Certification Problem

Input (TPDB TRS_Innermost/Transformed_CSR_innermost_04/Ex9_BLR02_iGM)

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 6 components.

• The 1^st component contains the pair

  mark#(cons(X1,X2))	→	active#(cons(mark(X1),X2))	(57)
  active#(filter(cons(X,Y),0,M))	→	mark#(cons(0,filter(Y,M,M)))	(30)
  mark#(cons(X1,X2))	→	mark#(X1)	(59)
  mark#(filter(X1,X2,X3))	→	active#(filter(mark(X1),mark(X2),mark(X3)))	(52)
  active#(filter(cons(X,Y),s(N),M))	→	mark#(cons(X,filter(Y,N,M)))	(33)
  active#(sieve(cons(0,Y)))	→	mark#(cons(0,sieve(Y)))	(36)
  active#(sieve(cons(s(N),Y)))	→	mark#(cons(s(N),sieve(filter(Y,N,N))))	(39)
  active#(nats(N))	→	mark#(cons(N,nats(s(N))))	(43)
  mark#(filter(X1,X2,X3))	→	mark#(X1)	(54)
  mark#(filter(X1,X2,X3))	→	mark#(X2)	(55)
  mark#(filter(X1,X2,X3))	→	mark#(X3)	(56)
  mark#(s(X))	→	active#(s(mark(X)))	(61)
  mark#(s(X))	→	mark#(X)	(63)
  mark#(sieve(X))	→	active#(sieve(mark(X)))	(64)
  mark#(sieve(X))	→	mark#(X)	(66)
  mark#(nats(X))	→	active#(nats(mark(X)))	(67)
  mark#(nats(X))	→	mark#(X)	(69)
  mark#(zprimes)	→	active#(zprimes)	(70)
  active#(zprimes)	→	mark#(sieve(nats(s(s(0)))))	(47)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [mark#(x1)]	=	1
  [cons(x1, x2)]	=	0
  [active#(x1)]	=	1 · x1
  [mark(x1)]	=	0
  [filter(x1, x2, x3)]	=	1
  [0]	=	0
  [s(x1)]	=	0
  [sieve(x1)]	=	1
  [nats(x1)]	=	1
  [zprimes]	=	1
  [active(x1)]	=	0

  together with the usable rules

  cons(X1,mark(X2))	→	cons(X1,X2)	(21)
  cons(mark(X1),X2)	→	cons(X1,X2)	(20)
  cons(active(X1),X2)	→	cons(X1,X2)	(22)
  cons(X1,active(X2))	→	cons(X1,X2)	(23)
  filter(X1,mark(X2),X3)	→	filter(X1,X2,X3)	(15)
  filter(mark(X1),X2,X3)	→	filter(X1,X2,X3)	(14)
  filter(X1,X2,mark(X3))	→	filter(X1,X2,X3)	(16)
  filter(active(X1),X2,X3)	→	filter(X1,X2,X3)	(17)
  filter(X1,active(X2),X3)	→	filter(X1,X2,X3)	(18)
  filter(X1,X2,active(X3))	→	filter(X1,X2,X3)	(19)
  s(active(X))	→	s(X)	(25)
  s(mark(X))	→	s(X)	(24)
  sieve(active(X))	→	sieve(X)	(27)
  sieve(mark(X))	→	sieve(X)	(26)
  nats(active(X))	→	nats(X)	(29)
  nats(mark(X))	→	nats(X)	(28)

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

  mark#(cons(X1,X2))	→	active#(cons(mark(X1),X2))	(57)
  mark#(s(X))	→	active#(s(mark(X)))	(61)

  could be deleted.

  1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [active#(x1)]	=	1 · x1
  [filter(x1, x2, x3)]	=	1 · x1 + 1 · x2 + 1 · x3
  [cons(x1, x2)]	=	1 · x1
  [0]	=	0
  [mark#(x1)]	=	1 · x1
  [mark(x1)]	=	1 · x1
  [s(x1)]	=	1 · x1
  [sieve(x1)]	=	1 · x1
  [nats(x1)]	=	1 · x1
  [zprimes]	=	1
  [active(x1)]	=	1 · x1

  the pair

  active#(zprimes)

  →

  mark#(sieve(nats(s(s(0)))))

  (47)

