Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-19)

The rewrite relation of the following TRS is considered.

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

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(a(b(x1))))) b(a(b(a(b(x1))))) (4)
b(a(a(a(a(x1))))) b(b(b(b(b(x1))))) (5)
b(b(b(a(b(x1))))) b(b(b(a(a(x1))))) (6)
a(b(a(a(b(x1))))) a(a(b(a(b(x1))))) (7)
a(a(a(a(a(x1))))) a(b(b(b(b(x1))))) (8)
a(b(b(a(b(x1))))) a(b(b(a(a(x1))))) (9)

1.1 Closure Under Flat Contexts

Using the flat contexts

{b(), a()}

We obtain the transformed TRS
b(b(b(a(a(b(x1)))))) b(b(a(b(a(b(x1)))))) (10)
b(b(a(a(a(a(x1)))))) b(b(b(b(b(b(x1)))))) (11)
b(b(b(b(a(b(x1)))))) b(b(b(b(a(a(x1)))))) (12)
b(a(b(a(a(b(x1)))))) b(a(a(b(a(b(x1)))))) (13)
b(a(a(a(a(a(x1)))))) b(a(b(b(b(b(x1)))))) (14)
b(a(b(b(a(b(x1)))))) b(a(b(b(a(a(x1)))))) (15)
a(b(b(a(a(b(x1)))))) a(b(a(b(a(b(x1)))))) (16)
a(b(a(a(a(a(x1)))))) a(b(b(b(b(b(x1)))))) (17)
a(b(b(b(a(b(x1)))))) a(b(b(b(a(a(x1)))))) (18)
a(a(b(a(a(b(x1)))))) a(a(a(b(a(b(x1)))))) (19)
a(a(a(a(a(a(x1)))))) a(a(b(b(b(b(x1)))))) (20)
a(a(b(b(a(b(x1)))))) a(a(b(b(a(a(x1)))))) (21)

1.1.1 Closure Under Flat Contexts

Using the flat contexts

{b(), a()}

We obtain the transformed TRS
b(b(b(b(a(a(b(x1))))))) b(b(b(a(b(a(b(x1))))))) (22)
b(b(b(a(a(a(a(x1))))))) b(b(b(b(b(b(b(x1))))))) (23)
b(b(b(b(b(a(b(x1))))))) b(b(b(b(b(a(a(x1))))))) (24)
b(b(a(b(a(a(b(x1))))))) b(b(a(a(b(a(b(x1))))))) (25)
b(b(a(a(a(a(a(x1))))))) b(b(a(b(b(b(b(x1))))))) (26)
b(b(a(b(b(a(b(x1))))))) b(b(a(b(b(a(a(x1))))))) (27)
b(a(b(b(a(a(b(x1))))))) b(a(b(a(b(a(b(x1))))))) (28)
b(a(b(a(a(a(a(x1))))))) b(a(b(b(b(b(b(x1))))))) (29)
b(a(b(b(b(a(b(x1))))))) b(a(b(b(b(a(a(x1))))))) (30)
b(a(a(b(a(a(b(x1))))))) b(a(a(a(b(a(b(x1))))))) (31)
b(a(a(a(a(a(a(x1))))))) b(a(a(b(b(b(b(x1))))))) (32)
b(a(a(b(b(a(b(x1))))))) b(a(a(b(b(a(a(x1))))))) (33)
a(b(b(b(a(a(b(x1))))))) a(b(b(a(b(a(b(x1))))))) (34)
a(b(b(a(a(a(a(x1))))))) a(b(b(b(b(b(b(x1))))))) (35)
a(b(b(b(b(a(b(x1))))))) a(b(b(b(b(a(a(x1))))))) (36)
a(b(a(b(a(a(b(x1))))))) a(b(a(a(b(a(b(x1))))))) (37)
a(b(a(a(a(a(a(x1))))))) a(b(a(b(b(b(b(x1))))))) (38)
a(b(a(b(b(a(b(x1))))))) a(b(a(b(b(a(a(x1))))))) (39)
a(a(b(b(a(a(b(x1))))))) a(a(b(a(b(a(b(x1))))))) (40)
a(a(b(a(a(a(a(x1))))))) a(a(b(b(b(b(b(x1))))))) (41)
a(a(b(b(b(a(b(x1))))))) a(a(b(b(b(a(a(x1))))))) (42)
a(a(a(b(a(a(b(x1))))))) a(a(a(a(b(a(b(x1))))))) (43)
a(a(a(a(a(a(a(x1))))))) a(a(a(b(b(b(b(x1))))))) (44)
a(a(a(b(b(a(b(x1))))))) a(a(a(b(b(a(a(x1))))))) (45)

