Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-369)

The rewrite relation of the following TRS is considered.

b(b(b(b(x1)))) → b(a(a(a(x1)))) (1)
a(a(b(a(x1)))) → b(b(b(a(x1)))) (2)
b(b(a(b(x1)))) → a(a(b(b(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(b(b(b(x1))))) → b(b(a(a(a(x1))))) (4)
b(a(a(b(a(x1))))) → b(b(b(b(a(x1))))) (5)
b(b(b(a(b(x1))))) → b(a(a(b(b(x1))))) (6)
a(b(b(b(b(x1))))) → a(b(a(a(a(x1))))) (7)
a(a(a(b(a(x1))))) → a(b(b(b(a(x1))))) (8)
a(b(b(a(b(x1))))) → a(a(a(b(b(x1))))) (9)

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
b0(b0(b0(b0(b0(x1))))) → b0(b1(a1(a1(a0(x1))))) (10)
b0(b0(b0(b0(b1(x1))))) → b0(b1(a1(a1(a1(x1))))) (11)
a0(b0(b0(b0(b0(x1))))) → a0(b1(a1(a1(a0(x1))))) (12)
a0(b0(b0(b0(b1(x1))))) → a0(b1(a1(a1(a1(x1))))) (13)
b1(a1(a0(b1(a0(x1))))) → b0(b0(b0(b1(a0(x1))))) (14)
b1(a1(a0(b1(a1(x1))))) → b0(b0(b0(b1(a1(x1))))) (15)
a1(a1(a0(b1(a0(x1))))) → a0(b0(b0(b1(a0(x1))))) (16)
a1(a1(a0(b1(a1(x1))))) → a0(b0(b0(b1(a1(x1))))) (17)
b0(b0(b1(a0(b0(x1))))) → b1(a1(a0(b0(b0(x1))))) (18)
b0(b0(b1(a0(b1(x1))))) → b1(a1(a0(b0(b1(x1))))) (19)
a0(b0(b1(a0(b0(x1))))) → a1(a1(a0(b0(b0(x1))))) (20)
a0(b0(b1(a0(b1(x1))))) → a1(a1(a0(b0(b1(x1))))) (21)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b0#(b0(b0(b0(b0(x1))))) → b0#(b1(a1(a1(a0(x1))))) (22)
b0#(b0(b0(b0(b0(x1))))) → b1#(a1(a1(a0(x1)))) (23)
b0#(b0(b0(b0(b0(x1))))) → a0#(x1) (24)
b0#(b0(b0(b0(b0(x1))))) → a1#(a0(x1)) (25)
b0#(b0(b0(b0(b0(x1))))) → a1#(a1(a0(x1))) (26)
b0#(b0(b0(b0(b1(x1))))) → b0#(b1(a1(a1(a1(x1))))) (27)
b0#(b0(b0(b0(b1(x1))))) → b1#(a1(a1(a1(x1)))) (28)
b0#(b0(b0(b0(b1(x1))))) → a1#(x1) (29)
b0#(b0(b0(b0(b1(x1))))) → a1#(a1(x1)) (30)
b0#(b0(b0(b0(b1(x1))))) → a1#(a1(a1(x1))) (31)
b0#(b0(b1(a0(b0(x1))))) → b0#(b0(x1)) (32)
b0#(b0(b1(a0(b0(x1))))) → b1#(a1(a0(b0(b0(x1))))) (33)
b0#(b0(b1(a0(b0(x1))))) → a0#(b0(b0(x1))) (34)
b0#(b0(b1(a0(b0(x1))))) → a1#(a0(b0(b0(x1)))) (35)
b0#(b0(b1(a0(b1(x1))))) → b0#(b1(x1)) (36)
b0#(b0(b1(a0(b1(x1))))) → b1#(a1(a0(b0(b1(x1))))) (37)
