WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs 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:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          implies#(x,or(y,z)) -> c_1(implies#(x,z))
          implies#(not(x),y) -> c_2()
          implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),y) -> c_2()
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - 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,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1,3}.
        Here rules are labelled as follows:
          1: implies#(x,or(y,z)) -> c_1(implies#(x,z))
          2: implies#(not(x),y) -> c_2()
          3: implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Weak DPs:
            implies#(not(x),y) -> c_2()
        - 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,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:implies#(x,or(y,z)) -> c_1(implies#(x,z))
             -->_1 implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z))):2
             -->_1 implies#(not(x),y) -> c_2():3
             -->_1 implies#(x,or(y,z)) -> c_1(implies#(x,z)):1
          
          2:S:implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
             -->_1 implies#(not(x),y) -> c_2():3
             -->_1 implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z))):2
             -->_1 implies#(x,or(y,z)) -> c_1(implies#(x,z)):1
          
          3:W:implies#(not(x),y) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: implies#(not(x),y) -> c_2()
* Step 4: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - 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,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          implies#(x,or(y,z)) -> c_1(implies#(x,z))
          implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
* Step 5: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
             p(implies) = [0]                  
                 p(not) = [1] x1 + [1]         
                  p(or) = [1] x1 + [1] x2 + [1]
            p(implies#) = [9] x1 + [9] x2 + [9]
                 p(c_1) = [1] x1 + [3]         
                 p(c_2) = [0]                  
                 p(c_3) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
               implies#(x,or(y,z)) = [9] x + [9] y + [9] z + [18]
                                   > [9] x + [9] z + [12]        
                                   = c_1(implies#(x,z))          
          
          implies#(not(x),or(y,z)) = [9] x + [9] y + [9] z + [27]
                                   > [9] x + [9] y + [9] z + [18]
                                   = c_3(implies#(y,or(x,z)))    
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - 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(?,O(n^1))