Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_7_56_Bor03_GM)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Reduction Pair Processor

1.1.1 Reduction Pair Processor

1.1.1.1 Reduction Pair Processor

1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  mark#(from(X))	→	a__from#(mark(X))	(16)
  a__from#(X)	→	mark#(X)	(11)
  mark#(from(X))	→	mark#(X)	(17)
  mark#(cons(X1,X2))	→	mark#(X1)	(21)
  mark#(s(X))	→	mark#(X)	(22)

  1.1.1.1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

  [mark#(x1)]	=	2 + 0 · x1
  [from(x1)]	=	3 + 3 · x1
  [a__from#(x1)]	=	2 + 3 · x1
  [mark(x1)]	=	0 + 0 · x1
  [cons(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
  [s(x1)]	=	-∞ + 0 · x1
  [a__from(x1)]	=	3 + 3 · x1
  [after(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
  [a__after(x1, x2)]	=	0 + 0 · x1 + 0 · x2
  [0]	=	0

  the pair

  mark#(from(X))

  →

  mark#(X)

  (17)

  could be deleted.

  1.1.1.1.1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

  [mark#(x1)]	=	-∞ + 0 · x1
  [from(x1)]	=	0 + 1 · x1
  [a__from#(x1)]	=	-∞ + 0 · x1
  [mark(x1)]	=	-∞ + 0 · x1
  [cons(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
  [s(x1)]	=	-∞ + 0 · x1
  [a__from(x1)]	=	0 + 1 · x1
  [after(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
  [a__after(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
  [0]	=	2

  the pair

  mark#(from(X))

  →

  a__from#(mark(X))

  (16)

  could be deleted.

  1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    mark#(s(X))	→	mark#(X)	(22)
    mark#(cons(X1,X2))	→	mark#(X1)	(21)

    1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over the naturals

    [s(x1)]	=	1 · x1
    [cons(x1, x2)]	=	1 · x1 + 1 · x2
    [mark#(x1)]	=	1 · x1

    having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

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

    mark#(s(X))	→	mark#(X)	(22)
    1	>	1
    mark#(cons(X1,X2))	→	mark#(X1)	(21)
    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__after#(s(N),cons(X,XS))

  →

  a__after#(mark(N),mark(XS))

  (13)

  1.1.1.1.1.1.1.2 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [a__after#(x1, x2)]	=	2 + 2 · x1 + x2
  [mark(x1)]	=	x1
  [from(x1)]	=	-2
  [a__from(x1)]	=	-2
  [after(x1, x2)]	=	2 · x2
  [a__after(x1, x2)]	=	2 · x2
  [0]	=	0
  [s(x1)]	=	2 + 2 · x1
  [cons(x1, x2)]	=	2 · x2

  the pair

  a__after#(s(N),cons(X,XS))

  →

  a__after#(mark(N),mark(XS))

  (13)

  could be deleted.

  1.1.1.1.1.1.1.2.1 P is empty

  There are no pairs anymore.