Certification Problem

Input (TPDB SRS_Relative/Mixed_relative_SRS/un03)

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

b(a(b(a(b(x1))))) → b(a(a(b(a(a(a(b(x1)))))))) (1)
b(a(a(b(a(a(b(x1))))))) → b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))) (2)
b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))) → b(b(a(a(b(x1))))) (3)

and S is the following TRS.

b(b(b(x1))) → b(a(b(a(a(b(x1)))))) (4)
b(a(b(a(a(b(x1)))))) → b(b(b(x1))) (5)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{b(☐), a(☐)}

We obtain the transformed TRS
b(b(a(b(a(b(x1)))))) → b(b(a(a(b(a(a(a(b(x1))))))))) (6)
b(b(a(a(b(a(a(b(x1)))))))) → b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))) (7)
b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))) → b(b(b(a(a(b(x1)))))) (8)
a(b(a(b(a(b(x1)))))) → a(b(a(a(b(a(a(a(b(x1))))))))) (9)
a(b(a(a(b(a(a(b(x1)))))))) → a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))) (10)
a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))) → a(b(b(a(a(b(x1)))))) (11)
b(b(b(b(x1)))) → b(b(a(b(a(a(b(x1))))))) (12)
b(b(a(b(a(a(b(x1))))))) → b(b(b(b(x1)))) (13)
a(b(b(b(x1)))) → a(b(a(b(a(a(b(x1))))))) (14)
a(b(a(b(a(a(b(x1))))))) → a(b(b(b(x1)))) (15)

1.1 Closure Under Flat Contexts

Using the flat contexts

{b(☐), a(☐)}

We obtain the transformed TRS
b(b(b(a(b(a(b(x1))))))) → b(b(b(a(a(b(a(a(a(b(x1)))))))))) (16)
b(b(b(a(a(b(a(a(b(x1))))))))) → b(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))))) (17)
b(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))))) → b(b(b(b(a(a(b(x1))))))) (18)
b(a(b(a(b(a(b(x1))))))) → b(a(b(a(a(b(a(a(a(b(x1)))))))))) (19)
b(a(b(a(a(b(a(a(b(x1))))))))) → b(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))))) (20)
b(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))))) → b(a(b(b(a(a(b(x1))))))) (21)
a(b(b(a(b(a(b(x1))))))) → a(b(b(a(a(b(a(a(a(b(x1)))))))))) (22)
a(b(b(a(a(b(a(a(b(x1))))))))) → a(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))))) (23)
a(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))))) → a(b(b(b(a(a(b(x1))))))) (24)
a(a(b(a(b(a(b(x1))))))) → a(a(b(a(a(b(a(a(a(b(x1)))))))))) (25)
a(a(b(a(a(b(a(a(b(x1))))))))) → a(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))))) (26)
a(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1))))))))))))))) → a(a(b(b(a(a(b(x1))))))) (27)
b(b(b(b(b(x1))))) → b(b(b(a(b(a(a(b(x1)))))))) (28)
b(b(b(a(b(a(a(b(x1)))))))) → b(b(b(b(b(x1))))) (29)
b(a(b(b(b(x1))))) → b(a(b(a(b(a(a(b(x1)))))))) (30)
b(a(b(a(b(a(a(b(x1)))))))) → b(a(b(b(b(x1))))) (31)
a(b(b(b(b(x1))))) → a(b(b(a(b(a(a(b(x1)))))))) (32)
a(b(b(a(b(a(a(b(x1)))))))) → a(b(b(b(b(x1))))) (33)
a(a(b(b(b(x1))))) → a(a(b(a(b(a(a(b(x1)))))))) (34)
a(a(b(a(b(a(a(b(x1)))))))) → a(a(b(b(b(x1))))) (35)

