Certification Problem

Input (TPDB TRS_Standard/AProVE_07/thiemann07)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

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

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

• The 1^st component contains the pair

  if2#(false,l)	→	prod#(cons(times(car(l),cadr(l)),cddr(l)))	(39)
  prod#(l)	→	if#(shorter(l,0),shorter(l,s(0)),l)	(34)
  if#(false,b,l)	→	if2#(b,l)	(37)

  1.1.1.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  shorter(nil,y)	→	true	(16)
  shorter(cons(x,l),0)	→	false	(17)
  shorter(cons(x,l),s(y))	→	shorter(l,y)	(18)
  car(cons(x,l))	→	x	(1)
  cadr(cons(x,cons(y,l)))	→	y	(5)
  times(x,y)	→	iftimes(isZero(x),x,y)	(11)
  cddr(nil)	→	nil	(2)
  cddr(cons(x,nil))	→	nil	(3)
  cddr(cons(x,cons(y,l)))	→	l	(4)
  isZero(0)	→	true	(6)
  isZero(s(x))	→	false	(7)
  iftimes(true,x,y)	→	0	(12)
  iftimes(false,x,y)	→	plus(y,times(p(x),y))	(13)
  p(s(x))	→	x	(14)
  p(0)	→	0	(15)
  plus(x,y)	→	ifplus(isZero(x),x,y)	(8)
  ifplus(true,x,y)	→	y	(9)
  ifplus(false,x,y)	→	s(plus(p(x),y))	(10)

  1.1.1.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  car(cons(x0,x1))

  cddr(nil)

  cddr(cons(x0,nil))

  cddr(cons(x0,cons(x1,x2)))

  cadr(cons(x0,cons(x1,x2)))

  isZero(0)

  isZero(s(x0))

  plus(x0,x1)

  ifplus(true,x0,x1)

  ifplus(false,x0,x1)

  times(x0,x1)

  iftimes(true,x0,x1)

  iftimes(false,x0,x1)

  p(s(x0))

  p(0)

  shorter(nil,x0)

  shorter(cons(x0,x1),0)

  shorter(cons(x0,x1),s(x2))

  1.1.1.1.1.1 Rewriting Processor

  We rewrite the right hand side of the pair resulting in

  →

  1.1.1.1.1.1.1 Narrowing Processor

  We consider all narrowings of the pair below position 1 to get the following set of pairs

  prod#(nil)	→	if#(true,shorter(nil,s(0)),nil)	(45)
  prod#(cons(x0,x1))	→	if#(false,shorter(cons(x0,x1),s(0)),cons(x0,x1))	(46)

  1.1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    if2#(false,l)	→	prod#(cons(iftimes(isZero(car(l)),car(l),cadr(l)),cddr(l)))	(44)
    prod#(cons(x0,x1))	→	if#(false,shorter(cons(x0,x1),s(0)),cons(x0,x1))	(46)
    if#(false,b,l)	→	if2#(b,l)	(37)

    1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

    We restrict the rewrite rules to the following usable rules of the DP problem.

    shorter(nil,y)	→	true	(16)
    shorter(cons(x,l),0)	→	false	(17)
    car(cons(x,l))	→	x	(1)
    isZero(0)	→	true	(6)
    isZero(s(x))	→	false	(7)
    cadr(cons(x,cons(y,l)))	→	y	(5)
    iftimes(true,x,y)	→	0	(12)
    iftimes(false,x,y)	→	plus(y,times(p(x),y))	(13)
    cddr(nil)	→	nil	(2)
    cddr(cons(x,nil))	→	nil	(3)
    cddr(cons(x,cons(y,l)))	→	l	(4)
    p(s(x))	→	x	(14)
    p(0)	→	0	(15)
    times(x,y)	→	iftimes(isZero(x),x,y)	(11)
    plus(x,y)	→	ifplus(isZero(x),x,y)	(8)
    ifplus(true,x,y)	→	y	(9)
    ifplus(false,x,y)	→	s(plus(p(x),y))	(10)

    1.1.1.1.1.1.1.1.1.1.1 Instantiation Processor

    We instantiate the pair to the following set of pairs

    if#(false,y_0,cons(z0,z1))

