Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212364)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

1.1 Dependency Pair Transformation

There are 137 ruless (increase limit for explicit display).

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  0#(1(0(x1)))	→	1#(x1)	(107)
  1#(0(3(1(x1))))	→	0#(1(1(x1)))	(151)
  0#(1(0(x1)))	→	1#(2(x1))	(125)
  1#(2(0(3(1(x1)))))	→	1#(1(2(0(x1))))	(211)
  1#(0(3(1(x1))))	→	1#(1(x1))	(152)
  1#(0(3(1(x1))))	→	1#(1(5(x1)))	(154)
  1#(0(3(1(x1))))	→	1#(5(x1))	(155)
  1#(0(3(1(x1))))	→	5#(x1)	(156)
  5#(5(0(3(1(x1)))))	→	5#(0(x1))	(218)
  5#(5(0(3(1(x1)))))	→	0#(x1)	(219)
  0#(1(0(1(x1))))	→	1#(1(x1))	(149)
  1#(0(2(1(0(x1)))))	→	1#(0(0(x1)))	(206)
  1#(0(2(1(0(x1)))))	→	0#(0(x1))	(207)
  0#(1(0(3(x1))))	→	1#(0(x1))	(158)
  1#(2(0(3(1(x1)))))	→	1#(2(0(x1)))	(212)
  1#(2(0(3(1(x1)))))	→	0#(x1)	(213)
  0#(1(0(3(x1))))	→	0#(x1)	(159)
  0#(0(1(3(x1))))	→	0#(1(x1))	(169)
  0#(0(1(3(x1))))	→	1#(x1)	(170)
  0#(5(1(3(x1))))	→	1#(1(x1))	(184)
  0#(5(1(3(x1))))	→	1#(x1)	(185)
  0#(5(1(3(x1))))	→	0#(x1)	(189)
  0#(5(1(3(x1))))	→	5#(1(x1))	(194)
  0#(5(5(0(3(x1)))))	→	0#(5(x1))	(226)
  0#(5(5(0(3(x1)))))	→	5#(x1)	(227)
  0#(0(4(1(3(x1)))))	→	0#(0(1(x1)))	(229)
  0#(0(4(1(3(x1)))))	→	0#(1(x1))	(230)
  0#(0(4(1(3(x1)))))	→	1#(x1)	(231)
  0#(5(4(1(3(x1)))))	→	1#(5(0(x1)))	(233)
  0#(5(4(1(3(x1)))))	→	5#(0(x1))	(234)
  0#(5(4(1(3(x1)))))	→	0#(x1)	(235)

  1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	1 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	0 0 0 + 0 0 0 0 -∞ -∞ 0 -∞ -∞ · x1
  [0(x1)]	=	1 1 1 + 0 0 0 0 0 0 0 0 0 · x1
  [1#(x1)]	=	0 -∞ -∞ + -∞ 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [3(x1)]	=	0 0 0 + -∞ 0 0 -∞ 0 0 0 -∞ 0 · x1
  [2(x1)]	=	0 0 -∞ + 0 0 0 0 0 0 0 0 0 · x1
  [5(x1)]	=	0 0 0 + -∞ -∞ -∞ 0 0 0 0 0 -∞ · x1
  [5#(x1)]	=	1 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [4(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1

  the pair

  1#(0(3(1(x1))))

  →

  1#(1(5(x1)))

  (154)

  could be deleted.

  1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	0 0 0 + 0 0 -∞ 0 0 0 0 0 0 · x1
  [0(x1)]	=	-∞ 1 0 + 0 0 0 0 0 0 0 0 0 · x1
  [1#(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [3(x1)]	=	0 0 0 + 0 0 0 0 0 0 -∞ 0 0 · x1
  [2(x1)]	=	0 0 0 + 0 0 -∞ 0 0 0 0 0 0 · x1
  [5(x1)]	=	0 0 1 + -∞ -∞ -∞ 0 0 0 0 0 0 · x1
  [5#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [4(x1)]	=	0 -∞ 0 + -∞ -∞ -∞ -∞ -∞ -∞ 0 -∞ -∞ · x1

  the pair

  0#(5(4(1(3(x1)))))

  →

  1#(5(0(x1)))

  (233)

  could be deleted.

