/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem:
 +(x,0()) -> x
 +(0(),x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 +(+(x,y),z) -> +(x,+(y,z))
 *(x,0()) -> 0()
 *(0(),x) -> 0()
 *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
 *(*(x,y),z) -> *(x,*(y,z))
 app(nil(),l) -> l
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 sum(nil()) -> 0()
 sum(cons(x,l)) -> +(x,sum(l))
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 prod(nil()) -> s(0())
 prod(cons(x,l)) -> *(x,prod(l))
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))

Proof:
 DP Processor:
  DPs:
   +#(s(x),s(y)) -> +#(x,y)
   +#(+(x,y),z) -> +#(y,z)
   +#(+(x,y),z) -> +#(x,+(y,z))
   *#(s(x),s(y)) -> +#(x,y)
   *#(s(x),s(y)) -> *#(x,y)
   *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
   *#(*(x,y),z) -> *#(y,z)
   *#(*(x,y),z) -> *#(x,*(y,z))
   app#(cons(x,l1),l2) -> app#(l1,l2)
   sum#(cons(x,l)) -> sum#(l)
   sum#(cons(x,l)) -> +#(x,sum(l))
   sum#(app(l1,l2)) -> sum#(l2)
   sum#(app(l1,l2)) -> sum#(l1)
   sum#(app(l1,l2)) -> +#(sum(l1),sum(l2))
   prod#(cons(x,l)) -> prod#(l)
   prod#(cons(x,l)) -> *#(x,prod(l))
   prod#(app(l1,l2)) -> prod#(l2)
   prod#(app(l1,l2)) -> prod#(l1)
   prod#(app(l1,l2)) -> *#(prod(l1),prod(l2))
  TRS:
   +(x,0()) -> x
   +(0(),x) -> x
   +(s(x),s(y)) -> s(s(+(x,y)))
   +(+(x,y),z) -> +(x,+(y,z))
   *(x,0()) -> 0()
   *(0(),x) -> 0()
   *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
   *(*(x,y),z) -> *(x,*(y,z))
   app(nil(),l) -> l
   app(cons(x,l1),l2) -> cons(x,app(l1,l2))
   sum(nil()) -> 0()
   sum(cons(x,l)) -> +(x,sum(l))
   sum(app(l1,l2)) -> +(sum(l1),sum(l2))
   prod(nil()) -> s(0())
   prod(cons(x,l)) -> *(x,prod(l))
   prod(app(l1,l2)) -> *(prod(l1),prod(l2))
  TDG Processor:
   DPs:
    +#(s(x),s(y)) -> +#(x,y)
    +#(+(x,y),z) -> +#(y,z)
    +#(+(x,y),z) -> +#(x,+(y,z))
    *#(s(x),s(y)) -> +#(x,y)
    *#(s(x),s(y)) -> *#(x,y)
    *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
    *#(*(x,y),z) -> *#(y,z)
    *#(*(x,y),z) -> *#(x,*(y,z))
    app#(cons(x,l1),l2) -> app#(l1,l2)
    sum#(cons(x,l)) -> sum#(l)
    sum#(cons(x,l)) -> +#(x,sum(l))
    sum#(app(l1,l2)) -> sum#(l2)
    sum#(app(l1,l2)) -> sum#(l1)
    sum#(app(l1,l2)) -> +#(sum(l1),sum(l2))
    prod#(cons(x,l)) -> prod#(l)
    prod#(cons(x,l)) -> *#(x,prod(l))
    prod#(app(l1,l2)) -> prod#(l2)
    prod#(app(l1,l2)) -> prod#(l1)
    prod#(app(l1,l2)) -> *#(prod(l1),prod(l2))
   TRS:
    +(x,0()) -> x
    +(0(),x) -> x
    +(s(x),s(y)) -> s(s(+(x,y)))
    +(+(x,y),z) -> +(x,+(y,z))
    *(x,0()) -> 0()
    *(0(),x) -> 0()
    *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
    *(*(x,y),z) -> *(x,*(y,z))
    app(nil(),l) -> l
    app(cons(x,l1),l2) -> cons(x,app(l1,l2))
    sum(nil()) -> 0()
    sum(cons(x,l)) -> +(x,sum(l))
    sum(app(l1,l2)) -> +(sum(l1),sum(l2))
    prod(nil()) -> s(0())
    prod(cons(x,l)) -> *(x,prod(l))
    prod(app(l1,l2)) -> *(prod(l1),prod(l2))
   graph:
    prod#(cons(x,l)) -> prod#(l) ->
    prod#(app(l1,l2)) -> *#(prod(l1),prod(l2))
