Certification Problem

Input (TPDB TRS_Equational/AProVE_AC_04/AC05)

The rewrite relation of the following equational TRS is considered.

Associative symbols: plus, times

Commutative symbols: plus, times

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 AC Dependency Pair Transformation

The following set of (strict) dependency pairs is constructed for the TRS.

1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

• The 1^st component contains the pair

  times#(times(x,0),ext)	→	times#(0,ext)	(31)
  times#(x,s(y))	→	times#(x,y)	(22)
  times#(times(x,s(y)),ext)	→	times#(plus(x,times(x,y)),ext)	(32)
  times#(times(x,s(y)),ext)	→	times#(x,y)	(34)
  times#(times(x,y),z)	→	times#(y,z)	(19)
  times#(x,y)	→	times#(y,x)	(15)
  times#(times(x,y),z)	→	times#(x,times(y,z))	(17)

  1.1.1 AC Reduction Pair Processor with Usable Rules

  Using the non-linear polynomial interpretation over the naturals

  [times#(x1, x2)]	=	1 · x2 + 1 · x1 + 1 · x1 · x2
  [times(x1, x2)]	=	1 · x2 + 1 · x1 + 1 · x1 · x2
  [s(x1)]	=	1 + 1 · x1
  [plus(x1, x2)]	=	1 · x2 + 1 · x1
  [0]	=	0

  together with the usable rules

  plus(x,s(y))	→	s(plus(x,y))	(3)
  plus(plus(x,s(y)),ext)	→	plus(s(plus(x,y)),ext)	(36)
  plus(x,0)	→	x	(2)
  plus(plus(x,0),ext)	→	plus(x,ext)	(35)
  times(x,s(y))	→	plus(x,times(x,y))	(5)
  times(times(x,s(y)),ext)	→	times(plus(x,times(x,y)),ext)	(38)
  times(times(x,0),ext)	→	times(0,ext)	(37)
  times(x,0)	→	0	(4)
  plus(plus(x,y),z)	→	plus(x,plus(y,z))	(12)
  plus(x,y)	→	plus(y,x)	(10)
  times(x,y)	→	times(y,x)	(11)
  times(times(x,y),z)	→	times(x,times(y,z))	(13)

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

  times#(times(x,s(y)),ext)	→	times#(plus(x,times(x,y)),ext)	(32)
  times#(times(x,s(y)),ext)	→	times#(x,y)	(34)
  times#(x,s(y))	→	times#(x,y)	(22)

  could be deleted.

  1.1.1.1 AC Reduction Pair Processor with Usable Rules

  Using the non-linear polynomial interpretation over the naturals

  [times#(x1, x2)]	=	2 · x2 + 2 · x1 + 1 · x1 · x2
  [times(x1, x2)]	=	2 + 2 · x2 + 2 · x1 + 1 · x1 · x2
  [0]	=	0
  [s(x1)]	=	1 + 1 · x1
  [plus(x1, x2)]	=	1 · x2 + 1 · x1

  together with the usable rules

  times(x,s(y))	→	plus(x,times(x,y))	(5)
  times(times(x,s(y)),ext)	→	times(plus(x,times(x,y)),ext)	(38)
  times(times(x,0),ext)	→	times(0,ext)	(37)
  times(x,0)	→	0	(4)
  plus(x,s(y))	→	s(plus(x,y))	(3)
  plus(plus(x,s(y)),ext)	→	plus(s(plus(x,y)),ext)	(36)
  plus(x,0)	→	x	(2)
  plus(plus(x,0),ext)	→	plus(x,ext)	(35)
  times(x,y)	→	times(y,x)	(11)
  times(times(x,y),z)	→	times(x,times(y,z))	(13)
  plus(plus(x,y),z)	→	plus(x,plus(y,z))	(12)
  plus(x,y)	→	plus(y,x)	(10)

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

  times#(times(x,y),z)	→	times#(y,z)	(19)
  times#(times(x,0),ext)	→	times#(0,ext)	(31)

  could be deleted.

  1.1.1.1.1 AC Dependency Pair Problem is trivial

  There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.
• The 2^nd component contains the pair

  plus#(plus(x,0),ext)	→	plus#(x,ext)	(28)
  plus#(x,s(y))	→	plus#(x,y)	(20)
  plus#(plus(x,s(y)),ext)	→	plus#(s(plus(x,y)),ext)	(29)
  plus#(plus(x,s(y)),ext)	→	plus#(x,y)	(30)
  plus#(plus(x,y),z)	→	plus#(y,z)	(18)
  plus#(x,y)	→	plus#(y,x)	(14)
  plus#(plus(x,y),z)	→	plus#(x,plus(y,z))	(16)

  1.1.2 AC Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [plus#(x1, x2)]	=	2 · x1 + 2 · x2
  [plus(x1, x2)]	=	1 · x1 + 1 · x2
  [s(x1)]	=	1 · x1
  [0]	=	0

  together with the usable rules

  plus(x,s(y))	→	s(plus(x,y))	(3)
  plus(plus(x,s(y)),ext)	→	plus(s(plus(x,y)),ext)	(36)
  plus(x,0)	→	x	(2)
  plus(plus(x,0),ext)	→	plus(x,ext)	(35)
  plus(plus(x,y),z)	→	plus(x,plus(y,z))	(12)
  plus(x,y)	→	plus(y,x)	(10)

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

  plus#(plus(x,0),ext)

  →

  plus#(x,ext)

  (28)

  and the rules

  p(s(x))	→	x	(1)
  plus(x,0)	→	x	(2)
  times(x,0)	→	0	(4)
  times(x,s(y))	→	plus(x,times(x,y))	(5)
  minus(x,0)	→	x	(6)
  minus(s(x),s(y))	→	minus(p(s(x)),p(s(y)))	(7)
  div(0,s(y))	→	0	(8)
  div(s(x),s(y))	→	s(div(minus(x,y),s(y)))	(9)
  plus(plus(x,0),ext)	→	plus(x,ext)	(35)
  times(times(x,0),ext)	→	times(0,ext)	(37)
  times(times(x,s(y)),ext)	→	times(plus(x,times(x,y)),ext)	(38)

  could be deleted.