    →

    if2#(y_0,cons(z0,z1))

    (48)

    1.1.1.1.1.1.1.1.1.1.1.1 Instantiation Processor

    We instantiate the pair to the following set of pairs

    if2#(false,cons(z1,z2))

    →

    prod#(cons(iftimes(isZero(car(cons(z1,z2))),car(cons(z1,z2)),cadr(cons(z1,z2))),cddr(cons(z1,z2))))

    (49)

    1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

    We restrict the rewrite rules to the following usable rules of the DP problem.

    car(cons(x,l))	→	x	(1)
    isZero(0)	→	true	(6)
    isZero(s(x))	→	false	(7)
    cadr(cons(x,cons(y,l)))	→	y	(5)
    iftimes(true,x,y)	→	0	(12)
    iftimes(false,x,y)	→	plus(y,times(p(x),y))	(13)
    cddr(cons(x,nil))	→	nil	(3)
    cddr(cons(x,cons(y,l)))	→	l	(4)
    p(s(x))	→	x	(14)
    p(0)	→	0	(15)
    times(x,y)	→	iftimes(isZero(x),x,y)	(11)
    plus(x,y)	→	ifplus(isZero(x),x,y)	(8)
    ifplus(true,x,y)	→	y	(9)
    ifplus(false,x,y)	→	s(plus(p(x),y))	(10)
    shorter(nil,y)	→	true	(16)
    shorter(cons(x,l),0)	→	false	(17)

    1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

    We restrict the rewrite rules to the following usable rules of the DP problem.

    isZero(0)	→	true	(6)
    isZero(s(x))	→	false	(7)
    cadr(cons(x,cons(y,l)))	→	y	(5)
    iftimes(true,x,y)	→	0	(12)
    iftimes(false,x,y)	→	plus(y,times(p(x),y))	(13)
    cddr(cons(x,nil))	→	nil	(3)
    cddr(cons(x,cons(y,l)))	→	l	(4)
    p(s(x))	→	x	(14)
    p(0)	→	0	(15)
    times(x,y)	→	iftimes(isZero(x),x,y)	(11)
    plus(x,y)	→	ifplus(isZero(x),x,y)	(8)
    ifplus(true,x,y)	→	y	(9)
    ifplus(false,x,y)	→	s(plus(p(x),y))	(10)
    shorter(nil,y)	→	true	(16)
    shorter(cons(x,l),0)	→	false	(17)

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

    We restrict the innermost strategy to the following left hand sides.

    cddr(nil)

    cddr(cons(x0,nil))

    cddr(cons(x0,cons(x1,x2)))

    cadr(cons(x0,cons(x1,x2)))

    isZero(0)

    isZero(s(x0))

    plus(x0,x1)

    ifplus(true,x0,x1)

    ifplus(false,x0,x1)

    times(x0,x1)

    iftimes(true,x0,x1)

    iftimes(false,x0,x1)

    p(s(x0))

    p(0)

    shorter(nil,x0)

    shorter(cons(x0,x1),0)

    shorter(cons(x0,x1),s(x2))

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Forward Instantiation Processor

    We instantiate the pair to the following set of pairs

    if#(false,false,cons(x1,x2))

    →

    if2#(false,cons(x1,x2))

    (52)

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

    We consider all narrowings of the pair below position 2 to get the following set of pairs

    prod#(cons(y0,nil))	→	if#(false,true,cons(y0,nil))	(53)
    prod#(cons(y0,cons(x0,x1)))	→	if#(false,false,cons(y0,cons(x0,x1)))	(54)

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      prod#(cons(y0,cons(x0,x1)))	→	if#(false,false,cons(y0,cons(x0,x1)))	(54)
      if#(false,false,cons(x1,x2))	→	if2#(false,cons(x1,x2))	(52)
      if2#(false,cons(z1,z2))	→	prod#(cons(iftimes(isZero(z1),z1,cadr(cons(z1,z2))),cddr(cons(z1,z2))))	(51)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

      We restrict the rewrite rules to the following usable rules of the DP problem.

