Certification Problem

Input (TPDB TRS_Innermost/raML/queue.raml)

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

  breadth#1#(tuple#2(@queue',@elem))	→	breadth#2#(@elem,@queue')	(38)
  breadth#2#(::(@z,@_@9),@queue')	→	breadth#3#(breadth#4(@z),@queue')	(39)
  breadth#3#(tuple#2(@x,@ys),@queue')	→	breadth#5#(enqueues(@ys,@queue'))	(41)
  breadth#5#(tuple#2(@breadth@7,@breadth@8))	→	breadth#(@breadth@7,@breadth@8)	(44)
  breadth#(@breadth@1,@breadth@2)	→	breadth#1#(dequeue(@breadth@1,@breadth@2))	(36)

  1.1.1 Usable Rules Processor

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

  dequeue#1(tuple#2(@inq,@outq))	→	dequeue#2(@outq,@inq)	(20)
  dequeue#2(nil,@inq)	→	dequeue#3(@inq)	(22)
  dequeue#3(::(@x,@xs))	→	dequeue#4(copyover(::(@x,@xs),nil))	(23)
  dequeue#4(tuple#2(@dequeue@3,@dequeue@4))	→	dequeue(@dequeue@3,@dequeue@4)	(25)
  dequeue(@dequeue@1,@dequeue@2)	→	dequeue#1(tuple#2(@dequeue@1,@dequeue@2))	(19)
  dequeue#2(::(@y,@ys),@inq)	→	tuple#2(tuple#2(@inq,@ys),::(@y,nil))	(21)
  dequeue#3(nil)	→	tuple#2(tuple#2(nil,nil),nil)	(24)
  copyover#1(tuple#2(@inq,@outq))	→	copyover#2(@inq,@outq)	(16)
  copyover#2(::(@x,@xs),@outq)	→	copyover(@xs,::(@x,@outq))	(17)
  copyover(@copyover@1,@copyover@2)	→	copyover#1(tuple#2(@copyover@1,@copyover@2))	(15)
  copyover#2(nil,@outq)	→	tuple#2(nil,@outq)	(18)
  enqueues#1(::(@x,@xs),@queue)	→	enqueues(@xs,enqueue(@x,@queue))	(30)
  enqueues(@l,@queue)	→	enqueues#1(@l,@queue)	(29)
  enqueue(@x,@queue)	→	enqueue#1(@queue,@x)	(27)
  enqueues#1(nil,@queue)	→	@queue	(31)
  enqueue#1(tuple#2(@inq,@outq),@x)	→	tuple#2(::(@x,@inq),@outq)	(28)
  breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6))	→	children(@children@3,@children@4,@children@5,@children@6)	(6)
  children(@a,@b,@l1,@l2)	→	tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))	(8)
  children#1(::(@x,@xs),@b,@l2)	→	children#3(@l2,@b,@x,@xs)	(9)
  children#1(nil,@b,@l2)	→	children#2(@l2,@b)	(10)
  children#2(::(@y,@ys),@b)	→	::(tuple#4(@y,@b,nil,@ys),nil)	(11)
  children#2(nil,@b)	→	nil	(12)
  children#3(::(@y,@ys),@b,@x,@xs)	→	::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))	(13)
  children#3(nil,@b,@x,@xs)	→	nil	(14)

  1.1.1.1 Innermost Lhss Removal Processor

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

  breadth#4(tuple#4(x0,x1,x2,x3))

  children(x0,x1,x2,x3)

  children#1(::(x0,x1),x2,x3)

  children#1(nil,x0,x1)

  children#2(::(x0,x1),x2)

  children#2(nil,x0)

  children#3(::(x0,x1),x2,x3,x4)

  children#3(nil,x0,x1,x2)

  copyover(x0,x1)

  copyover#1(tuple#2(x0,x1))

  copyover#2(::(x0,x1),x2)

  copyover#2(nil,x0)

  dequeue(x0,x1)

  dequeue#1(tuple#2(x0,x1))

  dequeue#2(::(x0,x1),x2)

  dequeue#2(nil,x0)

  dequeue#3(::(x0,x1))

  dequeue#3(nil)

  dequeue#4(tuple#2(x0,x1))

  enqueue(x0,x1)

  enqueue#1(tuple#2(x0,x1),x2)

  enqueues(x0,x1)

  enqueues#1(::(x0,x1),x2)

  enqueues#1(nil,x0)

