Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z100)

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 4 components.

• The 1^st component contains the pair

  b#(l2(x1))	→	b#(r1(x1))	(26)
  b#(l1(x1))	→	b#(r2(x1))	(24)

  1.1.1 Reduction Pair Processor

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

  [b#(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [l2(x1)]	=	1 0 0 + 0 1 0 -∞ 0 -∞ -∞ 0 -∞ · x1
  [r1(x1)]	=	0 0 0 + 0 1 0 1 0 0 0 0 1 · x1
  [l1(x1)]	=	1 1 -∞ + 0 1 0 1 0 0 0 0 0 · x1
  [r2(x1)]	=	0 0 0 + -∞ 0 -∞ 0 -∞ -∞ 0 -∞ 0 · x1
  [a(x1)]	=	0 -∞ -∞ + -∞ 0 -∞ 0 -∞ -∞ 0 0 0 · x1
  [b(x1)]	=	0 1 0 + 0 -∞ -∞ 1 0 0 0 -∞ -∞ · x1

  the pair

  b#(l1(x1))

  →

  b#(r2(x1))

  (24)

  could be deleted.

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [b#(x1)]	=	1 · x1
  [l2(x1)]	=	1
  [r1(x1)]	=	0
  [a(x1)]	=	1 · x1
  [b(x1)]	=	1 + 1 · x1
  [l1(x1)]	=	0
  [r2(x1)]	=	1 + 1 · x1

  together with the usable rules

  r1(a(x1))	→	a(a(a(r1(x1))))	(1)
  r1(b(x1))	→	l1(b(x1))	(5)
  a(l1(x1))	→	l1(a(a(a(x1))))	(3)
  a(a(l2(x1)))	→	l2(a(a(x1)))	(4)
  a(a(x1))	→	x1	(9)

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

  b#(l2(x1))

  →

  b#(r1(x1))

  (26)

  could be deleted.

  1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  r1#(a(x1))

  →

  r1#(x1)

  (13)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [a(x1)]	=	1 · x1
  [r1#(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.

  r1#(a(x1))	→	r1#(x1)	(13)
  1	>	1

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

• The 3^rd component contains the pair

  r2#(a(x1))

  →

  r2#(x1)

  (17)

  1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [a(x1)]	=	1 · x1
  [r2#(x1)]	=	1 · x1

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

  1.1.3.1 Size-Change Termination

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

  r2#(a(x1))	→	r2#(x1)	(17)
  1	>	1

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

• The 4^th component contains the pair

  a#(l1(x1))	→	a#(a(x1))	(19)
  a#(l1(x1))	→	a#(a(a(x1)))	(18)
  a#(l1(x1))	→	a#(x1)	(20)
  a#(a(l2(x1)))	→	a#(a(x1))	(21)
  a#(a(l2(x1)))	→	a#(x1)	(22)

  1.1.4 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [a(x1)]	=	1 · x1
  [l1(x1)]	=	1 · x1
  [l2(x1)]	=	1 · x1
  [a#(x1)]	=	1 · x1

  together with the usable rules

  a(l1(x1))	→	l1(a(a(a(x1))))	(3)
  a(a(l2(x1)))	→	l2(a(a(x1)))	(4)
  a(a(x1))	→	x1	(9)

  (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.4.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [a(x1)]	=	1 · x1
  [l1(x1)]	=	3 + 2 · x1
  [l2(x1)]	=	2 + 2 · x1
  [a#(x1)]	=	1 · x1

  the pairs

  a#(l1(x1))	→	a#(a(x1))	(19)
  a#(l1(x1))	→	a#(a(a(x1)))	(18)
  a#(l1(x1))	→	a#(x1)	(20)
  a#(a(l2(x1)))	→	a#(a(x1))	(21)
  a#(a(l2(x1)))	→	a#(x1)	(22)

  and no rules could be deleted.

  1.1.4.1.1 P is empty

  There are no pairs anymore.