      isZero(0)	→	true	(6)
      isZero(s(x))	→	false	(7)
      cadr(cons(x,cons(y,l)))	→	y	(5)
      iftimes(true,x,y)	→	0	(12)
      iftimes(false,x,y)	→	plus(y,times(p(x),y))	(13)
      cddr(cons(x,nil))	→	nil	(3)
      cddr(cons(x,cons(y,l)))	→	l	(4)
      p(s(x))	→	x	(14)
      p(0)	→	0	(15)
      times(x,y)	→	iftimes(isZero(x),x,y)	(11)
      plus(x,y)	→	ifplus(isZero(x),x,y)	(8)
      ifplus(true,x,y)	→	y	(9)
      ifplus(false,x,y)	→	s(plus(p(x),y))	(10)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

      We restrict the innermost strategy to the following left hand sides.

      cddr(nil)

      cddr(cons(x0,nil))

      cddr(cons(x0,cons(x1,x2)))

      cadr(cons(x0,cons(x1,x2)))

      isZero(0)

      isZero(s(x0))

      plus(x0,x1)

      ifplus(true,x0,x1)

      ifplus(false,x0,x1)

      times(x0,x1)

      iftimes(true,x0,x1)

      iftimes(false,x0,x1)

      p(s(x0))

      p(0)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

      We consider all narrowings of the pair below position 1.2 to get the following set of pairs

      if2#(false,cons(x0,nil))	→	prod#(cons(iftimes(isZero(x0),x0,cadr(cons(x0,nil))),nil))	(55)
      if2#(false,cons(x0,cons(x1,x2)))	→	prod#(cons(iftimes(isZero(x0),x0,cadr(cons(x0,cons(x1,x2)))),x2))	(56)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 1 component.

      • The 1^st component contains the pair

        if#(false,false,cons(x1,x2))	→	if2#(false,cons(x1,x2))	(52)
        if2#(false,cons(x0,cons(x1,x2)))	→	prod#(cons(iftimes(isZero(x0),x0,cadr(cons(x0,cons(x1,x2)))),x2))	(56)
        prod#(cons(y0,cons(x0,x1)))	→	if#(false,false,cons(y0,cons(x0,x1)))	(54)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

        We restrict the rewrite rules to the following usable rules of the DP problem.

        isZero(0)	→	true	(6)
        isZero(s(x))	→	false	(7)
        cadr(cons(x,cons(y,l)))	→	y	(5)
        iftimes(true,x,y)	→	0	(12)
        iftimes(false,x,y)	→	plus(y,times(p(x),y))	(13)
        p(s(x))	→	x	(14)
        p(0)	→	0	(15)
        times(x,y)	→	iftimes(isZero(x),x,y)	(11)
        plus(x,y)	→	ifplus(isZero(x),x,y)	(8)
        ifplus(true,x,y)	→	y	(9)
        ifplus(false,x,y)	→	s(plus(p(x),y))	(10)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

        We restrict the innermost strategy to the following left hand sides.

        cadr(cons(x0,cons(x1,x2)))

        isZero(0)

        isZero(s(x0))

        plus(x0,x1)

        ifplus(true,x0,x1)

        ifplus(false,x0,x1)

        times(x0,x1)

        iftimes(true,x0,x1)

        iftimes(false,x0,x1)

        p(s(x0))

        p(0)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

        We rewrite the right hand side of the pair resulting in

        →

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

        We restrict the rewrite rules to the following usable rules of the DP problem.

        isZero(0)	→	true	(6)
        isZero(s(x))	→	false	(7)
        iftimes(true,x,y)	→	0	(12)
        iftimes(false,x,y)	→	plus(y,times(p(x),y))	(13)
        p(s(x))	→	x	(14)
        p(0)	→	0	(15)
        times(x,y)	→	iftimes(isZero(x),x,y)	(11)
        plus(x,y)	→	ifplus(isZero(x),x,y)	(8)
        ifplus(true,x,y)	→	y	(9)
        ifplus(false,x,y)	→	s(plus(p(x),y))	(10)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

        We restrict the innermost strategy to the following left hand sides.

        isZero(0)

        isZero(s(x0))

        plus(x0,x1)

        ifplus(true,x0,x1)

        ifplus(false,x0,x1)

        times(x0,x1)

        iftimes(true,x0,x1)

        iftimes(false,x0,x1)

        p(s(x0))

        p(0)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Instantiation Processor

        We instantiate the pair to the following set of pairs

        if#(false,false,cons(z0,cons(z1,z2)))

