Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/211978)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

There are 138 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  0#(1(5(0(x1))))	→	5#(x1)	(94)
  0#(3(0(1(x1))))	→	0#(x1)	(87)
  5#(1(0(x1)))	→	5#(x1)	(153)
  5#(3(1(0(x1))))	→	5#(x1)	(150)
  5#(0(1(4(0(x1)))))	→	0#(0(x1))	(139)
  5#(3(1(0(x1))))	→	5#(0(x1))	(131)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

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

  together with the usable rules

  5(1(0(x1)))	→	0(5(0(2(2(1(x1))))))	(18)
  5(5(1(0(0(x1)))))	→	5(5(0(4(1(0(x1))))))	(50)
  0(0(1(x1)))	→	4(0(5(4(0(1(x1))))))	(4)
  5(0(1(x1)))	→	5(0(4(3(0(1(x1))))))	(15)
  0(1(1(x1)))	→	1(0(3(4(1(x1)))))	(8)
  0(0(1(x1)))	→	0(2(3(0(1(x1)))))	(1)
  0(0(1(x1)))	→	2(1(0(0(3(4(x1))))))	(3)
  5(1(0(x1)))	→	5(0(2(2(1(x1)))))	(16)
  5(1(0(x1)))	→	4(4(1(0(4(5(x1))))))	(21)
  5(1(2(0(x1))))	→	5(0(4(2(2(1(x1))))))	(36)
  5(1(1(x1)))	→	3(5(2(3(1(1(x1))))))	(26)
  5(1(0(x1)))	→	1(4(0(5(2(3(x1))))))	(19)
  0(3(1(0(x1))))	→	0(0(2(3(1(x1)))))	(32)
  5(1(0(x1)))	→	5(0(5(4(1(x1)))))	(17)
  5(1(1(x1)))	→	4(1(2(1(5(4(x1))))))	(27)
  5(0(1(0(x1))))	→	5(0(0(4(1(3(x1))))))	(34)
  5(1(1(x1)))	→	1(1(5(4(x1))))	(22)
  0(1(3(0(x1))))	→	0(2(0(2(1(3(x1))))))	(28)
  5(3(1(0(x1))))	→	1(3(0(4(3(5(x1))))))	(44)
  0(1(0(x1)))	→	0(0(2(1(2(x1)))))	(5)
  0(3(1(1(x1))))	→	5(1(1(0(3(4(x1))))))	(33)
  5(0(1(x1)))	→	0(5(4(1(x1))))	(10)
  5(1(5(1(x1))))	→	5(4(1(5(1(x1)))))	(39)
  0(1(0(x1)))	→	0(0(2(5(4(1(x1))))))	(7)
  5(1(0(x1)))	→	1(5(0(4(4(2(x1))))))	(20)
  5(1(1(x1)))	→	1(1(4(5(4(4(x1))))))	(25)
  5(0(1(4(0(x1)))))	→	1(4(5(4(0(0(x1))))))	(49)
  0(1(5(0(x1))))	→	0(5(4(2(1(0(x1))))))	(30)
  5(0(1(x1)))	→	0(5(4(1(4(4(x1))))))	(14)
  0(3(0(1(x1))))	→	0(0(4(1(3(0(x1))))))	(31)
  5(0(1(x1)))	→	5(0(2(1(2(x1)))))	(12)
  5(3(1(1(x1))))	→	1(1(5(3(3(4(x1))))))	(45)
  5(1(1(x1)))	→	5(4(1(1(x1))))	(23)
  5(1(1(x1)))	→	1(5(3(4(1(x1)))))	(24)
  5(0(1(x1)))	→	2(5(4(0(1(x1)))))	(11)
  0(1(1(x1)))	→	5(0(3(4(1(1(x1))))))	(9)
  5(0(1(x1)))	→	0(1(4(5(4(4(x1))))))	(13)
  5(3(0(1(x1))))	→	0(1(5(2(3(x1)))))	(40)
  0(1(0(x1)))	→	1(0(0(5(4(x1)))))	(6)
  5(1(4(0(x1))))	→	4(5(2(1(3(0(x1))))))	(38)
  1(4(5(1(0(x1)))))	→	5(4(2(1(1(0(x1))))))	(48)
  0(1(4(2(0(x1)))))	→	1(0(4(2(3(0(x1))))))	(47)
  5(1(4(0(x1))))	→	1(5(4(0(2(3(x1))))))	(37)
  5(3(1(0(x1))))	→	1(4(3(5(0(x1)))))	(41)
  5(3(1(0(x1))))	→	1(5(0(4(3(x1)))))	(42)
  0(1(2(5(0(x1)))))	→	1(5(4(0(2(0(x1))))))	(46)
  5(1(2(0(x1))))	→	1(4(0(5(4(2(x1))))))	(35)
  0(1(5(0(x1))))	→	0(0(5(4(1(5(x1))))))	(29)
  5(3(1(0(x1))))	→	5(4(3(1(0(x1)))))	(43)
  0(0(1(x1)))	→	0(4(0(5(4(1(x1))))))	(2)

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

  0#(1(5(0(x1))))	→	5#(x1)	(94)
  0#(3(0(1(x1))))	→	0#(x1)	(87)
  5#(1(0(x1)))	→	5#(x1)	(153)
  5#(3(1(0(x1))))	→	5#(x1)	(150)
  5#(0(1(4(0(x1)))))	→	0#(0(x1))	(139)
  5#(3(1(0(x1))))	→	5#(0(x1))	(131)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

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

  →

  1#(1(0(x1)))

