Certification Problem

Input (TPDB SRS_Standard/Zantema_06/03)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  a#(a(b(b(x1))))	→	a#(a(x1))	(5)
  a#(a(b(b(x1))))	→	a#(a(a(x1)))	(4)
  a#(a(b(b(x1))))	→	a#(x1)	(6)
  a#(c(x1))	→	a#(x1)	(8)

  1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [a#(x1)]	=	1 · x1
  [a(x1)]	=	1 · x1
  [b(x1)]	=	1 · x1
  [c(x1)]	=	1 + 1 · x1

  the pair

  a#(c(x1))

  →

  a#(x1)

  (8)

  could be deleted.

  1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [a#(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [a(x1)]	=	0 0 0 + -∞ -∞ 0 1 -∞ 0 0 -∞ -∞ · x1
  [b(x1)]	=	0 0 1 + -∞ -∞ -∞ 1 -∞ 0 0 0 -∞ · x1
  [c(x1)]	=	-∞ -∞ 0 + 0 -∞ -∞ 0 0 -∞ -∞ -∞ 0 · x1

  the pairs

  a#(a(b(b(x1))))	→	a#(a(a(x1)))	(4)
  a#(a(b(b(x1))))	→	a#(x1)	(6)

  could be deleted.

  1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 5 with strict dimension 1 over the integers

  [a#(x1)]	=	0 0 0 0 0 + 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x1
  [a(x1)]	=	0 0 0 0 0 + 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 · x1
  [b(x1)]	=	0 0 0 1 0 + 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 · x1
  [c(x1)]	=	0 0 0 0 0 + 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 · x1

  the pair

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

  →

  a#(a(x1))

  (5)

  could be deleted.

  1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  c#(b(x1))

  →

  c#(x1)

  (9)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [b(x1)]	=	1 · x1
  [c#(x1)]	=	1 · x1

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.2.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  c#(b(x1))	→	c#(x1)	(9)
  1	>	1

  As there is no critical graph in the transitive closure, there are no infinite chains.