Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/213051)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

There are 119 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  4#(4(1(0(5(x1)))))	→	4#(1(x1))	(165)
  4#(1(5(5(0(x1)))))	→	0#(5(4(x1)))	(156)
  0#(5(5(4(1(x1)))))	→	4#(1(0(5(5(3(x1))))))	(150)
  4#(1(0(x1)))	→	0#(4(x1))	(66)
  0#(1(0(5(0(x1)))))	→	0#(0(x1))	(137)
  0#(0(5(5(1(x1)))))	→	0#(0(3(5(5(x1)))))	(136)
  0#(4(4(1(x1))))	→	4#(0(3(4(1(x1)))))	(96)
  4#(1(5(5(0(x1)))))	→	4#(x1)	(154)
  4#(3(1(5(0(x1)))))	→	4#(x1)	(162)
  4#(3(1(1(0(x1)))))	→	0#(1(4(3(x1))))	(161)
  0#(1(5(0(x1))))	→	0#(0(3(x1)))	(82)
  0#(0(4(1(x1))))	→	4#(0(3(0(3(x1)))))	(73)
  4#(1(5(0(0(x1)))))	→	0#(4(5(0(x1))))	(153)
  0#(0(2(1(x1))))	→	0#(1(x1))	(69)
  0#(0(1(1(x1))))	→	0#(0(3(1(x1))))	(68)
  0#(4(1(x1)))	→	4#(0(3(1(x1))))	(57)
  4#(1(5(0(0(x1)))))	→	4#(5(0(x1)))	(152)
  4#(1(5(0(x1))))	→	4#(0(3(5(x1))))	(119)
  4#(1(5(0(x1))))	→	0#(5(4(x1)))	(116)
  4#(1(5(0(x1))))	→	4#(x1)	(114)
  4#(3(1(1(0(x1)))))	→	4#(3(x1))	(160)
  4#(3(1(0(1(x1)))))	→	0#(4(x1))	(159)
  4#(3(1(0(1(x1)))))	→	4#(x1)	(158)
  4#(3(1(0(x1))))	→	4#(x1)	(122)
  4#(1(0(1(x1))))	→	0#(4(x1))	(113)
  4#(1(0(1(x1))))	→	4#(x1)	(112)
  4#(1(0(0(x1))))	→	0#(4(1(2(0(x1)))))	(111)
  4#(1(0(x1)))	→	4#(x1)	(65)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the

  prec(4#)	=	0		stat(4#)	=	lex
  prec(0#)	=	0		stat(0#)	=	lex
  prec(5)	=	0		stat(5)	=	lex
  prec(4)	=	0		stat(4)	=	lex
  prec(2)	=	0		stat(2)	=	lex
  prec(3)	=	0		stat(3)	=	lex
  prec(0)	=	0		stat(0)	=	lex
  prec(1)	=	0		stat(1)	=	lex

  π(4#)	=	1
  π(0#)	=	1
  π(5)	=	1
  π(4)	=	1
  π(2)	=	1
  π(3)	=	1
  π(0)	=	1
  π(1)	=	[1]