    prod#(cons(x,l)) -> prod#(l) -> prod#(app(l1,l2)) -> prod#(l1)
    prod#(cons(x,l)) -> prod#(l) -> prod#(app(l1,l2)) -> prod#(l2)
    prod#(cons(x,l)) -> prod#(l) -> prod#(cons(x,l)) -> *#(x,prod(l))
    prod#(cons(x,l)) -> prod#(l) -> prod#(cons(x,l)) -> prod#(l)
    prod#(cons(x,l)) -> *#(x,prod(l)) ->
    *#(*(x,y),z) -> *#(x,*(y,z))
    prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(*(x,y),z) -> *#(y,z)
    prod#(cons(x,l)) -> *#(x,prod(l)) ->
    *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
    prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(s(x),s(y)) -> *#(x,y)
    prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(s(x),s(y)) -> +#(x,y)
    prod#(app(l1,l2)) -> prod#(l2) ->
    prod#(app(l1,l2)) -> *#(prod(l1),prod(l2))
    prod#(app(l1,l2)) -> prod#(l2) -> prod#(app(l1,l2)) -> prod#(l1)
    prod#(app(l1,l2)) -> prod#(l2) -> prod#(app(l1,l2)) -> prod#(l2)
    prod#(app(l1,l2)) -> prod#(l2) ->
    prod#(cons(x,l)) -> *#(x,prod(l))
    prod#(app(l1,l2)) -> prod#(l2) -> prod#(cons(x,l)) -> prod#(l)
    prod#(app(l1,l2)) -> prod#(l1) ->
    prod#(app(l1,l2)) -> *#(prod(l1),prod(l2))
    prod#(app(l1,l2)) -> prod#(l1) -> prod#(app(l1,l2)) -> prod#(l1)
    prod#(app(l1,l2)) -> prod#(l1) -> prod#(app(l1,l2)) -> prod#(l2)
    prod#(app(l1,l2)) -> prod#(l1) ->
    prod#(cons(x,l)) -> *#(x,prod(l))
    prod#(app(l1,l2)) -> prod#(l1) ->
    prod#(cons(x,l)) -> prod#(l)
    prod#(app(l1,l2)) -> *#(prod(l1),prod(l2)) ->
    *#(*(x,y),z) -> *#(x,*(y,z))
    prod#(app(l1,l2)) -> *#(prod(l1),prod(l2)) ->
    *#(*(x,y),z) -> *#(y,z)
    prod#(app(l1,l2)) -> *#(prod(l1),prod(l2)) ->
    *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
    prod#(app(l1,l2)) -> *#(prod(l1),prod(l2)) ->
    *#(s(x),s(y)) -> *#(x,y)
    prod#(app(l1,l2)) -> *#(prod(l1),prod(l2)) ->
    *#(s(x),s(y)) -> +#(x,y)
    sum#(cons(x,l)) -> sum#(l) -> sum#(app(l1,l2)) -> +#(sum(l1),sum(l2))
    sum#(cons(x,l)) -> sum#(l) -> sum#(app(l1,l2)) -> sum#(l1)
    sum#(cons(x,l)) -> sum#(l) -> sum#(app(l1,l2)) -> sum#(l2)
    sum#(cons(x,l)) -> sum#(l) -> sum#(cons(x,l)) -> +#(x,sum(l))
    sum#(cons(x,l)) -> sum#(l) -> sum#(cons(x,l)) -> sum#(l)
    sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(+(x,y),z) -> +#(x,+(y,z))
    sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(+(x,y),z) -> +#(y,z)
    sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(s(x),s(y)) -> +#(x,y)
    sum#(app(l1,l2)) -> sum#(l2) ->
    sum#(app(l1,l2)) -> +#(sum(l1),sum(l2))
    sum#(app(l1,l2)) -> sum#(l2) -> sum#(app(l1,l2)) -> sum#(l1)
    sum#(app(l1,l2)) -> sum#(l2) -> sum#(app(l1,l2)) -> sum#(l2)
    sum#(app(l1,l2)) -> sum#(l2) -> sum#(cons(x,l)) -> +#(x,sum(l))
    sum#(app(l1,l2)) -> sum#(l2) -> sum#(cons(x,l)) -> sum#(l)
    sum#(app(l1,l2)) -> sum#(l1) ->
    sum#(app(l1,l2)) -> +#(sum(l1),sum(l2))
    sum#(app(l1,l2)) -> sum#(l1) -> sum#(app(l1,l2)) -> sum#(l1)
    sum#(app(l1,l2)) -> sum#(l1) -> sum#(app(l1,l2)) -> sum#(l2)
    sum#(app(l1,l2)) -> sum#(l1) -> sum#(cons(x,l)) -> +#(x,sum(l))