  1.1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	0 0 0 + 0 0 0 0 0 0 0 0 -∞ · x1
  [0(x1)]	=	1 1 -∞ + 0 0 0 0 -∞ 0 0 0 0 · x1
  [1#(x1)]	=	0 -∞ -∞ + -∞ 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [3(x1)]	=	0 0 0 + -∞ -∞ 0 0 0 0 -∞ -∞ -∞ · x1
  [2(x1)]	=	0 -∞ -∞ + 0 0 0 0 0 0 0 0 0 · x1
  [5(x1)]	=	0 0 0 + 0 -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [5#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [4(x1)]	=	1 -∞ 0 + -∞ -∞ 0 0 -∞ 0 -∞ -∞ 0 · x1

  the pair

  1#(0(3(1(x1))))

  →

  1#(5(x1))

  (155)

  could be deleted.

  1.1.1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	1 -∞ -∞ + 0 -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	0 0 0 + 0 -∞ 0 0 0 0 -∞ -∞ -∞ · x1
  [0(x1)]	=	1 -∞ -∞ + 0 -∞ -∞ 0 0 0 0 0 -∞ · x1
  [1#(x1)]	=	1 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [3(x1)]	=	0 -∞ 0 + 0 -∞ 0 -∞ -∞ 0 0 0 0 · x1
  [2(x1)]	=	0 0 0 + -∞ 0 -∞ -∞ -∞ -∞ -∞ 0 -∞ · x1
  [5#(x1)]	=	0 -∞ -∞ + -∞ -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [5(x1)]	=	0 1 0 + 0 0 0 0 0 0 0 0 0 · x1
  [4(x1)]	=	0 -∞ -∞ + -∞ -∞ -∞ -∞ 0 -∞ -∞ -∞ -∞ · x1

  the pair

  0#(5(1(3(x1))))

  →

  5#(1(x1))

  (194)

  could be deleted.

  1.1.1.1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	0 0 0 + -∞ 0 -∞ -∞ -∞ -∞ 0 0 0 · x1
  [0(x1)]	=	-∞ -∞ 1 + 0 0 0 0 0 0 0 0 0 · x1
  [1#(x1)]	=	0 -∞ -∞ + -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [3(x1)]	=	-∞ 0 0 + 0 -∞ -∞ 0 0 0 0 0 -∞ · x1
  [2(x1)]	=	0 0 0 + 0 -∞ 0 0 -∞ -∞ -∞ 0 -∞ · x1
  [5#(x1)]	=	0 -∞ -∞ + 0 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [5(x1)]	=	0 0 0 + -∞ -∞ 0 -∞ -∞ -∞ 0 0 -∞ · x1
  [4(x1)]	=	0 0 -∞ + 0 -∞ -∞ 0 -∞ -∞ -∞ -∞ -∞ · x1

  the pair

  0#(1(0(1(x1))))

  →

  1#(1(x1))

  (149)

  could be deleted.

  1.1.1.1.1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	0 0 0 + 0 0 0 0 -∞ 0 -∞ -∞ -∞ · x1
  [0(x1)]	=	1 0 0 + 0 -∞ 0 0 0 -∞ 0 -∞ 0 · x1
  [1#(x1)]	=	0 -∞ -∞ + -∞ -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [3(x1)]	=	0 0 0 + -∞ -∞ 0 -∞ -∞ -∞ 0 0 0 · x1
  [2(x1)]	=	0 0 0 + -∞ 0 -∞ -∞ 0 0 0 0 -∞ · x1
  [5#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [5(x1)]	=	0 1 1 + 0 0 0 0 0 0 0 0 0 · x1
  [4(x1)]	=	0 1 0 + -∞ -∞ -∞ -∞ 0 -∞ 0 -∞ -∞ · x1

  the pair

  1#(2(0(3(1(x1)))))

  →

  1#(1(2(0(x1))))

  (211)

  could be deleted.

