* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(e(x),e(y)) -> e(g(x,y))
            h(e(x),y) -> h(d(x,y),s(y))
        - Signature:
            {g/2,h/2} / {d/2,e/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g,h} and constructors {d,e,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          g#(e(x),e(y)) -> c_1(g#(x,y))
          h#(e(x),y) -> c_2(h#(d(x,y),s(y)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(e(x),e(y)) -> c_1(g#(x,y))
            h#(e(x),y) -> c_2(h#(d(x,y),s(y)))
        - Strict TRS:
            g(e(x),e(y)) -> e(g(x,y))
            h(e(x),y) -> h(d(x,y),s(y))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {d/2,e/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {d,e,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g#(e(x),e(y)) -> c_1(g#(x,y))
          h#(e(x),y) -> c_2(h#(d(x,y),s(y)))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(e(x),e(y)) -> c_1(g#(x,y))
            h#(e(x),y) -> c_2(h#(d(x,y),s(y)))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {d/2,e/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {d,e,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {}.
        Here rules are labelled as follows:
          1: g#(e(x),e(y)) -> c_1(g#(x,y))
          2: h#(e(x),y) -> c_2(h#(d(x,y),s(y)))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(e(x),e(y)) -> c_1(g#(x,y))
        - Weak DPs:
            h#(e(x),y) -> c_2(h#(d(x,y),s(y)))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {d/2,e/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {d,e,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:g#(e(x),e(y)) -> c_1(g#(x,y))
             -->_1 g#(e(x),e(y)) -> c_1(g#(x,y)):1
          
          2:W:h#(e(x),y) -> c_2(h#(d(x,y),s(y)))
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: h#(e(x),y) -> c_2(h#(d(x,y),s(y)))
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(e(x),e(y)) -> c_1(g#(x,y))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {d/2,e/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {d,e,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g#(e(x),e(y)) -> c_1(g#(x,y))
          
        The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(e(x),e(y)) -> c_1(g#(x,y))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {d/2,e/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {d,e,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1}
        
        Following symbols are considered usable:
          {g#,h#}
        TcT has computed the following interpretation:
            p(d) = [1] x2 + [0]         
            p(e) = [1] x1 + [4]         
            p(g) = [2]                  
            p(h) = [1] x1 + [8]         
            p(s) = [0]                  
           p(g#) = [4] x2 + [0]         
           p(h#) = [1] x1 + [2] x2 + [1]
          p(c_1) = [1] x1 + [15]        
          p(c_2) = [2] x1 + [2]         
        
        Following rules are strictly oriented:
        g#(e(x),e(y)) = [4] y + [16]
                      > [4] y + [15]
                      = c_1(g#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 5.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(e(x),e(y)) -> c_1(g#(x,y))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {d/2,e/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {d,e,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(e(x),e(y)) -> c_1(g#(x,y))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {d/2,e/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {d,e,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:g#(e(x),e(y)) -> c_1(g#(x,y))
             -->_1 g#(e(x),e(y)) -> c_1(g#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g#(e(x),e(y)) -> c_1(g#(x,y))
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {d/2,e/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {d,e,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))