Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z084)

The rewrite relation of the following TRS is considered.

b(c(x1))	→	a(x1)	(1)
b(b(x1))	→	a(c(x1))	(2)
a(x1)	→	c(b(x1))	(3)
c(c(c(x1)))	→	b(x1)	(4)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

b^#(b(x1))	→	c^#(x1)	(5)
c^#(c(c(x1)))	→	b^#(x1)	(6)
b^#(c(x1))	→	a^#(x1)	(7)
a^#(x1)	→	c^#(b(x1))	(8)
b^#(b(x1))	→	a^#(c(x1))	(9)
a^#(x1)	→	b^#(x1)	(10)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(x1) → b^#(x1) (10)

b^#(b(x1)) → a^#(c(x1)) (9)

a^#(x1) → c^#(b(x1)) (8)

c^#(c(c(x1))) → b^#(x1) (6)

b^#(c(x1)) → a^#(x1) (7)

b^#(b(x1)) → c^#(x1) (5)

1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation

[a(x₁)] = x₁ + 63723

[b(x₁)] = x₁ + 42482

[c(x₁)] = x₁ + 21241

[c^#(x₁)] = x₁ + 0

[a^#(x₁)] = x₁ + 42483

[b^#(x₁)] = x₁ + 21243

together with the usable rules

c(c(c(x1))) → b(x1) (4)

b(c(x1)) → a(x1) (1)

a(x1) → c(b(x1)) (3)

b(b(x1)) → a(c(x1)) (2)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

a^#(x1) → b^#(x1) (10)

b^#(b(x1)) → a^#(c(x1)) (9)

a^#(x1) → c^#(b(x1)) (8)

c^#(c(c(x1))) → b^#(x1) (6)

b^#(c(x1)) → a^#(x1) (7)

b^#(b(x1)) → c^#(x1) (5)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.