Certification Problem

Input (TPDB TRS_Standard/AProVE_07/kabasci02)

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 4 components.

• The 1^st component contains the pair

  div#(x,y)	→	quot#(x,y,y)	(39)
  quot#(s(x),s(y),z)	→	quot#(x,y,z)	(40)
  quot#(x,0,s(z))	→	div#(x,s(z))	(41)
  div#(div(x,y),z)	→	div#(x,times(zero(y),z))	(42)
  div#(div(x,y),z)	→	zero#(y)	(44)
  zero#(s(x))	→	if#(eq(x,s(0)),plus(zero(0),0),s(plus(0,zero(0))))	(53)
  if#(false,x,y)	→	pr#(x,y)	(52)
  pr#(x,s(s(y)))	→	if#(divides(s(s(y)),x),x,s(y))	(50)
  pr#(x,s(s(y)))	→	divides#(s(s(y)),x)	(51)
  divides#(y,x)	→	div#(x,y)	(48)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [div#(x1, x2)]	=	x1
  [times(x1, x2)]	=	-2
  [if#(x1, x2, x3)]	=	-1 + x1 + x2
  [div(x1, x2)]	=	2 + x1
  [zero(x1)]	=	2
  [divides(x1, x2)]	=	1
  [quot(x1, x2, x3)]	=	2 + x1
  [s(x1)]	=	x1
  [if(x1, x2, x3)]	=	-1 + 2 · x1
  [eq(x1, x2)]	=	1
  [0]	=	0
  [plus(x1, x2)]	=	x1
  [false]	=	1
  [true]	=	1
  [pr(x1, x2)]	=	1
  [p(x1)]	=	-2
  [quot#(x1, x2, x3)]	=	x1
  [zero#(x1)]	=	2
  [pr#(x1, x2)]	=	x1
  [divides#(x1, x2)]	=	x2

  together with the usable rules

  zero(s(x))	→	if(eq(x,s(0)),plus(zero(0),0),s(plus(0,zero(0))))	(31)
  eq(0,s(y))	→	false	(19)
  eq(s(x),s(y))	→	eq(x,y)	(20)
  plus(x,0)	→	x	(3)
  divides(y,x)	→	eq(x,times(div(x,y),y))	(21)
  if(true,x,y)	→	false	(25)
  pr(x,s(s(y)))	→	if(divides(s(s(y)),x),x,s(y))	(24)
  if(false,x,y)	→	pr(x,y)	(26)
  pr(x,s(0))	→	true	(23)
  div(0,y)	→	0	(11)
  eq(0,0)	→	true	(17)
  eq(s(x),0)	→	false	(18)
  div(x,y)	→	quot(x,y,y)	(12)
  quot(zero(y),s(y),z)	→	0	(13)
  quot(s(x),s(y),z)	→	quot(x,y,z)	(14)
  quot(x,0,s(z))	→	s(div(x,s(z)))	(15)
  div(div(x,y),z)	→	div(x,times(zero(y),z))	(16)

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

  div#(div(x,y),z)

  →

  div#(x,times(zero(y),z))

  (42)

  could be deleted.

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [if#(x1, x2, x3)]	=	2 · x2 + x3
  [eq(x1, x2)]	=	2 + 2 · x2
  [0]	=	0
  [s(x1)]	=	2 + 2 · x1
  [false]	=	0
  [div(x1, x2)]	=	2 + x2
  [if(x1, x2, x3)]	=	x3
  [plus(x1, x2)]	=	x1 + 2 · x2
  [divides(x1, x2)]	=	-2 + x2
  [times(x1, x2)]	=	2 + 2 · x1 + 2 · x2
  [zero(x1)]	=	-2 + 2 · x1
  [true]	=	1
  [pr(x1, x2)]	=	1 + 2 · x1
  [quot(x1, x2, x3)]	=	x2 + 2 · x3
  [p(x1)]	=	-2
  [div#(x1, x2)]	=	2 · x1
  [quot#(x1, x2, x3)]	=	2 · x1
  [zero#(x1)]	=	2 · x1
  [pr#(x1, x2)]	=	2 · x1 + x2
  [divides#(x1, x2)]	=	x1 + 2 · x2

  together with the usable rules

  plus(x,0)	→	x	(3)
  plus(0,y)	→	y	(4)

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

  quot#(s(x),s(y),z)	→	quot#(x,y,z)	(40)
  div#(div(x,y),z)	→	zero#(y)	(44)
  zero#(s(x))	→	if#(eq(x,s(0)),plus(zero(0),0),s(plus(0,zero(0))))	(53)
  pr#(x,s(s(y)))	→	if#(divides(s(s(y)),x),x,s(y))	(50)

  could be deleted.

  1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    quot#(x,0,s(z))	→	div#(x,s(z))	(41)
    div#(x,y)	→	quot#(x,y,y)	(39)

    1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    prec(0)	=	1		weight(0)	=	2
    prec(s)	=	0		weight(s)	=	1

    in combination with the following argument filter

    π(quot#)	=	2
    π(0)	=	[]
    π(div#)	=	2
    π(s)	=	[]

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

    quot#(x,0,s(z))

    →

    div#(x,s(z))

    (41)

    could be deleted.

    1.1.1.1.1.1.1 Size-Change Termination

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

    div#(x,y)	→	quot#(x,y,y)	(39)
    1	≥	1
    2	≥	2
    2	≥	3

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

• The 2^nd component contains the pair

  times#(s(x),y)

  →

  times#(x,y)

  (38)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [times#(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.

  1.1.2.1 Size-Change Termination

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

  times#(s(x),y)	→	times#(x,y)	(38)
  1	>	1
  2	≥	2

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

• The 3^rd component contains the pair

  plus#(s(x),y)	→	plus#(p(s(x)),y)	(33)
  plus#(s(x),y)	→	plus#(x,y)	(32)
  plus#(x,s(y))	→	plus#(x,p(s(y)))	(35)

  1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rule

  p(s(x))

  →

  x

  (2)

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

  1.1.3.1 Switch to Innermost Termination

  The TRS does not have overlaps with the pairs and is locally confluent:

  20

  Hence, it suffices to show innermost termination in the following.

  1.1.3.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

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

  the pair

  plus#(s(x),y)

  →

  plus#(x,y)

  (32)

  and the rule

  p(s(x))

  →

  x

  (2)

  could be deleted.

  1.1.3.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 4^th component contains the pair

  eq#(s(x),s(y))

  →

  eq#(x,y)

  (45)

  1.1.4 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [eq#(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.

  1.1.4.1 Size-Change Termination

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

  eq#(s(x),s(y))	→	eq#(x,y)	(45)
  1	>	1
  2	>	2

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