  1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	0 1 0 + 0 -∞ 0 1 1 1 0 0 -∞ · x1
  [0(x1)]	=	0 0 0 + -∞ -∞ 0 0 0 -∞ 0 0 -∞ · x1
  [1#(x1)]	=	0 -∞ -∞ + -∞ -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [3(x1)]	=	0 0 0 + -∞ -∞ -∞ 0 -∞ 0 0 0 0 · x1
  [2(x1)]	=	0 0 -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ 0 -∞ · x1
  [5#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [5(x1)]	=	0 0 0 + 0 -∞ -∞ 0 0 -∞ 0 0 -∞ · x1
  [4(x1)]	=	0 -∞ 0 + 0 -∞ -∞ 0 -∞ -∞ 0 -∞ -∞ · x1

  the pairs

  5#(5(0(3(1(x1)))))	→	5#(0(x1))	(218)
  5#(5(0(3(1(x1)))))	→	0#(x1)	(219)

  could be deleted.

  1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    1#(0(3(1(x1))))	→	0#(1(1(x1)))	(151)
    0#(1(0(x1)))	→	1#(x1)	(107)
    1#(0(3(1(x1))))	→	1#(1(x1))	(152)
    1#(0(2(1(0(x1)))))	→	1#(0(0(x1)))	(206)
    1#(0(2(1(0(x1)))))	→	0#(0(x1))	(207)
    0#(1(0(x1)))	→	1#(2(x1))	(125)
    1#(2(0(3(1(x1)))))	→	1#(2(0(x1)))	(212)
    1#(2(0(3(1(x1)))))	→	0#(x1)	(213)
    0#(1(0(3(x1))))	→	1#(0(x1))	(158)
    0#(1(0(3(x1))))	→	0#(x1)	(159)
    0#(0(1(3(x1))))	→	0#(1(x1))	(169)
    0#(0(1(3(x1))))	→	1#(x1)	(170)
    0#(5(1(3(x1))))	→	1#(1(x1))	(184)
    0#(5(1(3(x1))))	→	1#(x1)	(185)
    0#(5(1(3(x1))))	→	0#(x1)	(189)
    0#(5(5(0(3(x1)))))	→	0#(5(x1))	(226)
    0#(0(4(1(3(x1)))))	→	0#(0(1(x1)))	(229)
    0#(0(4(1(3(x1)))))	→	0#(1(x1))	(230)
    0#(0(4(1(3(x1)))))	→	1#(x1)	(231)
    0#(5(4(1(3(x1)))))	→	0#(x1)	(235)

    1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [1#(x1)]	=	0 -∞ -∞ + -∞ 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [0(x1)]	=	-∞ 1 0 + 0 0 0 0 0 0 0 0 0 · x1
    [3(x1)]	=	1 -∞ -∞ + 0 0 0 0 -∞ -∞ 0 0 0 · x1
    [1(x1)]	=	0 -∞ 0 + -∞ -∞ -∞ 0 -∞ -∞ 0 0 0 · x1
    [0#(x1)]	=	0 -∞ -∞ + 0 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [2(x1)]	=	0 -∞ 0 + -∞ 0 -∞ 0 0 -∞ 0 0 -∞ · x1
    [5(x1)]	=	0 0 0 + 0 0 0 0 0 0 0 0 0 · x1
    [4(x1)]	=	0 0 0 + -∞ 0 -∞ -∞ 0 -∞ -∞ 0 -∞ · x1

    the pair

    1#(0(3(1(x1))))

    →

    0#(1(1(x1)))

