Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/acbabacba-abacbacbabac.srs)

The rewrite relation of the following TRS is considered.

a(c(b(a(b(a(c(b(a(x1))))))))) → a(b(a(c(b(a(c(b(a(b(a(c(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

{c(☐), b(☐), a(☐)}

We obtain the transformed TRS
c(a(c(b(a(b(a(c(b(a(x1)))))))))) → c(a(b(a(c(b(a(c(b(a(b(a(c(x1))))))))))))) (2)
b(a(c(b(a(b(a(c(b(a(x1)))))))))) → b(a(b(a(c(b(a(c(b(a(b(a(c(x1))))))))))))) (3)
a(a(c(b(a(b(a(c(b(a(x1)))))))))) → a(a(b(a(c(b(a(c(b(a(b(a(c(x1))))))))))))) (4)

1.1 Semantic Labeling

The following interpretations form a model of the rules.

As carrier we take the set {0,1,2}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 3):

[c(x1)] = 3x1 + 0
[b(x1)] = 3x1 + 1
[a(x1)] = 3x1 + 2

We obtain the labeled TRS
a2(a0(c1(b2(a1(b2(a0(c1(b2(a2(x1)))))))))) → a2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c2(x1))))))))))))) (5)
a2(a0(c1(b2(a1(b2(a0(c1(b2(a0(x1)))))))))) → a2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c0(x1))))))))))))) (6)
a2(a0(c1(b2(a1(b2(a0(c1(b2(a1(x1)))))))))) → a2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c1(x1))))))))))))) (7)
c2(a0(c1(b2(a1(b2(a0(c1(b2(a2(x1)))))))))) → c2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c2(x1))))))))))))) (8)
c2(a0(c1(b2(a1(b2(a0(c1(b2(a0(x1)))))))))) → c2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c0(x1))))))))))))) (9)
c2(a0(c1(b2(a1(b2(a0(c1(b2(a1(x1)))))))))) → c2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c1(x1))))))))))))) (10)
b2(a0(c1(b2(a1(b2(a0(c1(b2(a2(x1)))))))))) → b2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c2(x1))))))))))))) (11)
b2(a0(c1(b2(a1(b2(a0(c1(b2(a0(x1)))))))))) → b2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c0(x1))))))))))))) (12)
b2(a0(c1(b2(a1(b2(a0(c1(b2(a1(x1)))))))))) → b2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c1(x1))))))))))))) (13)

1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c0(x1)] = x1 +
0
[c1(x1)] = x1 +
0
[c2(x1)] = x1 +
0
[b2(x1)] = x1 +
0
[a0(x1)] = x1 +
0
[a1(x1)] = x1 +
0
[a2(x1)] = x1 +
1
all of the following rules can be deleted.
a2(a0(c1(b2(a1(b2(a0(c1(b2(a2(x1)))))))))) → a2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c2(x1))))))))))))) (5)
c2(a0(c1(b2(a1(b2(a0(c1(b2(a2(x1)))))))))) → c2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c2(x1))))))))))))) (8)
b2(a0(c1(b2(a1(b2(a0(c1(b2(a2(x1)))))))))) → b2(a1(b2(a0(c1(b2(a0(c1(b2(a1(b2(a0(c2(x1))))))))))))) (11)

1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a0(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → c0(a0(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(a2(x1))))))))))))) (14)
a1(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → c1(a0(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(a2(x1))))))))))))) (15)
a0(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → c0(a0(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(c2(x1))))))))))))) (16)
a1(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → c1(a0(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(c2(x1))))))))))))) (17)
a0(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → c0(a0(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(b2(x1))))))))))))) (18)
a1(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → c1(a0(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(b2(x1))))))))))))) (19)

1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a0#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a0#(b2(c1(a0(b2(a1(c2(x1))))))) (20)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a0#(b2(a1(c2(x1)))) (21)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a0#(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(c2(x1)))))))))))) (22)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a1#(c2(x1)) (23)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a1#(b2(c1(a0(b2(c1(a0(b2(a1(c2(x1)))))))))) (24)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a0#(b2(c1(a0(b2(a1(b2(x1))))))) (25)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a0#(b2(a1(b2(x1)))) (26)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a0#(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(b2(x1)))))))))))) (27)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a1#(b2(x1)) (28)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a1#(b2(c1(a0(b2(c1(a0(b2(a1(b2(x1)))))))))) (29)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a0#(b2(c1(a0(b2(a1(a2(x1))))))) (30)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a0#(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(a2(x1)))))))))))) (31)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a0#(b2(a1(a2(x1)))) (32)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a1#(b2(c1(a0(b2(c1(a0(b2(a1(a2(x1)))))))))) (33)
a0#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a1#(a2(x1)) (34)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a0#(b2(c1(a0(b2(a1(c2(x1))))))) (35)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a0#(b2(a1(c2(x1)))) (36)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a0#(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(c2(x1)))))))))))) (37)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a1#(c2(x1)) (38)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(c2(x1)))))))))) → a1#(b2(c1(a0(b2(c1(a0(b2(a1(c2(x1)))))))))) (39)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a0#(b2(c1(a0(b2(a1(b2(x1))))))) (40)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a0#(b2(a1(b2(x1)))) (41)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a0#(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(b2(x1)))))))))))) (42)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a1#(b2(x1)) (43)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(b2(x1)))))))))) → a1#(b2(c1(a0(b2(c1(a0(b2(a1(b2(x1)))))))))) (44)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a0#(b2(c1(a0(b2(a1(a2(x1))))))) (45)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a0#(b2(a1(b2(c1(a0(b2(c1(a0(b2(a1(a2(x1)))))))))))) (46)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a0#(b2(a1(a2(x1)))) (47)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a1#(b2(c1(a0(b2(c1(a0(b2(a1(a2(x1)))))))))) (48)
a1#(b2(c1(a0(b2(a1(b2(c1(a0(a2(x1)))))))))) → a1#(a2(x1)) (49)

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.