Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-61)

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 1 component.

• The 1^st component contains the pair

  c#(b(x1))	→	c#(x1)	(8)
  c#(b(x1))	→	a#(c(x1))	(9)
  a#(a(x1))	→	c#(x1)	(5)
  a#(a(x1))	→	a#(b(c(x1)))	(7)

  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

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

  together with the usable rules

  a(x1)	→	x1	(1)
  a(a(x1))	→	a(b(c(x1)))	(2)
  b(x1)	→	x1	(3)
  c(b(x1))	→	b(a(c(x1)))	(4)

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

  a#(a(x1))

  →

  c#(x1)

  (5)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    c#(b(x1))

    →

    c#(x1)

    (8)

    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.

    c#(b(x1))	→	c#(x1)	(8)
    1	>	1

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

  • The 2^nd component contains the pair

    a#(a(x1))

    →

    a#(b(c(x1)))

    (7)

    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

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

    together with the usable rules

    a(x1)	→	x1	(1)
    a(a(x1))	→	a(b(c(x1)))	(2)
    b(x1)	→	x1	(3)
    c(b(x1))	→	b(a(c(x1)))	(4)

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

    a#(a(x1))

    →

    a#(b(c(x1)))

    (7)

    could be deleted.

    1.1.1.1.2.1 P is empty

    There are no pairs anymore.