    sum#(app(l1,l2)) -> sum#(l1) ->
    sum#(cons(x,l)) -> sum#(l)
    sum#(app(l1,l2)) -> +#(sum(l1),sum(l2)) ->
    +#(+(x,y),z) -> +#(x,+(y,z))
    sum#(app(l1,l2)) -> +#(sum(l1),sum(l2)) ->
    +#(+(x,y),z) -> +#(y,z)
    sum#(app(l1,l2)) -> +#(sum(l1),sum(l2)) ->
    +#(s(x),s(y)) -> +#(x,y)
    app#(cons(x,l1),l2) -> app#(l1,l2) -> app#(cons(x,l1),l2) -> app#(l1,l2)
    *#(*(x,y),z) -> *#(y,z) -> *#(*(x,y),z) -> *#(x,*(y,z))
    *#(*(x,y),z) -> *#(y,z) -> *#(*(x,y),z) -> *#(y,z)
    *#(*(x,y),z) -> *#(y,z) -> *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
    *#(*(x,y),z) -> *#(y,z) -> *#(s(x),s(y)) -> *#(x,y)
    *#(*(x,y),z) -> *#(y,z) -> *#(s(x),s(y)) -> +#(x,y)
    *#(*(x,y),z) -> *#(x,*(y,z)) -> *#(*(x,y),z) -> *#(x,*(y,z))
    *#(*(x,y),z) -> *#(x,*(y,z)) -> *#(*(x,y),z) -> *#(y,z)
    *#(*(x,y),z) -> *#(x,*(y,z)) -> *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
    *#(*(x,y),z) -> *#(x,*(y,z)) -> *#(s(x),s(y)) -> *#(x,y)
    *#(*(x,y),z) -> *#(x,*(y,z)) -> *#(s(x),s(y)) -> +#(x,y)
    *#(s(x),s(y)) -> *#(x,y) -> *#(*(x,y),z) -> *#(x,*(y,z))
    *#(s(x),s(y)) -> *#(x,y) -> *#(*(x,y),z) -> *#(y,z)
    *#(s(x),s(y)) -> *#(x,y) -> *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
    *#(s(x),s(y)) -> *#(x,y) -> *#(s(x),s(y)) -> *#(x,y)
    *#(s(x),s(y)) -> *#(x,y) -> *#(s(x),s(y)) -> +#(x,y)
    *#(s(x),s(y)) -> +#(*(x,y),+(x,y)) ->
    +#(+(x,y),z) -> +#(x,+(y,z))
    *#(s(x),s(y)) -> +#(*(x,y),+(x,y)) -> +#(+(x,y),z) -> +#(y,z)
    *#(s(x),s(y)) -> +#(*(x,y),+(x,y)) -> +#(s(x),s(y)) -> +#(x,y)
    *#(s(x),s(y)) -> +#(x,y) -> +#(+(x,y),z) -> +#(x,+(y,z))
    *#(s(x),s(y)) -> +#(x,y) -> +#(+(x,y),z) -> +#(y,z)
    *#(s(x),s(y)) -> +#(x,y) -> +#(s(x),s(y)) -> +#(x,y)
    +#(s(x),s(y)) -> +#(x,y) -> +#(+(x,y),z) -> +#(x,+(y,z))
    +#(s(x),s(y)) -> +#(x,y) -> +#(+(x,y),z) -> +#(y,z)
    +#(s(x),s(y)) -> +#(x,y) -> +#(s(x),s(y)) -> +#(x,y)
    +#(+(x,y),z) -> +#(y,z) -> +#(+(x,y),z) -> +#(x,+(y,z))
    +#(+(x,y),z) -> +#(y,z) -> +#(+(x,y),z) -> +#(y,z)
    +#(+(x,y),z) -> +#(y,z) -> +#(s(x),s(y)) -> +#(x,y)
    +#(+(x,y),z) -> +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,+(y,z))
    +#(+(x,y),z) -> +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(y,z)
    +#(+(x,y),z) -> +#(x,+(y,z)) -> +#(s(x),s(y)) -> +#(x,y)
   SCC Processor:
    #sccs: 5
    #rules: 13
    #arcs: 77/361
    DPs:
     app#(cons(x,l1),l2) -> app#(l1,l2)
    TRS:
     +(x,0()) -> x
     +(0(),x) -> x
     +(s(x),s(y)) -> s(s(+(x,y)))
     +(+(x,y),z) -> +(x,+(y,z))
     *(x,0()) -> 0()
     *(0(),x) -> 0()
     *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
     *(*(x,y),z) -> *(x,*(y,z))
     app(nil(),l) -> l
     app(cons(x,l1),l2) -> cons(x,app(l1,l2))
     sum(nil()) -> 0()
     sum(cons(x,l)) -> +(x,sum(l))
     sum(app(l1,l2)) -> +(sum(l1),sum(l2))
     prod(nil()) -> s(0())
     prod(cons(x,l)) -> *(x,prod(l))
     prod(app(l1,l2)) -> *(prod(l1),prod(l2))
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 0
     problem:
      DPs:
       
