/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:
 min(x,0()) -> 0()
 min(0(),y) -> 0()
 min(s(x),s(y)) -> s(min(x,y))
 max(x,0()) -> x
 max(0(),y) -> y
 max(s(x),s(y)) -> s(max(x,y))
 -(x,0()) -> x
 -(s(x),s(y)) -> -(x,y)
 gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
 gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
 gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
 gcd(x,0(),0()) -> x
 gcd(0(),y,0()) -> y
 gcd(0(),0(),z) -> z

Proof:
 DP Processor:
  DPs:
   min#(s(x),s(y)) -> min#(x,y)
   max#(s(x),s(y)) -> max#(x,y)
   -#(s(x),s(y)) -> -#(x,y)
   gcd#(s(x),s(y),z) -> min#(x,y)
   gcd#(s(x),s(y),z) -> max#(x,y)
   gcd#(s(x),s(y),z) -> -#(max(x,y),min(x,y))
   gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
   gcd#(x,s(y),s(z)) -> min#(y,z)
   gcd#(x,s(y),s(z)) -> max#(y,z)
   gcd#(x,s(y),s(z)) -> -#(max(y,z),min(y,z))
   gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
   gcd#(s(x),y,s(z)) -> min#(x,z)
   gcd#(s(x),y,s(z)) -> max#(x,z)
   gcd#(s(x),y,s(z)) -> -#(max(x,z),min(x,z))
   gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
  TRS:
   min(x,0()) -> 0()
   min(0(),y) -> 0()
   min(s(x),s(y)) -> s(min(x,y))
   max(x,0()) -> x
   max(0(),y) -> y
   max(s(x),s(y)) -> s(max(x,y))
   -(x,0()) -> x
   -(s(x),s(y)) -> -(x,y)
   gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
   gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
   gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
   gcd(x,0(),0()) -> x
   gcd(0(),y,0()) -> y
   gcd(0(),0(),z) -> z
  TDG Processor:
   DPs:
    min#(s(x),s(y)) -> min#(x,y)
    max#(s(x),s(y)) -> max#(x,y)
    -#(s(x),s(y)) -> -#(x,y)
    gcd#(s(x),s(y),z) -> min#(x,y)
    gcd#(s(x),s(y),z) -> max#(x,y)
    gcd#(s(x),s(y),z) -> -#(max(x,y),min(x,y))
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
    gcd#(x,s(y),s(z)) -> min#(y,z)
    gcd#(x,s(y),s(z)) -> max#(y,z)
    gcd#(x,s(y),s(z)) -> -#(max(y,z),min(y,z))
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
    gcd#(s(x),y,s(z)) -> min#(x,z)
    gcd#(s(x),y,s(z)) -> max#(x,z)
    gcd#(s(x),y,s(z)) -> -#(max(x,z),min(x,z))
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
   TRS:
    min(x,0()) -> 0()
    min(0(),y) -> 0()
    min(s(x),s(y)) -> s(min(x,y))
    max(x,0()) -> x
    max(0(),y) -> y
    max(s(x),s(y)) -> s(max(x,y))
    -(x,0()) -> x
    -(s(x),s(y)) -> -(x,y)
    gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
    gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
    gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
    gcd(x,0(),0()) -> x
    gcd(0(),y,0()) -> y
    gcd(0(),0(),z) -> z
   graph:
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(s(x),y,s(z)) -> -#(max(x,z),min(x,z))
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(s(x),y,s(z)) -> max#(x,z)
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(s(x),y,s(z)) -> min#(x,z)
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(x,s(y),s(z)) -> -#(max(y,z),min(y,z))
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(x,s(y),s(z)) -> max#(y,z)
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(x,s(y),s(z)) -> min#(y,z)
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(s(x),s(y),z) -> -#(max(x,y),min(x,y))
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(s(x),s(y),z) -> max#(x,y)
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z) ->
    gcd#(s(x),s(y),z) -> min#(x,y)
    gcd#(s(x),s(y),z) -> -#(max(x,y),min(x,y)) ->
    -#(s(x),s(y)) -> -#(x,y)
