Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z058)

The rewrite relation of the following TRS is considered.

b(a(b(a(b(b(x1)))))) → a(b(b(a(b(b(a(x1))))))) (1)

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

1.1 Closure Under Flat Contexts

Using the flat contexts

{b(☐), a(☐)}

We obtain the transformed TRS
b(b(b(a(b(a(b(b(x1)))))))) → b(b(a(b(b(a(b(b(a(x1))))))))) (4)
b(a(b(a(b(a(b(b(x1)))))))) → b(a(a(b(b(a(b(b(a(x1))))))))) (5)
a(b(b(a(b(a(b(b(x1)))))))) → a(b(a(b(b(a(b(b(a(x1))))))))) (6)
a(a(b(a(b(a(b(b(x1)))))))) → a(a(a(b(b(a(b(b(a(x1))))))))) (7)

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(b(x1))))))))) → b(b(b(a(b(b(a(b(b(a(x1)))))))))) (8)
b(b(a(b(a(b(a(b(b(x1))))))))) → b(b(a(a(b(b(a(b(b(a(x1)))))))))) (9)
b(a(b(b(a(b(a(b(b(x1))))))))) → b(a(b(a(b(b(a(b(b(a(x1)))))))))) (10)
b(a(a(b(a(b(a(b(b(x1))))))))) → b(a(a(a(b(b(a(b(b(a(x1)))))))))) (11)
a(b(b(b(a(b(a(b(b(x1))))))))) → a(b(b(a(b(b(a(b(b(a(x1)))))))))) (12)
a(b(a(b(a(b(a(b(b(x1))))))))) → a(b(a(a(b(b(a(b(b(a(x1)))))))))) (13)
a(a(b(b(a(b(a(b(b(x1))))))))) → a(a(b(a(b(b(a(b(b(a(x1)))))))))) (14)
a(a(a(b(a(b(a(b(b(x1))))))))) → a(a(a(a(b(b(a(b(b(a(x1)))))))))) (15)

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
b0(b4(b2(b5(a2(b1(a0(b0(b0(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (16)
b0(b4(b2(b5(a2(b1(a0(b0(b4(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (17)
b0(b4(b2(b5(a2(b1(a0(b4(b2(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (18)
b0(b4(b2(b5(a2(b1(a0(b4(b6(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (19)
b0(b4(b2(b5(a2(b1(a4(b2(b1(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (20)
b0(b4(b2(b5(a2(b1(a4(b2(b5(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (21)
b0(b4(b2(b5(a2(b1(a4(b6(b3(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (22)
b0(b4(b2(b5(a2(b1(a4(b6(b7(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (23)
b2(b5(a2(b5(a2(b1(a0(b0(b0(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (24)
b2(b5(a2(b5(a2(b1(a0(b0(b4(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (25)
b2(b5(a2(b5(a2(b1(a0(b4(b2(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (26)
b2(b5(a2(b5(a2(b1(a0(b4(b6(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (27)
b2(b5(a2(b5(a2(b1(a4(b2(b1(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (28)
b2(b5(a2(b5(a2(b1(a4(b2(b5(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (29)
b2(b5(a2(b5(a2(b1(a4(b6(b3(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (30)
b2(b5(a2(b5(a2(b1(a4(b6(b7(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (31)
b1(a4(b2(b5(a2(b1(a0(b0(b0(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (32)
b1(a4(b2(b5(a2(b1(a0(b0(b4(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (33)
b1(a4(b2(b5(a2(b1(a0(b4(b2(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (34)
b1(a4(b2(b5(a2(b1(a0(b4(b6(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (35)
b1(a4(b2(b5(a2(b1(a4(b2(b1(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (36)
b1(a4(b2(b5(a2(b1(a4(b2(b5(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (37)
b1(a4(b2(b5(a2(b1(a4(b6(b3(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (38)
b1(a4(b2(b5(a2(b1(a4(b6(b7(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (39)
b3(a5(a2(b5(a2(b1(a0(b0(b0(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (40)
