Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex49_GM04_Z)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  div#(s(X),n__s(Y))

  →

  div#(minus(X,activate(Y)),n__s(activate(Y)))

  (21)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [0#]	=	0
  [div#(x1, x2)]	=	x1 + x2 + 0
  [s(x1)]	=	x1 + 29246
  [minus(x1, x2)]	=	x1 + 29243
  [activate(x1)]	=	x1 + 2
  [geq#(x1, x2)]	=	0
  [activate#(x1)]	=	0
  [false]	=	0
  [div(x1, x2)]	=	0
  [geq(x1, x2)]	=	0
  [true]	=	0
  [n__s(x1)]	=	x1 + 29244
  [0]	=	41063
  [if(x1, x2, x3)]	=	0
  [s#(x1)]	=	0
  [n__0]	=	41063
  [minus#(x1, x2)]	=	0
  [if#(x1, x2, x3)]	=	0

  together with the usable rules

  minus(n__0,Y)	→	0	(1)
  0	→	n__0	(10)
  activate(X)	→	X	(14)
  activate(n__0)	→	0	(12)
  s(X)	→	n__s(X)	(11)
  activate(n__s(X))	→	s(X)	(13)
  minus(n__s(X),n__s(Y))	→	minus(activate(X),activate(Y))	(2)

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

  div#(s(X),n__s(Y))

  →

  div#(minus(X,activate(Y)),n__s(activate(Y)))

  (21)

  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

  geq#(n__s(X),n__s(Y))

  →

  geq#(activate(X),activate(Y))

  (19)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [0#]	=	0
  [div#(x1, x2)]	=	x1 + x2 + 0
  [s(x1)]	=	x1 + 3
  [minus(x1, x2)]	=	x1 + 29243
  [activate(x1)]	=	x1 + 1
  [geq#(x1, x2)]	=	x1 + x2 + 0
  [activate#(x1)]	=	0
  [false]	=	0
  [div(x1, x2)]	=	0
  [geq(x1, x2)]	=	0
  [true]	=	0
  [n__s(x1)]	=	x1 + 2
  [0]	=	41063
  [if(x1, x2, x3)]	=	0
  [s#(x1)]	=	0
  [n__0]	=	41063
  [minus#(x1, x2)]	=	0
  [if#(x1, x2, x3)]	=	0

  together with the usable rules

  minus(n__0,Y)	→	0	(1)
  0	→	n__0	(10)
  activate(X)	→	X	(14)
  activate(n__0)	→	0	(12)
  s(X)	→	n__s(X)	(11)
  activate(n__s(X))	→	s(X)	(13)
  minus(n__s(X),n__s(Y))	→	minus(activate(X),activate(Y))	(2)

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

  geq#(n__s(X),n__s(Y))

  →

  geq#(activate(X),activate(Y))

  (19)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  minus#(n__s(X),n__s(Y))

  →

  minus#(activate(X),activate(Y))

  (26)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [0#]	=	0
  [div#(x1, x2)]	=	x1 + x2 + 0
  [s(x1)]	=	x1 + 3
  [minus(x1, x2)]	=	x1 + 1
  [activate(x1)]	=	x1 + 1
  [geq#(x1, x2)]	=	x1 + x2 + 0
  [activate#(x1)]	=	0
  [false]	=	0
  [div(x1, x2)]	=	0
  [geq(x1, x2)]	=	0
  [true]	=	0
  [n__s(x1)]	=	x1 + 2
  [0]	=	38146
  [if(x1, x2, x3)]	=	0
  [s#(x1)]	=	0
  [n__0]	=	38145
  [minus#(x1, x2)]	=	x1 + x2 + 0
  [if#(x1, x2, x3)]	=	0

  together with the usable rules

  minus(n__0,Y)	→	0	(1)
  0	→	n__0	(10)
  activate(X)	→	X	(14)
  activate(n__0)	→	0	(12)
  s(X)	→	n__s(X)	(11)
  activate(n__s(X))	→	s(X)	(13)
  minus(n__s(X),n__s(Y))	→	minus(activate(X),activate(Y))	(2)

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

  minus#(n__s(X),n__s(Y))

  →

  minus#(activate(X),activate(Y))

  (26)

  could be deleted.

  1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.