Certification Problem

Input (TPDB TRS_Standard/Secret_06_TRS/gen-1)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  b#(x,b(z,y))	→	f#(b(f(f(z)),c(x,z,y)))	(5)
  f#(c(a,z,x))	→	b#(a,z)	(4)
  b#(x,b(z,y))	→	f#(f(z))	(7)
  b#(x,b(z,y))	→	f#(z)	(8)

  1.1.1 Instantiation Processor

  We instantiate the pair to the following set of pairs

  b#(a,b(x1,x2))

  →

  f#(b(f(f(x1)),c(a,x1,x2)))

  (9)

  1.1.1.1 Instantiation Processor

  We instantiate the pair to the following set of pairs

  b#(a,b(x1,x2))

  →

  f#(f(x1))

  (10)

  1.1.1.1.1 Instantiation Processor

  We instantiate the pair to the following set of pairs

  b#(a,b(x1,x2))

  →

  f#(x1)

  (11)

  1.1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

  [f#(x1)]	=	0 + 0 · x1
  [c(x1, x2, x3)]	=	-∞ + -∞ · x1 + 1 · x2 + 1 · x3
  [a]	=	0
  [b#(x1, x2)]	=	0 + -∞ · x1 + 1 · x2
  [b(x1, x2)]	=	1 + 1 · x1 + 1 · x2
  [f(x1)]	=	1 + 0 · x1

  the pairs

  b#(a,b(x1,x2))	→	f#(f(x1))	(10)
  b#(a,b(x1,x2))	→	f#(x1)	(11)

  could be deleted.

  1.1.1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

  [f#(x1)]	=	-∞ + 0 · x1
  [c(x1, x2, x3)]	=	-∞ + 1 · x1 + 1 · x2 + -∞ · x3
  [a]	=	0
  [b#(x1, x2)]	=	-∞ + -∞ · x1 + 0 · x2
  [b(x1, x2)]	=	1 + 1 · x1 + 0 · x2
  [f(x1)]	=	-∞ + 0 · x1

  the pair

  f#(c(a,z,x))

  →

  b#(a,z)

  (4)

  could be deleted.

  1.1.1.1.1.1.1.1 Instantiation Processor

  We instantiate the pair to the following set of pairs

  There are no rules.

  1.1.1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.