  1.1.2.1 AC Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [plus#(x1, x2)]	=	1 · x1 + 1 · x2
  [plus(x1, x2)]	=	1 · x1 + 1 · x2
  [times(x1, x2)]	=	1 · x1 + 1 · x2
  [s(x1)]	=	1 + 1 · x1

  together with the usable rules

  plus(x,s(y))	→	s(plus(x,y))	(3)
  plus(plus(x,s(y)),ext)	→	plus(s(plus(x,y)),ext)	(36)
  plus(x,y)	→	plus(y,x)	(10)
  times(x,y)	→	times(y,x)	(11)
  plus(plus(x,y),z)	→	plus(x,plus(y,z))	(12)
  times(times(x,y),z)	→	times(x,times(y,z))	(13)

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

  plus#(plus(x,s(y)),ext)	→	plus#(x,y)	(30)
  plus#(x,s(y))	→	plus#(x,y)	(20)

  and no rules could be deleted.

  1.1.2.1.1 AC Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [plus#(x1, x2)]	=	1 · x1 + 1 · x2
  [plus(x1, x2)]	=	2 + 1 · x1 + 1 · x2
  [times(x1, x2)]	=	1 · x1 + 1 · x2
  [s(x1)]	=	1 · x1

  together with the usable rules

  plus(x,s(y))	→	s(plus(x,y))	(3)
  plus(plus(x,s(y)),ext)	→	plus(s(plus(x,y)),ext)	(36)
  plus(x,y)	→	plus(y,x)	(10)
  times(x,y)	→	times(y,x)	(11)
  plus(plus(x,y),z)	→	plus(x,plus(y,z))	(12)
  times(times(x,y),z)	→	times(x,times(y,z))	(13)

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

  plus#(plus(x,y),z)

  →

  plus#(y,z)

  (18)

  and no rules could be deleted.

  1.1.2.1.1.1 AC Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [plus#(x1, x2)]	=	1 · x1 + 1 · x2
  [plus(x1, x2)]	=	1 + 1 · x1 + 1 · x2
  [s(x1)]	=	0

  together with the usable rules

  plus(x,s(y))	→	s(plus(x,y))	(3)
  plus(plus(x,s(y)),ext)	→	plus(s(plus(x,y)),ext)	(36)
  plus(plus(x,y),z)	→	plus(x,plus(y,z))	(12)
  plus(x,y)	→	plus(y,x)	(10)

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

  plus#(plus(x,s(y)),ext)

  →

  plus#(s(plus(x,y)),ext)

  (29)

  could be deleted.

  1.1.2.1.1.1.1 AC Dependency Pair Problem is trivial

  There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.
• The 3^rd component contains the pair

  div#(s(x),s(y))

  →

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

  (26)

  1.1.3 AC Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [div#(x1, x2)]	=	2 · x1
  [s(x1)]	=	2 + 2 · x1
  [minus(x1, x2)]	=	1 · x1
  [p(x1)]	=	1 · x1
  [0]	=	0

  together with the usable rules

  minus(s(x),s(y))	→	minus(p(s(x)),p(s(y)))	(7)
  minus(x,0)	→	x	(6)
  p(s(x))	→	x	(1)

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

  div#(s(x),s(y))

  →

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

  (26)

  could be deleted.

  1.1.3.1 AC Dependency Pair Problem is trivial

  There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.
• The 4^th component contains the pair

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

  →

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

  (23)

  1.1.4 AC Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [p(x1)]	=	1 · x1
  [s(x1)]	=	1 · x1
  [minus#(x1, x2)]	=	1 · x1 + 1 · x2

  together with the usable rule

  p(s(x))

  →

  x

  (1)

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

  plus(x,0)	→	x	(2)
  plus(x,s(y))	→	s(plus(x,y))	(3)
  times(x,0)	→	0	(4)
  times(x,s(y))	→	plus(x,times(x,y))	(5)
  minus(x,0)	→	x	(6)
  minus(s(x),s(y))	→	minus(p(s(x)),p(s(y)))	(7)
  div(0,s(y))	→	0	(8)
  div(s(x),s(y))	→	s(div(minus(x,y),s(y)))	(9)
  plus(plus(x,0),ext)	→	plus(x,ext)	(35)
  plus(plus(x,s(y)),ext)	→	plus(s(plus(x,y)),ext)	(36)
  times(times(x,0),ext)	→	times(0,ext)	(37)
  times(times(x,s(y)),ext)	→	times(plus(x,times(x,y)),ext)	(38)

  could be deleted.

  1.1.4.1 AC Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [plus(x1, x2)]	=	1 · x1 + 1 · x2
  [times(x1, x2)]	=	1 · x1 + 1 · x2
  [p(x1)]	=	1 · x1
  [s(x1)]	=	2 + 1 · x1
  [minus#(x1, x2)]	=	2 · x1 + 2 · x2

  together with the usable rules

  p(s(x))	→	x	(1)
  plus(x,y)	→	plus(y,x)	(10)
  times(x,y)	→	times(y,x)	(11)
  plus(plus(x,y),z)	→	plus(x,plus(y,z))	(12)
  times(times(x,y),z)	→	times(x,times(y,z))	(13)

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

  p(s(x))

  →

  x

  (1)

  could be deleted.

  1.1.4.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.