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 192 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 +
0
[b4(x1)] = x1 +
1
[b2(x1)] = x1 +
1
[b6(x1)] = x1 +
0
[b1(x1)] = x1 +
0
[b5(x1)] = x1 +
0
[b3(x1)] = x1 +
0
[b7(x1)] = x1 +
0
[a0(x1)] = x1 +
0
[a4(x1)] = x1 +
1
[a2(x1)] = x1 +
0
[a6(x1)] = x1 +
1
[a1(x1)] = x1 +
1
[a5(x1)] = x1 +
1
[a3(x1)] = x1 +
0
[a7(x1)] = x1 +
1
all of the following rules can be deleted.

There are 119 ruless (increase limit for explicit display).

1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
b0(a0(a1(b3(b6(b4(b0(x1))))))) b0(a0(b1(a2(b5(b2(b4(x1))))))) (238)
b4(a0(a1(b3(b6(b4(b0(x1))))))) b4(a0(b1(a2(b5(b2(b4(x1))))))) (239)
b2(a4(a1(b3(b6(b4(b0(x1))))))) b2(a4(b1(a2(b5(b2(b4(x1))))))) (240)
b6(a4(a1(b3(b6(b4(b0(x1))))))) b6(a4(b1(a2(b5(b2(b4(x1))))))) (241)
b1(a2(a5(b3(b6(b4(b0(x1))))))) b1(a2(b5(a2(b5(b2(b4(x1))))))) (242)
b5(a2(a5(b3(b6(b4(b0(x1))))))) b5(a2(b5(a2(b5(b2(b4(x1))))))) (243)
b3(a6(a5(b3(b6(b4(b0(x1))))))) b3(a6(b5(a2(b5(b2(b4(x1))))))) (244)
b7(a6(a5(b3(b6(b4(b0(x1))))))) b7(a6(b5(a2(b5(b2(b4(x1))))))) (245)
b0(a0(a1(b3(b6(b4(a0(x1))))))) b0(a0(b1(a2(b5(b2(a4(x1))))))) (246)
b4(a0(a1(b3(b6(b4(a0(x1))))))) b4(a0(b1(a2(b5(b2(a4(x1))))))) (247)
b2(a4(a1(b3(b6(b4(a0(x1))))))) b2(a4(b1(a2(b5(b2(a4(x1))))))) (248)
b6(a4(a1(b3(b6(b4(a0(x1))))))) b6(a4(b1(a2(b5(b2(a4(x1))))))) (249)
b1(a2(a5(b3(b6(b4(a0(x1))))))) b1(a2(b5(a2(b5(b2(a4(x1))))))) (250)
b5(a2(a5(b3(b6(b4(a0(x1))))))) b5(a2(b5(a2(b5(b2(a4(x1))))))) (251)
b3(a6(a5(b3(b6(b4(a0(x1))))))) b3(a6(b5(a2(b5(b2(a4(x1))))))) (252)
b7(a6(a5(b3(b6(b4(a0(x1))))))) b7(a6(b5(a2(b5(b2(a4(x1))))))) (253)
b0(a0(a1(b3(a6(b5(a2(x1))))))) b0(a0(b1(a2(a5(b3(a6(x1))))))) (254)
b4(a0(a1(b3(a6(b5(a2(x1))))))) b4(a0(b1(a2(a5(b3(a6(x1))))))) (255)
b2(a4(a1(b3(a6(b5(a2(x1))))))) b2(a4(b1(a2(a5(b3(a6(x1))))))) (256)