    gcd#(s(x),s(y),z) -> max#(x,y) -> max#(s(x),s(y)) -> max#(x,y)
    gcd#(s(x),s(y),z) -> min#(x,y) ->
    min#(s(x),s(y)) -> min#(x,y)
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(s(x),y,s(z)) -> -#(max(x,z),min(x,z))
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(s(x),y,s(z)) -> max#(x,z)
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(s(x),y,s(z)) -> min#(x,z)
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(x,s(y),s(z)) -> -#(max(y,z),min(y,z))
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(x,s(y),s(z)) -> max#(y,z)
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(x,s(y),s(z)) -> min#(y,z)
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(s(x),s(y),z) -> -#(max(x,y),min(x,y))
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(s(x),s(y),z) -> max#(x,y)
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z))) ->
    gcd#(s(x),s(y),z) -> min#(x,y)
    gcd#(s(x),y,s(z)) -> -#(max(x,z),min(x,z)) ->
    -#(s(x),s(y)) -> -#(x,y)
    gcd#(s(x),y,s(z)) -> max#(x,z) -> max#(s(x),s(y)) -> max#(x,y)
    gcd#(s(x),y,s(z)) -> min#(x,z) ->
    min#(s(x),s(y)) -> min#(x,y)
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(s(x),y,s(z)) -> -#(max(x,z),min(x,z))
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(s(x),y,s(z)) -> max#(x,z)
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(s(x),y,s(z)) -> min#(x,z)
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(x,s(y),s(z)) -> -#(max(y,z),min(y,z))
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(x,s(y),s(z)) -> max#(y,z)
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(x,s(y),s(z)) -> min#(y,z)
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(s(x),s(y),z) -> -#(max(x,y),min(x,y))
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(s(x),s(y),z) -> max#(x,y)
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z))) ->
    gcd#(s(x),s(y),z) -> min#(x,y)
    gcd#(x,s(y),s(z)) -> -#(max(y,z),min(y,z)) ->
    -#(s(x),s(y)) -> -#(x,y)
    gcd#(x,s(y),s(z)) -> max#(y,z) -> max#(s(x),s(y)) -> max#(x,y)
    gcd#(x,s(y),s(z)) -> min#(y,z) -> min#(s(x),s(y)) -> min#(x,y)
    -#(s(x),s(y)) -> -#(x,y) -> -#(s(x),s(y)) -> -#(x,y)
    max#(s(x),s(y)) -> max#(x,y) -> max#(s(x),s(y)) -> max#(x,y)
    min#(s(x),s(y)) -> min#(x,y) -> min#(s(x),s(y)) -> min#(x,y)
   SCC Processor:
    #sccs: 4
    #rules: 6
    #arcs: 48/225
    DPs:
     gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
     gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
     gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
    TRS:
     min(x,0()) -> 0()
     min(0(),y) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(x,0()) -> x
     max(0(),y) -> y
     max(s(x),s(y)) -> s(max(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
     gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
     gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
     gcd(x,0(),0()) -> x
     gcd(0(),y,0()) -> y
     gcd(0(),0(),z) -> z
    Usable Rule Processor:
     DPs:
      gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
      gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
      gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
     TRS:
      min(x,0()) -> 0()
      min(0(),y) -> 0()
      min(s(x),s(y)) -> s(min(x,y))
      max(x,0()) -> x
      max(0(),y) -> y
      max(s(x),s(y)) -> s(max(x,y))
      -(x,0()) -> x
      -(s(x),s(y)) -> -(x,y)
     Matrix Interpretation Processor: dim=1
      
