Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/aprove08)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

1.1 Rule Removal

1.1.1 Dependency Pair Transformation

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  0#(p(x1))

  →

  0#(x1)

  (32)

  1.1.1.1.1 Size-Change Termination

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

  0#(p(x1))	→	0#(x1)	(32)
  1	>	1

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

• The 2^nd component contains the pair

  s#(j(x1))	→	s#(s(p(p(s(s(s(s(x1))))))))	(29)
  s#(j(x1))	→	s#(s(s(s(x1))))	(27)
  s#(j(x1))	→	s#(s(s(x1)))	(26)
  s#(j(x1))	→	s#(s(x1))	(25)
  s#(j(x1))	→	s#(x1)	(24)
  s#(i(x1))	→	s#(s(s(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1))))))))))))))))))	(23)
  s#(i(x1))	→	s#(s(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1)))))))))))))))))	(22)
  s#(i(x1))	→	s#(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1))))))))))))))))	(21)
  s#(i(x1))	→	s#(s(p(s(p(x1)))))	(17)

  1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

  [j(x1)]	=	2 · x1 + 3
  [i(x1)]	=	2 · x1 + 3
  [p(x1)]	=	0 · x1 + 0
  [s#(x1)]	=	0 · x1 + 0
  [s(x1)]	=	0 · x1 + 0

  together with the usable rules

  s(i(x1))	→	s(s(s(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1))))))))))))))))))	(9)
  s(j(x1))	→	s(p(s(p(i(s(s(p(p(s(s(s(s(x1)))))))))))))	(11)
  s(p(p(x1)))	→	p(x1)	(12)
  s(p(x1))	→	x1	(13)

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

  s#(j(x1))	→	s#(s(p(p(s(s(s(s(x1))))))))	(29)
  s#(j(x1))	→	s#(s(s(s(x1))))	(27)
  s#(j(x1))	→	s#(s(s(x1)))	(26)
  s#(j(x1))	→	s#(s(x1))	(25)
  s#(j(x1))	→	s#(x1)	(24)
  s#(i(x1))	→	s#(s(p(s(p(x1)))))	(17)

  could be deleted.

  1.1.1.1.2.1 Reduction Pair Processor with Usable Rules

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

  [j(x1)]	=	0 0 0 0 · x1 + 0 0 2 0
  [i(x1)]	=	0 0 0 0 · x1 + 2 0 0 0
  [p(x1)]	=	0 1 1 0 · x1 + 0 0 0 0
  [s#(x1)]	=	2 0 0 0 · x1 + 0 0 0 0
  [s(x1)]	=	0 1 1 0 · x1 + 0 0 0 0

  together with the usable rules

  s(p(p(x1)))	→	p(x1)	(12)
  s(p(x1))	→	x1	(13)
  s(i(x1))	→	s(s(s(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1))))))))))))))))))	(9)
  s(j(x1))	→	s(p(s(p(i(s(s(p(p(s(s(s(s(x1)))))))))))))	(11)

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

  s#(i(x1))

  →

  s#(s(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1)))))))))))))))))

  (22)

  could be deleted.

  1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

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

  [j(x1)]	=	-∞ · x1 + 3
  [i(x1)]	=	-∞ · x1 + 4
  [p(x1)]	=	-1 · x1 + 1
  [s#(x1)]	=	1 · x1 + 4
  [s(x1)]	=	1 · x1 + 1

  together with the usable rules

  s(p(p(x1)))	→	p(x1)	(12)
  s(p(x1))	→	x1	(13)
  s(i(x1))	→	s(s(s(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1))))))))))))))))))	(9)
  s(j(x1))	→	s(p(s(p(i(s(s(p(p(s(s(s(s(x1)))))))))))))	(11)

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

  s#(i(x1))

  →

  s#(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1))))))))))))))))

  (21)

  could be deleted.

  1.1.1.1.2.1.1.1 Reduction Pair Processor with Usable Rules

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

  [j(x1)]	=	-∞ · x1 + 5
  [i(x1)]	=	-∞ · x1 + 6
  [p(x1)]	=	-1 · x1 + 0
  [s#(x1)]	=	-4 · x1 + 0
  [s(x1)]	=	1 · x1 + 0

  together with the usable rules

  s(p(p(x1)))	→	p(x1)	(12)
  s(p(x1))	→	x1	(13)
  s(i(x1))	→	s(s(s(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1))))))))))))))))))	(9)
  s(j(x1))	→	s(p(s(p(i(s(s(p(p(s(s(s(s(x1)))))))))))))	(11)

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

  s#(i(x1))

  →

  s#(s(s(s(p(p(p(p(s(p(s(p(j(s(s(p(s(p(x1))))))))))))))))))

  (23)

  could be deleted.

  1.1.1.1.2.1.1.1.1 P is empty

  There are no pairs anymore.