/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem:
 active(U11(tt(),M,N)) -> mark(U12(tt(),M,N))
 active(U12(tt(),M,N)) -> mark(s(plus(N,M)))
 active(plus(N,0())) -> mark(N)
 active(plus(N,s(M))) -> mark(U11(tt(),M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(s(X)) -> s(active(X))
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 s(mark(X)) -> mark(s(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(tt()) -> ok(tt())
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(s(X)) -> s(proper(X))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 s(ok(X)) -> ok(s(X))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [top](x0) = 2x0,
   
   [ok](x0) = x0,
   
   [proper](x0) = x0,
   
   [0] = 0,
   
   [s](x0) = x0 + 3,
   
   [plus](x0, x1) = 4x0 + 5x1,
   
   [mark](x0) = x0,
   
   [U12](x0, x1, x2) = 4x0 + 5x1 + 4x2,
   
   [active](x0) = x0,
   
   [U11](x0, x1, x2) = 7x0 + 5x1 + 4x2 + 5,
   
   [tt] = 1
  orientation:
   active(U11(tt(),M,N)) = 5M + 4N + 12 >= 5M + 4N + 4 = mark(U12(tt(),M,N))
   
   active(U12(tt(),M,N)) = 5M + 4N + 4 >= 5M + 4N + 3 = mark(s(plus(N,M)))
   
   active(plus(N,0())) = 4N >= N = mark(N)
   
   active(plus(N,s(M))) = 5M + 4N + 15 >= 5M + 4N + 12 = mark(U11(tt(),M,N))
   
   active(U11(X1,X2,X3)) = 7X1 + 5X2 + 4X3 + 5 >= 7X1 + 5X2 + 4X3 + 5 = U11(active(X1),X2,X3)
   
   active(U12(X1,X2,X3)) = 4X1 + 5X2 + 4X3 >= 4X1 + 5X2 + 4X3 = U12(active(X1),X2,X3)
   
   active(s(X)) = X + 3 >= X + 3 = s(active(X))
   
   active(plus(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = plus(active(X1),X2)
   
   active(plus(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = plus(X1,active(X2))
   
   U11(mark(X1),X2,X3) = 7X1 + 5X2 + 4X3 + 5 >= 7X1 + 5X2 + 4X3 + 5 = mark(U11(X1,X2,X3))
   
   U12(mark(X1),X2,X3) = 4X1 + 5X2 + 4X3 >= 4X1 + 5X2 + 4X3 = mark(U12(X1,X2,X3))
   
   s(mark(X)) = X + 3 >= X + 3 = mark(s(X))
   
   plus(mark(X1),X2) = 4X1 + 5X2 >= 4X1 + 5X2 = mark(plus(X1,X2))
   
   plus(X1,mark(X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = mark(plus(X1,X2))
   
   proper(U11(X1,X2,X3)) = 7X1 + 5X2 + 4X3 + 5 >= 7X1 + 5X2 + 4X3 + 5 = U11(proper(X1),proper(X2),proper(X3))
   
   proper(tt()) = 1 >= 1 = ok(tt())
   
   proper(U12(X1,X2,X3)) = 4X1 + 5X2 + 4X3 >= 4X1 + 5X2 + 4X3 = U12(proper(X1),proper(X2),proper(X3))
   
   proper(s(X)) = X + 3 >= X + 3 = s(proper(X))
   
   proper(plus(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = plus(proper(X1),proper(X2))
   
   proper(0()) = 0 >= 0 = ok(0())
   
   U11(ok(X1),ok(X2),ok(X3)) = 7X1 + 5X2 + 4X3 + 5 >= 7X1 + 5X2 + 4X3 + 5 = ok(U11(X1,X2,X3))
   
