/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR @_@9 @a @b @breadth@1 @breadth@2 @breadth@7 @breadth@8 @children@3 @children@4 @children@5 @children@6 @copyover@1 @copyover@2 @dequeue@1 @dequeue@2 @dequeue@3 @dequeue@4 @elem @inq @l @l1 @l2 @outq @queue @queue' @x @xs @y @ys @z)
(RULES
breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
breadth#2(nil,@queue') -> nil
breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
children#1(nil,@b,@l2) -> children#2(@l2,@b)
children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
children#2(nil,@b) -> nil
children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
children#3(nil,@b,@x,@xs) -> nil
copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
copyover#2(nil,@outq) -> tuple#2(nil,@outq)
dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
dequeue#2(nil,@inq) -> dequeue#3(@inq)
dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
empty(@x) -> tuple#2(nil,nil)
enqueue(@x,@queue) -> enqueue#1(@queue,@x)
enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
enqueues(@l,@queue) -> enqueues#1(@l,@queue)
enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
enqueues#1(nil,@queue) -> @queue
startBreadth(@xs) -> startBreadth#1(@xs)
startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
startBreadth#1(nil) -> nil
startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 BREADTH(@breadth@1,@breadth@2) -> BREADTH#1(dequeue(@breadth@1,@breadth@2))
 BREADTH(@breadth@1,@breadth@2) -> DEQUEUE(@breadth@1,@breadth@2)
 BREADTH#1(tuple#2(@queue',@elem)) -> BREADTH#2(@elem,@queue')
 BREADTH#2(::(@z,@_@9),@queue') -> BREADTH#3(breadth#4(@z),@queue')
 BREADTH#2(::(@z,@_@9),@queue') -> BREADTH#4(@z)
 BREADTH#3(tuple#2(@x,@ys),@queue') -> BREADTH#5(enqueues(@ys,@queue'))
 BREADTH#3(tuple#2(@x,@ys),@queue') -> ENQUEUES(@ys,@queue')
 BREADTH#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> CHILDREN(@children@3,@children@4,@children@5,@children@6)
 BREADTH#5(tuple#2(@breadth@7,@breadth@8)) -> BREADTH(@breadth@7,@breadth@8)
 CHILDREN(@a,@b,@l1,@l2) -> CHILDREN#1(@l1,@b,@l2)
 CHILDREN#1(::(@x,@xs),@b,@l2) -> CHILDREN#3(@l2,@b,@x,@xs)
 CHILDREN#1(nil,@b,@l2) -> CHILDREN#2(@l2,@b)
 COPYOVER(@copyover@1,@copyover@2) -> COPYOVER#1(tuple#2(@copyover@1,@copyover@2))
 COPYOVER#1(tuple#2(@inq,@outq)) -> COPYOVER#2(@inq,@outq)
 COPYOVER#2(::(@x,@xs),@outq) -> COPYOVER(@xs,::(@x,@outq))
 DEQUEUE(@dequeue@1,@dequeue@2) -> DEQUEUE#1(tuple#2(@dequeue@1,@dequeue@2))
 DEQUEUE#1(tuple#2(@inq,@outq)) -> DEQUEUE#2(@outq,@inq)
 DEQUEUE#2(nil,@inq) -> DEQUEUE#3(@inq)
 DEQUEUE#3(::(@x,@xs)) -> COPYOVER(::(@x,@xs),nil)
