Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-226)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  c#(c(x1))	→	c#(a(x1))	(10)
  c#(c(x1))	→	b#(c(a(x1)))	(11)
  b#(a(x1))	→	b#(x1)	(7)
  b#(a(x1))	→	c#(b(x1))	(8)

  1.1.1.1 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

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

  together with the usable rules

  a(x1)	→	x1	(1)
  b(a(x1))	→	c(b(x1))	(5)
  b(x1)	→	x1	(3)
  c(c(x1))	→	a(b(c(a(x1))))	(6)

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

  c#(c(x1))

  →

  c#(a(x1))

  (10)

  could be deleted.

  1.1.1.1.1 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

  [b#(x1)]	=	0 2 -∞ -∞ · x1 + -∞ -∞ -∞ -∞
  [b(x1)]	=	0 1 -∞ 0 · x1 + -∞ -∞ -∞ -∞
  [c#(x1)]	=	0 2 -∞ -∞ · x1 + -∞ -∞ -∞ -∞
  [a(x1)]	=	0 -∞ 0 1 · x1 + -∞ -∞ -∞ -∞
  [c(x1)]	=	1 -∞ 0 -∞ · x1 + -∞ -∞ -∞ -∞

  together with the usable rules

  a(x1)	→	x1	(1)
  b(a(x1))	→	c(b(x1))	(5)
  b(x1)	→	x1	(3)
  c(c(x1))	→	a(b(c(a(x1))))	(6)

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

  b#(a(x1))	→	b#(x1)	(7)
  b#(a(x1))	→	c#(b(x1))	(8)

  could be deleted.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.