   U12(ok(X1),ok(X2),ok(X3)) = 4X1 + 5X2 + 4X3 >= 4X1 + 5X2 + 4X3 = ok(U12(X1,X2,X3))
   
   s(ok(X)) = X + 3 >= X + 3 = ok(s(X))
   
   plus(ok(X1),ok(X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = ok(plus(X1,X2))
   
   top(mark(X)) = 2X >= 2X = top(proper(X))
   
   top(ok(X)) = 2X >= 2X = top(active(X))
  problem:
   active(plus(N,0())) -> mark(N)
   active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
   active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
   active(s(X)) -> s(active(X))
   active(plus(X1,X2)) -> plus(active(X1),X2)
   active(plus(X1,X2)) -> plus(X1,active(X2))
   U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
   U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1),X2) -> mark(plus(X1,X2))
   plus(X1,mark(X2)) -> mark(plus(X1,X2))
   proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
   proper(tt()) -> ok(tt())
   proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
   U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [top](x0) = x0 + 6,
    
    [ok](x0) = x0,
    
    [proper](x0) = x0,
    
    [0] = 2,
    
    [s](x0) = 2x0,
    
    [plus](x0, x1) = x0 + 2x1,
    
    [mark](x0) = x0 + 4,
    
    [U12](x0, x1, x2) = x0 + x1 + x2,
    
    [active](x0) = x0,
    
    [U11](x0, x1, x2) = x0 + x1 + 4x2,
    
    [tt] = 0
   orientation:
    active(plus(N,0())) = N + 4 >= N + 4 = mark(N)
    
    active(U11(X1,X2,X3)) = X1 + X2 + 4X3 >= X1 + X2 + 4X3 = U11(active(X1),X2,X3)
    
    active(U12(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U12(active(X1),X2,X3)
    
    active(s(X)) = 2X >= 2X = s(active(X))
    
    active(plus(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = plus(active(X1),X2)
    
    active(plus(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = plus(X1,active(X2))
    
    U11(mark(X1),X2,X3) = X1 + X2 + 4X3 + 4 >= X1 + X2 + 4X3 + 4 = mark(U11(X1,X2,X3))
    
    U12(mark(X1),X2,X3) = X1 + X2 + X3 + 4 >= X1 + X2 + X3 + 4 = mark(U12(X1,X2,X3))
    
    s(mark(X)) = 2X + 8 >= 2X + 4 = mark(s(X))
    
    plus(mark(X1),X2) = X1 + 2X2 + 4 >= X1 + 2X2 + 4 = mark(plus(X1,X2))
    
    plus(X1,mark(X2)) = X1 + 2X2 + 8 >= X1 + 2X2 + 4 = mark(plus(X1,X2))
    
    proper(U11(X1,X2,X3)) = X1 + X2 + 4X3 >= X1 + X2 + 4X3 = U11(proper(X1),proper(X2),proper(X3))
    
    proper(tt()) = 0 >= 0 = ok(tt())
    
    proper(U12(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U12(proper(X1),proper(X2),proper(X3))
    
    proper(s(X)) = 2X >= 2X = s(proper(X))
    
    proper(plus(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = plus(proper(X1),proper(X2))
    
    proper(0()) = 2 >= 2 = ok(0())
    
    U11(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 4X3 >= X1 + X2 + 4X3 = ok(U11(X1,X2,X3))
    
    U12(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U12(X1,X2,X3))
    
    s(ok(X)) = 2X >= 2X = ok(s(X))
    
    plus(ok(X1),ok(X2)) = X1 + 2X2 >= X1 + 2X2 = ok(plus(X1,X2))
    
    top(mark(X)) = X + 10 >= X + 6 = top(proper(X))
    
    top(ok(X)) = X + 6 >= X + 6 = top(active(X))
   problem:
    active(plus(N,0())) -> mark(N)
    active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
    active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
    active(s(X)) -> s(active(X))
    active(plus(X1,X2)) -> plus(active(X1),X2)
    active(plus(X1,X2)) -> plus(X1,active(X2))
    U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
    U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
    plus(mark(X1),X2) -> mark(plus(X1,X2))
    proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
    proper(tt()) -> ok(tt())
    proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
    proper(s(X)) -> s(proper(X))
    proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
    U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
    s(ok(X)) -> ok(s(X))
    plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
    top(ok(X)) -> top(active(X))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [top](x0) = x0,
     