b0#(b0(b1(a0(b1(x1))))) → a0#(b0(b1(x1))) (38)
b0#(b0(b1(a0(b1(x1))))) → a1#(a0(b0(b1(x1)))) (39)
b1#(a1(a0(b1(a0(x1))))) → b0#(b0(b0(b1(a0(x1))))) (40)
b1#(a1(a0(b1(a0(x1))))) → b0#(b0(b1(a0(x1)))) (41)
b1#(a1(a0(b1(a0(x1))))) → b0#(b1(a0(x1))) (42)
b1#(a1(a0(b1(a1(x1))))) → b0#(b0(b0(b1(a1(x1))))) (43)
b1#(a1(a0(b1(a1(x1))))) → b0#(b0(b1(a1(x1)))) (44)
b1#(a1(a0(b1(a1(x1))))) → b0#(b1(a1(x1))) (45)
a0#(b0(b0(b0(b0(x1))))) → b1#(a1(a1(a0(x1)))) (46)
a0#(b0(b0(b0(b0(x1))))) → a0#(x1) (47)
a0#(b0(b0(b0(b0(x1))))) → a0#(b1(a1(a1(a0(x1))))) (48)
a0#(b0(b0(b0(b0(x1))))) → a1#(a0(x1)) (49)
a0#(b0(b0(b0(b0(x1))))) → a1#(a1(a0(x1))) (50)
a0#(b0(b0(b0(b1(x1))))) → b1#(a1(a1(a1(x1)))) (51)
a0#(b0(b0(b0(b1(x1))))) → a0#(b1(a1(a1(a1(x1))))) (52)
a0#(b0(b0(b0(b1(x1))))) → a1#(x1) (53)
a0#(b0(b0(b0(b1(x1))))) → a1#(a1(x1)) (54)
a0#(b0(b0(b0(b1(x1))))) → a1#(a1(a1(x1))) (55)
a0#(b0(b1(a0(b0(x1))))) → b0#(b0(x1)) (56)
a0#(b0(b1(a0(b0(x1))))) → a0#(b0(b0(x1))) (57)
a0#(b0(b1(a0(b0(x1))))) → a1#(a0(b0(b0(x1)))) (58)
a0#(b0(b1(a0(b0(x1))))) → a1#(a1(a0(b0(b0(x1))))) (59)
a0#(b0(b1(a0(b1(x1))))) → b0#(b1(x1)) (60)
a0#(b0(b1(a0(b1(x1))))) → a0#(b0(b1(x1))) (61)
a0#(b0(b1(a0(b1(x1))))) → a1#(a0(b0(b1(x1)))) (62)
a0#(b0(b1(a0(b1(x1))))) → a1#(a1(a0(b0(b1(x1))))) (63)
a1#(a1(a0(b1(a0(x1))))) → b0#(b0(b1(a0(x1)))) (64)
a1#(a1(a0(b1(a0(x1))))) → b0#(b1(a0(x1))) (65)
a1#(a1(a0(b1(a0(x1))))) → a0#(b0(b0(b1(a0(x1))))) (66)
a1#(a1(a0(b1(a1(x1))))) → b0#(b0(b1(a1(x1)))) (67)
a1#(a1(a0(b1(a1(x1))))) → b0#(b1(a1(x1))) (68)
a1#(a1(a0(b1(a1(x1))))) → a0#(b0(b0(b1(a1(x1))))) (69)

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
b0(b0(b0(b0(b0(x1))))) → b0(b1(a1(a1(a0(x1))))) (10)
b0(b0(b0(b0(b1(x1))))) → b0(b1(a1(a1(a1(x1))))) (11)
a0(b0(b0(b0(b0(x1))))) → a0(b1(a1(a1(a0(x1))))) (12)
a0(b0(b0(b0(b1(x1))))) → a0(b1(a1(a1(a1(x1))))) (13)
b1(a1(a0(b1(a0(x1))))) → b0(b0(b0(b1(a0(x1))))) (14)
b1(a1(a0(b1(a1(x1))))) → b0(b0(b0(b1(a1(x1))))) (15)
a1(a1(a0(b1(a0(x1))))) → a0(b0(b0(b1(a0(x1))))) (16)
a1(a1(a0(b1(a1(x1))))) → a0(b0(b0(b1(a1(x1))))) (17)