 DEQUEUE#3(::(@x,@xs)) -> DEQUEUE#4(copyover(::(@x,@xs),nil))
 DEQUEUE#4(tuple#2(@dequeue@3,@dequeue@4)) -> DEQUEUE(@dequeue@3,@dequeue@4)
 ENQUEUE(@x,@queue) -> ENQUEUE#1(@queue,@x)
 ENQUEUES(@l,@queue) -> ENQUEUES#1(@l,@queue)
 ENQUEUES#1(::(@x,@xs),@queue) -> ENQUEUE(@x,@queue)
 ENQUEUES#1(::(@x,@xs),@queue) -> ENQUEUES(@xs,enqueue(@x,@queue))
 STARTBREADTH(@xs) -> STARTBREADTH#1(@xs)
 STARTBREADTH#1(::(@x,@xs)) -> EMPTY(#unit)
 STARTBREADTH#1(::(@x,@xs)) -> ENQUEUE(tuple#4(@x,@x,@xs,@xs),empty(#unit))
 STARTBREADTH#1(::(@x,@xs)) -> STARTBREADTH#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 STARTBREADTH#2(tuple#2(@breadth@1,@breadth@2)) -> BREADTH(@breadth@1,@breadth@2)
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)

Problem 1: 

SCC Processor:
-> Pairs:
 BREADTH(@breadth@1,@breadth@2) -> BREADTH#1(dequeue(@breadth@1,@breadth@2))
 BREADTH(@breadth@1,@breadth@2) -> DEQUEUE(@breadth@1,@breadth@2)
 BREADTH#1(tuple#2(@queue',@elem)) -> BREADTH#2(@elem,@queue')
 BREADTH#2(::(@z,@_@9),@queue') -> BREADTH#3(breadth#4(@z),@queue')
 BREADTH#2(::(@z,@_@9),@queue') -> BREADTH#4(@z)
 BREADTH#3(tuple#2(@x,@ys),@queue') -> BREADTH#5(enqueues(@ys,@queue'))
 BREADTH#3(tuple#2(@x,@ys),@queue') -> ENQUEUES(@ys,@queue')
 BREADTH#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> CHILDREN(@children@3,@children@4,@children@5,@children@6)
 BREADTH#5(tuple#2(@breadth@7,@breadth@8)) -> BREADTH(@breadth@7,@breadth@8)
 CHILDREN(@a,@b,@l1,@l2) -> CHILDREN#1(@l1,@b,@l2)
 CHILDREN#1(::(@x,@xs),@b,@l2) -> CHILDREN#3(@l2,@b,@x,@xs)
 CHILDREN#1(nil,@b,@l2) -> CHILDREN#2(@l2,@b)
 COPYOVER(@copyover@1,@copyover@2) -> COPYOVER#1(tuple#2(@copyover@1,@copyover@2))
 COPYOVER#1(tuple#2(@inq,@outq)) -> COPYOVER#2(@inq,@outq)
 COPYOVER#2(::(@x,@xs),@outq) -> COPYOVER(@xs,::(@x,@outq))
 DEQUEUE(@dequeue@1,@dequeue@2) -> DEQUEUE#1(tuple#2(@dequeue@1,@dequeue@2))
 DEQUEUE#1(tuple#2(@inq,@outq)) -> DEQUEUE#2(@outq,@inq)
 DEQUEUE#2(nil,@inq) -> DEQUEUE#3(@inq)
 DEQUEUE#3(::(@x,@xs)) -> COPYOVER(::(@x,@xs),nil)
 DEQUEUE#3(::(@x,@xs)) -> DEQUEUE#4(copyover(::(@x,@xs),nil))
 DEQUEUE#4(tuple#2(@dequeue@3,@dequeue@4)) -> DEQUEUE(@dequeue@3,@dequeue@4)
 ENQUEUE(@x,@queue) -> ENQUEUE#1(@queue,@x)
 ENQUEUES(@l,@queue) -> ENQUEUES#1(@l,@queue)
 ENQUEUES#1(::(@x,@xs),@queue) -> ENQUEUE(@x,@queue)
 ENQUEUES#1(::(@x,@xs),@queue) -> ENQUEUES(@xs,enqueue(@x,@queue))
 STARTBREADTH(@xs) -> STARTBREADTH#1(@xs)
 STARTBREADTH#1(::(@x,@xs)) -> EMPTY(#unit)
 STARTBREADTH#1(::(@x,@xs)) -> ENQUEUE(tuple#4(@x,@x,@xs,@xs),empty(#unit))