     [ok](x0) = x0,
     
     [proper](x0) = 4x0,
     
     [0] = 4,
     
     [s](x0) = 2x0,
     
     [plus](x0, x1) = 2x0 + x1,
     
     [mark](x0) = x0 + 4,
     
     [U12](x0, x1, x2) = 2x0 + x1 + 2x2,
     
     [active](x0) = x0,
     
     [U11](x0, x1, x2) = 4x0 + 2x1 + x2,
     
     [tt] = 0
    orientation:
     active(plus(N,0())) = 2N + 4 >= N + 4 = mark(N)
     
     active(U11(X1,X2,X3)) = 4X1 + 2X2 + X3 >= 4X1 + 2X2 + X3 = U11(active(X1),X2,X3)
     
     active(U12(X1,X2,X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = U12(active(X1),X2,X3)
     
     active(s(X)) = 2X >= 2X = s(active(X))
     
     active(plus(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = plus(active(X1),X2)
     
     active(plus(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = plus(X1,active(X2))
     
     U11(mark(X1),X2,X3) = 4X1 + 2X2 + X3 + 16 >= 4X1 + 2X2 + X3 + 4 = mark(U11(X1,X2,X3))
     
     U12(mark(X1),X2,X3) = 2X1 + X2 + 2X3 + 8 >= 2X1 + X2 + 2X3 + 4 = mark(U12(X1,X2,X3))
     
     plus(mark(X1),X2) = 2X1 + X2 + 8 >= 2X1 + X2 + 4 = mark(plus(X1,X2))
     
     proper(U11(X1,X2,X3)) = 16X1 + 8X2 + 4X3 >= 16X1 + 8X2 + 4X3 = U11(proper(X1),proper(X2),proper(X3))
     
     proper(tt()) = 0 >= 0 = ok(tt())
     
     proper(U12(X1,X2,X3)) = 8X1 + 4X2 + 8X3 >= 8X1 + 4X2 + 8X3 = U12(proper(X1),proper(X2),proper(X3))
     
     proper(s(X)) = 8X >= 8X = s(proper(X))
     
     proper(plus(X1,X2)) = 8X1 + 4X2 >= 8X1 + 4X2 = plus(proper(X1),proper(X2))
     
     proper(0()) = 16 >= 4 = ok(0())
     
     U11(ok(X1),ok(X2),ok(X3)) = 4X1 + 2X2 + X3 >= 4X1 + 2X2 + X3 = ok(U11(X1,X2,X3))
     
     U12(ok(X1),ok(X2),ok(X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = ok(U12(X1,X2,X3))
     
     s(ok(X)) = 2X >= 2X = ok(s(X))
     
     plus(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(plus(X1,X2))
     
     top(ok(X)) = X >= X = top(active(X))
    problem:
     active(plus(N,0())) -> mark(N)
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [top](x0) = x0,
      
      [ok](x0) = 4x0,
      
      [proper](x0) = 5x0,
      
      [0] = 0,
      
      [s](x0) = x0,
      
      [plus](x0, x1) = x0 + 2x1,
      
      [mark](x0) = 2x0,
      
      [U12](x0, x1, x2) = x0 + 4x1 + 4x2,
      
      [active](x0) = 4x0,
      
      [U11](x0, x1, x2) = x0 + x1 + x2,
      
      [tt] = 1
     orientation:
      active(plus(N,0())) = 4N >= 2N = mark(N)
      
      active(U11(X1,X2,X3)) = 4X1 + 4X2 + 4X3 >= 4X1 + X2 + X3 = U11(active(X1),X2,X3)
      
