Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z066)

The rewrite relation of the following TRS is considered.

a(b(c(x1))) → c(b(a(x1))) (1)
C(B(A(x1))) → A(B(C(x1))) (2)
b(a(C(x1))) → C(a(b(x1))) (3)
c(A(B(x1))) → B(A(c(x1))) (4)
A(c(b(x1))) → b(c(A(x1))) (5)
B(C(a(x1))) → a(C(B(x1))) (6)
a(A(x1)) → x1 (7)
A(a(x1)) → x1 (8)
b(B(x1)) → x1 (9)
B(b(x1)) → x1 (10)
c(C(x1)) → x1 (11)
C(c(x1)) → x1 (12)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[C(x1)] = x1 +
1
[B(x1)] = x1 +
1
[A(x1)] = x1 +
1
[c(x1)] = x1 +
1
[b(x1)] = x1 +
1
[a(x1)] = x1 +
1
all of the following rules can be deleted.
a(A(x1)) → x1 (7)
A(a(x1)) → x1 (8)
b(B(x1)) → x1 (9)
B(b(x1)) → x1 (10)
c(C(x1)) → x1 (11)
C(c(x1)) → x1 (12)

1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
c(b(a(x1))) → a(b(c(x1))) (13)
A(B(C(x1))) → C(B(A(x1))) (14)
C(a(b(x1))) → b(a(C(x1))) (15)
B(A(c(x1))) → c(A(B(x1))) (16)
b(c(A(x1))) → A(c(b(x1))) (17)
a(C(B(x1))) → B(C(a(x1))) (18)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
C#(a(b(x1))) → C#(x1) (19)
C#(a(b(x1))) → b#(a(C(x1))) (20)
C#(a(b(x1))) → a#(C(x1)) (21)
B#(A(c(x1))) → B#(x1) (22)
B#(A(c(x1))) → A#(B(x1)) (23)
B#(A(c(x1))) → c#(A(B(x1))) (24)
A#(B(C(x1))) → C#(B(A(x1))) (25)
A#(B(C(x1))) → B#(A(x1)) (26)
A#(B(C(x1))) → A#(x1) (27)
c#(b(a(x1))) → c#(x1) (28)
c#(b(a(x1))) → b#(c(x1)) (29)
c#(b(a(x1))) → a#(b(c(x1))) (30)
b#(c(A(x1))) → A#(c(b(x1))) (31)
b#(c(A(x1))) → c#(b(x1)) (32)
b#(c(A(x1))) → b#(x1) (33)
a#(C(B(x1))) → C#(a(x1)) (34)
a#(C(B(x1))) → B#(C(a(x1))) (35)
a#(C(B(x1))) → a#(x1) (36)

1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[C(x1)] = x1 +
1
[B(x1)] = x1 +
1
[A(x1)] = x1 +
1
[c(x1)] = x1 +
1
[b(x1)] = x1 +
1
[a(x1)] = x1 +
1
[C#(x1)] = x1 +
0
[B#(x1)] = x1 +
0
[A#(x1)] = x1 +
0
[c#(x1)] = x1 +
0
[b#(x1)] = x1 +
0
[a#(x1)] = x1 +
0
together with the usable rules
c(b(a(x1))) → a(b(c(x1))) (13)
A(B(C(x1))) → C(B(A(x1))) (14)
C(a(b(x1))) → b(a(C(x1))) (15)
B(A(c(x1))) → c(A(B(x1))) (16)
b(c(A(x1))) → A(c(b(x1))) (17)
a(C(B(x1))) → B(C(a(x1))) (18)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
C#(a(b(x1))) → C#(x1) (19)
C#(a(b(x1))) → a#(C(x1)) (21)
B#(A(c(x1))) → B#(x1) (22)
B#(A(c(x1))) → A#(B(x1)) (23)
A#(B(C(x1))) → B#(A(x1)) (26)
A#(B(C(x1))) → A#(x1) (27)
c#(b(a(x1))) → c#(x1) (28)
c#(b(a(x1))) → b#(c(x1)) (29)
b#(c(A(x1))) → c#(b(x1)) (32)
b#(c(A(x1))) → b#(x1) (33)
a#(C(B(x1))) → C#(a(x1)) (34)
a#(C(B(x1))) → a#(x1) (36)
and no rules could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.