Certification Problem

Input (TPDB TRS_Standard/SK90/4.60)

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

  msort#(.(x,y))

  →

  msort#(del(min(x,y),.(x,y)))

  (7)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [del#(x1, x2)]	=	0
  [msort#(x1)]	=	x1 + 0
  [<=(x1, x2)]	=	1
  [min#(x1, x2)]	=	0
  [msort(x1)]	=	0
  [if(x1, x2, x3)]	=	x1 + 0
  [del(x1, x2)]	=	x1 + 1
  [=(x1, x2)]	=	1
  [nil]	=	1
  [.(x1, x2)]	=	x1 + x2 + 23678
  [min(x1, x2)]	=	x1 + 23676

  together with the usable rules

  min(x,.(y,z))	→	if(<=(x,y),min(x,z),min(y,z))	(4)
  min(x,nil)	→	x	(3)
  del(x,nil)	→	nil	(5)
  del(x,.(y,z))	→	if(=(x,y),z,.(y,del(x,z)))	(6)

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

  msort#(.(x,y))

  →

  msort#(del(min(x,y),.(x,y)))

  (7)

  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

  del#(x,.(y,z))

  →

  del#(x,z)

  (10)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [del#(x1, x2)]	=	x2 + 0
  [msort#(x1)]	=	x1 + 0
  [<=(x1, x2)]	=	1
  [min#(x1, x2)]	=	0
  [msort(x1)]	=	0
  [if(x1, x2, x3)]	=	x1 + 0
  [del(x1, x2)]	=	x1 + 1
  [=(x1, x2)]	=	1
  [nil]	=	1
  [.(x1, x2)]	=	x1 + x2 + 1
  [min(x1, x2)]	=	x1 + 23676

  together with the usable rules

  min(x,.(y,z))	→	if(<=(x,y),min(x,z),min(y,z))	(4)
  min(x,nil)	→	x	(3)
  del(x,nil)	→	nil	(5)
  del(x,.(y,z))	→	if(=(x,y),z,.(y,del(x,z)))	(6)

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

  del#(x,.(y,z))

  →

  del#(x,z)

  (10)

  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

  min#(x,.(y,z))	→	min#(x,z)	(8)
  min#(x,.(y,z))	→	min#(y,z)	(9)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [del#(x1, x2)]	=	0
  [msort#(x1)]	=	x1 + 0
  [<=(x1, x2)]	=	1
  [min#(x1, x2)]	=	x2 + 0
  [msort(x1)]	=	0
  [if(x1, x2, x3)]	=	x1 + 8855
  [del(x1, x2)]	=	x1 + 8856
  [=(x1, x2)]	=	1
  [nil]	=	1
  [.(x1, x2)]	=	x1 + x2 + 1
  [min(x1, x2)]	=	x1 + 23676

  together with the usable rules

  min(x,.(y,z))	→	if(<=(x,y),min(x,z),min(y,z))	(4)
  min(x,nil)	→	x	(3)
  del(x,nil)	→	nil	(5)
  del(x,.(y,z))	→	if(=(x,y),z,.(y,del(x,z)))	(6)

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

  min#(x,.(y,z))	→	min#(x,z)	(8)
  min#(x,.(y,z))	→	min#(y,z)	(9)

  could be deleted.

  1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.