      active(U12(X1,X2,X3)) = 4X1 + 16X2 + 16X3 >= 4X1 + 4X2 + 4X3 = U12(active(X1),X2,X3)
      
      active(s(X)) = 4X >= 4X = s(active(X))
      
      active(plus(X1,X2)) = 4X1 + 8X2 >= 4X1 + 2X2 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = 4X1 + 8X2 >= X1 + 8X2 = plus(X1,active(X2))
      
      proper(U11(X1,X2,X3)) = 5X1 + 5X2 + 5X3 >= 5X1 + 5X2 + 5X3 = U11(proper(X1),proper(X2),proper(X3))
      
      proper(tt()) = 5 >= 4 = ok(tt())
      
      proper(U12(X1,X2,X3)) = 5X1 + 20X2 + 20X3 >= 5X1 + 20X2 + 20X3 = U12(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = 5X >= 5X = s(proper(X))
      
      proper(plus(X1,X2)) = 5X1 + 10X2 >= 5X1 + 10X2 = plus(proper(X1),proper(X2))
      
      U11(ok(X1),ok(X2),ok(X3)) = 4X1 + 4X2 + 4X3 >= 4X1 + 4X2 + 4X3 = ok(U11(X1,X2,X3))
      
      U12(ok(X1),ok(X2),ok(X3)) = 4X1 + 16X2 + 16X3 >= 4X1 + 16X2 + 16X3 = ok(U12(X1,X2,X3))
      
      s(ok(X)) = 4X >= 4X = ok(s(X))
      
      plus(ok(X1),ok(X2)) = 4X1 + 8X2 >= 4X1 + 8X2 = ok(plus(X1,X2))
      
      top(ok(X)) = 4X >= 4X = top(active(X))
     problem:
      active(plus(N,0())) -> mark(N)
      active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
      active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
      active(s(X)) -> s(active(X))
      active(plus(X1,X2)) -> plus(active(X1),X2)
      active(plus(X1,X2)) -> plus(X1,active(X2))
      proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
      proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
      proper(s(X)) -> s(proper(X))
      proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
      U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
      U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
      s(ok(X)) -> ok(s(X))
      plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
      top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [top](x0) = x0 + 1,
       
       [ok](x0) = 4x0,
       
       [proper](x0) = x0,
       
       [0] = 3,
       
       [s](x0) = 2x0,
       
       [plus](x0, x1) = 2x0 + 5x1,
       
       [mark](x0) = 2x0,
       
       [U12](x0, x1, x2) = 4x0 + 4x1 + 2x2,
       
       [active](x0) = 2x0,
       
       [U11](x0, x1, x2) = 4x0 + x1 + 4x2
      orientation:
       active(plus(N,0())) = 4N + 30 >= 2N = mark(N)
       
       active(U11(X1,X2,X3)) = 8X1 + 2X2 + 8X3 >= 8X1 + X2 + 4X3 = U11(active(X1),X2,X3)
       
       active(U12(X1,X2,X3)) = 8X1 + 8X2 + 4X3 >= 8X1 + 4X2 + 2X3 = U12(active(X1),X2,X3)
       
       active(s(X)) = 4X >= 4X = s(active(X))
       
       active(plus(X1,X2)) = 4X1 + 10X2 >= 4X1 + 5X2 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = 4X1 + 10X2 >= 2X1 + 10X2 = plus(X1,active(X2))
       
       proper(U11(X1,X2,X3)) = 4X1 + X2 + 4X3 >= 4X1 + X2 + 4X3 = U11(proper(X1),proper(X2),proper(X3))
       
       proper(U12(X1,X2,X3)) = 4X1 + 4X2 + 2X3 >= 4X1 + 4X2 + 2X3 = U12(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = 2X >= 2X = s(proper(X))
       
       proper(plus(X1,X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = plus(proper(X1),proper(X2))
       