  (151)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the argument filter

  π(4)	=	1
  π(3)	=	1

  in combination with the following Weighted Path Order with the following precedence and status

  prec(0#)	=	0		status(0#)	=	[]		list-extension(0#)	=	Lex
  prec(1)	=	1		status(1)	=	[]		list-extension(1)	=	Lex
  prec(5)	=	2		status(5)	=	[]		list-extension(5)	=	Lex
  prec(0)	=	2		status(0)	=	[]		list-extension(0)	=	Lex
  prec(5#)	=	0		status(5#)	=	[]		list-extension(5#)	=	Lex
  prec(2)	=	0		status(2)	=	[]		list-extension(2)	=	Lex
  prec(1#)	=	0		status(1#)	=	[1]		list-extension(1#)	=	Lex

  and the following Max-polynomial interpretation

  [0#(x1)]	=	1
  [1(x1)]	=	55976
  [5(x1)]	=	x1 + 0
  [0(x1)]	=	55976
  [5#(x1)]	=	1
  [2(x1)]	=	55975
  [1#(x1)]	=	x1 + 1

  together with the usable rules

  5(1(0(x1)))	→	0(5(0(2(2(1(x1))))))	(18)
  5(5(1(0(0(x1)))))	→	5(5(0(4(1(0(x1))))))	(50)
  0(0(1(x1)))	→	4(0(5(4(0(1(x1))))))	(4)
  5(0(1(x1)))	→	5(0(4(3(0(1(x1))))))	(15)
  0(1(1(x1)))	→	1(0(3(4(1(x1)))))	(8)
  0(0(1(x1)))	→	0(2(3(0(1(x1)))))	(1)
  0(0(1(x1)))	→	2(1(0(0(3(4(x1))))))	(3)
  5(1(0(x1)))	→	5(0(2(2(1(x1)))))	(16)
  5(1(0(x1)))	→	4(4(1(0(4(5(x1))))))	(21)
  5(1(2(0(x1))))	→	5(0(4(2(2(1(x1))))))	(36)
  5(1(1(x1)))	→	3(5(2(3(1(1(x1))))))	(26)
  5(1(0(x1)))	→	1(4(0(5(2(3(x1))))))	(19)
  0(3(1(0(x1))))	→	0(0(2(3(1(x1)))))	(32)
  5(1(0(x1)))	→	5(0(5(4(1(x1)))))	(17)
  5(1(1(x1)))	→	4(1(2(1(5(4(x1))))))	(27)
  5(0(1(0(x1))))	→	5(0(0(4(1(3(x1))))))	(34)
  5(1(1(x1)))	→	1(1(5(4(x1))))	(22)
  0(1(3(0(x1))))	→	0(2(0(2(1(3(x1))))))	(28)
  5(3(1(0(x1))))	→	1(3(0(4(3(5(x1))))))	(44)
  0(1(0(x1)))	→	0(0(2(1(2(x1)))))	(5)
  0(3(1(1(x1))))	→	5(1(1(0(3(4(x1))))))	(33)
  5(0(1(x1)))	→	0(5(4(1(x1))))	(10)
  5(1(5(1(x1))))	→	5(4(1(5(1(x1)))))	(39)
  0(1(0(x1)))	→	0(0(2(5(4(1(x1))))))	(7)
  5(1(0(x1)))	→	1(5(0(4(4(2(x1))))))	(20)
  5(1(1(x1)))	→	1(1(4(5(4(4(x1))))))	(25)
  5(0(1(4(0(x1)))))	→	1(4(5(4(0(0(x1))))))	(49)
  0(1(5(0(x1))))	→	0(5(4(2(1(0(x1))))))	(30)
  5(0(1(x1)))	→	0(5(4(1(4(4(x1))))))	(14)
  0(3(0(1(x1))))	→	0(0(4(1(3(0(x1))))))	(31)
  5(0(1(x1)))	→	5(0(2(1(2(x1)))))	(12)
  5(3(1(1(x1))))	→	1(1(5(3(3(4(x1))))))	(45)
  5(1(1(x1)))	→	5(4(1(1(x1))))	(23)
  5(1(1(x1)))	→	1(5(3(4(1(x1)))))	(24)
  5(0(1(x1)))	→	2(5(4(0(1(x1)))))	(11)
  0(1(1(x1)))	→	5(0(3(4(1(1(x1))))))	(9)
  5(0(1(x1)))	→	0(1(4(5(4(4(x1))))))	(13)
  5(3(0(1(x1))))	→	0(1(5(2(3(x1)))))	(40)
  0(1(0(x1)))	→	1(0(0(5(4(x1)))))	(6)
  5(1(4(0(x1))))	→	4(5(2(1(3(0(x1))))))	(38)
  1(4(5(1(0(x1)))))	→	5(4(2(1(1(0(x1))))))	(48)
  0(1(4(2(0(x1)))))	→	1(0(4(2(3(0(x1))))))	(47)
  5(1(4(0(x1))))	→	1(5(4(0(2(3(x1))))))	(37)
  5(3(1(0(x1))))	→	1(4(3(5(0(x1)))))	(41)
  5(3(1(0(x1))))	→	1(5(0(4(3(x1)))))	(42)
  0(1(2(5(0(x1)))))	→	1(5(4(0(2(0(x1))))))	(46)
  5(1(2(0(x1))))	→	1(4(0(5(4(2(x1))))))	(35)
  0(1(5(0(x1))))	→	0(0(5(4(1(5(x1))))))	(29)
  5(3(1(0(x1))))	→	5(4(3(1(0(x1)))))	(43)
  0(0(1(x1)))	→	0(4(0(5(4(1(x1))))))	(2)

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

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

  →

  1#(1(0(x1)))

  (151)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.