  1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers

  [breadth#1#(x1)]	=	0 0 + 1 0 0 0 · x1
  [tuple#2(x1, x2)]	=	0 0 + 1 0 0 0 · x1 + 1 0 1 0 · x2
  [breadth#2#(x1, x2)]	=	0 0 + 1 0 0 0 · x1 + 1 0 0 0 · x2
  [::(x1, x2)]	=	1 1 + 0 1 0 0 · x1 + 1 0 1 1 · x2
  [breadth#3#(x1, x2)]	=	0 0 + 0 1 0 0 · x1 + 1 0 0 0 · x2
  [breadth#4(x1)]	=	1 0 + 1 0 0 1 · x1
  [breadth#5#(x1)]	=	0 0 + 1 0 0 0 · x1
  [enqueues(x1, x2)]	=	0 0 + 1 0 0 0 · x1 + 1 0 0 1 · x2
  [breadth#(x1, x2)]	=	0 0 + 1 0 0 0 · x1 + 1 0 0 0 · x2
  [dequeue(x1, x2)]	=	0 0 + 1 0 1 0 · x1 + 1 0 1 0 · x2
  [tuple#4(x1,...,x4)]	=	0 0 + 1 0 0 0 · x1 + 1 0 0 0 · x2 + 0 1 0 1 · x3 + 1 0 1 0 · x4
  [children(x1,...,x4)]	=	1 0 + 1 0 0 0 · x1 + 1 0 0 0 · x2 + 0 1 0 1 · x3 + 1 0 1 0 · x4
  [enqueues#1(x1, x2)]	=	0 0 + 1 0 0 0 · x1 + 1 0 0 1 · x2
  [enqueue(x1, x2)]	=	1 0 + 0 1 0 0 · x1 + 1 0 0 1 · x2
  [dequeue#2(x1, x2)]	=	0 0 + 1 0 1 0 · x1 + 1 0 1 0 · x2
  [nil]	=	0 0
  [dequeue#3(x1)]	=	0 0 + 1 0 1 0 · x1
  [dequeue#4(x1)]	=	0 0 + 1 0 1 0 · x1
  [copyover(x1, x2)]	=	0 0 + 1 0 1 0 · x1 + 1 0 1 0 · x2
  [dequeue#1(x1)]	=	0 0 + 1 0 1 0 · x1
  [copyover#2(x1, x2)]	=	0 0 + 1 0 1 0 · x1 + 1 0 1 0 · x2
  [copyover#1(x1)]	=	0 0 + 1 0 1 0 · x1
  [enqueue#1(x1, x2)]	=	1 0 + 1 0 0 1 · x1 + 0 1 0 0 · x2
  [children#1(x1, x2, x3)]	=	0 1 + 0 1 0 1 · x1 + 0 0 1 1 · x2 + 1 0 1 1 · x3
  [children#3(x1,...,x4)]	=	1 1 + 1 0 1 1 · x1 + 0 0 0 1 · x2 + 0 0 0 0 · x3 + 1 1 1 1 · x4
  [children#2(x1, x2)]	=	0 1 + 1 0 0 0 · x1 + 0 0 1 1 · x2

  the pair

  breadth#2#(::(@z,@_@9),@queue')

  →

  breadth#3#(breadth#4(@z),@queue')

