Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z050)

The rewrite relation of the following TRS is considered.

a(b(b(a(a(b(x1))))))

→

a(a(b(b(a(b(a(x1)))))))

(1)

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.

a^#(b(b(a(a(b(x1))))))	→	a^#(b(b(a(b(a(x1))))))	(2)
a^#(b(b(a(a(b(x1))))))	→	a^#(b(a(x1)))	(3)
a^#(b(b(a(a(b(x1))))))	→	a^#(a(b(b(a(b(a(x1)))))))	(4)
a^#(b(b(a(a(b(x1))))))	→	a^#(x1)	(5)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(b(b(a(a(b(x1))))))	→	a^#(x1)	(5)
a^#(b(b(a(a(b(x1))))))	→	a^#(a(b(b(a(b(a(x1)))))))	(4)
a^#(b(b(a(a(b(x1))))))	→	a^#(b(a(x1)))	(3)
a^#(b(b(a(a(b(x1))))))	→	a^#(b(b(a(b(a(x1))))))	(2)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[b(x₁)]	=	x₁ + 28958
[a^#(x₁)]	=	x₁ + 0

together with the usable rule

a(b(b(a(a(b(x1))))))

→

a(a(b(b(a(b(a(x1)))))))

(1)

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

a^#(b(b(a(a(b(x1))))))	→	a^#(x1)	(5)
a^#(b(b(a(a(b(x1))))))	→	a^#(b(a(x1)))	(3)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(b(b(a(a(b(x1)))))) → a^#(b(b(a(b(a(x1)))))) (2)

a^#(b(b(a(a(b(x1)))))) → a^#(a(b(b(a(b(a(x1))))))) (4)

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

[a(x₁)] = 0

[b(x₁)] = x₁ + 8856

[a^#(x₁)] = x₁ + 0

together with the usable rule
a(b(b(a(a(b(x1)))))) → a(a(b(b(a(b(a(x1))))))) (1)
(w.r.t. the implicit argument filter of the reduction pair), the pair
a^#(b(b(a(a(b(x1)))))) → a^#(a(b(b(a(b(a(x1))))))) (4)
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.
- The 1^st component contains the pair
  a^#(b(b(a(a(b(x1)))))) → a^#(b(b(a(b(a(x1)))))) (2)
  
  1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
  Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals
  
  [a(x₁)] =
  
  1 0
  
  0 0
  
  · x₁ +
  
  1
  
  0
  
  [b(x₁)] =
  
  0 1
  
  1 1
  
  · x₁ +
  
  1
  
  9982
  
  [a^#(x₁)] =
  
  0 1
  
  1 1
  
  · x₁ +
  
  0
  
  0
  
  together with the usable rule
  a(b(b(a(a(b(x1)))))) → a(a(b(b(a(b(a(x1))))))) (1)
  (w.r.t. the implicit argument filter of the reduction pair), the pair
  a^#(b(b(a(a(b(x1)))))) → a^#(b(b(a(b(a(x1)))))) (2)
  could be deleted.
  1.1.1.1.1.1.1.1 Dependency Graph Processor
  
  The dependency pairs are split into 0 components.