b0(b0(b1(a0(b0(x1))))) → b1(a1(a0(b0(b0(x1))))) (18)
b0(b0(b1(a0(b1(x1))))) → b1(a1(a0(b0(b1(x1))))) (19)
a0(b0(b1(a0(b0(x1))))) → a1(a1(a0(b0(b0(x1))))) (20)
a0(b0(b1(a0(b1(x1))))) → a1(a1(a0(b0(b1(x1))))) (21)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b0#(b0(b0(b0(b0(x1))))) → b1#(a1(a1(a0(x1)))) (23)
b0#(b0(b0(b0(b0(x1))))) → a0#(x1) (24)
b0#(b0(b0(b0(b0(x1))))) → a1#(a0(x1)) (25)
b0#(b0(b0(b0(b0(x1))))) → a1#(a1(a0(x1))) (26)
b0#(b0(b0(b0(b1(x1))))) → b1#(a1(a1(a1(x1)))) (28)
b0#(b0(b0(b0(b1(x1))))) → a1#(x1) (29)
b0#(b0(b0(b0(b1(x1))))) → a1#(a1(x1)) (30)
b0#(b0(b0(b0(b1(x1))))) → a1#(a1(a1(x1))) (31)
b0#(b0(b1(a0(b0(x1))))) → b0#(b0(x1)) (32)
b0#(b0(b1(a0(b0(x1))))) → a0#(b0(b0(x1))) (34)
b0#(b0(b1(a0(b0(x1))))) → a1#(a0(b0(b0(x1)))) (35)
b0#(b0(b1(a0(b1(x1))))) → b0#(b1(x1)) (36)
b0#(b0(b1(a0(b1(x1))))) → a0#(b0(b1(x1))) (38)
b0#(b0(b1(a0(b1(x1))))) → a1#(a0(b0(b1(x1)))) (39)
b1#(a1(a0(b1(a0(x1))))) → b0#(b0(b1(a0(x1)))) (41)
b1#(a1(a0(b1(a0(x1))))) → b0#(b1(a0(x1))) (42)
b1#(a1(a0(b1(a1(x1))))) → b0#(b0(b1(a1(x1)))) (44)
b1#(a1(a0(b1(a1(x1))))) → b0#(b1(a1(x1))) (45)
a0#(b0(b0(b0(b0(x1))))) → b1#(a1(a1(a0(x1)))) (46)
a0#(b0(b0(b0(b0(x1))))) → a0#(x1) (47)
a0#(b0(b0(b0(b0(x1))))) → a1#(a0(x1)) (49)
a0#(b0(b0(b0(b0(x1))))) → a1#(a1(a0(x1))) (50)
a0#(b0(b0(b0(b1(x1))))) → b1#(a1(a1(a1(x1)))) (51)
a0#(b0(b0(b0(b1(x1))))) → a1#(x1) (53)
a0#(b0(b0(b0(b1(x1))))) → a1#(a1(x1)) (54)
a0#(b0(b0(b0(b1(x1))))) → a1#(a1(a1(x1))) (55)
a0#(b0(b1(a0(b0(x1))))) → b0#(b0(x1)) (56)
a0#(b0(b1(a0(b0(x1))))) → a0#(b0(b0(x1))) (57)
a0#(b0(b1(a0(b0(x1))))) → a1#(a0(b0(b0(x1)))) (58)
a0#(b0(b1(a0(b1(x1))))) → b0#(b1(x1)) (60)
a0#(b0(b1(a0(b1(x1))))) → a0#(b0(b1(x1))) (61)
a0#(b0(b1(a0(b1(x1))))) → a1#(a0(b0(b1(x1)))) (62)
a1#(a1(a0(b1(a0(x1))))) → b0#(b0(b1(a0(x1)))) (64)
a1#(a1(a0(b1(a0(x1))))) → b0#(b1(a0(x1))) (65)
a1#(a1(a0(b1(a1(x1))))) → b0#(b0(b1(a1(x1)))) (67)
a1#(a1(a0(b1(a1(x1))))) → b0#(b1(a1(x1))) (68)
and no rules could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.