 STARTBREADTH#1(::(@x,@xs)) -> STARTBREADTH#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 STARTBREADTH#2(tuple#2(@breadth@1,@breadth@2)) -> BREADTH(@breadth@1,@breadth@2)
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ENQUEUES(@l,@queue) -> ENQUEUES#1(@l,@queue)
 ENQUEUES#1(::(@x,@xs),@queue) -> ENQUEUES(@xs,enqueue(@x,@queue))
->->-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
->->Cycle:
->->-> Pairs:
 COPYOVER(@copyover@1,@copyover@2) -> COPYOVER#1(tuple#2(@copyover@1,@copyover@2))
 COPYOVER#1(tuple#2(@inq,@outq)) -> COPYOVER#2(@inq,@outq)
 COPYOVER#2(::(@x,@xs),@outq) -> COPYOVER(@xs,::(@x,@outq))
->->-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
->->Cycle:
->->-> Pairs:
 DEQUEUE(@dequeue@1,@dequeue@2) -> DEQUEUE#1(tuple#2(@dequeue@1,@dequeue@2))
 DEQUEUE#1(tuple#2(@inq,@outq)) -> DEQUEUE#2(@outq,@inq)
 DEQUEUE#2(nil,@inq) -> DEQUEUE#3(@inq)
 DEQUEUE#3(::(@x,@xs)) -> DEQUEUE#4(copyover(::(@x,@xs),nil))
 DEQUEUE#4(tuple#2(@dequeue@3,@dequeue@4)) -> DEQUEUE(@dequeue@3,@dequeue@4)
->->-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
->->Cycle:
->->-> Pairs:
 BREADTH(@breadth@1,@breadth@2) -> BREADTH#1(dequeue(@breadth@1,@breadth@2))
 BREADTH#1(tuple#2(@queue',@elem)) -> BREADTH#2(@elem,@queue')
 BREADTH#2(::(@z,@_@9),@queue') -> BREADTH#3(breadth#4(@z),@queue')
 BREADTH#3(tuple#2(@x,@ys),@queue') -> BREADTH#5(enqueues(@ys,@queue'))
 BREADTH#5(tuple#2(@breadth@7,@breadth@8)) -> BREADTH(@breadth@7,@breadth@8)
->->-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ENQUEUES(@l,@queue) -> ENQUEUES#1(@l,@queue)
 ENQUEUES#1(::(@x,@xs),@queue) -> ENQUEUES(@xs,enqueue(@x,@queue))
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
->Projection:
 pi(ENQUEUES) = 1
 pi(ENQUEUES#1) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 ENQUEUES(@l,@queue) -> ENQUEUES#1(@l,@queue)
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 COPYOVER(@copyover@1,@copyover@2) -> COPYOVER#1(tuple#2(@copyover@1,@copyover@2))
 COPYOVER#1(tuple#2(@inq,@outq)) -> COPYOVER#2(@inq,@outq)
 COPYOVER#2(::(@x,@xs),@outq) -> COPYOVER(@xs,::(@x,@outq))
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[::](X1,X2) = X1 + 2.X2 + 2
[tuple#2](X1,X2) = X1
[COPYOVER](X1,X2) = 2.X1 + 1
[COPYOVER#1](X) = 2.X
[COPYOVER#2](X1,X2) = 2.X1

Problem 1.2: 

SCC Processor:
-> Pairs:
 COPYOVER#1(tuple#2(@inq,@outq)) -> COPYOVER#2(@inq,@outq)
 COPYOVER#2(::(@x,@xs),@outq) -> COPYOVER(@xs,::(@x,@outq))
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 DEQUEUE(@dequeue@1,@dequeue@2) -> DEQUEUE#1(tuple#2(@dequeue@1,@dequeue@2))
 DEQUEUE#1(tuple#2(@inq,@outq)) -> DEQUEUE#2(@outq,@inq)
 DEQUEUE#2(nil,@inq) -> DEQUEUE#3(@inq)