b3(a5(a2(b5(a2(b1(a0(b0(b4(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (41)
b3(a5(a2(b5(a2(b1(a0(b4(b2(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (42)
b3(a5(a2(b5(a2(b1(a0(b4(b6(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (43)
b3(a5(a2(b5(a2(b1(a4(b2(b1(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (44)
b3(a5(a2(b5(a2(b1(a4(b2(b5(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (45)
b3(a5(a2(b5(a2(b1(a4(b6(b3(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (46)
b3(a5(a2(b5(a2(b1(a4(b6(b7(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (47)
a0(b4(b2(b5(a2(b1(a0(b0(b0(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (48)
a0(b4(b2(b5(a2(b1(a0(b0(b4(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (49)
a0(b4(b2(b5(a2(b1(a0(b4(b2(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (50)
a0(b4(b2(b5(a2(b1(a0(b4(b6(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (51)
a0(b4(b2(b5(a2(b1(a4(b2(b1(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (52)
a0(b4(b2(b5(a2(b1(a4(b2(b5(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (53)
a0(b4(b2(b5(a2(b1(a4(b6(b3(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (54)
a0(b4(b2(b5(a2(b1(a4(b6(b7(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (55)
a2(b5(a2(b5(a2(b1(a0(b0(b0(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (56)
a2(b5(a2(b5(a2(b1(a0(b0(b4(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (57)
a2(b5(a2(b5(a2(b1(a0(b4(b2(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (58)
a2(b5(a2(b5(a2(b1(a0(b4(b6(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (59)
a2(b5(a2(b5(a2(b1(a4(b2(b1(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (60)
a2(b5(a2(b5(a2(b1(a4(b2(b5(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (61)
a2(b5(a2(b5(a2(b1(a4(b6(b3(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (62)
a2(b5(a2(b5(a2(b1(a4(b6(b7(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (63)
a1(a4(b2(b5(a2(b1(a0(b0(b0(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (64)
a1(a4(b2(b5(a2(b1(a0(b0(b4(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (65)
a1(a4(b2(b5(a2(b1(a0(b4(b2(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (66)
a1(a4(b2(b5(a2(b1(a0(b4(b6(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (67)
a1(a4(b2(b5(a2(b1(a4(b2(b1(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (68)
a1(a4(b2(b5(a2(b1(a4(b2(b5(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (69)
a1(a4(b2(b5(a2(b1(a4(b6(b3(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (70)
a1(a4(b2(b5(a2(b1(a4(b6(b7(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (71)
a3(a5(a2(b5(a2(b1(a0(b0(b0(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (72)
a3(a5(a2(b5(a2(b1(a0(b0(b4(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (73)
a3(a5(a2(b5(a2(b1(a0(b4(b2(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (74)
a3(a5(a2(b5(a2(b1(a0(b4(b6(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (75)
a3(a5(a2(b5(a2(b1(a4(b2(b1(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (76)
a3(a5(a2(b5(a2(b1(a4(b2(b5(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (77)
a3(a5(a2(b5(a2(b1(a4(b6(b3(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (78)
a3(a5(a2(b5(a2(b1(a4(b6(b7(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (79)