        →

        if2#(false,cons(z0,cons(z1,z2)))

        (58)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

        prec(cons)	=	1		weight(cons)	=	1
        prec(false)	=	0		weight(false)	=	2

        in combination with the following argument filter

        π(prod#)	=	1
        π(cons)	=	[2]
        π(if#)	=	3
        π(if2#)	=	2
        π(false)	=	[]

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

        if2#(false,cons(x0,cons(x1,x2)))

        →

        prod#(cons(iftimes(isZero(x0),x0,x1),x2))

        (57)

        could be deleted.

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

        The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  iftimes#(false,x,y)	→	times#(p(x),y)	(31)
  times#(x,y)	→	iftimes#(isZero(x),x,y)	(28)

  1.1.1.2 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  isZero(0)	→	true	(6)
  isZero(s(x))	→	false	(7)
  p(s(x))	→	x	(14)
  p(0)	→	0	(15)

  1.1.1.2.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  isZero(0)

  isZero(s(x0))

  p(s(x0))

  p(0)

  1.1.1.2.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the rationals with delta = 1/16

  [iftimes#(x1, x2, x3)]	=	0 + 1/4 · x1 + 1/4 · x2 + 0 · x3
  [false]	=	1/4
  [times#(x1, x2)]	=	0 + 1 · x1 + 0 · x2
  [p(x1)]	=	0 + 1/4 · x1
  [isZero(x1)]	=	0 + 1/2 · x1
  [s(x1)]	=	2 + 4 · x1
  [0]	=	0
  [true]	=	0

  the pair

  iftimes#(false,x,y)

  →

  times#(p(x),y)

  (31)

  could be deleted.

  1.1.1.2.1.1.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  isZero(0)	→	true	(6)
  isZero(s(x))	→	false	(7)

  1.1.1.2.1.1.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  isZero(0)

  isZero(s(x0))

  1.1.1.2.1.1.1.1.1 Size-Change Termination

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

  times#(x,y)	→	iftimes#(isZero(x),x,y)	(28)
  1	≥	2
  2	≥	3

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

• The 3^rd component contains the pair

  ifplus#(false,x,y)	→	plus#(p(x),y)	(26)
  plus#(x,y)	→	ifplus#(isZero(x),x,y)	(24)

  1.1.1.3 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  isZero(0)	→	true	(6)
  isZero(s(x))	→	false	(7)
  p(s(x))	→	x	(14)
  p(0)	→	0	(15)

  1.1.1.3.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  isZero(0)

  isZero(s(x0))

  p(s(x0))

  p(0)

  1.1.1.3.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the rationals with delta = 1/16

  [ifplus#(x1, x2, x3)]	=	0 + 1/4 · x1 + 1/4 · x2 + 0 · x3
  [false]	=	1/4
  [plus#(x1, x2)]	=	0 + 1 · x1 + 0 · x2
  [p(x1)]	=	0 + 1/4 · x1
  [isZero(x1)]	=	0 + 1/2 · x1
  [s(x1)]	=	2 + 4 · x1
  [0]	=	0
  [true]	=	0

  the pair

  ifplus#(false,x,y)

  →

  plus#(p(x),y)

  (26)

  could be deleted.

  1.1.1.3.1.1.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  isZero(0)	→	true	(6)
  isZero(s(x))	→	false	(7)

  1.1.1.3.1.1.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  isZero(0)

  isZero(s(x0))

  1.1.1.3.1.1.1.1.1 Size-Change Termination

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

  plus#(x,y)	→	ifplus#(isZero(x),x,y)	(24)
  1	≥	2
  2	≥	3

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

• The 4^th component contains the pair

  shorter#(cons(x,l),s(y))

  →

  shorter#(l,y)

  (33)

  1.1.1.4 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.1.4.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  There are no lhss.

  1.1.1.4.1.1 Size-Change Termination

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

  shorter#(cons(x,l),s(y))	→	shorter#(l,y)	(33)
  1	>	1
  2	>	2

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