b6(a4(a1(b3(a6(b5(a2(x1))))))) b6(a4(b1(a2(a5(b3(a6(x1))))))) (257)
b1(a2(a5(b3(a6(b5(a2(x1))))))) b1(a2(b5(a2(a5(b3(a6(x1))))))) (258)
b5(a2(a5(b3(a6(b5(a2(x1))))))) b5(a2(b5(a2(a5(b3(a6(x1))))))) (259)
b3(a6(a5(b3(a6(b5(a2(x1))))))) b3(a6(b5(a2(a5(b3(a6(x1))))))) (260)
b7(a6(a5(b3(a6(b5(a2(x1))))))) b7(a6(b5(a2(a5(b3(a6(x1))))))) (261)
a2(a5(a3(a7(a7(b7(b6(x1))))))) b2(b4(b0(b0(a0(b1(b2(x1))))))) (262)
b0(a0(b1(b2(b4(b0(b0(x1))))))) a0(a1(b3(b6(b4(b0(b0(x1))))))) (263)
b4(a0(b1(b2(b4(b0(b0(x1))))))) a4(a1(b3(b6(b4(b0(b0(x1))))))) (264)
b6(a4(b1(b2(b4(b0(b0(x1))))))) a6(a5(b3(b6(b4(b0(b0(x1))))))) (265)
b1(a2(b5(b2(b4(b0(b0(x1))))))) a1(a3(b7(b6(b4(b0(b0(x1))))))) (266)
b5(a2(b5(b2(b4(b0(b0(x1))))))) a5(a3(b7(b6(b4(b0(b0(x1))))))) (267)
b7(a6(b5(b2(b4(b0(b0(x1))))))) a7(a7(b7(b6(b4(b0(b0(x1))))))) (268)
b0(a0(b1(b2(a4(b1(b2(x1))))))) a0(a1(b3(b6(a4(b1(b2(x1))))))) (269)
b4(a0(b1(b2(a4(b1(b2(x1))))))) a4(a1(b3(b6(a4(b1(b2(x1))))))) (270)
b6(a4(b1(b2(a4(b1(b2(x1))))))) a6(a5(b3(b6(a4(b1(b2(x1))))))) (271)
b1(a2(b5(b2(a4(b1(b2(x1))))))) a1(a3(b7(b6(a4(b1(b2(x1))))))) (272)
b5(a2(b5(b2(a4(b1(b2(x1))))))) a5(a3(b7(b6(a4(b1(b2(x1))))))) (273)
b7(a6(b5(b2(a4(b1(b2(x1))))))) a7(a7(b7(b6(a4(b1(b2(x1))))))) (274)
b0(a0(b1(b2(b4(a0(b1(x1))))))) a0(a1(b3(b6(b4(a0(b1(x1))))))) (275)
b4(a0(b1(b2(b4(a0(b1(x1))))))) a4(a1(b3(b6(b4(a0(b1(x1))))))) (276)
b6(a4(b1(b2(b4(a0(b1(x1))))))) a6(a5(b3(b6(b4(a0(b1(x1))))))) (277)
b1(a2(b5(b2(b4(a0(b1(x1))))))) a1(a3(b7(b6(b4(a0(b1(x1))))))) (278)
b5(a2(b5(b2(b4(a0(b1(x1))))))) a5(a3(b7(b6(b4(a0(b1(x1))))))) (279)
b7(a6(b5(b2(b4(a0(b1(x1))))))) a7(a7(b7(b6(b4(a0(b1(x1))))))) (280)
b0(a0(b1(b2(a4(a1(b3(x1))))))) a0(a1(b3(b6(a4(a1(b3(x1))))))) (281)
b4(a0(b1(b2(a4(a1(b3(x1))))))) a4(a1(b3(b6(a4(a1(b3(x1))))))) (282)
b6(a4(b1(b2(a4(a1(b3(x1))))))) a6(a5(b3(b6(a4(a1(b3(x1))))))) (283)