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 +
7
[b4(x1)] = x1 +
7
[b2(x1)] = x1 +
1
[b6(x1)] = x1 +
0
[b1(x1)] = x1 +
1
[b5(x1)] = x1 +
3
[b3(x1)] = x1 +
1
[b7(x1)] = x1 +
0
[a0(x1)] = x1 +
7
[a4(x1)] = x1 +
0
[a2(x1)] = x1 +
3
[a6(x1)] = x1 +
0
[a1(x1)] = x1 +
0
[a5(x1)] = x1 +
0
[a3(x1)] = x1 +
0
[a7(x1)] = x1 +
0
all of the following rules can be deleted.
b0(b4(b2(b5(a2(b1(a0(b0(b0(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (16)
b0(b4(b2(b5(a2(b1(a0(b0(b4(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (17)
b0(b4(b2(b5(a2(b1(a0(b4(b2(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (18)
b0(b4(b2(b5(a2(b1(a0(b4(b6(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (19)
b0(b4(b2(b5(a2(b1(a4(b2(b1(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (20)
b0(b4(b2(b5(a2(b1(a4(b2(b5(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (21)
b0(b4(b2(b5(a2(b1(a4(b6(b3(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (22)
b0(b4(b2(b5(a2(b1(a4(b6(b7(x1))))))))) → b4(b2(b1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (23)
b2(b5(a2(b5(a2(b1(a0(b0(b0(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (24)
b2(b5(a2(b5(a2(b1(a0(b0(b4(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (25)
b2(b5(a2(b5(a2(b1(a0(b4(b2(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (26)
b2(b5(a2(b5(a2(b1(a0(b4(b6(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (27)
b2(b5(a2(b5(a2(b1(a4(b2(b1(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (28)
b2(b5(a2(b5(a2(b1(a4(b2(b5(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (29)
b2(b5(a2(b5(a2(b1(a4(b6(b3(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (30)
b2(b5(a2(b5(a2(b1(a4(b6(b7(x1))))))))) → b6(b3(a1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (31)
b1(a4(b2(b5(a2(b1(a0(b0(b0(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (32)
b1(a4(b2(b5(a2(b1(a0(b0(b4(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (33)
b1(a4(b2(b5(a2(b1(a0(b4(b2(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (34)
b1(a4(b2(b5(a2(b1(a0(b4(b6(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (35)
b1(a4(b2(b5(a2(b1(a4(b2(b1(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (36)
b1(a4(b2(b5(a2(b1(a4(b2(b5(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (37)
b1(a4(b2(b5(a2(b1(a4(b6(b3(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (38)
b3(a5(a2(b5(a2(b1(a0(b0(b0(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (40)
b3(a5(a2(b5(a2(b1(a0(b0(b4(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (41)
b3(a5(a2(b5(a2(b1(a0(b4(b2(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (42)
b3(a5(a2(b5(a2(b1(a0(b4(b6(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (43)
b3(a5(a2(b5(a2(b1(a4(b2(b1(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (44)
b3(a5(a2(b5(a2(b1(a4(b2(b5(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (45)
b3(a5(a2(b5(a2(b1(a4(b6(b3(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (46)
b3(a5(a2(b5(a2(b1(a4(b6(b7(x1))))))))) → b7(a3(a1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (47)
a0(b4(b2(b5(a2(b1(a0(b0(b0(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (48)
a0(b4(b2(b5(a2(b1(a0(b0(b4(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (49)
a0(b4(b2(b5(a2(b1(a0(b4(b2(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (50)
a0(b4(b2(b5(a2(b1(a0(b4(b6(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (51)
a0(b4(b2(b5(a2(b1(a4(b2(b1(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (52)
a0(b4(b2(b5(a2(b1(a4(b2(b5(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (53)
a0(b4(b2(b5(a2(b1(a4(b6(b3(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (54)
a0(b4(b2(b5(a2(b1(a4(b6(b7(x1))))))))) → a4(b2(b1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (55)
a2(b5(a2(b5(a2(b1(a0(b0(b0(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (56)
a2(b5(a2(b5(a2(b1(a0(b0(b4(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (57)
a2(b5(a2(b5(a2(b1(a0(b4(b2(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (58)
a2(b5(a2(b5(a2(b1(a0(b4(b6(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (59)
a2(b5(a2(b5(a2(b1(a4(b2(b1(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (60)
a2(b5(a2(b5(a2(b1(a4(b2(b5(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (61)
a2(b5(a2(b5(a2(b1(a4(b6(b3(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (62)
a2(b5(a2(b5(a2(b1(a4(b6(b7(x1))))))))) → a6(b3(a1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (63)
a1(a4(b2(b5(a2(b1(a0(b0(b0(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (64)
a1(a4(b2(b5(a2(b1(a0(b0(b4(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (65)
a1(a4(b2(b5(a2(b1(a0(b4(b2(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (66)
a1(a4(b2(b5(a2(b1(a0(b4(b6(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (67)
a1(a4(b2(b5(a2(b1(a4(b2(b1(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (68)
a1(a4(b2(b5(a2(b1(a4(b2(b5(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (69)
a1(a4(b2(b5(a2(b1(a4(b6(b3(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (70)
a1(a4(b2(b5(a2(b1(a4(b6(b7(x1))))))))) → a5(a2(b1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (71)
a3(a5(a2(b5(a2(b1(a0(b0(b0(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b2(b1(a0(x1)))))))))) (72)
a3(a5(a2(b5(a2(b1(a0(b0(b4(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b2(b1(a4(x1)))))))))) (73)
a3(a5(a2(b5(a2(b1(a0(b4(b2(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b2(b5(a2(x1)))))))))) (74)
a3(a5(a2(b5(a2(b1(a0(b4(b6(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b2(b5(a6(x1)))))))))) (75)
a3(a5(a2(b5(a2(b1(a4(b2(b1(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b6(b3(a1(x1)))))))))) (76)
a3(a5(a2(b5(a2(b1(a4(b2(b5(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b6(b3(a5(x1)))))))))) (77)
a3(a5(a2(b5(a2(b1(a4(b6(b3(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b6(b7(a3(x1)))))))))) (78)
a3(a5(a2(b5(a2(b1(a4(b6(b7(x1))))))))) → a7(a3(a1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (79)

1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the naturals
[b2(x1)] =
2
· x1 +
0
[b6(x1)] =
2
· x1 +
1
[b1(x1)] =
2
· x1 +
1
[b5(x1)] =
2
· x1 +
0
[b7(x1)] =
2
· x1 +
1
[a4(x1)] =
2
· x1 +
1
[a2(x1)] =
1
· x1 +
1
[a7(x1)] =
1
· x1 +
0
all of the following rules can be deleted.
b1(a4(b2(b5(a2(b1(a4(b6(b7(x1))))))))) → b5(a2(b1(a4(b2(b1(a4(b6(b7(a7(x1)))))))))) (39)

1.1.1.1.1.1.1 R is empty

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