WORST_CASE(?,O(1))
* Step 1: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            gcd(x,0()) -> x
            gcd(0(),y) -> y
            gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))
        - Signature:
            {gcd/2} / {-/2,0/0,</2,if/3,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd} and constructors {-,0,<,if,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          gcd#(x,0()) -> c_1()
          gcd#(0(),y) -> c_2()
          gcd#(s(x),s(y)) -> c_3(gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            gcd#(x,0()) -> c_1()
            gcd#(0(),y) -> c_2()
            gcd#(s(x),s(y)) -> c_3(gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y)))
        - Weak TRS:
            gcd(x,0()) -> x
            gcd(0(),y) -> y
            gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))
        - Signature:
            {gcd/2,gcd#/2} / {-/2,0/0,</2,if/3,s/1,c_1/0,c_2/0,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#} and constructors {-,0,<,if,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          gcd#(x,0()) -> c_1()
          gcd#(0(),y) -> c_2()
          gcd#(s(x),s(y)) -> c_3(gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y)))
* Step 3: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            gcd#(x,0()) -> c_1()
            gcd#(0(),y) -> c_2()
            gcd#(s(x),s(y)) -> c_3(gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y)))
        - Signature:
            {gcd/2,gcd#/2} / {-/2,0/0,</2,if/3,s/1,c_1/0,c_2/0,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#} and constructors {-,0,<,if,s}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:gcd#(x,0()) -> c_1()
             
          
          2:S:gcd#(0(),y) -> c_2()
             
          
          3:S:gcd#(s(x),s(y)) -> c_3(gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y)))
             
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {gcd/2,gcd#/2} / {-/2,0/0,</2,if/3,s/1,c_1/0,c_2/0,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#} and constructors {-,0,<,if,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))