  together with the usable rules

  0(0(1(x1)))	→	0(1(2(0(3(x1)))))	(1)
  0(0(1(x1)))	→	0(1(2(0(3(2(x1))))))	(2)
  0(0(1(x1)))	→	1(2(0(0(3(2(x1))))))	(3)
  0(1(0(x1)))	→	2(0(3(1(2(0(x1))))))	(4)
  0(4(1(x1)))	→	4(0(3(1(x1))))	(5)
  0(4(1(x1)))	→	2(4(0(3(2(1(x1))))))	(6)
  0(4(1(x1)))	→	2(4(0(5(3(1(x1))))))	(7)
  0(4(1(x1)))	→	4(2(1(2(0(3(x1))))))	(8)
  4(1(0(x1)))	→	1(2(0(4(x1))))	(9)
  0(0(1(1(x1))))	→	1(2(0(0(3(1(x1))))))	(10)
  0(0(2(1(x1))))	→	2(2(0(3(0(1(x1))))))	(11)
  0(0(4(1(x1))))	→	1(4(0(3(0(3(x1))))))	(12)
  0(0(5(1(x1))))	→	0(0(3(1(2(5(x1))))))	(13)
  0(1(1(0(x1))))	→	1(2(0(0(1(2(x1))))))	(14)
  0(1(4(1(x1))))	→	0(1(2(4(1(2(x1))))))	(15)
  0(1(5(0(x1))))	→	1(2(5(0(0(3(x1))))))	(16)
  0(2(4(1(x1))))	→	4(0(5(3(1(2(x1))))))	(17)
  0(4(1(0(x1))))	→	0(4(0(1(2(x1)))))	(18)
  0(4(2(1(x1))))	→	4(0(3(2(2(1(x1))))))	(19)
  0(4(2(1(x1))))	→	4(0(3(5(1(2(x1))))))	(20)
  0(4(4(1(x1))))	→	2(4(0(3(4(1(x1))))))	(21)
  0(5(0(1(x1))))	→	0(2(0(3(5(1(x1))))))	(22)
  0(5(0(1(x1))))	→	5(2(0(3(0(1(x1))))))	(23)
  0(5(1(0(x1))))	→	0(1(2(0(3(5(x1))))))	(24)
  0(5(1(0(x1))))	→	2(0(1(3(5(0(x1))))))	(25)
  4(0(5(1(x1))))	→	1(4(0(5(3(2(x1))))))	(26)
  4(1(0(0(x1))))	→	0(4(1(2(0(x1)))))	(27)
  4(1(0(1(x1))))	→	1(1(2(0(4(x1)))))	(28)
  4(1(5(0(x1))))	→	1(2(0(5(4(x1)))))	(29)
  4(1(5(0(x1))))	→	1(4(0(3(5(x1)))))	(30)
  4(3(1(0(x1))))	→	1(4(2(0(3(x1)))))	(31)
  4(3(1(0(x1))))	→	2(0(2(1(3(4(x1))))))	(32)
  4(3(1(0(x1))))	→	2(1(4(2(0(3(x1))))))	(33)
  4(3(1(0(x1))))	→	2(2(4(0(1(3(x1))))))	(34)
  5(0(1(0(x1))))	→	0(3(5(1(2(0(x1))))))	(35)
  5(0(1(0(x1))))	→	1(5(2(0(3(0(x1))))))	(36)
  5(4(1(0(x1))))	→	0(1(2(4(5(2(x1))))))	(37)
  0(0(5(5(1(x1)))))	→	1(0(0(3(5(5(x1))))))	(38)
  0(1(0(5(0(x1)))))	→	0(1(5(2(0(0(x1))))))	(39)
  0(2(5(0(1(x1)))))	→	2(0(3(5(0(1(x1))))))	(40)
  0(3(1(0(0(x1)))))	→	0(0(3(0(1(2(x1))))))	(41)
  0(4(1(4(1(x1)))))	→	4(4(0(1(2(1(x1))))))	(42)
  0(5(5(4(1(x1)))))	→	4(1(0(5(5(3(x1))))))	(43)
  4(1(5(0(0(x1)))))	→	1(2(0(4(5(0(x1))))))	(44)
  4(1(5(5(0(x1)))))	→	5(1(3(0(5(4(x1))))))	(45)
  4(3(1(0(1(x1)))))	→	1(1(2(3(0(4(x1))))))	(46)
  4(3(1(1(0(x1)))))	→	1(2(0(1(4(3(x1))))))	(47)
  4(3(1(5(0(x1)))))	→	5(1(0(3(2(4(x1))))))	(48)
  4(4(1(0(5(x1)))))	→	4(0(5(3(4(1(x1))))))	(49)

