Certification Problem

Input (TPDB TRS_Standard/AProVE_04/Liveness_WRS)

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

  tp#(mark(x))

  →

  tp#(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x)))

  (37)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

  [tp(x1)]	=	0 0
  [y]	=	6830 62746
  [no(x1)]	=	x1 + 58214 58214
  [mat#(x1, x2)]	=	0 0
  [chk(x1)]	=	0 1 0 1 · x1 + 86386 38642
  [c]	=	105958 58214
  [f(x1)]	=	0 1 1 0 · x1 + 19134 0
  [mark(x1)]	=	0 1 1 0 · x1 + 96856 77722
  [f#(x1)]	=	0 0
  [tp#(x1)]	=	0 1 0 0 · x1 + 0 0
  [active(x1)]	=	x1 + 96856 96856
  [mat(x1, x2)]	=	1 1 1 0 · x2 + 0 10470
  [active#(x1)]	=	0 0
  [X]	=	25947 4581
  [chk#(x1)]	=	0 0

  together with the usable rules

  chk(no(c))	→	active(c)	(4)
  f(mark(x))	→	mark(f(x))	(8)
  active(f(x))	→	mark(f(f(x)))	(1)
  mat(f(x),f(y))	→	f(mat(x,y))	(3)
  mat(f(x),c)	→	no(c)	(5)
  f(no(x))	→	no(f(x))	(7)
  f(active(x))	→	active(f(x))	(6)
  chk(no(f(x)))	→	f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x)))	(2)

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

  tp#(mark(x))

  →

  tp#(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x)))

  (37)

  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

  chk#(no(f(x)))

  →

  chk#(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x))

  (10)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [tp(x1)]	=	0
  [y]	=	1
  [no(x1)]	=	x1 + 27828
  [mat#(x1, x2)]	=	0
  [chk(x1)]	=	36229
  [c]	=	2447
  [f(x1)]	=	x1 + 16908
  [mark(x1)]	=	37070
  [f#(x1)]	=	0
  [tp#(x1)]	=	0
  [active(x1)]	=	37070
  [mat(x1, x2)]	=	x2 + 44735
  [active#(x1)]	=	0
  [X]	=	39825
  [chk#(x1)]	=	x1 + 0

  together with the usable rules

  f(mark(x))	→	mark(f(x))	(8)
  active(f(x))	→	mark(f(f(x)))	(1)
  mat(f(x),f(y))	→	f(mat(x,y))	(3)
  mat(f(x),c)	→	no(c)	(5)
  f(no(x))	→	no(f(x))	(7)
  f(active(x))	→	active(f(x))	(6)

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

  chk#(no(f(x)))

  →

  chk#(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x))

  (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

  f#(active(x))	→	active#(f(x))	(20)
  f#(active(x))	→	f#(x)	(41)
  active#(f(x))	→	f#(f(x))	(33)
  f#(mark(x))	→	f#(x)	(26)
  f#(no(x))	→	f#(x)	(11)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [tp(x1)]	=	0
  [y]	=	12213
  [no(x1)]	=	x1 + 28224
  [mat#(x1, x2)]	=	0
  [chk(x1)]	=	17067
  [c]	=	38472
  [f(x1)]	=	x1 + 1
  [mark(x1)]	=	x1 + 2
  [f#(x1)]	=	x1 + 0
  [tp#(x1)]	=	0
  [active(x1)]	=	x1 + 3
  [mat(x1, x2)]	=	x2 + 28224
  [active#(x1)]	=	x1 + 1
  [X]	=	62709
  [chk#(x1)]	=	x1 + 0

  together with the usable rules

  f(mark(x))	→	mark(f(x))	(8)
  active(f(x))	→	mark(f(f(x)))	(1)
  mat(f(x),f(y))	→	f(mat(x,y))	(3)
  mat(f(x),c)	→	no(c)	(5)
  f(no(x))	→	no(f(x))	(7)
  f(active(x))	→	active(f(x))	(6)

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

  f#(active(x))	→	active#(f(x))	(20)
  f#(active(x))	→	f#(x)	(41)
  active#(f(x))	→	f#(f(x))	(33)
  f#(mark(x))	→	f#(x)	(26)
  f#(no(x))	→	f#(x)	(11)

  could be deleted.

  1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.