Certification Problem

Input (TPDB SRS_Standard/Mixed_SRS/02)

The rewrite relation of the following TRS is considered.

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

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(a(a(a(a(x1))))) → b(b(a(a(b(x1))))) (3)
b(b(a(b(x1)))) → b(a(b(a(x1)))) (4)
a(a(a(a(a(x1))))) → a(b(a(a(b(x1))))) (5)
a(b(a(b(x1)))) → a(a(b(a(x1)))) (6)

1.1 Semantic Labeling

The following interpretations form a model of the rules.

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

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

We obtain the labeled TRS
a1(a1(a1(a1(a1(x1))))) → a0(b1(a1(a0(b1(x1))))) (7)
a1(a1(a1(a1(a0(x1))))) → a0(b1(a1(a0(b0(x1))))) (8)
b1(a1(a1(a1(a1(x1))))) → b0(b1(a1(a0(b1(x1))))) (9)
b1(a1(a1(a1(a0(x1))))) → b0(b1(a1(a0(b0(x1))))) (10)
a0(b1(a0(b1(x1)))) → a1(a0(b1(a1(x1)))) (11)
a0(b1(a0(b0(x1)))) → a1(a0(b1(a0(x1)))) (12)
b0(b1(a0(b1(x1)))) → b1(a0(b1(a1(x1)))) (13)
b0(b1(a0(b0(x1)))) → b1(a0(b1(a0(x1)))) (14)

1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a1(a1(a1(a1(a1(x1))))) → b1(a0(a1(b1(a0(x1))))) (15)
a0(a1(a1(a1(a1(x1))))) → b0(a0(a1(b1(a0(x1))))) (16)
a1(a1(a1(a1(b1(x1))))) → b1(a0(a1(b1(b0(x1))))) (17)
a0(a1(a1(a1(b1(x1))))) → b0(a0(a1(b1(b0(x1))))) (18)
b1(a0(b1(a0(x1)))) → a1(b1(a0(a1(x1)))) (19)
b0(a0(b1(a0(x1)))) → a0(b1(a0(a1(x1)))) (20)
b1(a0(b1(b0(x1)))) → a1(b1(a0(b1(x1)))) (21)
b0(a0(b1(b0(x1)))) → a0(b1(a0(b1(x1)))) (22)

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b0#(a0(b1(b0(x1)))) → b1#(x1) (23)
b0#(a0(b1(b0(x1)))) → b1#(a0(b1(x1))) (24)
b0#(a0(b1(b0(x1)))) → a0#(b1(x1)) (25)
b0#(a0(b1(b0(x1)))) → a0#(b1(a0(b1(x1)))) (26)
b0#(a0(b1(a0(x1)))) → b1#(a0(a1(x1))) (27)
b0#(a0(b1(a0(x1)))) → a0#(b1(a0(a1(x1)))) (28)
b0#(a0(b1(a0(x1)))) → a0#(a1(x1)) (29)
b0#(a0(b1(a0(x1)))) → a1#(x1) (30)
b1#(a0(b1(b0(x1)))) → b1#(x1) (31)
b1#(a0(b1(b0(x1)))) → b1#(a0(b1(x1))) (32)
b1#(a0(b1(b0(x1)))) → a0#(b1(x1)) (33)
b1#(a0(b1(b0(x1)))) → a1#(b1(a0(b1(x1)))) (34)
b1#(a0(b1(a0(x1)))) → b1#(a0(a1(x1))) (35)
b1#(a0(b1(a0(x1)))) → a0#(a1(x1)) (36)
b1#(a0(b1(a0(x1)))) → a1#(x1) (37)
b1#(a0(b1(a0(x1)))) → a1#(b1(a0(a1(x1)))) (38)
a0#(a1(a1(a1(b1(x1))))) → b0#(x1) (39)
a0#(a1(a1(a1(b1(x1))))) → b0#(a0(a1(b1(b0(x1))))) (40)
a0#(a1(a1(a1(b1(x1))))) → b1#(b0(x1)) (41)
a0#(a1(a1(a1(b1(x1))))) → a0#(a1(b1(b0(x1)))) (42)
a0#(a1(a1(a1(b1(x1))))) → a1#(b1(b0(x1))) (43)
a0#(a1(a1(a1(a1(x1))))) → b0#(a0(a1(b1(a0(x1))))) (44)
a0#(a1(a1(a1(a1(x1))))) → b1#(a0(x1)) (45)
a0#(a1(a1(a1(a1(x1))))) → a0#(x1) (46)
a0#(a1(a1(a1(a1(x1))))) → a0#(a1(b1(a0(x1)))) (47)
a0#(a1(a1(a1(a1(x1))))) → a1#(b1(a0(x1))) (48)
a1#(a1(a1(a1(b1(x1))))) → b0#(x1) (49)
a1#(a1(a1(a1(b1(x1))))) → b1#(b0(x1)) (50)
a1#(a1(a1(a1(b1(x1))))) → b1#(a0(a1(b1(b0(x1))))) (51)
a1#(a1(a1(a1(b1(x1))))) → a0#(a1(b1(b0(x1)))) (52)
a1#(a1(a1(a1(b1(x1))))) → a1#(b1(b0(x1))) (53)
a1#(a1(a1(a1(a1(x1))))) → b1#(a0(x1)) (54)
a1#(a1(a1(a1(a1(x1))))) → b1#(a0(a1(b1(a0(x1))))) (55)
a1#(a1(a1(a1(a1(x1))))) → a0#(x1) (56)
a1#(a1(a1(a1(a1(x1))))) → a0#(a1(b1(a0(x1)))) (57)
a1#(a1(a1(a1(a1(x1))))) → a1#(b1(a0(x1))) (58)

