Certification Problem

Input (TPDB SRS_Standard/Zantema_06/11)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  r#(a(c(x1)))	→	r#(x1)	(13)
  r#(a(c(x1)))	→	a#(r(x1))	(14)
  a#(r(l(x1)))	→	r#(c(c(l(a(x1)))))	(19)
  a#(r(l(x1)))	→	a#(x1)	(15)

  1.1.1.1 Subterm Criterion Processor

  We use the projection to multisets

  π(r#)	=	{ 1, 1 }
  π(a#)	=	{ 1, 1 }
  π(r)	=	{ 1, 1 }
  π(c)	=	{ 1 }
  π(a)	=	{ 1, 1 }
  π(l)	=	{ 1 }

  to remove the pairs:

  r#(a(c(x1)))	→	r#(x1)	(13)
  a#(r(l(x1)))	→	a#(x1)	(15)

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

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

  together with the usable rules

  l(a(x1))	→	a(l(x1))	(5)
  c(a(x1))	→	a(c(x1))	(6)
  r(a(c(x1)))	→	a(r(x1))	(7)
  a(r(l(x1)))	→	r(c(c(l(a(x1)))))	(8)

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

  a#(r(l(x1)))

  →

  r#(c(c(l(a(x1)))))

  (19)

  could be deleted.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  l#(a(x1))

  →

  l#(x1)

  (9)

  1.1.1.2 Size-Change Termination

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

  l#(a(x1))	→	l#(x1)	(9)
  1	>	1

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

• The 3^rd component contains the pair

  c#(a(x1))

  →

  c#(x1)

  (11)

  1.1.1.3 Size-Change Termination

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

  c#(a(x1))	→	c#(x1)	(11)
  1	>	1

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