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

  4#(1(5(5(0(x1)))))	→	0#(5(4(x1)))	(156)
  4#(1(0(x1)))	→	0#(4(x1))	(66)
  0#(1(0(5(0(x1)))))	→	0#(0(x1))	(137)
  0#(0(5(5(1(x1)))))	→	0#(0(3(5(5(x1)))))	(136)
  4#(1(5(5(0(x1)))))	→	4#(x1)	(154)
  4#(3(1(5(0(x1)))))	→	4#(x1)	(162)
  4#(3(1(1(0(x1)))))	→	0#(1(4(3(x1))))	(161)
  0#(1(5(0(x1))))	→	0#(0(3(x1)))	(82)
  0#(0(4(1(x1))))	→	4#(0(3(0(3(x1)))))	(73)
  4#(1(5(0(0(x1)))))	→	0#(4(5(0(x1))))	(153)
  0#(0(1(1(x1))))	→	0#(0(3(1(x1))))	(68)
  4#(1(5(0(0(x1)))))	→	4#(5(0(x1)))	(152)
  4#(1(5(0(x1))))	→	4#(0(3(5(x1))))	(119)
  4#(1(5(0(x1))))	→	0#(5(4(x1)))	(116)
  4#(1(5(0(x1))))	→	4#(x1)	(114)
  4#(3(1(1(0(x1)))))	→	4#(3(x1))	(160)
  4#(3(1(0(1(x1)))))	→	0#(4(x1))	(159)
  4#(3(1(0(1(x1)))))	→	4#(x1)	(158)
  4#(3(1(0(x1))))	→	4#(x1)	(122)
  4#(1(0(1(x1))))	→	0#(4(x1))	(113)
  4#(1(0(1(x1))))	→	4#(x1)	(112)
  4#(1(0(x1)))	→	4#(x1)	(65)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    4#(4(1(0(5(x1)))))	→	4#(1(x1))	(165)
    4#(1(0(0(x1))))	→	0#(4(1(2(0(x1)))))	(111)
    0#(4(1(x1)))	→	4#(0(3(1(x1))))	(57)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the

    prec(4#)	=	0		stat(4#)	=	lex
    prec(0#)	=	1		stat(0#)	=	lex
    prec(5)	=	1		stat(5)	=	lex
    prec(4)	=	6		stat(4)	=	lex
    prec(2)	=	0		stat(2)	=	lex
    prec(3)	=	0		stat(3)	=	lex
    prec(0)	=	0		stat(0)	=	lex
    prec(1)	=	4		stat(1)	=	lex

    π(4#)	=	1
    π(0#)	=	[]
    π(5)	=	[1]
    π(4)	=	[]
    π(2)	=	1
    π(3)	=	[]
    π(0)	=	1
    π(1)	=	[]

