Certification Problem

Input (TPDB TRS_Standard/AProVE_07/wiehe11)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ 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))

  (27)

  1.1.1 Subterm Criterion Processor

  We use the projection to multisets

  π(quot#)	=	{ 1 }
  π(minus)	=	{ 1 }

  to remove the pairs:

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

  →

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

  (27)

  1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  minus#(minus(x,y),z)	→	minus#(x,plus(y,z))	(30)
  minus#(s(x),s(y))	→	minus#(x,y)	(25)

  1.1.2 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))	(30)
  1	>	1
  minus#(s(x),s(y))	→	minus#(x,y)	(25)
  2	>	2
  1	>	1

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

• The 3^rd component contains the pair

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

  →

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

  (36)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [plus(x1, x2)]	=	3 · x1 + 0 · x2 + 0
  [false]	=	2
  [sum(x1)]	=	0 · x1 + 1
  [nil]	=	0
  [if(x1, x2, x3)]	=	2 · x1 + 2 · x2 + 0 · x3 + 0
  [0]	=	0
  [cons(x1, x2)]	=	0 · x1 + 1 · x2 + 1
  [sum#(x1)]	=	1 · x1 + 0
  [true]	=	0
  [id(x1)]	=	0 · x1 + 1
  [not(x1)]	=	0 · x1 + 0
  [gt(x1, x2)]	=	0 · x1 + 5 · x2 + 1
  [zero]	=	3
  [app(x1, x2)]	=	4 · x1 + 1 · x2 + 0
  [s(x1)]	=	0 · x1 + 6

  together with the usable rules

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

  (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)))))

  (36)

  could be deleted.

  1.1.3.1 P is empty

  There are no pairs anymore.

• The 4^th component contains the pair

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

  →

  app#(l,k)

  (31)

  1.1.4 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)
  2	≥	2
  1	>	1

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

• The 5^th component contains the pair

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

  →

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

  (33)

  1.1.5 Reduction Pair Processor with Usable Rules

  Using the

  prec(sum#)	=	0		stat(sum#)	=	lex
  prec(false)	=	0		stat(false)	=	lex
  prec(true)	=	0		stat(true)	=	lex
  prec(zero)	=	0		stat(zero)	=	lex
  prec(id)	=	0		stat(id)	=	lex
  prec(not)	=	0		stat(not)	=	lex
  prec(if)	=	0		stat(if)	=	lex
  prec(gt)	=	0		stat(gt)	=	lex
  prec(cons)	=	0		stat(cons)	=	lex
  prec(plus)	=	0		stat(plus)	=	lex
  prec(s)	=	0		stat(s)	=	lex
  prec(0)	=	0		stat(0)	=	lex

  π(sum#)	=	[1]
  π(false)	=	[]
  π(true)	=	[]
  π(zero)	=	[]
  π(id)	=	[]
  π(not)	=	1
  π(if)	=	3
  π(gt)	=	1
  π(cons)	=	[2]
  π(plus)	=	2
  π(s)	=	[]
  π(0)	=	[]

  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))

  (33)

  could be deleted.

  1.1.5.1 P is empty

  There are no pairs anymore.

• The 6^th component contains the pair

  plus#(id(x),s(y))	→	plus#(x,if(gt(s(y),y),y,s(y)))	(50)
  plus#(s(x),x)	→	plus#(if(gt(x,x),id(x),id(x)),s(x))	(47)
  plus#(s(x),s(y))	→	plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(43)
  plus#(s(x),y)	→	plus#(x,y)	(28)

  1.1.6 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [plus#(x1, x2)]	=	0 · x1 + 0 · x2 + -∞
  [false]	=	0
  [if(x1, x2, x3)]	=	-∞ · x1 + 0 · x2 + 0 · x3 + -∞
  [true]	=	0
  [id(x1)]	=	0 · x1 + -∞
  [not(x1)]	=	0 · x1 + 0
  [gt(x1, x2)]	=	-∞ · x1 + 0 · x2 + 0
  [zero]	=	2
  [s(x1)]	=	2 · x1 + 4

  together with the usable rules

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

  (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)))

  (43)

  could be deleted.

  1.1.6.1 Reduction Pair Processor with Usable Rules

  Using the

  prec(plus#)	=	0		stat(plus#)	=	lex
  prec(false)	=	0		stat(false)	=	lex
  prec(true)	=	0		stat(true)	=	lex
  prec(zero)	=	0		stat(zero)	=	lex
  prec(id)	=	0		stat(id)	=	lex
  prec(if)	=	0		stat(if)	=	lex
  prec(gt)	=	0		stat(gt)	=	lex
  prec(s)	=	4		stat(s)	=	lex

  π(plus#)	=	[1]
  π(false)	=	[]
  π(true)	=	[]
  π(zero)	=	[]
  π(id)	=	1
  π(if)	=	[2,3]
  π(gt)	=	1
  π(s)	=	[1]

  together with the usable rules

  if(true,x,y)	→	x	(19)
  if(false,x,y)	→	y	(20)
  id(x)	→	x	(18)

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

  plus#(s(x),x)	→	plus#(if(gt(x,x),id(x),id(x)),s(x))	(47)
  plus#(s(x),y)	→	plus#(x,y)	(28)

  could be deleted.

  1.1.6.1.1 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)))	(50)
  1	>	1

  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 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)
  2	>	2
  1	>	1

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