WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          implies(x,z){z -> or(y,z)} =
            implies(x,or(y,z)) ->^+ or(y,implies(x,z))
              = C[implies(x,z) = implies(x,z){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(or) = {2}
        
        Following symbols are considered usable:
          {implies}
        TcT has computed the following interpretation:
          p(implies) = 13 + 8*x2
              p(not) = 0        
               p(or) = 2 + x2   
        
        Following rules are strictly oriented:
        implies(x,or(y,z)) = 29 + 8*z          
                           > 15 + 8*z          
                           = or(y,implies(x,z))
        
         implies(not(x),y) = 13 + 8*y          
                           > 2 + y             
                           = or(x,y)           
        
        
        Following rules are (at-least) weakly oriented:
        implies(not(x),or(y,z)) =  29 + 8*z          
                                >= 29 + 8*z          
                                =  implies(y,or(x,z))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Weak TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(or) = {2}
        
        Following symbols are considered usable:
          {implies}
        TcT has computed the following interpretation:
          p(implies) = 6 + 2*x1 + 2*x2
              p(not) = 1 + x1         
               p(or) = x1 + x2        
        
        Following rules are strictly oriented:
        implies(not(x),or(y,z)) = 8 + 2*x + 2*y + 2*z
                                > 6 + 2*x + 2*y + 2*z
                                = implies(y,or(x,z)) 
        
        
        Following rules are (at-least) weakly oriented:
        implies(x,or(y,z)) =  6 + 2*x + 2*y + 2*z
                           >= 6 + 2*x + y + 2*z  
                           =  or(y,implies(x,z)) 
        
         implies(not(x),y) =  8 + 2*x + 2*y      
                           >= x + y              
                           =  or(x,y)            
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))