b1(a2(b5(b2(a4(a1(b3(x1))))))) a1(a3(b7(b6(a4(a1(b3(x1))))))) (284)
b5(a2(b5(b2(a4(a1(b3(x1))))))) a5(a3(b7(b6(a4(a1(b3(x1))))))) (285)
b7(a6(b5(b2(a4(a1(b3(x1))))))) a7(a7(b7(b6(a4(a1(b3(x1))))))) (286)
b0(a0(b1(b2(b4(b0(a0(x1))))))) a0(a1(b3(b6(b4(b0(a0(x1))))))) (287)
b4(a0(b1(b2(b4(b0(a0(x1))))))) a4(a1(b3(b6(b4(b0(a0(x1))))))) (288)
b6(a4(b1(b2(b4(b0(a0(x1))))))) a6(a5(b3(b6(b4(b0(a0(x1))))))) (289)
b1(a2(b5(b2(b4(b0(a0(x1))))))) a1(a3(b7(b6(b4(b0(a0(x1))))))) (290)
b5(a2(b5(b2(b4(b0(a0(x1))))))) a5(a3(b7(b6(b4(b0(a0(x1))))))) (291)
b7(a6(b5(b2(b4(b0(a0(x1))))))) a7(a7(b7(b6(b4(b0(a0(x1))))))) (292)
b0(a0(b1(b2(a4(b1(a2(x1))))))) a0(a1(b3(b6(a4(b1(a2(x1))))))) (293)
b4(a0(b1(b2(a4(b1(a2(x1))))))) a4(a1(b3(b6(a4(b1(a2(x1))))))) (294)
b6(a4(b1(b2(a4(b1(a2(x1))))))) a6(a5(b3(b6(a4(b1(a2(x1))))))) (295)
b1(a2(b5(b2(a4(b1(a2(x1))))))) a1(a3(b7(b6(a4(b1(a2(x1))))))) (296)
b5(a2(b5(b2(a4(b1(a2(x1))))))) a5(a3(b7(b6(a4(b1(a2(x1))))))) (297)
b7(a6(b5(b2(a4(b1(a2(x1))))))) a7(a7(b7(b6(a4(b1(a2(x1))))))) (298)
b0(a0(b1(b2(b4(a0(a1(x1))))))) a0(a1(b3(b6(b4(a0(a1(x1))))))) (299)
b4(a0(b1(b2(b4(a0(a1(x1))))))) a4(a1(b3(b6(b4(a0(a1(x1))))))) (300)
b6(a4(b1(b2(b4(a0(a1(x1))))))) a6(a5(b3(b6(b4(a0(a1(x1))))))) (301)
b1(a2(b5(b2(b4(a0(a1(x1))))))) a1(a3(b7(b6(b4(a0(a1(x1))))))) (302)
b5(a2(b5(b2(b4(a0(a1(x1))))))) a5(a3(b7(b6(b4(a0(a1(x1))))))) (303)
b7(a6(b5(b2(b4(a0(a1(x1))))))) a7(a7(b7(b6(b4(a0(a1(x1))))))) (304)
b0(a0(b1(b2(a4(a1(a3(x1))))))) a0(a1(b3(b6(a4(a1(a3(x1))))))) (305)
b4(a0(b1(b2(a4(a1(a3(x1))))))) a4(a1(b3(b6(a4(a1(a3(x1))))))) (306)
b6(a4(b1(b2(a4(a1(a3(x1))))))) a6(a5(b3(b6(a4(a1(a3(x1))))))) (307)
b1(a2(b5(b2(a4(a1(a3(x1))))))) a1(a3(b7(b6(a4(a1(a3(x1))))))) (308)
b5(a2(b5(b2(a4(a1(a3(x1))))))) a5(a3(b7(b6(a4(a1(a3(x1))))))) (309)
b7(a6(b5(b2(a4(a1(a3(x1))))))) a7(a7(b7(b6(a4(a1(a3(x1))))))) (310)

