/export/starexec/sandbox/solver/bin/starexec_run_ttt2 /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem:
 -(x,0()) -> x
 -(s(x),s(y)) -> -(x,y)
 *(x,0()) -> 0()
 *(x,s(y)) -> +(*(x,y),x)
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 odd(0()) -> false()
 odd(s(0())) -> true()
 odd(s(s(x))) -> odd(x)
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))
 if(true(),x,y) -> true()
 if(false(),x,y) -> false()
 pow(x,y) -> f(x,y,s(0()))
 f(x,0(),z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))

Proof:
 DP Processor:
  DPs:
   -#(s(x),s(y)) -> -#(x,y)
   *#(x,s(y)) -> *#(x,y)
   odd#(s(s(x))) -> odd#(x)
   half#(s(s(x))) -> half#(x)
   pow#(x,y) -> f#(x,y,s(0()))
   f#(x,s(y),z) -> half#(s(y))
   f#(x,s(y),z) -> *#(x,x)
   f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z)
   f#(x,s(y),z) -> *#(x,z)
   f#(x,s(y),z) -> f#(x,y,*(x,z))
   f#(x,s(y),z) -> odd#(s(y))
   f#(x,s(y),z) -> if#(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
  TRS:
   -(x,0()) -> x
   -(s(x),s(y)) -> -(x,y)
   *(x,0()) -> 0()
   *(x,s(y)) -> +(*(x,y),x)
   if(true(),x,y) -> x
   if(false(),x,y) -> y
   odd(0()) -> false()
   odd(s(0())) -> true()
   odd(s(s(x))) -> odd(x)
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
   if(true(),x,y) -> true()
   if(false(),x,y) -> false()
   pow(x,y) -> f(x,y,s(0()))
   f(x,0(),z) -> z
   f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
  TDG Processor:
   DPs:
    -#(s(x),s(y)) -> -#(x,y)
    *#(x,s(y)) -> *#(x,y)
    odd#(s(s(x))) -> odd#(x)
    half#(s(s(x))) -> half#(x)
    pow#(x,y) -> f#(x,y,s(0()))
    f#(x,s(y),z) -> half#(s(y))
    f#(x,s(y),z) -> *#(x,x)
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z)
    f#(x,s(y),z) -> *#(x,z)
    f#(x,s(y),z) -> f#(x,y,*(x,z))
    f#(x,s(y),z) -> odd#(s(y))
    f#(x,s(y),z) -> if#(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
   TRS:
    -(x,0()) -> x
    -(s(x),s(y)) -> -(x,y)
    *(x,0()) -> 0()
    *(x,s(y)) -> +(*(x,y),x)
    if(true(),x,y) -> x
    if(false(),x,y) -> y
    odd(0()) -> false()
    odd(s(0())) -> true()
    odd(s(s(x))) -> odd(x)
    half(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(x))) -> s(half(x))
    if(true(),x,y) -> true()
    if(false(),x,y) -> false()
    pow(x,y) -> f(x,y,s(0()))
    f(x,0(),z) -> z
    f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
   graph:
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z) ->
    f#(x,s(y),z) -> if#(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z) ->
    f#(x,s(y),z) -> odd#(s(y))
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z) ->
    f#(x,s(y),z) -> f#(x,y,*(x,z))
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z) ->
    f#(x,s(y),z) -> *#(x,z)
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z) ->
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z)
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z) ->
    f#(x,s(y),z) -> *#(x,x)
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z) ->
    f#(x,s(y),z) -> half#(s(y))
    f#(x,s(y),z) -> f#(x,y,*(x,z)) ->
    f#(x,s(y),z) -> if#(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
    f#(x,s(y),z) -> f#(x,y,*(x,z)) -> f#(x,s(y),z) -> odd#(s(y))
    f#(x,s(y),z) -> f#(x,y,*(x,z)) -> f#(x,s(y),z) -> f#(x,y,*(x,z))
    f#(x,s(y),z) -> f#(x,y,*(x,z)) -> f#(x,s(y),z) -> *#(x,z)
    f#(x,s(y),z) -> f#(x,y,*(x,z)) ->
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z)
    f#(x,s(y),z) -> f#(x,y,*(x,z)) -> f#(x,s(y),z) -> *#(x,x)
    f#(x,s(y),z) -> f#(x,y,*(x,z)) -> f#(x,s(y),z) -> half#(s(y))
    f#(x,s(y),z) -> half#(s(y)) -> half#(s(s(x))) -> half#(x)
    f#(x,s(y),z) -> odd#(s(y)) -> odd#(s(s(x))) -> odd#(x)
    f#(x,s(y),z) -> *#(x,z) -> *#(x,s(y)) -> *#(x,y)
    f#(x,s(y),z) -> *#(x,x) -> *#(x,s(y)) -> *#(x,y)
    pow#(x,y) -> f#(x,y,s(0())) ->
    f#(x,s(y),z) -> if#(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
    pow#(x,y) -> f#(x,y,s(0())) -> f#(x,s(y),z) -> odd#(s(y))
    pow#(x,y) -> f#(x,y,s(0())) -> f#(x,s(y),z) -> f#(x,y,*(x,z))
    pow#(x,y) -> f#(x,y,s(0())) -> f#(x,s(y),z) -> *#(x,z)
    pow#(x,y) -> f#(x,y,s(0())) ->
    f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z)
    pow#(x,y) -> f#(x,y,s(0())) -> f#(x,s(y),z) -> *#(x,x)
    pow#(x,y) -> f#(x,y,s(0())) -> f#(x,s(y),z) -> half#(s(y))
    half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> half#(x)
    odd#(s(s(x))) -> odd#(x) -> odd#(s(s(x))) -> odd#(x)
    *#(x,s(y)) -> *#(x,y) -> *#(x,s(y)) -> *#(x,y)
    -#(s(x),s(y)) -> -#(x,y) -> -#(s(x),s(y)) -> -#(x,y)
   SCC Processor:
    #sccs: 5
    #rules: 6
    #arcs: 29/144
    DPs:
     -#(s(x),s(y)) -> -#(x,y)
    TRS:
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(*(x,y),x)
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     odd(0()) -> false()
     odd(s(0())) -> true()
     odd(s(s(x))) -> odd(x)
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     if(true(),x,y) -> true()
     if(false(),x,y) -> false()
     pow(x,y) -> f(x,y,s(0()))
     f(x,0(),z) -> z
     f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
    Subterm Criterion Processor:
     simple projection:
      pi(-#) = 0
     problem:
      DPs:
       