      TRS:
       +(x,0()) -> x
       +(0(),x) -> x
       +(s(x),s(y)) -> s(s(+(x,y)))
       +(+(x,y),z) -> +(x,+(y,z))
       *(x,0()) -> 0()
       *(0(),x) -> 0()
       *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
       *(*(x,y),z) -> *(x,*(y,z))
       app(nil(),l) -> l
       app(cons(x,l1),l2) -> cons(x,app(l1,l2))
       sum(nil()) -> 0()
       sum(cons(x,l)) -> +(x,sum(l))
       sum(app(l1,l2)) -> +(sum(l1),sum(l2))
       prod(nil()) -> s(0())
       prod(cons(x,l)) -> *(x,prod(l))
       prod(app(l1,l2)) -> *(prod(l1),prod(l2))
     Qed
    
    DPs:
     sum#(cons(x,l)) -> sum#(l)
     sum#(app(l1,l2)) -> sum#(l2)
     sum#(app(l1,l2)) -> sum#(l1)
    TRS:
     +(x,0()) -> x
     +(0(),x) -> x
     +(s(x),s(y)) -> s(s(+(x,y)))
     +(+(x,y),z) -> +(x,+(y,z))
     *(x,0()) -> 0()
     *(0(),x) -> 0()
     *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
     *(*(x,y),z) -> *(x,*(y,z))
     app(nil(),l) -> l
     app(cons(x,l1),l2) -> cons(x,app(l1,l2))
     sum(nil()) -> 0()
     sum(cons(x,l)) -> +(x,sum(l))
     sum(app(l1,l2)) -> +(sum(l1),sum(l2))
     prod(nil()) -> s(0())
     prod(cons(x,l)) -> *(x,prod(l))
     prod(app(l1,l2)) -> *(prod(l1),prod(l2))
    Subterm Criterion Processor:
     simple projection:
      pi(sum#) = 0
     problem:
      DPs:
       
      TRS:
       +(x,0()) -> x
       +(0(),x) -> x
       +(s(x),s(y)) -> s(s(+(x,y)))
       +(+(x,y),z) -> +(x,+(y,z))
       *(x,0()) -> 0()
       *(0(),x) -> 0()
       *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
       *(*(x,y),z) -> *(x,*(y,z))
       app(nil(),l) -> l
       app(cons(x,l1),l2) -> cons(x,app(l1,l2))
       sum(nil()) -> 0()
       sum(cons(x,l)) -> +(x,sum(l))
       sum(app(l1,l2)) -> +(sum(l1),sum(l2))
       prod(nil()) -> s(0())
       prod(cons(x,l)) -> *(x,prod(l))
       prod(app(l1,l2)) -> *(prod(l1),prod(l2))
     Qed
    
    DPs:
     prod#(cons(x,l)) -> prod#(l)
     prod#(app(l1,l2)) -> prod#(l2)
     prod#(app(l1,l2)) -> prod#(l1)
    TRS:
     +(x,0()) -> x
     +(0(),x) -> x
     +(s(x),s(y)) -> s(s(+(x,y)))
     +(+(x,y),z) -> +(x,+(y,z))
     *(x,0()) -> 0()
     *(0(),x) -> 0()
     *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
     *(*(x,y),z) -> *(x,*(y,z))
     app(nil(),l) -> l
     app(cons(x,l1),l2) -> cons(x,app(l1,l2))
     sum(nil()) -> 0()
     sum(cons(x,l)) -> +(x,sum(l))
     sum(app(l1,l2)) -> +(sum(l1),sum(l2))
     prod(nil()) -> s(0())
     prod(cons(x,l)) -> *(x,prod(l))
     prod(app(l1,l2)) -> *(prod(l1),prod(l2))
    Subterm Criterion Processor:
     simple projection:
      pi(prod#) = 0
     problem:
      DPs:
       
