* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a(y,x) -> y
            a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
            b(x,y) -> c(a(c(y),a(0(),x)))
        - Signature:
            {a/2,b/2} / {0/0,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,b} and constructors {0,c}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: InnermostRuleRemoval WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a(y,x) -> y
            a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
            b(x,y) -> c(a(c(y),a(0(),x)))
        - Signature:
            {a/2,b/2} / {0/0,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,b} and constructors {0,c}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
        All above mentioned rules can be savely removed.
* Step 3: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a(y,x) -> y
            b(x,y) -> c(a(c(y),a(0(),x)))
        - Signature:
            {a/2,b/2} / {0/0,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,b} and constructors {0,c}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a#(y,x) -> c_1()
          b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 4: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            a#(y,x) -> c_1()
            b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
        - Weak TRS:
            a(y,x) -> y
            b(x,y) -> c(a(c(y),a(0(),x)))
        - Signature:
            {a/2,b/2,a#/2,b#/2} / {0/0,c/1,c_1/0,c_2/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,b#} and constructors {0,c}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          a(y,x) -> y
          a#(y,x) -> c_1()
          b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
* Step 5: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            a#(y,x) -> c_1()
            b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
        - Weak TRS:
            a(y,x) -> y
        - Signature:
            {a/2,b/2,a#/2,b#/2} / {0/0,c/1,c_1/0,c_2/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,b#} and constructors {0,c}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:a#(y,x) -> c_1()
             
          
          2:S:b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
             -->_2 a#(y,x) -> c_1():1
             -->_1 a#(y,x) -> c_1():1
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(y,x) -> y
        - Signature:
            {a/2,b/2,a#/2,b#/2} / {0/0,c/1,c_1/0,c_2/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,b#} and constructors {0,c}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))