1.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the naturals
[b0(x1)] =
2
· x1 +
1
[b4(x1)] =
2
· x1 +
0
[b2(x1)] =
2
· x1 +
1
[b6(x1)] =
2
· x1 +
1
[b1(x1)] =
2
· x1 +
0
[b5(x1)] =
2
· x1 +
1
[b3(x1)] =
2
· x1 +
0
[b7(x1)] =
2
· x1 +
0
[a0(x1)] =
2
· x1 +
1
[a4(x1)] =
2
· x1 +
0
[a2(x1)] =
2
· x1 +
1
[a6(x1)] =
2
· x1 +
0
[a1(x1)] =
2
· x1 +
0
[a5(x1)] =
2
· x1 +
0
[a3(x1)] =
2
· x1 +
1
[a7(x1)] =
2
· x1 +
1
all of the following rules can be deleted.
b0(a0(a1(b3(b6(b4(b0(x1))))))) b0(a0(b1(a2(b5(b2(b4(x1))))))) (238)
b4(a0(a1(b3(b6(b4(b0(x1))))))) b4(a0(b1(a2(b5(b2(b4(x1))))))) (239)
b2(a4(a1(b3(b6(b4(b0(x1))))))) b2(a4(b1(a2(b5(b2(b4(x1))))))) (240)
b6(a4(a1(b3(b6(b4(b0(x1))))))) b6(a4(b1(a2(b5(b2(b4(x1))))))) (241)
b1(a2(a5(b3(b6(b4(b0(x1))))))) b1(a2(b5(a2(b5(b2(b4(x1))))))) (242)
b5(a2(a5(b3(b6(b4(b0(x1))))))) b5(a2(b5(a2(b5(b2(b4(x1))))))) (243)
b3(a6(a5(b3(b6(b4(b0(x1))))))) b3(a6(b5(a2(b5(b2(b4(x1))))))) (244)
b7(a6(a5(b3(b6(b4(b0(x1))))))) b7(a6(b5(a2(b5(b2(b4(x1))))))) (245)
b0(a0(a1(b3(b6(b4(a0(x1))))))) b0(a0(b1(a2(b5(b2(a4(x1))))))) (246)
b4(a0(a1(b3(b6(b4(a0(x1))))))) b4(a0(b1(a2(b5(b2(a4(x1))))))) (247)
b2(a4(a1(b3(b6(b4(a0(x1))))))) b2(a4(b1(a2(b5(b2(a4(x1))))))) (248)
b6(a4(a1(b3(b6(b4(a0(x1))))))) b6(a4(b1(a2(b5(b2(a4(x1))))))) (249)
b1(a2(a5(b3(b6(b4(a0(x1))))))) b1(a2(b5(a2(b5(b2(a4(x1))))))) (250)
b5(a2(a5(b3(b6(b4(a0(x1))))))) b5(a2(b5(a2(b5(b2(a4(x1))))))) (251)
b3(a6(a5(b3(b6(b4(a0(x1))))))) b3(a6(b5(a2(b5(b2(a4(x1))))))) (252)
b7(a6(a5(b3(b6(b4(a0(x1))))))) b7(a6(b5(a2(b5(b2(a4(x1))))))) (253)
b0(a0(a1(b3(a6(b5(a2(x1))))))) b0(a0(b1(a2(a5(b3(a6(x1))))))) (254)
b4(a0(a1(b3(a6(b5(a2(x1))))))) b4(a0(b1(a2(a5(b3(a6(x1))))))) (255)
b2(a4(a1(b3(a6(b5(a2(x1))))))) b2(a4(b1(a2(a5(b3(a6(x1))))))) (256)
b6(a4(a1(b3(a6(b5(a2(x1))))))) b6(a4(b1(a2(a5(b3(a6(x1))))))) (257)
b1(a2(a5(b3(a6(b5(a2(x1))))))) b1(a2(b5(a2(a5(b3(a6(x1))))))) (258)
b5(a2(a5(b3(a6(b5(a2(x1))))))) b5(a2(b5(a2(a5(b3(a6(x1))))))) (259)