1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[b0(x1)] = x1 +
2
[b1(x1)] = x1 +
2
[a0(x1)] = x1 +
2
[a1(x1)] = x1 +
2
[b0#(x1)] = x1 +
1
[b1#(x1)] = x1 +
1
[a0#(x1)] = x1 +
1
[a1#(x1)] = x1 +
1
together with the usable rules
a1(a1(a1(a1(a1(x1))))) → b1(a0(a1(b1(a0(x1))))) (15)
a0(a1(a1(a1(a1(x1))))) → b0(a0(a1(b1(a0(x1))))) (16)
a1(a1(a1(a1(b1(x1))))) → b1(a0(a1(b1(b0(x1))))) (17)
a0(a1(a1(a1(b1(x1))))) → b0(a0(a1(b1(b0(x1))))) (18)
b1(a0(b1(a0(x1)))) → a1(b1(a0(a1(x1)))) (19)
b0(a0(b1(a0(x1)))) → a0(b1(a0(a1(x1)))) (20)
b1(a0(b1(b0(x1)))) → a1(b1(a0(b1(x1)))) (21)
b0(a0(b1(b0(x1)))) → a0(b1(a0(b1(x1)))) (22)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b0#(a0(b1(b0(x1)))) → b1#(x1) (23)
b0#(a0(b1(b0(x1)))) → b1#(a0(b1(x1))) (24)
b0#(a0(b1(b0(x1)))) → a0#(b1(x1)) (25)
b0#(a0(b1(a0(x1)))) → b1#(a0(a1(x1))) (27)
b0#(a0(b1(a0(x1)))) → a0#(a1(x1)) (29)
b0#(a0(b1(a0(x1)))) → a1#(x1) (30)
b1#(a0(b1(b0(x1)))) → b1#(x1) (31)
b1#(a0(b1(b0(x1)))) → b1#(a0(b1(x1))) (32)
b1#(a0(b1(b0(x1)))) → a0#(b1(x1)) (33)
b1#(a0(b1(a0(x1)))) → b1#(a0(a1(x1))) (35)
b1#(a0(b1(a0(x1)))) → a0#(a1(x1)) (36)
b1#(a0(b1(a0(x1)))) → a1#(x1) (37)
a0#(a1(a1(a1(b1(x1))))) → b0#(x1) (39)
a0#(a1(a1(a1(b1(x1))))) → b1#(b0(x1)) (41)
a0#(a1(a1(a1(b1(x1))))) → a0#(a1(b1(b0(x1)))) (42)
a0#(a1(a1(a1(b1(x1))))) → a1#(b1(b0(x1))) (43)
a0#(a1(a1(a1(a1(x1))))) → b1#(a0(x1)) (45)
a0#(a1(a1(a1(a1(x1))))) → a0#(x1) (46)
a0#(a1(a1(a1(a1(x1))))) → a0#(a1(b1(a0(x1)))) (47)
a0#(a1(a1(a1(a1(x1))))) → a1#(b1(a0(x1))) (48)
a1#(a1(a1(a1(b1(x1))))) → b0#(x1) (49)
a1#(a1(a1(a1(b1(x1))))) → b1#(b0(x1)) (50)
a1#(a1(a1(a1(b1(x1))))) → a0#(a1(b1(b0(x1)))) (52)
a1#(a1(a1(a1(b1(x1))))) → a1#(b1(b0(x1))) (53)
a1#(a1(a1(a1(a1(x1))))) → b1#(a0(x1)) (54)
a1#(a1(a1(a1(a1(x1))))) → a0#(x1) (56)
a1#(a1(a1(a1(a1(x1))))) → a0#(a1(b1(a0(x1)))) (57)
a1#(a1(a1(a1(a1(x1))))) → a1#(b1(a0(x1))) (58)
and no rules could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.