Certification Problem

Input (TPDB TRS_Standard/MNZ_10/8)

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

  k#(s(s(0)))	→	s#(s(s(s(s(s(s(s(0))))))))	(35)
  s#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k#(s(s(0)))	(29)
  k#(s(s(0)))	→	s#(s(s(s(s(s(s(0)))))))	(34)
  s#(s(s(0)))	→	k#(s(0))	(28)
  k#(s(s(0)))	→	s#(s(s(s(s(s(0))))))	(33)
  s#(s(0))	→	f#(s(0))	(15)
  f#(g(x))	→	g#(g(f(x)))	(21)
  g#(s(x))	→	s#(s(g(x)))	(24)
  g#(s(x))	→	s#(g(x))	(23)
  g#(s(x))	→	g#(x)	(22)
  g#(x)	→	h#(x,x)	(16)
  h#(k(x),g(x))	→	k#(s(x))	(37)
  k#(s(s(0)))	→	s#(s(s(s(s(0)))))	(32)
  k#(s(s(0)))	→	s#(s(s(s(0))))	(31)
  k#(s(s(0)))	→	s#(s(s(0)))	(30)
  h#(k(x),g(x))	→	s#(x)	(36)
  h#(f(x),g(x))	→	f#(s(x))	(26)
  f#(g(x))	→	g#(f(x))	(20)
  f#(g(x))	→	f#(x)	(19)
  h#(f(x),g(x))	→	s#(x)	(25)

  1.1.1 Reduction Pair Processor with Usable Rules

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

  [h(x1, x2)]	=	0 · x1 + 0 · x2 + 0
  [g#(x1)]	=	0 · x1 + 4
  [s(x1)]	=	0 · x1 + 0
  [g(x1)]	=	0 · x1 + 4
  [k(x1)]	=	-∞ · x1 + 0
  [k#(x1)]	=	0 · x1 + 1
  [0]	=	0
  [f#(x1)]	=	0 · x1 + 0
  [h#(x1, x2)]	=	-∞ · x1 + 0 · x2 + 0
  [s#(x1)]	=	0 · x1 + 1
  [f(x1)]	=	0 · x1 + 0

  together with the usable rules

  s(s(0))	→	f(s(0))	(1)
  g(x)	→	h(x,x)	(2)
  s(x)	→	h(x,0)	(3)
  s(x)	→	h(0,x)	(4)
  f(g(x))	→	g(g(f(x)))	(5)
  g(s(x))	→	s(s(g(x)))	(6)
  h(f(x),g(x))	→	f(s(x))	(7)
  s(s(0))	→	k(0)	(8)
  k(0)	→	0	(9)
  s(s(s(0)))	→	k(s(0))	(10)
  k(s(0))	→	s(0)	(11)
  s(s(s(s(s(s(s(s(s(s(0))))))))))	→	k(s(s(0)))	(12)
  k(s(s(0)))	→	s(s(s(s(s(s(s(s(0))))))))	(13)
  h(k(x),g(x))	→	k(s(x))	(14)

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

  s#(s(0))

  →

  f#(s(0))

  (15)

  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

    h#(f(x),g(x))	→	f#(s(x))	(26)
    f#(g(x))	→	g#(g(f(x)))	(21)
    g#(s(x))	→	g#(x)	(22)
    g#(x)	→	h#(x,x)	(16)
    f#(g(x))	→	g#(f(x))	(20)
    f#(g(x))	→	f#(x)	(19)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [h(x1, x2)]	=	-∞ · x1 + -∞ · x2 + 2
    [g#(x1)]	=	-∞ · x1 + 5
    [s(x1)]	=	-∞ · x1 + 2
    [g(x1)]	=	0 · x1 + 4
    [k(x1)]	=	0 · x1 + 0
    [0]	=	0
    [f#(x1)]	=	1 · x1 + 4
    [h#(x1, x2)]	=	-∞ · x1 + -∞ · x2 + 4
    [f(x1)]	=	0 · x1 + 0

    together with the usable rules

    s(s(0))	→	f(s(0))	(1)
    g(x)	→	h(x,x)	(2)
    s(x)	→	h(x,0)	(3)
    s(x)	→	h(0,x)	(4)
    f(g(x))	→	g(g(f(x)))	(5)
    g(s(x))	→	s(s(g(x)))	(6)
    h(f(x),g(x))	→	f(s(x))	(7)
    s(s(0))	→	k(0)	(8)
    k(0)	→	0	(9)
    s(s(s(0)))	→	k(s(0))	(10)
    k(s(0))	→	s(0)	(11)
    s(s(s(s(s(s(s(s(s(s(0))))))))))	→	k(s(s(0)))	(12)
    k(s(s(0)))	→	s(s(s(s(s(s(s(s(0))))))))	(13)
    h(k(x),g(x))	→	k(s(x))	(14)