 DEQUEUE#3(::(@x,@xs)) -> DEQUEUE#4(copyover(::(@x,@xs),nil))
 DEQUEUE#4(tuple#2(@dequeue@3,@dequeue@4)) -> DEQUEUE(@dequeue@3,@dequeue@4)
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
-> Usable rules:
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[copyover](X1,X2) = 1
[copyover#1](X) = 1
[copyover#2](X1,X2) = 1
[::](X1,X2) = 2.X1 + 2
[nil] = 0
[tuple#2](X1,X2) = 2.X1 + 1
[DEQUEUE](X1,X2) = 2.X1 + 2
[DEQUEUE#1](X) = X
[DEQUEUE#2](X1,X2) = 2.X2 + 1
[DEQUEUE#3](X) = 2.X + 1
[DEQUEUE#4](X) = 2.X

Problem 1.3: 

SCC Processor:
-> Pairs:
 DEQUEUE#1(tuple#2(@inq,@outq)) -> DEQUEUE#2(@outq,@inq)
 DEQUEUE#2(nil,@inq) -> DEQUEUE#3(@inq)
 DEQUEUE#3(::(@x,@xs)) -> DEQUEUE#4(copyover(::(@x,@xs),nil))
 DEQUEUE#4(tuple#2(@dequeue@3,@dequeue@4)) -> DEQUEUE(@dequeue@3,@dequeue@4)
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 BREADTH(@breadth@1,@breadth@2) -> BREADTH#1(dequeue(@breadth@1,@breadth@2))
 BREADTH#1(tuple#2(@queue',@elem)) -> BREADTH#2(@elem,@queue')
 BREADTH#2(::(@z,@_@9),@queue') -> BREADTH#3(breadth#4(@z),@queue')
 BREADTH#3(tuple#2(@x,@ys),@queue') -> BREADTH#5(enqueues(@ys,@queue'))
 BREADTH#5(tuple#2(@breadth@7,@breadth@8)) -> BREADTH(@breadth@7,@breadth@8)
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
-> Usable rules:
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[breadth#4](X) = [1 0;0 1].X + [1;0]
[children](X1,X2,X3,X4) = [1 1;0 0].X1 + [1 1;0 1].X2 + [1 1;0 1].X3 + [1 1;1 0].X4 + [1;0]
[children#1](X1,X2,X3) = [0 1;1 1].X1 + [0 1;1 1].X2 + [1 0;1 1].X3 + [0;1]
[children#2](X1,X2) = [1 0;1 1].X1 + [0 1;1 1].X2 + [0;1]
[children#3](X1,X2,X3,X4) = [1 0;1 1].X1 + [0 1;1 1].X2 + [0 0;0 1].X3 + [1 1;0 1].X4 + [1;1]
[copyover](X1,X2) = [1 0;1 0].X1 + [1 0;1 0].X2
[copyover#1](X) = [1 0;1 0].X
[copyover#2](X1,X2) = [1 0;1 0].X1 + [1 0;1 0].X2
[dequeue](X1,X2) = [1 0;1 0].X1 + [1 0;1 0].X2 + [0;1]
[dequeue#1](X) = [1 0;1 0].X + [0;1]
[dequeue#2](X1,X2) = [1 0;1 0].X1 + [1 0;1 0].X2 + [0;1]
[dequeue#3](X) = [1 0;1 0].X + [0;1]
[dequeue#4](X) = [1 0;1 0].X + [0;1]
[enqueue](X1,X2) = [0 1;0 0].X1 + [1 0;0 1].X2 + [1;1]
[enqueue#1](X1,X2) = [1 0;0 1].X1 + [0 1;0 0].X2 + [1;0]
[enqueues](X1,X2) = [1 0;0 1].X1 + [1 0;0 1].X2
[enqueues#1](X1,X2) = [1 0;0 1].X1 + [1 0;0 1].X2
[::](X1,X2) = [0 1;0 0].X1 + [1 0;1 1].X2 + [1;1]
[nil] = 0
[tuple#2](X1,X2) = [1 0;0 0].X1 + [1 0;1 0].X2
[tuple#4](X1,X2,X3,X4) = [1 1;0 0].X1 + [1 1;0 1].X2 + [1 1;0 1].X3 + [1 1;1 0].X4
[BREADTH](X1,X2) = [1 0;1 0].X1 + [1 0;1 0].X2 + [1;1]
[BREADTH#1](X) = [1 0;1 0].X + [0;1]
[BREADTH#2](X1,X2) = [1 0;1 0].X1 + [1 0;1 0].X2 + [0;1]
