Certification Problem

Input (TPDB TRS_Standard/Mixed_TRS/gcd)

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 4 components.

• The 1^st component contains the pair

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

  →

  gcd#(-(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 + 35544
  [gcd(x1, x2)]	=	max(0)
  [min#(x1, x2)]	=	max(0)
  [0]	=	1
  [max(x1, x2)]	=	max(x1 + 1, x2 + 2, 0)
  [max#(x1, x2)]	=	max(0)
  [-(x1, x2)]	=	max(x1 + 34723, x2 + 251, 0)
  [min(x1, x2)]	=	max(x1 + 817, 0)
  [-#(x1, x2)]	=	max(0)
  [gcd#(x1, x2)]	=	max(x1 + 818, x2 + 0, 0)

  together with the usable rules

  max(x,0)	→	x	(4)
  -(s(x),s(y))	→	-(x,y)	(8)
  min(x,0)	→	0	(1)
  min(s(x),s(y))	→	s(min(x,y))	(3)
  max(0,y)	→	y	(5)
  -(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#(-(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

  -#(s(x),s(y))

  →

  -#(x,y)

  (17)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [gcd(x1, x2)]	=	0
  [min#(x1, x2)]	=	0
  [0]	=	35232
  [max(x1, x2)]	=	15922
  [max#(x1, x2)]	=	0
  [-(x1, x2)]	=	0
  [min(x1, x2)]	=	35231
  [-#(x1, x2)]	=	x1 + 0
  [gcd#(x1, x2)]	=	0

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

  -#(s(x),s(y))

  →

  -#(x,y)

  (17)

  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

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

  →

  max#(x,y)

  (16)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [gcd(x1, x2)]	=	0
  [min#(x1, x2)]	=	0
  [0]	=	35232
  [max(x1, x2)]	=	52868
  [max#(x1, x2)]	=	x2 + 0
  [-(x1, x2)]	=	0
  [min(x1, x2)]	=	35231
  [-#(x1, x2)]	=	0
  [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)

  (16)

  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

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

  →

  min#(x,y)

  (14)

  1.1.4 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [gcd(x1, x2)]	=	0
  [min#(x1, x2)]	=	x2 + 0
  [0]	=	35232
  [max(x1, x2)]	=	44364
  [max#(x1, x2)]	=	0
  [-(x1, x2)]	=	0
  [min(x1, x2)]	=	35231
  [-#(x1, x2)]	=	0
  [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)

  (14)

  could be deleted.

  1.1.4.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.