Certification Problem

Input (TPDB TRS_Standard/Secret_05_TRS/tpa2)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ 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

  f#(s(x),y)	→	f#(p(-(s(x),y)),p(-(y,s(x))))	(11)
  f#(x,s(y))	→	f#(p(-(x,s(y))),p(-(s(y),x)))	(16)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [-(x1, x2)]	=	0 · x1 + -∞ · x2 + 8
  [p(x1)]	=	-4 · x1 + 0
  [0]	=	0
  [f#(x1, x2)]	=	-4 · x1 + -∞ · x2 + 1
  [s(x1)]	=	4 · x1 + 6

  together with the usable rules

  -(s(x),s(y))	→	-(x,y)	(2)
  -(x,0)	→	x	(1)
  p(s(x))	→	x	(3)

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

  f#(s(x),y)

  →

  f#(p(-(s(x),y)),p(-(y,s(x))))

  (11)

  could be deleted.

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [-(x1, x2)]	=	1 · x1 + 0 · x2 + -16
  [p(x1)]	=	-5 · x1 + 0
  [0]	=	2
  [f#(x1, x2)]	=	2 · x1 + 0 · x2 + 0
  [s(x1)]	=	5 · x1 + 3

  together with the usable rules

  -(x,0)	→	x	(1)
  -(s(x),s(y))	→	-(x,y)	(2)
  p(s(x))	→	x	(3)

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

  f#(x,s(y))

  →

  f#(p(-(x,s(y))),p(-(s(y),x)))

  (16)

  could be deleted.

  1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  -#(s(x),s(y))

  →

  -#(x,y)

  (6)

  1.1.2 Size-Change Termination

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

  -#(s(x),s(y))	→	-#(x,y)	(6)
  2	>	2
  1	>	1

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