)
0(1(2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))))) 0(1(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))))) (84)
0(0(1(2(1(1(0(0(1(0(2(1(1(0(2(x1))))))))))))))) 0(0(1(2(2(1(0(0(2(0(1(2(2(0(1(0(0(2(2(x1))))))))))))))))))) (85)
0(0(2(0(1(2(1(2(1(2(1(0(0(1(0(x1))))))))))))))) 0(0(2(2(0(0(2(1(1(1(0(0(0(1(1(1(2(2(0(x1))))))))))))))))))) (86)
0(0(2(1(0(0(2(1(1(1(0(1(2(2(0(x1))))))))))))))) 0(0(0(0(2(1(0(0(1(1(2(2(2(2(2(1(2(1(0(x1))))))))))))))))))) (87)
0(0(2(2(0(0(1(0(2(1(1(1(0(0(1(x1))))))))))))))) 0(0(2(2(1(1(2(0(0(1(2(2(0(2(0(0(2(2(2(x1))))))))))))))))))) (88)
0(0(2(2(1(0(1(0(1(1(0(0(0(2(0(x1))))))))))))))) 0(0(0(2(2(1(0(0(1(0(0(1(1(1(1(0(0(2(0(x1))))))))))))))))))) (89)
0(0(2(2(1(1(0(2(0(0(0(2(2(0(0(x1))))))))))))))) 0(0(0(0(1(2(0(0(0(2(2(0(1(2(2(2(2(0(0(x1))))))))))))))))))) (90)
0(0(2(2(2(1(0(2(0(0(1(0(1(0(2(x1))))))))))))))) 0(0(0(2(2(0(0(2(2(2(1(2(0(0(2(2(0(2(2(x1))))))))))))))))))) (91)

1.1.1 Semantic Labeling

The following interpretations form a model of the rules.

As carrier we take the set {0,...,8}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 9):

[2(x1)] = 3x1 + 0
[1(x1)] = 3x1 + 1
[0(x1)] = 3x1 + 2

We obtain the labeled TRS

There are 567 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
[20(x1)] = x1 +
0
[23(x1)] = x1 +
0
[26(x1)] = x1 +
0
[21(x1)] = x1 +
0
[24(x1)] = x1 +
0
[27(x1)] = x1 +
0
[22(x1)] = x1 +
1
[25(x1)] = x1 +
1
[28(x1)] = x1 +
0
[10(x1)] = x1 +
0
[13(x1)] = x1 +
0
[16(x1)] = x1 +
0
[11(x1)] = x1 +
0
[14(x1)] = x1 +
0
[17(x1)] = x1 +
0
[12(x1)] = x1 +
4
[15(x1)] = x1 +
5
[18(x1)] = x1 +
0
[00(x1)] = x1 +
0
[03(x1)] = x1 +
0
[06(x1)] = x1 +
0
[01(x1)] = x1 +
1
[04(x1)] = x1 +
0
[07(x1)] = x1 +
0
[02(x1)] = x1 +
0
[05(x1)] = x1 +
0
[08(x1)] = x1 +
0
all of the following rules can be deleted.

There are 567 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.