  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

    mark#(cons(X1,X2))	→	mark#(X1)	(59)
    mark#(filter(X1,X2,X3))	→	active#(filter(mark(X1),mark(X2),mark(X3)))	(52)
    active#(filter(cons(X,Y),0,M))	→	mark#(cons(0,filter(Y,M,M)))	(30)
    active#(filter(cons(X,Y),s(N),M))	→	mark#(cons(X,filter(Y,N,M)))	(33)
    active#(sieve(cons(0,Y)))	→	mark#(cons(0,sieve(Y)))	(36)
    active#(sieve(cons(s(N),Y)))	→	mark#(cons(s(N),sieve(filter(Y,N,N))))	(39)
    active#(nats(N))	→	mark#(cons(N,nats(s(N))))	(43)
    mark#(filter(X1,X2,X3))	→	mark#(X1)	(54)
    mark#(filter(X1,X2,X3))	→	mark#(X2)	(55)
    mark#(filter(X1,X2,X3))	→	mark#(X3)	(56)
    mark#(s(X))	→	mark#(X)	(63)
    mark#(sieve(X))	→	active#(sieve(mark(X)))	(64)
    mark#(sieve(X))	→	mark#(X)	(66)
    mark#(nats(X))	→	active#(nats(mark(X)))	(67)
    mark#(nats(X))	→	mark#(X)	(69)

    1.1.1.1.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [mark#(x1)]	=	1 · x1
    [cons(x1, x2)]	=	1 · x1
    [filter(x1, x2, x3)]	=	1 · x1 + 1 · x2 + 1 · x3
    [active#(x1)]	=	1 · x1
    [mark(x1)]	=	1 · x1
    [0]	=	0
    [s(x1)]	=	1 · x1
    [sieve(x1)]	=	1 · x1
    [nats(x1)]	=	1 + 1 · x1
    [active(x1)]	=	1 · x1
    [zprimes]	=	1

    the pairs

    active#(nats(N))	→	mark#(cons(N,nats(s(N))))	(43)
    mark#(nats(X))	→	mark#(X)	(69)

    could be deleted.

    1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over the naturals

    [mark#(x1)]	=	1
    [cons(x1, x2)]	=	0
    [filter(x1, x2, x3)]	=	1
    [active#(x1)]	=	1 · x1
    [mark(x1)]	=	0
    [0]	=	0
    [s(x1)]	=	0
    [sieve(x1)]	=	1
    [nats(x1)]	=	0
    [active(x1)]	=	0
    [zprimes]	=	0

    together with the usable rules

    filter(X1,mark(X2),X3)	→	filter(X1,X2,X3)	(15)
    filter(mark(X1),X2,X3)	→	filter(X1,X2,X3)	(14)
    filter(X1,X2,mark(X3))	→	filter(X1,X2,X3)	(16)
    filter(active(X1),X2,X3)	→	filter(X1,X2,X3)	(17)
    filter(X1,active(X2),X3)	→	filter(X1,X2,X3)	(18)
    filter(X1,X2,active(X3))	→	filter(X1,X2,X3)	(19)
    sieve(active(X))	→	sieve(X)	(27)
    sieve(mark(X))	→	sieve(X)	(26)
    nats(active(X))	→	nats(X)	(29)
    nats(mark(X))	→	nats(X)	(28)

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

    mark#(nats(X))

    →

    active#(nats(mark(X)))

    (67)

    could be deleted.

    1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [mark#(x1)]	=	1 · x1
    [cons(x1, x2)]	=	1 · x1
    [filter(x1, x2, x3)]	=	1 + 1 · x1 + 1 · x2 + 1 · x3
    [active#(x1)]	=	1 · x1
    [mark(x1)]	=	1 · x1
    [0]	=	0
    [s(x1)]	=	1 · x1
    [sieve(x1)]	=	1 · x1
    [active(x1)]	=	1 · x1
    [nats(x1)]	=	1 · x1
    [zprimes]	=	0

    the pairs

    active#(filter(cons(X,Y),0,M))	→	mark#(cons(0,filter(Y,M,M)))	(30)
    active#(filter(cons(X,Y),s(N),M))	→	mark#(cons(X,filter(Y,N,M)))	(33)
    mark#(filter(X1,X2,X3))	→	mark#(X1)	(54)
    mark#(filter(X1,X2,X3))	→	mark#(X2)	(55)
    mark#(filter(X1,X2,X3))	→	mark#(X3)	(56)

    could be deleted.