      usable rules:
       min(x,0()) -> 0()
       min(0(),y) -> 0()
       min(s(x),s(y)) -> s(min(x,y))
       max(x,0()) -> x
       max(0(),y) -> y
       max(s(x),s(y)) -> s(max(x,y))
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
      interpretation:
       [gcd#](x0, x1, x2) = 2x0 + 2x1 + 2x2 + 3,
       
       [-](x0, x1) = x0 + 1/2,
       
       [max](x0, x1) = x0 + x1,
       
       [s](x0) = 2x0 + 2,
       
       [min](x0, x1) = 1/2x0 + 1/2x1,
       
       [0] = 0
      orientation:
       gcd#(s(x),s(y),z) = 4x + 4y + 2z + 11 >= 4x + 4y + 2z + 8 = gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
       
       gcd#(x,s(y),s(z)) = 2x + 4y + 4z + 11 >= 2x + 4y + 4z + 8 = gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
       
       gcd#(s(x),y,s(z)) = 4x + 2y + 4z + 11 >= 4x + 2y + 4z + 8 = gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
       
       min(x,0()) = 1/2x >= 0 = 0()
       
       min(0(),y) = 1/2y >= 0 = 0()
       
       min(s(x),s(y)) = x + y + 2 >= x + y + 2 = s(min(x,y))
       
       max(x,0()) = x >= x = x
       
       max(0(),y) = y >= y = y
       
       max(s(x),s(y)) = 2x + 2y + 4 >= 2x + 2y + 2 = s(max(x,y))
       
       -(x,0()) = x + 1/2 >= x = x
       
       -(s(x),s(y)) = 2x + 5/2 >= x + 1/2 = -(x,y)
      problem:
       DPs:
        
       TRS:
        min(x,0()) -> 0()
        min(0(),y) -> 0()
        min(s(x),s(y)) -> s(min(x,y))
        max(x,0()) -> x
        max(0(),y) -> y
        max(s(x),s(y)) -> s(max(x,y))
        -(x,0()) -> x
        -(s(x),s(y)) -> -(x,y)
      Qed
    
    DPs:
     -#(s(x),s(y)) -> -#(x,y)
    TRS:
     min(x,0()) -> 0()
     min(0(),y) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(x,0()) -> x
     max(0(),y) -> y
     max(s(x),s(y)) -> s(max(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
     gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
     gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
     gcd(x,0(),0()) -> x
     gcd(0(),y,0()) -> y
     gcd(0(),0(),z) -> z
    Subterm Criterion Processor:
     simple projection:
      pi(-#) = 0
     problem:
      DPs:
       
      TRS:
       min(x,0()) -> 0()
       min(0(),y) -> 0()
       min(s(x),s(y)) -> s(min(x,y))
       max(x,0()) -> x
       max(0(),y) -> y
       max(s(x),s(y)) -> s(max(x,y))
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
       gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
       gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
       gcd(x,0(),0()) -> x
       gcd(0(),y,0()) -> y
       gcd(0(),0(),z) -> z
     Qed
    
    DPs:
     max#(s(x),s(y)) -> max#(x,y)
    TRS:
     min(x,0()) -> 0()
     min(0(),y) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(x,0()) -> x
     max(0(),y) -> y
     max(s(x),s(y)) -> s(max(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
     gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
     gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
     gcd(x,0(),0()) -> x
     gcd(0(),y,0()) -> y
     gcd(0(),0(),z) -> z
    Subterm Criterion Processor:
     simple projection:
      pi(max#) = 0
     problem:
      DPs:
       
      TRS:
       min(x,0()) -> 0()
       min(0(),y) -> 0()
       min(s(x),s(y)) -> s(min(x,y))
       max(x,0()) -> x
       max(0(),y) -> y
       max(s(x),s(y)) -> s(max(x,y))
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
       gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
       gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
       gcd(x,0(),0()) -> x
       gcd(0(),y,0()) -> y
       gcd(0(),0(),z) -> z
     Qed
    
    DPs:
     min#(s(x),s(y)) -> min#(x,y)
    TRS:
     min(x,0()) -> 0()
     min(0(),y) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(x,0()) -> x
     max(0(),y) -> y
     max(s(x),s(y)) -> s(max(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
     gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
     gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
     gcd(x,0(),0()) -> x
     gcd(0(),y,0()) -> y
     gcd(0(),0(),z) -> z
    Subterm Criterion Processor:
     simple projection:
      pi(min#) = 0
     problem:
      DPs:
       
      TRS:
       min(x,0()) -> 0()
       min(0(),y) -> 0()
       min(s(x),s(y)) -> s(min(x,y))
       max(x,0()) -> x
       max(0(),y) -> y
       max(s(x),s(y)) -> s(max(x,y))
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
       gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
       gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
       gcd(x,0(),0()) -> x
       gcd(0(),y,0()) -> y
       gcd(0(),0(),z) -> z
     Qed