    (151)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [0#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [1(x1)]	=	0 1 -∞ + 0 -∞ -∞ -∞ 0 -∞ 1 0 0 · x1
    [0(x1)]	=	-∞ -∞ -∞ + 0 0 0 0 0 0 0 0 0 · x1
    [1#(x1)]	=	1 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [3(x1)]	=	-∞ -∞ -∞ + 0 0 0 -∞ 0 -∞ -∞ 0 -∞ · x1
    [2(x1)]	=	0 0 0 + -∞ 0 0 -∞ -∞ -∞ -∞ 0 -∞ · x1
    [5(x1)]	=	0 -∞ 1 + -∞ -∞ -∞ -∞ 0 0 -∞ -∞ -∞ · x1
    [4(x1)]	=	0 0 0 + -∞ 0 -∞ -∞ -∞ -∞ 0 0 0 · x1

    the pairs

    1#(0(2(1(0(x1)))))	→	0#(0(x1))	(207)
    0#(1(0(3(x1))))	→	0#(x1)	(159)
    0#(5(1(3(x1))))	→	0#(x1)	(189)
    0#(5(4(1(3(x1)))))	→	0#(x1)	(235)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [0#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [1(x1)]	=	0 0 0 + -∞ -∞ -∞ -∞ -∞ -∞ 0 0 0 · x1
    [0(x1)]	=	0 0 1 + 0 0 0 0 0 0 0 0 0 · x1
    [1#(x1)]	=	0 -∞ -∞ + 0 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [3(x1)]	=	0 0 -∞ + 0 0 0 0 0 -∞ 0 0 -∞ · x1
    [2(x1)]	=	0 0 0 + -∞ 0 -∞ -∞ 0 -∞ 0 0 0 · x1
    [5(x1)]	=	0 1 0 + 0 0 0 0 0 0 0 0 0 · x1
    [4(x1)]	=	0 0 -∞ + -∞ 0 0 -∞ 0 0 -∞ 0 0 · x1

    the pair

    0#(5(1(3(x1))))

    →

    1#(1(x1))

    (184)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [0#(x1)]	=	1 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [1(x1)]	=	-∞ 0 1 + 0 0 0 -∞ -∞ -∞ 0 0 -∞ · x1
    [0(x1)]	=	0 -∞ 1 + 0 -∞ 0 0 -∞ 0 0 0 -∞ · x1
    [1#(x1)]	=	0 -∞ -∞ + -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [3(x1)]	=	-∞ -∞ 0 + 0 -∞ 0 -∞ 0 0 -∞ -∞ -∞ · x1
    [2(x1)]	=	0 0 0 + -∞ -∞ 0 -∞ -∞ 0 0 -∞ 0 · x1
    [5(x1)]	=	0 0 0 + 0 -∞ -∞ 0 -∞ -∞ -∞ -∞ 0 · x1
    [4(x1)]	=	0 0 -∞ + -∞ -∞ 0 0 0 0 -∞ -∞ 0 · x1

    the pair

    1#(0(3(1(x1))))

    →

    1#(1(x1))

    (152)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [0#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [1(x1)]	=	0 1 0 + 0 0 0 1 0 0 0 0 0 · x1
    [0(x1)]	=	-∞ -∞ -∞ + 0 0 0 0 0 0 0 0 0 · x1
    [1#(x1)]	=	1 -∞ -∞ + 0 0 1 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [2(x1)]	=	0 0 -∞ + -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [3(x1)]	=	-∞ 0 0 + 0 0 0 -∞ -∞ -∞ 0 0 -∞ · x1
    [5(x1)]	=	0 0 0 + -∞ 0 0 -∞ -∞ 0 0 -∞ 0 · x1
    [4(x1)]	=	0 0 0 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ 0 -∞ · x1

    the pair

    1#(0(2(1(0(x1)))))

    →

    1#(0(0(x1)))