      TRS:
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(*(x,y),x)
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       odd(0()) -> false()
       odd(s(0())) -> true()
       odd(s(s(x))) -> odd(x)
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       if(true(),x,y) -> true()
       if(false(),x,y) -> false()
       pow(x,y) -> f(x,y,s(0()))
       f(x,0(),z) -> z
       f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
     Qed
    
    DPs:
     f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z)
     f#(x,s(y),z) -> f#(x,y,*(x,z))
    TRS:
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(*(x,y),x)
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     odd(0()) -> false()
     odd(s(0())) -> true()
     odd(s(s(x))) -> odd(x)
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     if(true(),x,y) -> true()
     if(false(),x,y) -> false()
     pow(x,y) -> f(x,y,s(0()))
     f(x,0(),z) -> z
     f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
    Subterm Criterion Processor:
     simple projection:
      pi(s) = [0,0]
      pi(+) = [0,0,0]
      pi(half) = 0
      pi(f#) = [1,1]
     problem:
      DPs:
       f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z)
      TRS:
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(*(x,y),x)
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       odd(0()) -> false()
       odd(s(0())) -> true()
       odd(s(s(x))) -> odd(x)
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       if(true(),x,y) -> true()
       if(false(),x,y) -> false()
       pow(x,y) -> f(x,y,s(0()))
       f(x,0(),z) -> z
       f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
     Usable Rule Processor:
      DPs:
       f#(x,s(y),z) -> f#(*(x,x),half(s(y)),z)
      TRS:
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       half(0()) -> 0()
       *(x,0()) -> 0()
       *(x,s(y)) -> +(*(x,y),x)
      Matrix Interpretation Processor: dim=1
       
       usable rules:
        half(s(0())) -> 0()
        half(s(s(x))) -> s(half(x))
        half(0()) -> 0()
        *(x,0()) -> 0()
        *(x,s(y)) -> +(*(x,y),x)
       interpretation:
        [f#](x0, x1, x2) = x0 + 5/2x1 + 1,
        