    together with the usable rules

    0(0(1(x1)))	→	0(1(2(0(3(x1)))))	(1)
    0(0(1(x1)))	→	0(1(2(0(3(2(x1))))))	(2)
    0(0(1(x1)))	→	1(2(0(0(3(2(x1))))))	(3)
    0(1(0(x1)))	→	2(0(3(1(2(0(x1))))))	(4)
    0(4(1(x1)))	→	4(0(3(1(x1))))	(5)
    0(4(1(x1)))	→	2(4(0(3(2(1(x1))))))	(6)
    0(4(1(x1)))	→	2(4(0(5(3(1(x1))))))	(7)
    0(4(1(x1)))	→	4(2(1(2(0(3(x1))))))	(8)
    4(1(0(x1)))	→	1(2(0(4(x1))))	(9)
    0(0(1(1(x1))))	→	1(2(0(0(3(1(x1))))))	(10)
    0(0(2(1(x1))))	→	2(2(0(3(0(1(x1))))))	(11)
    0(0(4(1(x1))))	→	1(4(0(3(0(3(x1))))))	(12)
    0(0(5(1(x1))))	→	0(0(3(1(2(5(x1))))))	(13)
    0(1(1(0(x1))))	→	1(2(0(0(1(2(x1))))))	(14)
    0(1(4(1(x1))))	→	0(1(2(4(1(2(x1))))))	(15)
    0(1(5(0(x1))))	→	1(2(5(0(0(3(x1))))))	(16)
    0(2(4(1(x1))))	→	4(0(5(3(1(2(x1))))))	(17)
    0(4(1(0(x1))))	→	0(4(0(1(2(x1)))))	(18)
    0(4(2(1(x1))))	→	4(0(3(2(2(1(x1))))))	(19)
    0(4(2(1(x1))))	→	4(0(3(5(1(2(x1))))))	(20)
    0(4(4(1(x1))))	→	2(4(0(3(4(1(x1))))))	(21)
    0(5(0(1(x1))))	→	0(2(0(3(5(1(x1))))))	(22)
    0(5(0(1(x1))))	→	5(2(0(3(0(1(x1))))))	(23)
    0(5(1(0(x1))))	→	0(1(2(0(3(5(x1))))))	(24)
    0(5(1(0(x1))))	→	2(0(1(3(5(0(x1))))))	(25)
    4(0(5(1(x1))))	→	1(4(0(5(3(2(x1))))))	(26)
    4(1(0(0(x1))))	→	0(4(1(2(0(x1)))))	(27)
    4(1(0(1(x1))))	→	1(1(2(0(4(x1)))))	(28)
    4(1(5(0(x1))))	→	1(2(0(5(4(x1)))))	(29)
    4(1(5(0(x1))))	→	1(4(0(3(5(x1)))))	(30)
    4(3(1(0(x1))))	→	1(4(2(0(3(x1)))))	(31)
    4(3(1(0(x1))))	→	2(0(2(1(3(4(x1))))))	(32)
    4(3(1(0(x1))))	→	2(1(4(2(0(3(x1))))))	(33)
    4(3(1(0(x1))))	→	2(2(4(0(1(3(x1))))))	(34)
    5(0(1(0(x1))))	→	0(3(5(1(2(0(x1))))))	(35)
    5(0(1(0(x1))))	→	1(5(2(0(3(0(x1))))))	(36)
    5(4(1(0(x1))))	→	0(1(2(4(5(2(x1))))))	(37)
    0(0(5(5(1(x1)))))	→	1(0(0(3(5(5(x1))))))	(38)
    0(1(0(5(0(x1)))))	→	0(1(5(2(0(0(x1))))))	(39)
    0(2(5(0(1(x1)))))	→	2(0(3(5(0(1(x1))))))	(40)
    0(3(1(0(0(x1)))))	→	0(0(3(0(1(2(x1))))))	(41)
    0(4(1(4(1(x1)))))	→	4(4(0(1(2(1(x1))))))	(42)
    0(5(5(4(1(x1)))))	→	4(1(0(5(5(3(x1))))))	(43)
    4(1(5(0(0(x1)))))	→	1(2(0(4(5(0(x1))))))	(44)
    4(1(5(5(0(x1)))))	→	5(1(3(0(5(4(x1))))))	(45)
    4(3(1(0(1(x1)))))	→	1(1(2(3(0(4(x1))))))	(46)
    4(3(1(1(0(x1)))))	→	1(2(0(1(4(3(x1))))))	(47)
    4(3(1(5(0(x1)))))	→	5(1(0(3(2(4(x1))))))	(48)
    4(4(1(0(5(x1)))))	→	4(0(5(3(4(1(x1))))))	(49)

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

    4#(4(1(0(5(x1)))))	→	4#(1(x1))	(165)
    4#(1(0(0(x1))))	→	0#(4(1(2(0(x1)))))	(111)
    0#(4(1(x1)))	→	4#(0(3(1(x1))))	(57)

    could be deleted.

    1.1.1.1.1.1 P is empty

    There are no pairs anymore.