/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(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            concat(cons(U,V),Y) -> cons(U,concat(V,Y))
            concat(leaf(),Y) -> Y
            lessleaves(X,leaf()) -> false()
            lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z))
            lessleaves(leaf(),cons(W,Z)) -> true()
        - Signature:
            {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            concat(cons(U,V),Y) -> cons(U,concat(V,Y))
            concat(leaf(),Y) -> Y
            lessleaves(X,leaf()) -> false()
            lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z))
            lessleaves(leaf(),cons(W,Z)) -> true()
        - Signature:
            {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            concat(cons(U,V),Y) -> cons(U,concat(V,Y))
            concat(leaf(),Y) -> Y
            lessleaves(X,leaf()) -> false()
            lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z))
            lessleaves(leaf(),cons(W,Z)) -> true()
        - Signature:
            {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          concat(y,z){y -> cons(x,y)} =
            concat(cons(x,y),z) ->^+ cons(x,concat(y,z))
              = C[concat(y,z) = concat(y,z){}]

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            concat(cons(U,V),Y) -> cons(U,concat(V,Y))
            concat(leaf(),Y) -> Y
            lessleaves(X,leaf()) -> false()
            lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z))
            lessleaves(leaf(),cons(W,Z)) -> true()
        - Signature:
            {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf,true}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
** Step 1.b:2: Bounds.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            concat(cons(U,V),Y) -> cons(U,concat(V,Y))
            concat(leaf(),Y) -> Y
            lessleaves(X,leaf()) -> false()
            lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z))
            lessleaves(leaf(),cons(W,Z)) -> true()
        - Signature:
            {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf
            ,true}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 2.
        The enriched problem is compatible with follwoing automaton.
          concat_0(2,2) -> 1
          concat_1(2,2) -> 3
          concat_1(2,2) -> 4
          concat_1(2,2) -> 5
          concat_1(2,3) -> 3
          concat_1(2,3) -> 4
          concat_1(2,3) -> 5
          concat_2(2,3) -> 4
          concat_2(2,3) -> 5
          cons_0(2,2) -> 1
          cons_0(2,2) -> 2
          cons_0(2,2) -> 3
          cons_0(2,2) -> 4
          cons_0(2,2) -> 5
          cons_1(2,3) -> 1
          cons_1(2,3) -> 3
          cons_1(2,3) -> 4
          cons_1(2,3) -> 5
          false_0() -> 1
          false_0() -> 2
          false_0() -> 3
          false_0() -> 4
          false_0() -> 5
          false_1() -> 1
          leaf_0() -> 1
          leaf_0() -> 2
          leaf_0() -> 3
          leaf_0() -> 4
          leaf_0() -> 5
          lessleaves_0(2,2) -> 1
          lessleaves_1(3,3) -> 1
          lessleaves_2(4,5) -> 1
          true_0() -> 1
          true_0() -> 2
          true_0() -> 3
          true_0() -> 4
          true_0() -> 5
          true_1() -> 1
          2 -> 1
          2 -> 3
          2 -> 4
          2 -> 5
          3 -> 4
          3 -> 5
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            concat(cons(U,V),Y) -> cons(U,concat(V,Y))
            concat(leaf(),Y) -> Y
            lessleaves(X,leaf()) -> false()
            lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z))
            lessleaves(leaf(),cons(W,Z)) -> true()
        - Signature:
            {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))