       U11(ok(X1),ok(X2),ok(X3)) = 16X1 + 4X2 + 16X3 >= 16X1 + 4X2 + 16X3 = ok(U11(X1,X2,X3))
       
       U12(ok(X1),ok(X2),ok(X3)) = 16X1 + 16X2 + 8X3 >= 16X1 + 16X2 + 8X3 = ok(U12(X1,X2,X3))
       
       s(ok(X)) = 8X >= 8X = ok(s(X))
       
       plus(ok(X1),ok(X2)) = 8X1 + 20X2 >= 8X1 + 20X2 = ok(plus(X1,X2))
       
       top(ok(X)) = 4X + 1 >= 2X + 1 = top(active(X))
      problem:
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
       active(s(X)) -> s(active(X))
       active(plus(X1,X2)) -> plus(active(X1),X2)
       active(plus(X1,X2)) -> plus(X1,active(X2))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(ok(X)) -> top(active(X))
      Matrix Interpretation Processor: dim=3
       
       interpretation:
                    [1 0 0]  
        [top](x0) = [0 0 0]x0
                    [0 0 0]  ,
        
                   [1 1 0]     [0]
        [ok](x0) = [0 0 1]x0 + [0]
                   [0 0 0]     [1],
        
                         
        [proper](x0) = x0
                         ,
        
                    
        [s](x0) = x0
                    ,
        
                         [1 0 0]       
        [plus](x0, x1) = [0 1 0]x0 + x1
                         [0 0 0]       ,
        
                                 [1 0 0]     [1 0 0]     [1]
        [U12](x0, x1, x2) = x0 + [0 0 0]x1 + [0 0 0]x2 + [0]
                                 [0 0 0]     [0 0 0]     [0],
        
                       [1 1 0]  
        [active](x0) = [0 1 0]x0
                       [0 0 1]  ,
        
                            [1 0 0]     [1 0 0]     [1 1 1]     [0]
        [U11](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [1]
                            [0 0 0]     [0 0 0]     [0 0 0]     [1]
       orientation:
                                [1 1 0]     [1 0 0]     [1 1 1]     [1]    [1 1 0]     [1 0 0]     [1 1 1]     [0]                        
        active(U11(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = U11(active(X1),X2,X3)
                                [0 0 0]     [0 0 0]     [0 0 0]     [1]    [0 0 0]     [0 0 0]     [0 0 0]     [1]                        
        
                                [1 1 0]     [1 0 0]     [1 0 0]     [1]    [1 1 0]     [1 0 0]     [1 0 0]     [1]                        
        active(U12(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U12(active(X1),X2,X3)
                                [0 0 1]     [0 0 0]     [0 0 0]     [0]    [0 0 1]     [0 0 0]     [0 0 0]     [0]                        
        
                       [1 1 0]     [1 1 0]                
        active(s(X)) = [0 1 0]X >= [0 1 0]X = s(active(X))
                       [0 0 1]     [0 0 1]                
        
                              [1 1 0]     [1 1 0]      [1 1 0]                             
        active(plus(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + X2 = plus(active(X1),X2)
                              [0 0 0]     [0 0 1]      [0 0 0]                             
        
                              [1 1 0]     [1 1 0]      [1 0 0]     [1 1 0]                        
        active(plus(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = plus(X1,active(X2))
                              [0 0 0]     [0 0 1]      [0 0 0]     [0 0 1]                        
        
                                [1 0 0]     [1 0 0]     [1 1 1]     [0]    [1 0 0]     [1 0 0]     [1 1 1]     [0]                                        
        proper(U11(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = U11(proper(X1),proper(X2),proper(X3))
                                [0 0 0]     [0 0 0]     [0 0 0]     [1]    [0 0 0]     [0 0 0]     [0 0 0]     [1]                                        
        
