WORST_CASE(?,O(n^2))
* Step 1: LocalSizeboundsProc WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B,C) -> f1(A,1 + B,C) [A >= 1 + B]           (?,1)
          1. f3(A,B,C) -> f1(A,B,C)     [B >= 1 + C]           (1,1)
          2. f1(A,B,C) -> f1(A,B,1 + C) [B >= 2 + C && B >= A] (?,1)
        Signature:
          {(f1,3);(f3,3)}
        Flow Graph:
          [0->{0,2},1->{0,2},2->{0,2}]
        
    + Applied Processor:
        LocalSizeboundsProc
    + Details:
        LocalSizebounds generated; rvgraph
          (<0,0,A>, A, .= 0) (<0,0,B>, 1 + B, .+ 1) (<0,0,C>,     C, .= 0) 
          (<1,0,A>, A, .= 0) (<1,0,B>,     B, .= 0) (<1,0,C>,     C, .= 0) 
          (<2,0,A>, A, .= 0) (<2,0,B>,     B, .= 0) (<2,0,C>, 1 + C, .+ 1) 
* Step 2: SizeboundsProc WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B,C) -> f1(A,1 + B,C) [A >= 1 + B]           (?,1)
          1. f3(A,B,C) -> f1(A,B,C)     [B >= 1 + C]           (1,1)
          2. f1(A,B,C) -> f1(A,B,1 + C) [B >= 2 + C && B >= A] (?,1)
        Signature:
          {(f1,3);(f3,3)}
        Flow Graph:
          [0->{0,2},1->{0,2},2->{0,2}]
        Sizebounds:
          (<0,0,A>, ?) (<0,0,B>, ?) (<0,0,C>, ?) 
          (<1,0,A>, ?) (<1,0,B>, ?) (<1,0,C>, ?) 
          (<2,0,A>, ?) (<2,0,B>, ?) (<2,0,C>, ?) 
    + Applied Processor:
        SizeboundsProc
    + Details:
        Sizebounds computed:
          (<0,0,A>, A) (<0,0,B>,     A) (<0,0,C>, A + B + C) 
          (<1,0,A>, A) (<1,0,B>,     B) (<1,0,C>,         C) 
          (<2,0,A>, A) (<2,0,B>, A + B) (<2,0,C>,     A + B) 
* Step 3: UnsatPaths WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B,C) -> f1(A,1 + B,C) [A >= 1 + B]           (?,1)
          1. f3(A,B,C) -> f1(A,B,C)     [B >= 1 + C]           (1,1)
          2. f1(A,B,C) -> f1(A,B,1 + C) [B >= 2 + C && B >= A] (?,1)
        Signature:
          {(f1,3);(f3,3)}
        Flow Graph:
          [0->{0,2},1->{0,2},2->{0,2}]
        Sizebounds:
          (<0,0,A>, A) (<0,0,B>,     A) (<0,0,C>, A + B + C) 
          (<1,0,A>, A) (<1,0,B>,     B) (<1,0,C>,         C) 
          (<2,0,A>, A) (<2,0,B>, A + B) (<2,0,C>,     A + B) 
    + Applied Processor:
        UnsatPaths
    + Details:
        We remove following edges from the transition graph: [(2,0)]
* Step 4: PolyRank WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B,C) -> f1(A,1 + B,C) [A >= 1 + B]           (?,1)
          1. f3(A,B,C) -> f1(A,B,C)     [B >= 1 + C]           (1,1)
          2. f1(A,B,C) -> f1(A,B,1 + C) [B >= 2 + C && B >= A] (?,1)
        Signature:
          {(f1,3);(f3,3)}
        Flow Graph:
          [0->{0,2},1->{0,2},2->{2}]
        Sizebounds:
          (<0,0,A>, A) (<0,0,B>,     A) (<0,0,C>, A + B + C) 
          (<1,0,A>, A) (<1,0,B>,     B) (<1,0,C>,         C) 
          (<2,0,A>, A) (<2,0,B>, A + B) (<2,0,C>,     A + B) 
    + Applied Processor:
        PolyRank {useFarkas = True, withSizebounds = [], shape = Linear}
    + Details:
        We apply a polynomial interpretation of shape linear:
          p(f1) = 1 + x1 + -1*x2
          p(f3) = 1 + x1 + -1*x2
        
