Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-402)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  a#(a(a(b(x1))))	→	a#(a(x1))	(9)
  a#(a(a(b(x1))))	→	a#(x1)	(8)

  1.1.1.1 Subterm Criterion Processor

  We use the projection to multisets

  π(a#)	=	{ 1, 1 }
  π(a)	=	{ 1, 1 }
  π(b)	=	{ 1, 1 }

  to remove the pairs:

  a#(a(a(b(x1))))	→	a#(a(x1))	(9)
  a#(a(a(b(x1))))	→	a#(x1)	(8)

  1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  b#(b(b(a(x1))))	→	b#(a(b(a(x1))))	(7)
  b#(b(b(a(x1))))	→	a#(b(a(x1)))	(6)
  a#(b(b(b(x1))))	→	b#(b(b(b(x1))))	(5)

  1.1.1.2 Subterm Criterion Processor

  We use the projection to multisets

  π(b#)	=	{ 1, 1 }
  π(a#)	=	{ 1, 1 }
  π(a)	=	{ 1, 1 }
  π(b)	=	{ 1, 1 }

  to remove the pairs:

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

  →

  a#(b(a(x1)))

  (6)

  1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

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

    →

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

    (7)

    1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

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

    together with the usable rules

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

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

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

    →

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

    (7)

    could be deleted.

    1.1.1.2.1.1.1 P is empty

    There are no pairs anymore.