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

    g#(x)

    →

    h#(x,x)

    (16)

    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

      f#(g(x))

      →

      f#(x)

      (19)

      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.

      f#(g(x))	→	f#(x)	(19)
      1	>	1

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

    • The 2^nd component contains the pair

      g#(s(x))

      →

      g#(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.

      g#(s(x))	→	g#(x)	(22)
      1	>	1

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

  • The 2^nd component contains the pair

    k#(s(s(0)))	→	s#(s(s(s(s(s(s(s(0))))))))	(35)
    s#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k#(s(s(0)))	(29)
    k#(s(s(0)))	→	s#(s(s(s(s(s(s(0)))))))	(34)
    s#(s(s(0)))	→	k#(s(0))	(28)
    k#(s(s(0)))	→	s#(s(s(s(s(s(0))))))	(33)
    k#(s(s(0)))	→	s#(s(s(s(s(0)))))	(32)
    k#(s(s(0)))	→	s#(s(s(s(0))))	(31)
    k#(s(s(0)))	→	s#(s(s(0)))	(30)

    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 naturals

    [h(x1, x2)]	=	0 2 0 0 · x1 + 0 0 0 0 · x2 + 0 0 2 0
    [s(x1)]	=	0 2 0 0 · x1 + 0 0 2 0
    [g(x1)]	=	0 2 0 0 · x1 + 0 0 2 0
    [k(x1)]	=	0 2 0 0 · x1 + 0 0 2 0
    [k#(x1)]	=	1 0 0 0 · x1 + 0 0 0 0
    [0]	=	0 0 0 0
    [s#(x1)]	=	1 0 0 0 · x1 + 0 0 0 0
    [f(x1)]	=	0 2 0 0 · x1 + 0 0 2 0

    together with the usable rules

    s(s(0))	→	f(s(0))	(1)
    g(x)	→	h(x,x)	(2)
    s(x)	→	h(x,0)	(3)
    s(x)	→	h(0,x)	(4)
    f(g(x))	→	g(g(f(x)))	(5)
    g(s(x))	→	s(s(g(x)))	(6)
    h(f(x),g(x))	→	f(s(x))	(7)
    s(s(0))	→	k(0)	(8)
    k(0)	→	0	(9)
    s(s(s(0)))	→	k(s(0))	(10)
    k(s(0))	→	s(0)	(11)
    s(s(s(s(s(s(s(s(s(s(0))))))))))	→	k(s(s(0)))	(12)
    k(s(s(0)))	→	s(s(s(s(s(s(s(s(0))))))))	(13)
    h(k(x),g(x))	→	k(s(x))	(14)

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

    s#(s(s(0)))

    →

    k#(s(0))

    (28)

    could be deleted.

    1.1.1.1.2.1 Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the naturals

    [h(x1, x2)]	=	0 1 1 0 · x1 + 0 1 1 0 · x2 + 0 0 0 0
    [s(x1)]	=	0 1 1 0 · x1 + 0 0 1 0
    [g(x1)]	=	3 3 3 3 · x1 + 0 0 0 0
    [k(x1)]	=	3 0 3 0 · x1 + 1 0 1 0
    [k#(x1)]	=	0 2 0 0 · x1 + 1 0 0 0
    [0]	=	0 0 0 0
    [s#(x1)]	=	1 0 0 0 · x1 + 0 0 0 0
    [f(x1)]	=	0 0 0 0 · x1 + 0 0 0 0

    together with the usable rules

    s(s(0))	→	f(s(0))	(1)
    g(x)	→	h(x,x)	(2)
    s(x)	→	h(x,0)	(3)
    s(x)	→	h(0,x)	(4)
    f(g(x))	→	g(g(f(x)))	(5)
    g(s(x))	→	s(s(g(x)))	(6)
    h(f(x),g(x))	→	f(s(x))	(7)
    s(s(0))	→	k(0)	(8)
    k(0)	→	0	(9)
    s(s(s(0)))	→	k(s(0))	(10)
    k(s(0))	→	s(0)	(11)
    s(s(s(s(s(s(s(s(s(s(0))))))))))	→	k(s(s(0)))	(12)
    k(s(s(0)))	→	s(s(s(s(s(s(s(s(0))))))))	(13)
    h(k(x),g(x))	→	k(s(x))	(14)

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

    s#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k#(s(s(0)))	(29)
    k#(s(s(0)))	→	s#(s(s(s(s(s(0))))))	(33)
    k#(s(s(0)))	→	s#(s(s(s(s(0)))))	(32)
    k#(s(s(0)))	→	s#(s(s(s(0))))	(31)
    k#(s(s(0)))	→	s#(s(s(0)))	(30)

    could be deleted.

    1.1.1.1.2.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.