Certification Problem

Input (TPDB TRS_Standard/AProVE_07/wiehe11)

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 7 components.

• The 1^st component contains the pair

  quot#(s(x),s(y))

  →

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

  (26)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

  prec(s)

  =

  0

  weight(s)

  =

  1

  in combination with the following argument filter

  π(quot#)	=	1
  π(s)	=	[1]
  π(minus)	=	1

  together with the usable rules

  minus(x,0)	→	x	(1)
  minus(minus(x,y),z)	→	minus(x,plus(y,z))	(7)
  minus(s(x),s(y))	→	minus(x,y)	(2)

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

  quot#(s(x),s(y))

  →

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

  (26)

  could be deleted.

  1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  sum#(app(l,cons(x,cons(y,k))))

  →

  sum#(app(l,sum(cons(x,cons(y,k)))))

  (34)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

  prec(app)	=	2		weight(app)	=	1
  prec(cons)	=	1		weight(cons)	=	3
  prec(sum)	=	0		weight(sum)	=	5
  prec(nil)	=	3		weight(nil)	=	1

  in combination with the following argument filter

  π(sum#)	=	1
  π(app)	=	[1,2]
  π(cons)	=	[2]
  π(sum)	=	[]
  π(nil)	=	[]

  together with the usable rules

  sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(12)
  app(nil,k)	→	k	(8)
  app(l,nil)	→	l	(9)
  app(cons(x,l),k)	→	cons(x,app(l,k))	(10)
  sum(cons(x,nil))	→	cons(x,nil)	(11)

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

  sum#(app(l,cons(x,cons(y,k))))

  →

  sum#(app(l,sum(cons(x,cons(y,k)))))

  (34)

  could be deleted.

  1.1.2.1 P is empty

  There are no pairs anymore.

• The 3^rd component contains the pair

  minus#(minus(x,y),z)	→	minus#(x,plus(y,z))	(29)
  minus#(s(x),s(y))	→	minus#(x,y)	(25)

  1.1.3 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  minus#(minus(x,y),z)	→	minus#(x,plus(y,z))	(29)
  1	>	1
  minus#(s(x),s(y))	→	minus#(x,y)	(25)
  1	>	1
  2	>	2

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 4^th component contains the pair

  sum#(cons(x,cons(y,l)))

  →

  sum#(cons(plus(x,y),l))

  (32)

  1.1.4 Reduction Pair Processor with Usable Rules

  Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

  prec(cons)

  =

  1

  weight(cons)

  =

  1

  in combination with the following argument filter

  π(sum#)	=	1
  π(cons)	=	[2]

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

  sum#(cons(x,cons(y,l)))

  →

  sum#(cons(plus(x,y),l))

  (32)

  could be deleted.

  1.1.4.1 P is empty

  There are no pairs anymore.

• The 5^th component contains the pair

  plus#(s(x),s(y))	→	plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(37)
  plus#(s(x),y)	→	plus#(x,y)	(28)
  plus#(s(x),x)	→	plus#(if(gt(x,x),id(x),id(x)),s(x))	(44)
  plus#(id(x),s(y))	→	plus#(x,if(gt(s(y),y),y,s(y)))	(48)

  1.1.5 Reduction Pair Processor with Usable Rules

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

  [plus#(x1, x2)]	=	0 + 0 · x1 + 0 · x2
  [s(x1)]	=	5 + 5 · x1
  [if(x1, x2, x3)]	=	-∞ + 0 · x1 + 0 · x2 + 0 · x3
  [gt(x1, x2)]	=	-∞ + 0 · x1 + -∞ · x2
  [not(x1)]	=	4 + 3 · x1
  [id(x1)]	=	-∞ + 0 · x1
  [zero]	=	1
  [true]	=	0
  [false]	=	0

  together with the usable rules

  gt(s(x),zero)	→	true	(22)
  gt(zero,y)	→	false	(23)
  gt(s(x),s(y))	→	gt(x,y)	(24)
  if(true,x,y)	→	x	(19)
  if(false,x,y)	→	y	(20)
  not(x)	→	if(x,false,true)	(21)
  id(x)	→	x	(18)

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

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

  →

  plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))

  (37)

  could be deleted.

  1.1.5.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [plus#(x1, x2)]	=	x1
  [if(x1, x2, x3)]	=	-2 + 2 · x1 + x2 + x3
  [gt(x1, x2)]	=	1
  [s(x1)]	=	1 + 2 · x1
  [zero]	=	0
  [true]	=	1
  [false]	=	1
  [id(x1)]	=	x1

  together with the usable rules

  gt(s(x),zero)	→	true	(22)
  gt(zero,y)	→	false	(23)
  gt(s(x),s(y))	→	gt(x,y)	(24)
  id(x)	→	x	(18)
  if(true,x,y)	→	x	(19)
  if(false,x,y)	→	y	(20)

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

  plus#(s(x),y)	→	plus#(x,y)	(28)
  plus#(s(x),x)	→	plus#(if(gt(x,x),id(x),id(x)),s(x))	(44)

  could be deleted.

  1.1.5.1.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  plus#(id(x),s(y))	→	plus#(x,if(gt(s(y),y),y,s(y)))	(48)
  1	>	1

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 6^th component contains the pair

  app#(cons(x,l),k)

  →

  app#(l,k)

  (31)

  1.1.6 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [app#(x1, x2)]	=	1 · x1 + 1 · x2

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

  1.1.6.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  app#(cons(x,l),k)	→	app#(l,k)	(31)
  1	>	1
  2	≥	2

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 7^th component contains the pair

  gt#(s(x),s(y))

  →

  gt#(x,y)

  (52)

  1.1.7 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

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

  1.1.7.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  gt#(s(x),s(y))	→	gt#(x,y)	(52)
  1	>	1
  2	>	2

  As there is no critical graph in the transitive closure, there are no infinite chains.