Certification Problem

Input (TPDB TRS_Standard/AProVE_07/kabasci04)

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 5 components.

• The 1^st component contains the pair

  gcd#(s(x),s(y))

  →

  gcd#(minus(max(x,y),min(x,y)),s(min(x,y)))

  (12)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 30911
  [minus(x1, x2)]	=	max(x1 + 1, 0)
  [gcd(x1, x2)]	=	max(0)
  [any(x1)]	=	44022
  [min#(x1, x2)]	=	max(0)
  [any#(x1)]	=	0
  [0]	=	1
  [max(x1, x2)]	=	max(x1 + 10088, x2 + 10089, 0)
  [max#(x1, x2)]	=	max(0)
  [minus#(x1, x2)]	=	max(0)
  [min(x1, x2)]	=	max(x1 + 20819, 0)
  [gcd#(x1, x2)]	=	max(x1 + 20820, x2 + 0, 0)

  together with the usable rules

  max(x,0)	→	x	(4)
  minus(s(x),s(y))	→	s(minus(x,any(y)))	(8)
  min(x,0)	→	0	(1)
  min(s(x),s(y))	→	s(min(x,y))	(3)
  max(0,y)	→	y	(5)
  minus(x,0)	→	x	(7)
  max(s(x),s(y))	→	s(max(x,y))	(6)
  min(0,y)	→	0	(2)

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

  gcd#(s(x),s(y))

  →

  gcd#(minus(max(x,y),min(x,y)),s(min(x,y)))

  (12)

  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

  minus#(s(x),s(y))

  →

  minus#(x,any(y))

  (14)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [minus(x1, x2)]	=	x1 + 34725
  [gcd(x1, x2)]	=	0
  [any(x1)]	=	28882
  [min#(x1, x2)]	=	0
  [any#(x1)]	=	0
  [0]	=	10317
  [max(x1, x2)]	=	3081
  [max#(x1, x2)]	=	0
  [minus#(x1, x2)]	=	x1 + 0
  [min(x1, x2)]	=	10316
  [gcd#(x1, x2)]	=	0

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

  minus#(s(x),s(y))

  →

  minus#(x,any(y))

  (14)

  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

  any#(s(x))

  →

  any#(x)

  (15)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [minus(x1, x2)]	=	x1 + 34725
  [gcd(x1, x2)]	=	0
  [any(x1)]	=	40519
  [min#(x1, x2)]	=	0
  [any#(x1)]	=	x1 + 0
  [0]	=	10317
  [max(x1, x2)]	=	33855
  [max#(x1, x2)]	=	0
  [minus#(x1, x2)]	=	0
  [min(x1, x2)]	=	10316
  [gcd#(x1, x2)]	=	0

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

  any#(s(x))

  →

  any#(x)

  (15)

  could be deleted.

  1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 4^th component contains the pair

  max#(s(x),s(y))

  →

  max#(x,y)

  (18)

  1.1.4 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [minus(x1, x2)]	=	x1 + 49485
  [gcd(x1, x2)]	=	0
  [any(x1)]	=	27463
  [min#(x1, x2)]	=	0
  [any#(x1)]	=	0
  [0]	=	10317
  [max(x1, x2)]	=	40436
  [max#(x1, x2)]	=	x2 + 0
  [minus#(x1, x2)]	=	0
  [min(x1, x2)]	=	10316
  [gcd#(x1, x2)]	=	0

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

  max#(s(x),s(y))

  →

  max#(x,y)

  (18)

  could be deleted.

  1.1.4.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 5^th component contains the pair

  min#(s(x),s(y))

  →

  min#(x,y)

  (19)

  1.1.5 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [minus(x1, x2)]	=	x1 + 41684
  [gcd(x1, x2)]	=	0
  [any(x1)]	=	17067
  [min#(x1, x2)]	=	x1 + 0
  [any#(x1)]	=	0
  [0]	=	21136
  [max(x1, x2)]	=	40436
  [max#(x1, x2)]	=	0
  [minus#(x1, x2)]	=	0
  [min(x1, x2)]	=	21135
  [gcd#(x1, x2)]	=	0

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

  min#(s(x),s(y))

  →

  min#(x,y)

  (19)

  could be deleted.

  1.1.5.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.