b3(a6(a5(b3(a6(b5(a2(x1))))))) b3(a6(b5(a2(a5(b3(a6(x1))))))) (260)
b7(a6(a5(b3(a6(b5(a2(x1))))))) b7(a6(b5(a2(a5(b3(a6(x1))))))) (261)
b0(a0(b1(b2(b4(b0(b0(x1))))))) a0(a1(b3(b6(b4(b0(b0(x1))))))) (263)
b4(a0(b1(b2(b4(b0(b0(x1))))))) a4(a1(b3(b6(b4(b0(b0(x1))))))) (264)
b6(a4(b1(b2(b4(b0(b0(x1))))))) a6(a5(b3(b6(b4(b0(b0(x1))))))) (265)
b1(a2(b5(b2(b4(b0(b0(x1))))))) a1(a3(b7(b6(b4(b0(b0(x1))))))) (266)
b5(a2(b5(b2(b4(b0(b0(x1))))))) a5(a3(b7(b6(b4(b0(b0(x1))))))) (267)
b7(a6(b5(b2(b4(b0(b0(x1))))))) a7(a7(b7(b6(b4(b0(b0(x1))))))) (268)
b0(a0(b1(b2(a4(b1(b2(x1))))))) a0(a1(b3(b6(a4(b1(b2(x1))))))) (269)
b4(a0(b1(b2(a4(b1(b2(x1))))))) a4(a1(b3(b6(a4(b1(b2(x1))))))) (270)
b6(a4(b1(b2(a4(b1(b2(x1))))))) a6(a5(b3(b6(a4(b1(b2(x1))))))) (271)
b1(a2(b5(b2(a4(b1(b2(x1))))))) a1(a3(b7(b6(a4(b1(b2(x1))))))) (272)
b5(a2(b5(b2(a4(b1(b2(x1))))))) a5(a3(b7(b6(a4(b1(b2(x1))))))) (273)
b7(a6(b5(b2(a4(b1(b2(x1))))))) a7(a7(b7(b6(a4(b1(b2(x1))))))) (274)
b0(a0(b1(b2(b4(a0(b1(x1))))))) a0(a1(b3(b6(b4(a0(b1(x1))))))) (275)
b4(a0(b1(b2(b4(a0(b1(x1))))))) a4(a1(b3(b6(b4(a0(b1(x1))))))) (276)
b6(a4(b1(b2(b4(a0(b1(x1))))))) a6(a5(b3(b6(b4(a0(b1(x1))))))) (277)
b1(a2(b5(b2(b4(a0(b1(x1))))))) a1(a3(b7(b6(b4(a0(b1(x1))))))) (278)
b5(a2(b5(b2(b4(a0(b1(x1))))))) a5(a3(b7(b6(b4(a0(b1(x1))))))) (279)
b7(a6(b5(b2(b4(a0(b1(x1))))))) a7(a7(b7(b6(b4(a0(b1(x1))))))) (280)
b0(a0(b1(b2(a4(a1(b3(x1))))))) a0(a1(b3(b6(a4(a1(b3(x1))))))) (281)
b4(a0(b1(b2(a4(a1(b3(x1))))))) a4(a1(b3(b6(a4(a1(b3(x1))))))) (282)
b6(a4(b1(b2(a4(a1(b3(x1))))))) a6(a5(b3(b6(a4(a1(b3(x1))))))) (283)
b1(a2(b5(b2(a4(a1(b3(x1))))))) a1(a3(b7(b6(a4(a1(b3(x1))))))) (284)
b5(a2(b5(b2(a4(a1(b3(x1))))))) a5(a3(b7(b6(a4(a1(b3(x1))))))) (285)
b7(a6(b5(b2(a4(a1(b3(x1))))))) a7(a7(b7(b6(a4(a1(b3(x1))))))) (286)
b0(a0(b1(b2(b4(b0(a0(x1))))))) a0(a1(b3(b6(b4(b0(a0(x1))))))) (287)
b4(a0(b1(b2(b4(b0(a0(x1))))))) a4(a1(b3(b6(b4(b0(a0(x1))))))) (288)
b6(a4(b1(b2(b4(b0(a0(x1))))))) a6(a5(b3(b6(b4(b0(a0(x1))))))) (289)