      TRS:
       +(x,0()) -> x
       +(0(),x) -> x
       +(s(x),s(y)) -> s(s(+(x,y)))
       +(+(x,y),z) -> +(x,+(y,z))
       *(x,0()) -> 0()
       *(0(),x) -> 0()
       *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
       *(*(x,y),z) -> *(x,*(y,z))
       app(nil(),l) -> l
       app(cons(x,l1),l2) -> cons(x,app(l1,l2))
       sum(nil()) -> 0()
       sum(cons(x,l)) -> +(x,sum(l))
       sum(app(l1,l2)) -> +(sum(l1),sum(l2))
       prod(nil()) -> s(0())
       prod(cons(x,l)) -> *(x,prod(l))
       prod(app(l1,l2)) -> *(prod(l1),prod(l2))
     Qed
    
    DPs:
     *#(s(x),s(y)) -> *#(x,y)
     *#(*(x,y),z) -> *#(y,z)
     *#(*(x,y),z) -> *#(x,*(y,z))
    TRS:
     +(x,0()) -> x
     +(0(),x) -> x
     +(s(x),s(y)) -> s(s(+(x,y)))
     +(+(x,y),z) -> +(x,+(y,z))
     *(x,0()) -> 0()
     *(0(),x) -> 0()
     *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
     *(*(x,y),z) -> *(x,*(y,z))
     app(nil(),l) -> l
     app(cons(x,l1),l2) -> cons(x,app(l1,l2))
     sum(nil()) -> 0()
     sum(cons(x,l)) -> +(x,sum(l))
     sum(app(l1,l2)) -> +(sum(l1),sum(l2))
     prod(nil()) -> s(0())
     prod(cons(x,l)) -> *(x,prod(l))
     prod(app(l1,l2)) -> *(prod(l1),prod(l2))
    Subterm Criterion Processor:
     simple projection:
      pi(*#) = 0
     problem:
      DPs:
       
      TRS:
       +(x,0()) -> x
       +(0(),x) -> x
       +(s(x),s(y)) -> s(s(+(x,y)))
       +(+(x,y),z) -> +(x,+(y,z))
       *(x,0()) -> 0()
       *(0(),x) -> 0()
       *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
       *(*(x,y),z) -> *(x,*(y,z))
       app(nil(),l) -> l
       app(cons(x,l1),l2) -> cons(x,app(l1,l2))
       sum(nil()) -> 0()
       sum(cons(x,l)) -> +(x,sum(l))
       sum(app(l1,l2)) -> +(sum(l1),sum(l2))
       prod(nil()) -> s(0())
       prod(cons(x,l)) -> *(x,prod(l))
       prod(app(l1,l2)) -> *(prod(l1),prod(l2))
     Qed
    
    DPs:
     +#(s(x),s(y)) -> +#(x,y)
     +#(+(x,y),z) -> +#(y,z)
     +#(+(x,y),z) -> +#(x,+(y,z))
    TRS:
     +(x,0()) -> x
     +(0(),x) -> x
     +(s(x),s(y)) -> s(s(+(x,y)))
     +(+(x,y),z) -> +(x,+(y,z))
     *(x,0()) -> 0()
     *(0(),x) -> 0()
     *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
     *(*(x,y),z) -> *(x,*(y,z))
     app(nil(),l) -> l
     app(cons(x,l1),l2) -> cons(x,app(l1,l2))
     sum(nil()) -> 0()
     sum(cons(x,l)) -> +(x,sum(l))
     sum(app(l1,l2)) -> +(sum(l1),sum(l2))
     prod(nil()) -> s(0())
     prod(cons(x,l)) -> *(x,prod(l))
     prod(app(l1,l2)) -> *(prod(l1),prod(l2))
    Subterm Criterion Processor:
     simple projection:
      pi(+#) = 0
     problem:
      DPs:
       
      TRS:
       +(x,0()) -> x
       +(0(),x) -> x
       +(s(x),s(y)) -> s(s(+(x,y)))
       +(+(x,y),z) -> +(x,+(y,z))
       *(x,0()) -> 0()
       *(0(),x) -> 0()
       *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
       *(*(x,y),z) -> *(x,*(y,z))
       app(nil(),l) -> l
       app(cons(x,l1),l2) -> cons(x,app(l1,l2))
       sum(nil()) -> 0()
       sum(cons(x,l)) -> +(x,sum(l))
       sum(app(l1,l2)) -> +(sum(l1),sum(l2))
       prod(nil()) -> s(0())
       prod(cons(x,l)) -> *(x,prod(l))
       prod(app(l1,l2)) -> *(prod(l1),prod(l2))
     Qed