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 ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Reduction Pair Processor with Usable Rules

1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  a__after#(0,XS)	→	mark#(XS)	(12)
  mark#(after(X1,X2))	→	mark#(X2)	(18)
  mark#(after(X1,X2))	→	a__after#(mark(X1),mark(X2))	(20)
  a__after#(s(N),cons(X,XS))	→	mark#(XS)	(13)
  mark#(cons(X1,X2))	→	mark#(X1)	(21)
  mark#(s(X))	→	mark#(X)	(22)
  a__after#(s(N),cons(X,XS))	→	a__after#(mark(N),mark(XS))	(15)

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

  [cons(x1, x2)]	=	0 · x1 + 0 · x2 + -∞
  [mark(x1)]	=	0 · x1 + 0
  [from(x1)]	=	0 · x1 + -∞
  [0]	=	3
  [a__after#(x1, x2)]	=	-∞ · x1 + 0 · x2 + 0
  [a__from(x1)]	=	0 · x1 + 0
  [after(x1, x2)]	=	0 · x1 + 2 · x2 + 5
  [mark#(x1)]	=	0 · x1 + 0
  [a__after(x1, x2)]	=	0 · x1 + 2 · x2 + 5
  [s(x1)]	=	0 · x1 + -∞

  together with the usable rules

  a__from(X)	→	cons(mark(X),from(s(X)))	(1)
  a__after(0,XS)	→	mark(XS)	(2)
  a__after(s(N),cons(X,XS))	→	a__after(mark(N),mark(XS))	(3)
  mark(from(X))	→	a__from(mark(X))	(4)
  mark(after(X1,X2))	→	a__after(mark(X1),mark(X2))	(5)
  mark(cons(X1,X2))	→	cons(mark(X1),X2)	(6)
  mark(s(X))	→	s(mark(X))	(7)
  mark(0)	→	0	(8)
  a__from(X)	→	from(X)	(9)
  a__after(X1,X2)	→	after(X1,X2)	(10)

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

  mark#(after(X1,X2))	→	mark#(X2)	(18)
  mark#(after(X1,X2))	→	a__after#(mark(X1),mark(X2))	(20)

  could be deleted.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

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

    →

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

    (15)

    1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [cons(x1, x2)]	=	-∞ · x1 + 0 · x2 + 1
    [mark(x1)]	=	0 · x1 + 2
    [from(x1)]	=	-∞ · x1 + 0
    [0]	=	0
    [a__after#(x1, x2)]	=	0 · x1 + -∞ · x2 + 3
    [a__from(x1)]	=	-∞ · x1 + 1
    [after(x1, x2)]	=	-∞ · x1 + 0 · x2 + 4
    [a__after(x1, x2)]	=	-∞ · x1 + 0 · x2 + 4
    [s(x1)]	=	2 · x1 + 4

    together with the usable rules

    a__from(X)	→	cons(mark(X),from(s(X)))	(1)
    a__after(0,XS)	→	mark(XS)	(2)
    a__after(s(N),cons(X,XS))	→	a__after(mark(N),mark(XS))	(3)
    mark(from(X))	→	a__from(mark(X))	(4)
    mark(after(X1,X2))	→	a__after(mark(X1),mark(X2))	(5)
    mark(cons(X1,X2))	→	cons(mark(X1),X2)	(6)
    mark(s(X))	→	s(mark(X))	(7)
    mark(0)	→	0	(8)
    a__from(X)	→	from(X)	(9)
    a__after(X1,X2)	→	after(X1,X2)	(10)

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

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

    →

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

    (15)

    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

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

    1.1.1.1.1.1.2 Size-Change Termination

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

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

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