[BREADTH#3](X1,X2) = [0 1;0 1].X1 + [1 0;1 0].X2 + [1;1]
[BREADTH#5](X) = [1 0;1 0].X + [1;1]

Problem 1.4: 

SCC Processor:
-> Pairs:
 BREADTH#1(tuple#2(@queue',@elem)) -> BREADTH#2(@elem,@queue')
 BREADTH#2(::(@z,@_@9),@queue') -> BREADTH#3(breadth#4(@z),@queue')
 BREADTH#3(tuple#2(@x,@ys),@queue') -> BREADTH#5(enqueues(@ys,@queue'))
 BREADTH#5(tuple#2(@breadth@7,@breadth@8)) -> BREADTH(@breadth@7,@breadth@8)
-> Rules:
 breadth(@breadth@1,@breadth@2) -> breadth#1(dequeue(@breadth@1,@breadth@2))
 breadth#1(tuple#2(@queue',@elem)) -> breadth#2(@elem,@queue')
 breadth#2(::(@z,@_@9),@queue') -> breadth#3(breadth#4(@z),@queue')
 breadth#2(nil,@queue') -> nil
 breadth#3(tuple#2(@x,@ys),@queue') -> ::(@x,breadth#5(enqueues(@ys,@queue')))
 breadth#4(tuple#4(@children@3,@children@4,@children@5,@children@6)) -> children(@children@3,@children@4,@children@5,@children@6)
 breadth#5(tuple#2(@breadth@7,@breadth@8)) -> breadth(@breadth@7,@breadth@8)
 children(@a,@b,@l1,@l2) -> tuple#2(tuple#2(@a,@b),children#1(@l1,@b,@l2))
 children#1(::(@x,@xs),@b,@l2) -> children#3(@l2,@b,@x,@xs)
 children#1(nil,@b,@l2) -> children#2(@l2,@b)
 children#2(::(@y,@ys),@b) -> ::(tuple#4(@y,@b,nil,@ys),nil)
 children#2(nil,@b) -> nil
 children#3(::(@y,@ys),@b,@x,@xs) -> ::(tuple#4(@x,@b,nil,@xs),::(tuple#4(@x,@y,@xs,@ys),nil))
 children#3(nil,@b,@x,@xs) -> nil
 copyover(@copyover@1,@copyover@2) -> copyover#1(tuple#2(@copyover@1,@copyover@2))
 copyover#1(tuple#2(@inq,@outq)) -> copyover#2(@inq,@outq)
 copyover#2(::(@x,@xs),@outq) -> copyover(@xs,::(@x,@outq))
 copyover#2(nil,@outq) -> tuple#2(nil,@outq)
 dequeue(@dequeue@1,@dequeue@2) -> dequeue#1(tuple#2(@dequeue@1,@dequeue@2))
 dequeue#1(tuple#2(@inq,@outq)) -> dequeue#2(@outq,@inq)
 dequeue#2(::(@y,@ys),@inq) -> tuple#2(tuple#2(@inq,@ys),::(@y,nil))
 dequeue#2(nil,@inq) -> dequeue#3(@inq)
 dequeue#3(::(@x,@xs)) -> dequeue#4(copyover(::(@x,@xs),nil))
 dequeue#3(nil) -> tuple#2(tuple#2(nil,nil),nil)
 dequeue#4(tuple#2(@dequeue@3,@dequeue@4)) -> dequeue(@dequeue@3,@dequeue@4)
 empty(@x) -> tuple#2(nil,nil)
 enqueue(@x,@queue) -> enqueue#1(@queue,@x)
 enqueue#1(tuple#2(@inq,@outq),@x) -> tuple#2(::(@x,@inq),@outq)
 enqueues(@l,@queue) -> enqueues#1(@l,@queue)
 enqueues#1(::(@x,@xs),@queue) -> enqueues(@xs,enqueue(@x,@queue))
 enqueues#1(nil,@queue) -> @queue
 startBreadth(@xs) -> startBreadth#1(@xs)
 startBreadth#1(::(@x,@xs)) -> startBreadth#2(enqueue(tuple#4(@x,@x,@xs,@xs),empty(#unit)))
 startBreadth#1(nil) -> nil
 startBreadth#2(tuple#2(@breadth@1,@breadth@2)) -> breadth(@breadth@1,@breadth@2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.