  (39)

  could be deleted.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  dequeue#3#(::(@x,@xs))	→	dequeue#4#(copyover(::(@x,@xs),nil))	(54)
  dequeue#4#(tuple#2(@dequeue@3,@dequeue@4))	→	dequeue#(@dequeue@3,@dequeue@4)	(56)
  dequeue#(@dequeue@1,@dequeue@2)	→	dequeue#1#(tuple#2(@dequeue@1,@dequeue@2))	(51)
  dequeue#1#(tuple#2(@inq,@outq))	→	dequeue#2#(@outq,@inq)	(52)
  dequeue#2#(nil,@inq)	→	dequeue#3#(@inq)	(53)

  1.1.2 Usable Rules Processor

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

  copyover#1(tuple#2(@inq,@outq))	→	copyover#2(@inq,@outq)	(16)
  copyover#2(::(@x,@xs),@outq)	→	copyover(@xs,::(@x,@outq))	(17)
  copyover(@copyover@1,@copyover@2)	→	copyover#1(tuple#2(@copyover@1,@copyover@2))	(15)
  copyover#2(nil,@outq)	→	tuple#2(nil,@outq)	(18)

  1.1.2.1 Innermost Lhss Removal Processor

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

  copyover(x0,x1)

  copyover#1(tuple#2(x0,x1))

  copyover#2(::(x0,x1),x2)

  copyover#2(nil,x0)

  1.1.2.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [copyover#1(x1)]	=	1 · x1
  [tuple#2(x1, x2)]	=	2 · x1 + 1 · x2
  [copyover#2(x1, x2)]	=	2 · x1 + 1 · x2
  [::(x1, x2)]	=	2 + 2 · x1 + 1 · x2
  [copyover(x1, x2)]	=	2 · x1 + 1 · x2
  [nil]	=	0
  [dequeue#3#(x1)]	=	2 · x1
  [dequeue#4#(x1)]	=	1 · x1
  [dequeue#(x1, x2)]	=	2 · x1 + 1 · x2
  [dequeue#1#(x1)]	=	1 · x1
  [dequeue#2#(x1, x2)]	=	1 · x1 + 2 · x2

  the rules

  breadth(@breadth@1,@breadth@2)	→	breadth#1(dequeue(@breadth@1,@breadth@2))	(1)
  breadth#1(tuple#2(@queue',@elem))	→	breadth#2(@elem,@queue')	(2)
  breadth#2(::(@z,@_@9),@queue')	→	breadth#3(breadth#4(@z),@queue')	(3)
  breadth#2(nil,@queue')	→	nil	(4)
  breadth#3(tuple#2(@x,@ys),@queue')	→	::(@x,breadth#5(enqueues(@ys,@queue')))	(5)
  breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6))	→	children(@children@3,@children@4,@children@5,@children@6)	(6)
  breadth#5(tuple#2(@breadth@7,@breadth@8))	→	breadth(@breadth@7,@breadth@8)	(7)
  children(@a,@b,@l1,@l2)	→	tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))	(8)
  children#1(::(@x,@xs),@b,@l2)	→	children#3(@l2,@b,@x,@xs)	(9)
  children#1(nil,@b,@l2)	→	children#2(@l2,@b)	(10)
  children#2(::(@y,@ys),@b)	→	::(tuple#4(@y,@b,nil,@ys),nil)	(11)
  children#2(nil,@b)	→	nil	(12)
  children#3(::(@y,@ys),@b,@x,@xs)	→	::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))	(13)
  children#3(nil,@b,@x,@xs)	→	nil	(14)
  copyover#2(::(@x,@xs),@outq)	→	copyover(@xs,::(@x,@outq))	(17)
  dequeue(@dequeue@1,@dequeue@2)	→	dequeue#1(tuple#2(@dequeue@1,@dequeue@2))	(19)
  dequeue#1(tuple#2(@inq,@outq))	→	dequeue#2(@outq,@inq)	(20)
  dequeue#2(::(@y,@ys),@inq)	→	tuple#2(tuple#2(@inq,@ys),::(@y,nil))	(21)
  dequeue#2(nil,@inq)	→	dequeue#3(@inq)	(22)
  dequeue#3(::(@x,@xs))	→	dequeue#4(copyover(::(@x,@xs),nil))	(23)
  dequeue#3(nil)	→	tuple#2(tuple#2(nil,nil),nil)	(24)
  dequeue#4(tuple#2(@dequeue@3,@dequeue@4))	→	dequeue(@dequeue@3,@dequeue@4)	(25)
  empty(@x)	→	tuple#2(nil,nil)	(26)
  enqueue(@x,@queue)	→	enqueue#1(@queue,@x)	(27)
  enqueue#1(tuple#2(@inq,@outq),@x)	→	tuple#2(::(@x,@inq),@outq)	(28)
  enqueues(@l,@queue)	→	enqueues#1(@l,@queue)	(29)
  enqueues#1(::(@x,@xs),@queue)	→	enqueues(@xs,enqueue(@x,@queue))	(30)
  enqueues#1(nil,@queue)	→	@queue	(31)
  startBreadth(@xs)	→	startBreadth#1(@xs)	(32)
  startBreadth#1(::(@x,@xs))	→	startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))	(33)
  startBreadth#1(nil)	→	nil	(34)
  startBreadth#2(tuple#2(@breadth@1,@breadth@2))	→	breadth(@breadth@1,@breadth@2)	(35)