                                     [1 0 0]     [1 0 0]     [1]         [1 0 0]     [1 0 0]     [1]                                        
        proper(U12(X1,X2,X3)) = X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U12(proper(X1),proper(X2),proper(X3))
                                     [0 0 0]     [0 0 0]     [0]         [0 0 0]     [0 0 0]     [0]                                        
        
                                            
        proper(s(X)) = X >= X = s(proper(X))
                                            
        
                              [1 0 0]           [1 0 0]                                     
        proper(plus(X1,X2)) = [0 1 0]X1 + X2 >= [0 1 0]X1 + X2 = plus(proper(X1),proper(X2))
                              [0 0 0]           [0 0 0]                                     
        
                                    [1 1 0]     [1 1 0]     [1 1 1]     [1]    [1 1 0]     [1 0 0]     [1 1 1]     [1]                    
        U11(ok(X1),ok(X2),ok(X3)) = [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = ok(U11(X1,X2,X3))
                                    [0 0 0]     [0 0 0]     [0 0 0]     [1]    [0 0 0]     [0 0 0]     [0 0 0]     [1]                    
        
                                    [1 1 0]     [1 1 0]     [1 1 0]     [1]    [1 1 0]     [1 0 0]     [1 0 0]     [1]                    
        U12(ok(X1),ok(X2),ok(X3)) = [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = ok(U12(X1,X2,X3))
                                    [0 0 0]     [0 0 0]     [0 0 0]     [1]    [0 0 0]     [0 0 0]     [0 0 0]     [1]                    
        
                   [1 1 0]    [0]    [1 1 0]    [0]           
        s(ok(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = ok(s(X))
                   [0 0 0]    [1]    [0 0 0]    [1]           
        
                              [1 1 0]     [1 1 0]     [0]    [1 1 0]     [1 1 0]     [0]                  
        plus(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 1]X2 + [0] >= [0 0 0]X1 + [0 0 1]X2 + [0] = ok(plus(X1,X2))
                              [0 0 0]     [0 0 0]     [1]    [0 0 0]     [0 0 0]     [1]                  
        
                     [1 1 0]     [1 1 0]                  
        top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X))
                     [0 0 0]     [0 0 0]                  
       problem:
        active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
        active(s(X)) -> s(active(X))
        active(plus(X1,X2)) -> plus(active(X1),X2)
        active(plus(X1,X2)) -> plus(X1,active(X2))
        proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
        proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
        U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(ok(X)) -> top(active(X))
       Matrix Interpretation Processor: dim=1
        
        interpretation:
         [top](x0) = 6x0 + 1,
         
         [ok](x0) = 4x0 + 4,
         
         [proper](x0) = 6x0,
         
         [s](x0) = 3x0 + 2,
         
         [plus](x0, x1) = 4x0 + 2x1 + 4,
         
         [U12](x0, x1, x2) = 2x0 + 4x1 + x2,
         
         [active](x0) = x0,
         
         [U11](x0, x1, x2) = 2x0 + x1 + x2 + 1
        orientation:
         active(U12(X1,X2,X3)) = 2X1 + 4X2 + X3 >= 2X1 + 4X2 + X3 = U12(active(X1),X2,X3)
         
         active(s(X)) = 3X + 2 >= 3X + 2 = s(active(X))
         
         active(plus(X1,X2)) = 4X1 + 2X2 + 4 >= 4X1 + 2X2 + 4 = plus(active(X1),X2)
         
         active(plus(X1,X2)) = 4X1 + 2X2 + 4 >= 4X1 + 2X2 + 4 = plus(X1,active(X2))
         
         proper(U11(X1,X2,X3)) = 12X1 + 6X2 + 6X3 + 6 >= 12X1 + 6X2 + 6X3 + 1 = U11(proper(X1),proper(X2),proper(X3))
         
         proper(U12(X1,X2,X3)) = 12X1 + 24X2 + 6X3 >= 12X1 + 24X2 + 6X3 = U12(proper(X1),proper(X2),proper(X3))
         
         proper(s(X)) = 18X + 12 >= 18X + 2 = s(proper(X))
         
         proper(plus(X1,X2)) = 24X1 + 12X2 + 24 >= 24X1 + 12X2 + 4 = plus(proper(X1),proper(X2))
         
