/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(?,O(1))
* Step 1: Sum.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            b(u(x)) -> a(e(x))
            c(u(x)) -> b(x)
            d(x) -> e(u(x))
            d(u(x)) -> c(x)
            v(e(x)) -> x
        - Signature:
            {b/1,c/1,d/1,v/1} / {a/1,e/1,u/1}
        - Obligation:
             runtime complexity wrt. defined symbols {b,c,d,v} and constructors {a,e,u}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: ToInnermost.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            b(u(x)) -> a(e(x))
            c(u(x)) -> b(x)
            d(x) -> e(u(x))
            d(u(x)) -> c(x)
            v(e(x)) -> x
        - Signature:
            {b/1,c/1,d/1,v/1} / {a/1,e/1,u/1}
        - Obligation:
             runtime complexity wrt. defined symbols {b,c,d,v} and constructors {a,e,u}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 3: DependencyPairs.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            b(u(x)) -> a(e(x))
            c(u(x)) -> b(x)
            d(x) -> e(u(x))
            d(u(x)) -> c(x)
            v(e(x)) -> x
        - Signature:
            {b/1,c/1,d/1,v/1} / {a/1,e/1,u/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {b,c,d,v} and constructors {a,e,u}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          b#(u(x)) -> c_1()
          c#(u(x)) -> c_2(b#(x))
          d#(x) -> c_3()
          d#(u(x)) -> c_4(c#(x))
          v#(e(x)) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 4: UsableRules.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            b#(u(x)) -> c_1()
            c#(u(x)) -> c_2(b#(x))
            d#(x) -> c_3()
            d#(u(x)) -> c_4(c#(x))
            v#(e(x)) -> c_5()
        - Weak TRS:
            b(u(x)) -> a(e(x))
            c(u(x)) -> b(x)
            d(x) -> e(u(x))
            d(u(x)) -> c(x)
            v(e(x)) -> x
        - Signature:
            {b/1,c/1,d/1,v/1,b#/1,c#/1,d#/1,v#/1} / {a/1,e/1,u/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {b#,c#,d#,v#} and constructors {a,e,u}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          b#(u(x)) -> c_1()
          c#(u(x)) -> c_2(b#(x))
          d#(x) -> c_3()
          d#(u(x)) -> c_4(c#(x))
          v#(e(x)) -> c_5()
* Step 5: Trivial.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            b#(u(x)) -> c_1()
            c#(u(x)) -> c_2(b#(x))
            d#(x) -> c_3()
            d#(u(x)) -> c_4(c#(x))
            v#(e(x)) -> c_5()
        - Signature:
            {b/1,c/1,d/1,v/1,b#/1,c#/1,d#/1,v#/1} / {a/1,e/1,u/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {b#,c#,d#,v#} and constructors {a,e,u}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:b#(u(x)) -> c_1()
             
          
          2:S:c#(u(x)) -> c_2(b#(x))
             -->_1 b#(u(x)) -> c_1():1
          
          3:S:d#(x) -> c_3()
             
          
          4:S:d#(u(x)) -> c_4(c#(x))
             -->_1 c#(u(x)) -> c_2(b#(x)):2
          
          5:S:v#(e(x)) -> c_5()
             
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 6: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {b/1,c/1,d/1,v/1,b#/1,c#/1,d#/1,v#/1} / {a/1,e/1,u/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {b#,c#,d#,v#} and constructors {a,e,u}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))