Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup10)

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 Rule Removal

1.1.1.1 Dependency Pair Transformation

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(B(B(c(c(b(b(c(c(B(B(x1)))))))))))	(25)
  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(B(B(x1)))	(17)
  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(c(B(B(x1))))	(18)
  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(b(b(c(c(B(B(x1)))))))	(21)
  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(c(b(b(c(c(B(B(x1))))))))	(22)
  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(c(B(B(c(c(b(b(c(c(B(B(x1))))))))))))	(26)

  1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the

  prec(c#)	=	0		stat(c#)	=	lex
  prec(B)	=	0		stat(B)	=	lex
  prec(b)	=	0		stat(b)	=	lex
  prec(c)	=	0		stat(c)	=	lex

  π(c#)	=	1
  π(B)	=	1
  π(b)	=	1
  π(c)	=	[1]

  together with the usable rules

  b(b(b(b(x1))))	→	B(B(x1))	(3)
  B(B(B(B(x1))))	→	b(b(x1))	(4)
  c(c(b(b(c(c(B(B(c(c(x1))))))))))	→	b(b(c(c(B(B(c(c(b(b(c(c(B(B(x1))))))))))))))	(10)
  B(B(b(b(x1))))	→	x1	(7)
  b(b(B(B(x1))))	→	x1	(6)

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

  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(B(B(c(c(b(b(c(c(B(B(x1)))))))))))	(25)
  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(B(B(x1)))	(17)
  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(c(B(B(x1))))	(18)
  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(b(b(c(c(B(B(x1)))))))	(21)
  c#(c(b(b(c(c(B(B(c(c(x1))))))))))	→	c#(c(b(b(c(c(B(B(x1))))))))	(22)

  could be deleted.

  1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

  [c(x1)]	=	1 0 0 0 1 0 1 0 0 · x1 + 0 0 0 1 0 0 0 0 0
  [B(x1)]	=	0 0 1 1 0 0 0 1 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
  [b(x1)]	=	0 1 0 0 0 1 1 0 0 · x1 + 0 0 0 0 0 0 0 0 0

  together with the usable rules

  b(b(b(b(x1))))	→	B(B(x1))	(3)
  B(B(B(B(x1))))	→	b(b(x1))	(4)
  c(c(b(b(c(c(B(B(c(c(x1))))))))))	→	b(b(c(c(B(B(c(c(b(b(c(c(B(B(x1))))))))))))))	(10)
  B(B(b(b(x1))))	→	x1	(7)
  b(b(B(B(x1))))	→	x1	(6)

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

  c#(c(b(b(c(c(B(B(c(c(x1))))))))))

  →

  c#(c(B(B(c(c(b(b(c(c(B(B(x1))))))))))))

  (26)

  could be deleted.

  1.1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  B#(B(B(B(x1))))	→	b#(x1)	(13)
  b#(b(b(b(x1))))	→	B#(x1)	(11)
  B#(B(B(B(x1))))	→	b#(b(x1))	(14)
  b#(b(b(b(x1))))	→	B#(B(x1))	(12)

  1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

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

  together with the usable rules

  b(b(b(b(x1))))	→	B(B(x1))	(3)
  b(b(B(B(x1))))	→	x1	(6)
  B(B(B(B(x1))))	→	b(b(x1))	(4)
  B(B(b(b(x1))))	→	x1	(7)

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

  B#(B(B(B(x1))))	→	b#(x1)	(13)
  b#(b(b(b(x1))))	→	B#(x1)	(11)
  B#(B(B(B(x1))))	→	b#(b(x1))	(14)
  b#(b(b(b(x1))))	→	B#(B(x1))	(12)

  could be deleted.

  1.1.1.1.1.2.1 P is empty

  There are no pairs anymore.