         U11(ok(X1),ok(X2),ok(X3)) = 8X1 + 4X2 + 4X3 + 17 >= 8X1 + 4X2 + 4X3 + 8 = ok(U11(X1,X2,X3))
         
         U12(ok(X1),ok(X2),ok(X3)) = 8X1 + 16X2 + 4X3 + 28 >= 8X1 + 16X2 + 4X3 + 4 = ok(U12(X1,X2,X3))
         
         s(ok(X)) = 12X + 14 >= 12X + 12 = ok(s(X))
         
         plus(ok(X1),ok(X2)) = 16X1 + 8X2 + 28 >= 16X1 + 8X2 + 20 = ok(plus(X1,X2))
         
         top(ok(X)) = 24X + 25 >= 6X + 1 = top(active(X))
        problem:
         active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
         active(s(X)) -> s(active(X))
         active(plus(X1,X2)) -> plus(active(X1),X2)
         active(plus(X1,X2)) -> plus(X1,active(X2))
         proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
        Matrix Interpretation Processor: dim=3
         
         interpretation:
                         [1 0 0]  
          [proper](x0) = [0 1 0]x0
                         [0 0 0]  ,
          
                    [1 0 0]     [0]
          [s](x0) = [0 1 0]x0 + [1]
                    [0 0 0]     [0],
          
                           [1 0 0]     [1 0 0]     [0]
          [plus](x0, x1) = [0 1 0]x0 + [0 1 1]x1 + [1]
                           [0 0 0]     [0 0 0]     [0],
          
                              [1 0 0]     [1 0 0]     [1 0 0]     [0]
          [U12](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [1]
                              [0 0 0]     [0 0 0]     [0 0 0]     [0],
          
                         [1 1 0]  
          [active](x0) = [0 1 0]x0
                         [0 0 0]  
         orientation:
                                  [1 1 0]     [1 0 0]     [1 0 0]     [1]    [1 1 0]     [1 0 0]     [1 0 0]     [0]                        
          active(U12(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = U12(active(X1),X2,X3)
                                  [0 0 0]     [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0 0 0]     [0]                        
          
                         [1 1 0]    [1]    [1 1 0]    [0]               
          active(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(active(X))
                         [0 0 0]    [0]    [0 0 0]    [0]               
          
                                [1 1 0]     [1 1 1]     [1]    [1 1 0]     [1 0 0]     [0]                      
          active(plus(X1,X2)) = [0 1 0]X1 + [0 1 1]X2 + [1] >= [0 1 0]X1 + [0 1 1]X2 + [1] = plus(active(X1),X2)
                                [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0]                      
          
                                [1 1 0]     [1 1 1]     [1]    [1 0 0]     [1 1 0]     [0]                      
          active(plus(X1,X2)) = [0 1 0]X1 + [0 1 1]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = plus(X1,active(X2))
                                [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0]                      
          
                                  [1 0 0]     [1 0 0]     [1 0 0]     [0]    [1 0 0]     [1 0 0]     [1 0 0]     [0]                                        
          proper(U12(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = U12(proper(X1),proper(X2),proper(X3))
                                  [0 0 0]     [0 0 0]     [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0 0 0]     [0]                                        
         problem:
          proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
         Matrix Interpretation Processor: dim=1
          
          interpretation:
           [proper](x0) = 2x0,
           
           [U12](x0, x1, x2) = 4x0 + 4x1 + 4x2 + 4
          orientation:
           proper(U12(X1,X2,X3)) = 8X1 + 8X2 + 8X3 + 8 >= 8X1 + 8X2 + 8X3 + 4 = U12(proper(X1),proper(X2),proper(X3))
          problem:
           
          Qed