        [+](x0, x1) = x0,
        
        [*](x0, x1) = 1,
        
        [0] = 0,
        
        [half](x0) = 1/2x0,
        
        [s](x0) = 2x0 + 2
       orientation:
        f#(x,s(y),z) = x + 5y + 6 >= 5/2y + 9/2 = f#(*(x,x),half(s(y)),z)
        
        half(s(0())) = 1 >= 0 = 0()
        
        half(s(s(x))) = 2x + 3 >= x + 2 = s(half(x))
        
        half(0()) = 0 >= 0 = 0()
        
        *(x,0()) = 1 >= 0 = 0()
        
        *(x,s(y)) = 1 >= 1 = +(*(x,y),x)
       problem:
        DPs:
         
        TRS:
         half(s(0())) -> 0()
         half(s(s(x))) -> s(half(x))
         half(0()) -> 0()
         *(x,0()) -> 0()
         *(x,s(y)) -> +(*(x,y),x)
       Qed
    
    DPs:
     odd#(s(s(x))) -> odd#(x)
    TRS:
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(*(x,y),x)
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     odd(0()) -> false()
     odd(s(0())) -> true()
     odd(s(s(x))) -> odd(x)
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     if(true(),x,y) -> true()
     if(false(),x,y) -> false()
     pow(x,y) -> f(x,y,s(0()))
     f(x,0(),z) -> z
     f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
    Subterm Criterion Processor:
     simple projection:
      pi(odd#) = 0
     problem:
      DPs:
       
      TRS:
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(*(x,y),x)
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       odd(0()) -> false()
       odd(s(0())) -> true()
       odd(s(s(x))) -> odd(x)
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       if(true(),x,y) -> true()
       if(false(),x,y) -> false()
       pow(x,y) -> f(x,y,s(0()))
       f(x,0(),z) -> z
       f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
     Qed
    
    DPs:
     *#(x,s(y)) -> *#(x,y)
    TRS:
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(*(x,y),x)
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     odd(0()) -> false()
     odd(s(0())) -> true()
     odd(s(s(x))) -> odd(x)
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     if(true(),x,y) -> true()
     if(false(),x,y) -> false()
     pow(x,y) -> f(x,y,s(0()))
     f(x,0(),z) -> z
     f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
    Subterm Criterion Processor:
     simple projection:
      pi(*#) = 1
     problem:
      DPs:
       
      TRS:
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(*(x,y),x)
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       odd(0()) -> false()
       odd(s(0())) -> true()
       odd(s(s(x))) -> odd(x)
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       if(true(),x,y) -> true()
       if(false(),x,y) -> false()
       pow(x,y) -> f(x,y,s(0()))
       f(x,0(),z) -> z
       f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
     Qed
    
    DPs:
     half#(s(s(x))) -> half#(x)
    TRS:
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(*(x,y),x)
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     odd(0()) -> false()
     odd(s(0())) -> true()
     odd(s(s(x))) -> odd(x)
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     if(true(),x,y) -> true()
     if(false(),x,y) -> false()
     pow(x,y) -> f(x,y,s(0()))
     f(x,0(),z) -> z
     f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
    Subterm Criterion Processor:
     simple projection:
      pi(half#) = 0
     problem:
      DPs:
       
      TRS:
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(*(x,y),x)
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       odd(0()) -> false()
       odd(s(0())) -> true()
       odd(s(s(x))) -> odd(x)
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       if(true(),x,y) -> true()
       if(false(),x,y) -> false()
       pow(x,y) -> f(x,y,s(0()))
       f(x,0(),z) -> z
       f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
     Qed