    (206)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [0#(x1)]	=	0 -∞ -∞ + -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [1(x1)]	=	1 1 0 + -∞ 1 -∞ -∞ 1 0 -∞ 0 1 · x1
    [0(x1)]	=	0 -∞ -∞ + -∞ 0 0 0 0 0 -∞ 0 -∞ · x1
    [1#(x1)]	=	1 -∞ -∞ + -∞ 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [2(x1)]	=	0 0 -∞ + -∞ -∞ 0 0 -∞ 0 -∞ -∞ 0 · x1
    [3(x1)]	=	0 0 -∞ + -∞ -∞ -∞ -∞ 0 0 0 -∞ -∞ · x1
    [5(x1)]	=	0 0 0 + 0 -∞ -∞ 0 -∞ -∞ 0 -∞ -∞ · x1
    [4(x1)]	=	0 0 0 + 0 -∞ -∞ 0 -∞ -∞ 0 -∞ -∞ · x1

    the pair

    1#(2(0(3(1(x1)))))

    →

    0#(x1)

    (213)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 2 components.

    • The 1^st component contains the pair

      0#(5(5(0(3(x1)))))	→	0#(5(x1))	(226)
      0#(0(1(3(x1))))	→	0#(1(x1))	(169)
      0#(0(4(1(3(x1)))))	→	0#(0(1(x1)))	(229)
      0#(0(4(1(3(x1)))))	→	0#(1(x1))	(230)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	-2 + 2 · x1
      [5(x1)]	=	1 + x1
      [0(x1)]	=	1 + x1
      [3(x1)]	=	-2 + x1
      [1(x1)]	=	2
      [4(x1)]	=	-2 + 2 · x1
      [2(x1)]	=	-2

      the pairs

      0#(0(1(3(x1))))	→	0#(1(x1))	(169)
      0#(0(4(1(3(x1)))))	→	0#(1(x1))	(230)

      could be deleted.

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	2 · x1
      [5(x1)]	=	1 + x1
      [0(x1)]	=	1 + x1
      [3(x1)]	=	-2 + x1
      [1(x1)]	=	2
      [4(x1)]	=	-1 + 2 · x1
      [2(x1)]	=	-1 + x1

      the pair

      0#(0(4(1(3(x1)))))

      →

      0#(0(1(x1)))

      (229)

      could be deleted.

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	1 · x1
      [5(x1)]	=	1 + 1 · x1
      [0(x1)]	=	1 · x1
      [3(x1)]	=	1 · x1
      [1(x1)]	=	0
      [4(x1)]	=	0
      [2(x1)]	=	0

      together with the usable rule

      5(5(0(3(1(x1)))))

      →

      1(5(3(4(5(0(x1))))))

      (93)

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

      0#(5(5(0(3(x1)))))

      →

      0#(5(x1))

      (226)

      could be deleted.

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty

      There are no pairs anymore.

    • The 2^nd component contains the pair

      1#(2(0(3(1(x1)))))

      →

      1#(2(0(x1)))

      (212)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 Reduction Pair Processor

      Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

      [1#(x1)]	=	0 -∞ -∞ + -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
      [2(x1)]	=	0 -∞ 0 + -∞ -∞ -∞ -∞ 0 -∞ -∞ -∞ 0 · x1
      [0(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ 0 -∞ 0 0 0 0 · x1
      [3(x1)]	=	0 0 -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
      [1(x1)]	=	1 1 -∞ + 1 0 1 -∞ 0 0 -∞ 0 0 · x1
      [4(x1)]	=	0 0 -∞ + -∞ -∞ -∞ -∞ 0 -∞ -∞ -∞ 0 · x1
      [5(x1)]	=	0 0 -∞ + -∞ 0 0 -∞ 0 0 -∞ 0 0 · x1

      the pair

      1#(2(0(3(1(x1)))))

      →

      1#(2(0(x1)))

      (212)

      could be deleted.

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 P is empty

      There are no pairs anymore.