    1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

    prec(0)	=	2		weight(0)	=	1
    prec(s)	=	0		weight(s)	=	1
    prec(zprimes)	=	1		weight(zprimes)	=	4

    in combination with the following argument filter

    π(mark#)	=	1
    π(cons)	=	1
    π(filter)	=	1
    π(active#)	=	1
    π(mark)	=	1
    π(sieve)	=	1
    π(0)	=	[]
    π(s)	=	[1]
    π(active)	=	1
    π(nats)	=	1
    π(zprimes)	=	[]

    the pair

    mark#(s(X))

    →

    mark#(X)

    (63)

    could be deleted.

    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(filter)	=	1		weight(filter)	=	4
    prec(active#)	=	3		weight(active#)	=	3
    prec(0)	=	2		weight(0)	=	2
    prec(s)	=	0		weight(s)	=	1
    prec(sieve)	=	4		weight(sieve)	=	3

    in combination with the following argument filter

    π(mark#)	=	1
    π(cons)	=	1
    π(filter)	=	[]
    π(active#)	=	[]
    π(0)	=	[]
    π(s)	=	[]
    π(sieve)	=	[1]
    π(mark)	=	1
    π(active)	=	1

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

    mark#(filter(X1,X2,X3))	→	active#(filter(mark(X1),mark(X2),mark(X3)))	(52)
    active#(sieve(cons(0,Y)))	→	mark#(cons(0,sieve(Y)))	(36)
    active#(sieve(cons(s(N),Y)))	→	mark#(cons(s(N),sieve(filter(Y,N,N))))	(39)
    mark#(sieve(X))	→	active#(sieve(mark(X)))	(64)
    mark#(sieve(X))	→	mark#(X)	(66)

    could be deleted.

    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.

    There are no rules.

    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.

    cons(mark(x0),x1)

    cons(x0,mark(x1))

    cons(active(x0),x1)

    cons(x0,active(x1))

    1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

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

    mark#(cons(X1,X2))	→	mark#(X1)	(59)
    1	>	1

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

• The 2^nd component contains the pair

  filter#(X1,mark(X2),X3)	→	filter#(X1,X2,X3)	(72)
  filter#(mark(X1),X2,X3)	→	filter#(X1,X2,X3)	(71)
  filter#(X1,X2,mark(X3))	→	filter#(X1,X2,X3)	(73)
  filter#(active(X1),X2,X3)	→	filter#(X1,X2,X3)	(74)
  filter#(X1,active(X2),X3)	→	filter#(X1,X2,X3)	(75)
  filter#(X1,X2,active(X3))	→	filter#(X1,X2,X3)	(76)

  1.1.2 Usable Rules Processor

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

  There are no rules.

  1.1.2.1 Innermost Lhss Removal Processor

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

  active(filter(cons(x0,x1),0,x2))

  active(filter(cons(x0,x1),s(x2),x3))

  active(sieve(cons(0,x0)))

  active(sieve(cons(s(x0),x1)))

  active(nats(x0))

  active(zprimes)

  mark(filter(x0,x1,x2))

  mark(cons(x0,x1))

  mark(0)

  mark(s(x0))

  mark(sieve(x0))

  mark(nats(x0))

  mark(zprimes)

  1.1.2.1.1 Size-Change Termination

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

  filter#(X1,mark(X2),X3)	→	filter#(X1,X2,X3)	(72)
  1	≥	1
  2	>	2
  3	≥	3
  filter#(mark(X1),X2,X3)	→	filter#(X1,X2,X3)	(71)
  1	>	1
  2	≥	2
  3	≥	3
  filter#(X1,X2,mark(X3))	→	filter#(X1,X2,X3)	(73)
  1	≥	1
  2	≥	2
  3	>	3
  filter#(active(X1),X2,X3)	→	filter#(X1,X2,X3)	(74)
  1	>	1
  2	≥	2
  3	≥	3
  filter#(X1,active(X2),X3)	→	filter#(X1,X2,X3)	(75)
  1	≥	1
  2	>	2
  3	≥	3
  filter#(X1,X2,active(X3))	→	filter#(X1,X2,X3)	(76)
  1	≥	1
  2	≥	2
  3	>	3

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

• The 3^rd component contains the pair

  cons#(X1,mark(X2))	→	cons#(X1,X2)	(78)
  cons#(mark(X1),X2)	→	cons#(X1,X2)	(77)
  cons#(active(X1),X2)	→	cons#(X1,X2)	(79)
  cons#(X1,active(X2))	→	cons#(X1,X2)	(80)

  1.1.3 Usable Rules Processor

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

  There are no rules.

