WORST_CASE(?,O(n^2))
* Step 1: LocalSizeboundsProc WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B) -> f3(A,A)           [A >= 1]           (1,1)
          1. f3(A,B) -> f3(A,-1 + B)      [B >= 1]           (?,1)
          2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (?,1)
        Signature:
          {(f1,2);(f3,2)}
        Flow Graph:
          [0->{1,2},1->{1,2},2->{1,2}]
        
    + Applied Processor:
        LocalSizeboundsProc
    + Details:
        LocalSizebounds generated; rvgraph
          (<0,0,A>,     A, .= 0) (<0,0,B>,     A, .= 0) 
          (<1,0,A>,     A, .= 0) (<1,0,B>, 1 + B, .+ 1) 
          (<2,0,A>, 1 + A, .+ 1) (<2,0,B>, 1 + A, .+ 1) 
* Step 2: SizeboundsProc WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B) -> f3(A,A)           [A >= 1]           (1,1)
          1. f3(A,B) -> f3(A,-1 + B)      [B >= 1]           (?,1)
          2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (?,1)
        Signature:
          {(f1,2);(f3,2)}
        Flow Graph:
          [0->{1,2},1->{1,2},2->{1,2}]
        Sizebounds:
          (<0,0,A>, ?) (<0,0,B>, ?) 
          (<1,0,A>, ?) (<1,0,B>, ?) 
          (<2,0,A>, ?) (<2,0,B>, ?) 
    + Applied Processor:
        SizeboundsProc
    + Details:
        Sizebounds computed:
          (<0,0,A>, A) (<0,0,B>, A) 
          (<1,0,A>, ?) (<1,0,B>, ?) 
          (<2,0,A>, ?) (<2,0,B>, ?) 
* Step 3: UnsatPaths WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B) -> f3(A,A)           [A >= 1]           (1,1)
          1. f3(A,B) -> f3(A,-1 + B)      [B >= 1]           (?,1)
          2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (?,1)
        Signature:
          {(f1,2);(f3,2)}
        Flow Graph:
          [0->{1,2},1->{1,2},2->{1,2}]
        Sizebounds:
          (<0,0,A>, A) (<0,0,B>, A) 
          (<1,0,A>, ?) (<1,0,B>, ?) 
          (<2,0,A>, ?) (<2,0,B>, ?) 
    + Applied Processor:
        UnsatPaths
    + Details:
        We remove following edges from the transition graph: [(0,2),(2,2)]
* Step 4: PolyRank WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B) -> f3(A,A)           [A >= 1]           (1,1)
          1. f3(A,B) -> f3(A,-1 + B)      [B >= 1]           (?,1)
          2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (?,1)
        Signature:
          {(f1,2);(f3,2)}
        Flow Graph:
          [0->{1},1->{1,2},2->{1}]
        Sizebounds:
          (<0,0,A>, A) (<0,0,B>, A) 
          (<1,0,A>, ?) (<1,0,B>, ?) 
          (<2,0,A>, ?) (<2,0,B>, ?) 
    + Applied Processor:
        PolyRank {useFarkas = True, withSizebounds = [], shape = Linear}
    + Details:
        We apply a polynomial interpretation of shape linear:
          p(f1) = x1     
          p(f3) = -1 + x1
        
        The following rules are strictly oriented:
                  [A >= 1] ==>                  
                   f1(A,B)   = A                
                             > -1 + A           
                             = f3(A,A)          
        
        [0 >= B && A >= 2] ==>                  
                   f3(A,B)   = -1 + A           
                             > -2 + A           
                             = f3(-1 + A,-1 + A)
        
        
        The following rules are weakly oriented:
         [B >= 1] ==>             
          f3(A,B)   = -1 + A      
                   >= -1 + A      
                    = f3(A,-1 + B)
        
        
* Step 5: SizeboundsProc WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B) -> f3(A,A)           [A >= 1]           (1,1)
          1. f3(A,B) -> f3(A,-1 + B)      [B >= 1]           (?,1)
          2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (A,1)
        Signature:
          {(f1,2);(f3,2)}
        Flow Graph:
          [0->{1},1->{1,2},2->{1}]
        Sizebounds:
          (<0,0,A>, A) (<0,0,B>, A) 
          (<1,0,A>, ?) (<1,0,B>, ?) 
          (<2,0,A>, ?) (<2,0,B>, ?) 
    + Applied Processor:
        SizeboundsProc
    + Details:
        Sizebounds computed:
          (<0,0,A>,   A) (<0,0,B>,       A) 
          (<1,0,A>, 2*A) (<1,0,B>,       ?) 
          (<2,0,A>, 2*A) (<2,0,B>, 1 + 2*A) 
* Step 6: PolyRank WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B) -> f3(A,A)           [A >= 1]           (1,1)
          1. f3(A,B) -> f3(A,-1 + B)      [B >= 1]           (?,1)
          2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (A,1)
        Signature:
          {(f1,2);(f3,2)}
        Flow Graph:
          [0->{1},1->{1,2},2->{1}]
        Sizebounds:
          (<0,0,A>,   A) (<0,0,B>,       A) 
          (<1,0,A>, 2*A) (<1,0,B>,       ?) 
          (<2,0,A>, 2*A) (<2,0,B>, 1 + 2*A) 
    + Applied Processor:
        PolyRank {useFarkas = True, withSizebounds = [1], shape = Linear}
    + Details:
        We apply a polynomial interpretation of shape linear:
          p(f3) = x2
        
        The following rules are strictly oriented:
         [B >= 1] ==>             
          f3(A,B)   = B           
                    > -1 + B      
                    = f3(A,-1 + B)
        
        
        The following rules are weakly oriented:
        
        We use the following global sizebounds:
        (<0,0,A>,   A) (<0,0,B>,       A) 
        (<1,0,A>, 2*A) (<1,0,B>,       ?) 
        (<2,0,A>, 2*A) (<2,0,B>, 1 + 2*A) 
* Step 7: KnowledgePropagation WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B) -> f3(A,A)           [A >= 1]           (1,1)          
          1. f3(A,B) -> f3(A,-1 + B)      [B >= 1]           (2*A + 2*A^2,1)
          2. f3(A,B) -> f3(-1 + A,-1 + A) [0 >= B && A >= 2] (A,1)          
        Signature:
          {(f1,2);(f3,2)}
        Flow Graph:
          [0->{1},1->{1,2},2->{1}]
        Sizebounds:
          (<0,0,A>,   A) (<0,0,B>,       A) 
          (<1,0,A>, 2*A) (<1,0,B>,       ?) 
          (<2,0,A>, 2*A) (<2,0,B>, 1 + 2*A) 
    + Applied Processor:
        KnowledgePropagation
    + Details:
        The problem is already solved.

WORST_CASE(?,O(n^2))