  could be deleted.

  1.1.2.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [dequeue#4#(x1)]	=	1 + 2 · x1
  [copyover(x1, x2)]	=	1
  [copyover#1(x1)]	=	1
  [tuple#2(x1, x2)]	=	x1
  [copyover#2(x1, x2)]	=	1
  [nil]	=	0
  [dequeue#3#(x1)]	=	2 · x1
  [::(x1, x2)]	=	2
  [dequeue#(x1, x2)]	=	1 + 2 · x1
  [dequeue#1#(x1)]	=	2 · x1
  [dequeue#2#(x1, x2)]	=	2 · x2

  the pairs

  dequeue#3#(::(@x,@xs))	→	dequeue#4#(copyover(::(@x,@xs),nil))	(54)
  dequeue#(@dequeue@1,@dequeue@2)	→	dequeue#1#(tuple#2(@dequeue@1,@dequeue@2))	(51)

  could be deleted.

  1.1.2.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  enqueues#1#(::(@x,@xs),@queue)	→	enqueues#(@xs,enqueue(@x,@queue))	(59)
  enqueues#(@l,@queue)	→	enqueues#1#(@l,@queue)	(58)

  1.1.3 Usable Rules Processor

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

  enqueue(@x,@queue)	→	enqueue#1(@queue,@x)	(27)
  enqueue#1(tuple#2(@inq,@outq),@x)	→	tuple#2(::(@x,@inq),@outq)	(28)

  1.1.3.1 Innermost Lhss Removal Processor

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

  enqueue(x0,x1)

  enqueue#1(tuple#2(x0,x1),x2)

  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.

  enqueues#(@l,@queue)	→	enqueues#1#(@l,@queue)	(58)
  1	≥	1
  2	≥	2
  enqueues#1#(::(@x,@xs),@queue)	→	enqueues#(@xs,enqueue(@x,@queue))	(59)
  1	>	1

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

• The 4^th component contains the pair

  copyover#1#(tuple#2(@inq,@outq))	→	copyover#2#(@inq,@outq)	(49)
  copyover#2#(::(@x,@xs),@outq)	→	copyover#(@xs,::(@x,@outq))	(50)
  copyover#(@copyover@1,@copyover@2)	→	copyover#1#(tuple#2(@copyover@1,@copyover@2))	(48)

  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.

  There are no lhss.

