Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-236)

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(b(x1)))	(11)
  c#(c(x1))	→	b#(x1)	(9)
  b#(a(x1))	→	b#(x1)	(6)
  b#(a(x1))	→	c#(b(x1))	(7)

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

  together with the usable rules

  a(x1)	→	x1	(1)
  b(a(x1))	→	a(c(b(x1)))	(4)
  c(c(x1))	→	a(c(a(b(x1))))	(5)

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

  c#(c(x1))

  →

  b#(x1)

  (9)

  could be deleted.

  1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    b#(a(x1))

    →

    b#(x1)

    (6)

    1.1.1.1.1.1 Size-Change Termination

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

    b#(a(x1))	→	b#(x1)	(6)
    1	>	1

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

  • The 2^nd component contains the pair

    c#(c(x1))

    →

    c#(a(b(x1)))

    (11)

    1.1.1.1.1.2 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 -∞ 0 -∞
    [c#(x1)]	=	0 -∞ -∞ -∞ · x1 + 0 -∞ -∞ -∞
    [a(x1)]	=	0 0 -∞ 1 · x1 + 0 -∞ 1 -∞
    [c(x1)]	=	1 1 0 -∞ · x1 + 1 -∞ 0 -∞

    together with the usable rules

    a(x1)	→	x1	(1)
    b(a(x1))	→	a(c(b(x1)))	(4)
    c(c(x1))	→	a(c(a(b(x1))))	(5)

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

    c#(c(x1))

    →

    c#(a(b(x1)))

    (11)

    could be deleted.

    1.1.1.1.1.2.1 P is empty

    There are no pairs anymore.