        The following rules are strictly oriented:
        [A >= 1 + B] ==>              
           f1(A,B,C)   = 1 + A + -1*B 
                       > A + -1*B     
                       = f1(A,1 + B,C)
        
        
        The following rules are weakly oriented:
                  [B >= 1 + C] ==>              
                     f3(A,B,C)   = 1 + A + -1*B 
                                >= 1 + A + -1*B 
                                 = f1(A,B,C)    
        
        [B >= 2 + C && B >= A] ==>              
                     f1(A,B,C)   = 1 + A + -1*B 
                                >= 1 + A + -1*B 
                                 = f1(A,B,1 + C)
        
        
* Step 5: PolyRank WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B,C) -> f1(A,1 + B,C) [A >= 1 + B]           (1 + A + B,1)
          1. f3(A,B,C) -> f1(A,B,C)     [B >= 1 + C]           (1,1)        
          2. f1(A,B,C) -> f1(A,B,1 + C) [B >= 2 + C && B >= A] (?,1)        
        Signature:
          {(f1,3);(f3,3)}
        Flow Graph:
          [0->{0,2},1->{0,2},2->{2}]
        Sizebounds:
          (<0,0,A>, A) (<0,0,B>,     A) (<0,0,C>, A + B + C) 
          (<1,0,A>, A) (<1,0,B>,     B) (<1,0,C>,         C) 
          (<2,0,A>, A) (<2,0,B>, A + B) (<2,0,C>,     A + B) 
    + Applied Processor:
        PolyRank {useFarkas = True, withSizebounds = [2], shape = Linear}
    + Details:
        We apply a polynomial interpretation of shape linear:
          p(f1) = -1 + x2 + -1*x3
        
        The following rules are strictly oriented:
        [B >= 2 + C && B >= A] ==>              
                     f1(A,B,C)   = -1 + B + -1*C
                                 > -2 + B + -1*C
                                 = f1(A,B,1 + C)
        
        
        The following rules are weakly oriented:
        
        We use the following global sizebounds:
        (<0,0,A>, A) (<0,0,B>,     A) (<0,0,C>, A + B + C) 
        (<1,0,A>, A) (<1,0,B>,     B) (<1,0,C>,         C) 
        (<2,0,A>, A) (<2,0,B>, A + B) (<2,0,C>,     A + B) 
* Step 6: KnowledgePropagation WORST_CASE(?,O(n^2))
    + Considered Problem:
        Rules:
          0. f1(A,B,C) -> f1(A,1 + B,C) [A >= 1 + B]           (1 + A + B,1)                                            
          1. f3(A,B,C) -> f1(A,B,C)     [B >= 1 + C]           (1,1)                                                    
          2. f1(A,B,C) -> f1(A,B,1 + C) [B >= 2 + C && B >= A] (2 + 3*A + 3*A*B + A*C + 2*A^2 + 3*B + B*C + B^2 + 2*C,1)
        Signature:
          {(f1,3);(f3,3)}
        Flow Graph:
          [0->{0,2},1->{0,2},2->{2}]
        Sizebounds:
          (<0,0,A>, A) (<0,0,B>,     A) (<0,0,C>, A + B + C) 
          (<1,0,A>, A) (<1,0,B>,     B) (<1,0,C>,         C) 
          (<2,0,A>, A) (<2,0,B>, A + B) (<2,0,C>,     A + B) 
    + Applied Processor:
        KnowledgePropagation
    + Details:
        The problem is already solved.

WORST_CASE(?,O(n^2))