b1(a2(b5(b2(b4(b0(a0(x1))))))) a1(a3(b7(b6(b4(b0(a0(x1))))))) (290)
b5(a2(b5(b2(b4(b0(a0(x1))))))) a5(a3(b7(b6(b4(b0(a0(x1))))))) (291)
b7(a6(b5(b2(b4(b0(a0(x1))))))) a7(a7(b7(b6(b4(b0(a0(x1))))))) (292)
b0(a0(b1(b2(a4(b1(a2(x1))))))) a0(a1(b3(b6(a4(b1(a2(x1))))))) (293)
b4(a0(b1(b2(a4(b1(a2(x1))))))) a4(a1(b3(b6(a4(b1(a2(x1))))))) (294)
b6(a4(b1(b2(a4(b1(a2(x1))))))) a6(a5(b3(b6(a4(b1(a2(x1))))))) (295)
b1(a2(b5(b2(a4(b1(a2(x1))))))) a1(a3(b7(b6(a4(b1(a2(x1))))))) (296)
b5(a2(b5(b2(a4(b1(a2(x1))))))) a5(a3(b7(b6(a4(b1(a2(x1))))))) (297)
b7(a6(b5(b2(a4(b1(a2(x1))))))) a7(a7(b7(b6(a4(b1(a2(x1))))))) (298)
b0(a0(b1(b2(b4(a0(a1(x1))))))) a0(a1(b3(b6(b4(a0(a1(x1))))))) (299)
b4(a0(b1(b2(b4(a0(a1(x1))))))) a4(a1(b3(b6(b4(a0(a1(x1))))))) (300)
b6(a4(b1(b2(b4(a0(a1(x1))))))) a6(a5(b3(b6(b4(a0(a1(x1))))))) (301)
b1(a2(b5(b2(b4(a0(a1(x1))))))) a1(a3(b7(b6(b4(a0(a1(x1))))))) (302)
b5(a2(b5(b2(b4(a0(a1(x1))))))) a5(a3(b7(b6(b4(a0(a1(x1))))))) (303)
b7(a6(b5(b2(b4(a0(a1(x1))))))) a7(a7(b7(b6(b4(a0(a1(x1))))))) (304)
b0(a0(b1(b2(a4(a1(a3(x1))))))) a0(a1(b3(b6(a4(a1(a3(x1))))))) (305)
b4(a0(b1(b2(a4(a1(a3(x1))))))) a4(a1(b3(b6(a4(a1(a3(x1))))))) (306)
b6(a4(b1(b2(a4(a1(a3(x1))))))) a6(a5(b3(b6(a4(a1(a3(x1))))))) (307)
b1(a2(b5(b2(a4(a1(a3(x1))))))) a1(a3(b7(b6(a4(a1(a3(x1))))))) (308)
b5(a2(b5(b2(a4(a1(a3(x1))))))) a5(a3(b7(b6(a4(a1(a3(x1))))))) (309)
b7(a6(b5(b2(a4(a1(a3(x1))))))) a7(a7(b7(b6(a4(a1(a3(x1))))))) (310)

1.1.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[b0(x1)] = x1 +
0
[b4(x1)] = x1 +
0
[b2(x1)] = x1 +
0
[b6(x1)] = x1 +
1
[b1(x1)] = x1 +
0
[b7(x1)] = x1 +
1
[a0(x1)] = x1 +
0
[a2(x1)] = x1 +
1
[a5(x1)] = x1 +
1
[a3(x1)] = x1 +
1
[a7(x1)] = x1 +
1
all of the following rules can be deleted.
a2(a5(a3(a7(a7(b7(b6(x1))))))) b2(b4(b0(b0(a0(b1(b2(x1))))))) (262)

1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

There are no rules.

1.1.1.1.1.1.1.1.1.1 R is empty

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