  1.1.3.1 Innermost Lhss Removal Processor

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

  active(filter(cons(x0,x1),0,x2))

  active(filter(cons(x0,x1),s(x2),x3))

  active(sieve(cons(0,x0)))

  active(sieve(cons(s(x0),x1)))

  active(nats(x0))

  active(zprimes)

  mark(filter(x0,x1,x2))

  mark(cons(x0,x1))

  mark(0)

  mark(s(x0))

  mark(sieve(x0))

  mark(nats(x0))

  mark(zprimes)

  1.1.3.1.1 Size-Change Termination

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

  cons#(X1,mark(X2))	→	cons#(X1,X2)	(78)
  1	≥	1
  2	>	2
  cons#(mark(X1),X2)	→	cons#(X1,X2)	(77)
  1	>	1
  2	≥	2
  cons#(active(X1),X2)	→	cons#(X1,X2)	(79)
  1	>	1
  2	≥	2
  cons#(X1,active(X2))	→	cons#(X1,X2)	(80)
  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

  s#(active(X))	→	s#(X)	(82)
  s#(mark(X))	→	s#(X)	(81)

  1.1.4 Usable Rules Processor

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

  There are no rules.

  1.1.4.1 Innermost Lhss Removal Processor

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

  active(filter(cons(x0,x1),0,x2))

  active(filter(cons(x0,x1),s(x2),x3))

  active(sieve(cons(0,x0)))

  active(sieve(cons(s(x0),x1)))

  active(nats(x0))

  active(zprimes)

  mark(filter(x0,x1,x2))

  mark(cons(x0,x1))

  mark(0)

  mark(s(x0))

  mark(sieve(x0))

  mark(nats(x0))

  mark(zprimes)

  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.

  s#(active(X))	→	s#(X)	(82)
  1	>	1
  s#(mark(X))	→	s#(X)	(81)
  1	>	1

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

• The 5^th component contains the pair

  sieve#(active(X))	→	sieve#(X)	(84)
  sieve#(mark(X))	→	sieve#(X)	(83)

  1.1.5 Usable Rules Processor

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

  There are no rules.

  1.1.5.1 Innermost Lhss Removal Processor

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

  active(filter(cons(x0,x1),0,x2))

  active(filter(cons(x0,x1),s(x2),x3))

  active(sieve(cons(0,x0)))

  active(sieve(cons(s(x0),x1)))

  active(nats(x0))

  active(zprimes)

  mark(filter(x0,x1,x2))

  mark(cons(x0,x1))

  mark(0)

  mark(s(x0))

  mark(sieve(x0))

  mark(nats(x0))

  mark(zprimes)

  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.

  sieve#(active(X))	→	sieve#(X)	(84)
  1	>	1
  sieve#(mark(X))	→	sieve#(X)	(83)
  1	>	1

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

• The 6^th component contains the pair

  nats#(active(X))	→	nats#(X)	(86)
  nats#(mark(X))	→	nats#(X)	(85)

  1.1.6 Usable Rules Processor

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

  There are no rules.

  1.1.6.1 Innermost Lhss Removal Processor

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

  active(filter(cons(x0,x1),0,x2))

  active(filter(cons(x0,x1),s(x2),x3))

  active(sieve(cons(0,x0)))

  active(sieve(cons(s(x0),x1)))

  active(nats(x0))

  active(zprimes)

  mark(filter(x0,x1,x2))

  mark(cons(x0,x1))

  mark(0)

  mark(s(x0))

  mark(sieve(x0))

  mark(nats(x0))

  mark(zprimes)

  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.

  nats#(active(X))	→	nats#(X)	(86)
  1	>	1
  nats#(mark(X))	→	nats#(X)	(85)
  1	>	1

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