Certification Problem

Input (TPDB TRS_Standard/MNZ_10/0)

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

  h#(f(x),g(x))	→	f#(s(x))	(30)
  f#(g(x))	→	g#(g(f(x)))	(25)
  g#(s(x))	→	s#(s(g(x)))	(28)
  s#(s(s(s(s(s(s(s(0))))))))	→	f#(s(s(0)))	(15)
  f#(g(x))	→	g#(f(x))	(24)
  g#(s(x))	→	s#(g(x))	(27)
  s#(s(s(0)))	→	f#(s(0))	(14)
  f#(g(x))	→	f#(x)	(23)
  f#(s(s(0)))	→	s#(s(s(s(s(s(0))))))	(19)
  f#(s(s(0)))	→	s#(s(s(s(s(0)))))	(18)
  f#(s(s(0)))	→	s#(s(s(s(0))))	(17)
  f#(s(s(0)))	→	s#(s(s(0)))	(16)
  g#(s(x))	→	g#(x)	(26)
  g#(x)	→	h#(x,x)	(20)
  h#(f(x),g(x))	→	s#(x)	(29)

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

  together with the usable rules

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

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

  h#(f(x),g(x))

  →

  s#(x)

  (29)

  could be deleted.

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

  together with the usable rules

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

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

  g#(s(x))	→	s#(s(g(x)))	(28)
  g#(s(x))	→	s#(g(x))	(27)

  could be deleted.

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

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

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

  f#(g(x))	→	g#(g(f(x)))	(25)
  f#(g(x))	→	g#(f(x))	(24)
  f#(g(x))	→	f#(x)	(23)
  g#(x)	→	h#(x,x)	(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

    g#(s(x))

    →

    g#(x)

    (26)

    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.

    g#(s(x))	→	g#(x)	(26)
    1	>	1

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

  • The 2^nd component contains the pair

    f#(s(s(0)))	→	s#(s(s(s(s(s(0))))))	(19)
    s#(s(s(s(s(s(s(s(0))))))))	→	f#(s(s(0)))	(15)
    f#(s(s(0)))	→	s#(s(s(s(s(0)))))	(18)
    s#(s(s(0)))	→	f#(s(0))	(14)
    f#(s(s(0)))	→	s#(s(s(s(0))))	(17)
    f#(s(s(0)))	→	s#(s(s(0)))	(16)

    1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

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

    together with the usable rules

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

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

    s#(s(s(0)))

    →

    f#(s(0))

    (14)

    could be deleted.

    1.1.1.1.1.1.2.1 Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals

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

    together with the usable rules

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

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

    f#(s(s(0)))

    →

    s#(s(s(0)))

    (16)

    could be deleted.

    1.1.1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers

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

    together with the usable rules

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

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

    f#(s(s(0)))

    →

    s#(s(s(s(s(0)))))

    (18)

    could be deleted.

    1.1.1.1.1.1.2.1.1.1 Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers

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

    together with the usable rules

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

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

    f#(s(s(0)))

    →

    s#(s(s(s(0))))

    (17)

    could be deleted.

    1.1.1.1.1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the arctic semiring over the integers

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

    together with the usable rules

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

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

    f#(s(s(0)))

    →

    s#(s(s(s(s(s(0))))))

    (19)

    could be deleted.

    1.1.1.1.1.1.2.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.