1.1.1 Closure Under Flat Contexts

Using the flat contexts

{b(☐), a(☐)}

We obtain the transformed TRS
b(b(b(b(a(b(a(b(x1)))))))) → b(b(b(b(a(a(b(a(a(a(b(x1))))))))))) (36)
b(b(b(b(a(a(b(a(a(b(x1)))))))))) → b(b(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) (37)
b(b(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) → b(b(b(b(b(a(a(b(x1)))))))) (38)
b(b(a(b(a(b(a(b(x1)))))))) → b(b(a(b(a(a(b(a(a(a(b(x1))))))))))) (39)
b(b(a(b(a(a(b(a(a(b(x1)))))))))) → b(b(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) (40)
b(b(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) → b(b(a(b(b(a(a(b(x1)))))))) (41)
b(a(b(b(a(b(a(b(x1)))))))) → b(a(b(b(a(a(b(a(a(a(b(x1))))))))))) (42)
b(a(b(b(a(a(b(a(a(b(x1)))))))))) → b(a(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) (43)
b(a(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) → b(a(b(b(b(a(a(b(x1)))))))) (44)
b(a(a(b(a(b(a(b(x1)))))))) → b(a(a(b(a(a(b(a(a(a(b(x1))))))))))) (45)
b(a(a(b(a(a(b(a(a(b(x1)))))))))) → b(a(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) (46)
b(a(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) → b(a(a(b(b(a(a(b(x1)))))))) (47)
a(b(b(b(a(b(a(b(x1)))))))) → a(b(b(b(a(a(b(a(a(a(b(x1))))))))))) (48)
a(b(b(b(a(a(b(a(a(b(x1)))))))))) → a(b(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) (49)
a(b(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) → a(b(b(b(b(a(a(b(x1)))))))) (50)
a(b(a(b(a(b(a(b(x1)))))))) → a(b(a(b(a(a(b(a(a(a(b(x1))))))))))) (51)
a(b(a(b(a(a(b(a(a(b(x1)))))))))) → a(b(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) (52)
a(b(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) → a(b(a(b(b(a(a(b(x1)))))))) (53)
a(a(b(b(a(b(a(b(x1)))))))) → a(a(b(b(a(a(b(a(a(a(b(x1))))))))))) (54)
a(a(b(b(a(a(b(a(a(b(x1)))))))))) → a(a(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) (55)
a(a(b(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) → a(a(b(b(b(a(a(b(x1)))))))) (56)
a(a(a(b(a(b(a(b(x1)))))))) → a(a(a(b(a(a(b(a(a(a(b(x1))))))))))) (57)
a(a(a(b(a(a(b(a(a(b(x1)))))))))) → a(a(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) (58)
a(a(a(b(a(a(a(b(a(a(a(b(a(a(a(b(x1)))))))))))))))) → a(a(a(b(b(a(a(b(x1)))))))) (59)
b(b(b(b(b(b(x1)))))) → b(b(b(b(a(b(a(a(b(x1))))))))) (60)
b(b(b(b(a(b(a(a(b(x1))))))))) → b(b(b(b(b(b(x1)))))) (61)
b(b(a(b(b(b(x1)))))) → b(b(a(b(a(b(a(a(b(x1))))))))) (62)
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(a(b(b(b(x1)))))) (63)
b(a(b(b(b(b(x1)))))) → b(a(b(b(a(b(a(a(b(x1))))))))) (64)
b(a(b(b(a(b(a(a(b(x1))))))))) → b(a(b(b(b(b(x1)))))) (65)
b(a(a(b(b(b(x1)))))) → b(a(a(b(a(b(a(a(b(x1))))))))) (66)
b(a(a(b(a(b(a(a(b(x1))))))))) → b(a(a(b(b(b(x1)))))) (67)
a(b(b(b(b(b(x1)))))) → a(b(b(b(a(b(a(a(b(x1))))))))) (68)
a(b(b(b(a(b(a(a(b(x1))))))))) → a(b(b(b(b(b(x1)))))) (69)
a(b(a(b(b(b(x1)))))) → a(b(a(b(a(b(a(a(b(x1))))))))) (70)
a(b(a(b(a(b(a(a(b(x1))))))))) → a(b(a(b(b(b(x1)))))) (71)
a(a(b(b(b(b(x1)))))) → a(a(b(b(a(b(a(a(b(x1))))))))) (72)
a(a(b(b(a(b(a(a(b(x1))))))))) → a(a(b(b(b(b(x1)))))) (73)
a(a(a(b(b(b(x1)))))) → a(a(a(b(a(b(a(a(b(x1))))))))) (74)
a(a(a(b(a(b(a(a(b(x1))))))))) → a(a(a(b(b(b(x1)))))) (75)

1.1.1.1 Semantic Labeling

The following interpretations form a model of the rules.

As carrier we take the set {0,...,7}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 8):

[b(x1)] = 2x1 + 0
[a(x1)] = 2x1 + 1

We obtain the labeled TRS

There are 320 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[b0(x1)] = x1 +
34
[b4(x1)] = x1 +
34
[b2(x1)] = x1 +
63/2
[b6(x1)] = x1 +
34
[b1(x1)] = x1 +
67/2
[b5(x1)] = x1 +
31
[b3(x1)] = x1 +
69/2
[b7(x1)] = x1 +
62/3
[a0(x1)] = x1 +
0
[a4(x1)] = x1 +
0
[a2(x1)] = x1 +
0
[a6(x1)] = x1 +
5/2
[a1(x1)] = x1 +
5/2
[a5(x1)] = x1 +
0
[a3(x1)] = x1 +
5/6
all of the following rules can be deleted.

There are 192 ruless (increase limit for explicit display).

1.1.1.1.1.1 R is empty

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