  1.1.4.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [copyover#1#(x1)]	=	1 · x1
  [tuple#2(x1, x2)]	=	2 · x1 + 1 · x2
  [copyover#2#(x1, x2)]	=	2 · x1 + 1 · x2
  [::(x1, x2)]	=	2 + 2 · x1 + 1 · x2
  [copyover#(x1, x2)]	=	1 + 2 · x1 + 1 · x2

  the pairs

  copyover#2#(::(@x,@xs),@outq)	→	copyover#(@xs,::(@x,@outq))	(50)
  copyover#(@copyover@1,@copyover@2)	→	copyover#1#(tuple#2(@copyover@1,@copyover@2))	(48)

  and the rules

  breadth(@breadth@1,@breadth@2)	→	breadth#1(dequeue(@breadth@1,@breadth@2))	(1)
  breadth#1(tuple#2(@queue',@elem))	→	breadth#2(@elem,@queue')	(2)
  breadth#2(::(@z,@_@9),@queue')	→	breadth#3(breadth#4(@z),@queue')	(3)
  breadth#2(nil,@queue')	→	nil	(4)
  breadth#3(tuple#2(@x,@ys),@queue')	→	::(@x,breadth#5(enqueues(@ys,@queue')))	(5)
  breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6))	→	children(@children@3,@children@4,@children@5,@children@6)	(6)
  breadth#5(tuple#2(@breadth@7,@breadth@8))	→	breadth(@breadth@7,@breadth@8)	(7)
  children(@a,@b,@l1,@l2)	→	tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))	(8)
  children#1(::(@x,@xs),@b,@l2)	→	children#3(@l2,@b,@x,@xs)	(9)
  children#1(nil,@b,@l2)	→	children#2(@l2,@b)	(10)
  children#2(::(@y,@ys),@b)	→	::(tuple#4(@y,@b,nil,@ys),nil)	(11)
  children#2(nil,@b)	→	nil	(12)
  children#3(::(@y,@ys),@b,@x,@xs)	→	::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))	(13)
  children#3(nil,@b,@x,@xs)	→	nil	(14)
  copyover(@copyover@1,@copyover@2)	→	copyover#1(tuple#2(@copyover@1,@copyover@2))	(15)
  copyover#1(tuple#2(@inq,@outq))	→	copyover#2(@inq,@outq)	(16)
  copyover#2(::(@x,@xs),@outq)	→	copyover(@xs,::(@x,@outq))	(17)
  copyover#2(nil,@outq)	→	tuple#2(nil,@outq)	(18)
  dequeue(@dequeue@1,@dequeue@2)	→	dequeue#1(tuple#2(@dequeue@1,@dequeue@2))	(19)
  dequeue#1(tuple#2(@inq,@outq))	→	dequeue#2(@outq,@inq)	(20)
  dequeue#2(::(@y,@ys),@inq)	→	tuple#2(tuple#2(@inq,@ys),::(@y,nil))	(21)
  dequeue#2(nil,@inq)	→	dequeue#3(@inq)	(22)
  dequeue#3(::(@x,@xs))	→	dequeue#4(copyover(::(@x,@xs),nil))	(23)
  dequeue#3(nil)	→	tuple#2(tuple#2(nil,nil),nil)	(24)
  dequeue#4(tuple#2(@dequeue@3,@dequeue@4))	→	dequeue(@dequeue@3,@dequeue@4)	(25)
  empty(@x)	→	tuple#2(nil,nil)	(26)
  enqueue(@x,@queue)	→	enqueue#1(@queue,@x)	(27)
  enqueue#1(tuple#2(@inq,@outq),@x)	→	tuple#2(::(@x,@inq),@outq)	(28)
  enqueues(@l,@queue)	→	enqueues#1(@l,@queue)	(29)
  enqueues#1(::(@x,@xs),@queue)	→	enqueues(@xs,enqueue(@x,@queue))	(30)
  enqueues#1(nil,@queue)	→	@queue	(31)
  startBreadth(@xs)	→	startBreadth#1(@xs)	(32)
  startBreadth#1(::(@x,@xs))	→	startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))	(33)
  startBreadth#1(nil)	→	nil	(34)
  startBreadth#2(tuple#2(@breadth@1,@breadth@2))	→	breadth(@breadth@1,@breadth@2)	(35)

  could be deleted.

  1.1.4.1.1.1 Size-Change Termination

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

  copyover#1#(tuple#2(@inq,@outq))	→	copyover#2#(@inq,@outq)	(49)
  1	>	1
  1	>	2

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