/bin/sh: line 1:  2727 Quit                    (core dumped) z3 -T:14.25 -smt2 /tmp/SMTP2596-28 > /tmp/SMTS2596-29
/bin/sh: line 1:  2734 Quit                    (core dumped) z3 -T:14.25 -smt2 /tmp/SMTP2596-44 > /tmp/SMTS2596-45
/bin/sh: line 1:  2747 Quit                    (core dumped) z3 -T:14.25 -smt2 /tmp/SMTP2596-70 > /tmp/SMTS2596-71
WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            eq0(0(),0()) -> S(0())
            eq0(0(),S(x)) -> 0()
            eq0(S(x),0()) -> 0()
            eq0(S(x'),S(x)) -> eq0(x',x)
        - Signature:
            {eq0/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq0} and constructors {0,S}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            eq0(0(),0()) -> S(0())
            eq0(0(),S(x)) -> 0()
            eq0(S(x),0()) -> 0()
            eq0(S(x'),S(x)) -> eq0(x',x)
        - Signature:
            {eq0/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq0} and constructors {0,S}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            eq0(0(),0()) -> S(0())
            eq0(0(),S(x)) -> 0()
            eq0(S(x),0()) -> 0()
            eq0(S(x'),S(x)) -> eq0(x',x)
        - Signature:
            {eq0/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq0} and constructors {0,S}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          eq0(x,y){x -> S(x),y -> S(y)} =
            eq0(S(x),S(y)) ->^+ eq0(x,y)
              = C[eq0(x,y) = eq0(x,y){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            eq0(0(),0()) -> S(0())
            eq0(0(),S(x)) -> 0()
            eq0(S(x),0()) -> 0()
            eq0(S(x'),S(x)) -> eq0(x',x)
        - Signature:
            {eq0/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq0} and constructors {0,S}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          eq0#(0(),0()) -> c_1()
          eq0#(0(),S(x)) -> c_2()
          eq0#(S(x),0()) -> c_3()
          eq0#(S(x'),S(x)) -> c_4(eq0#(x',x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            eq0#(0(),0()) -> c_1()
            eq0#(0(),S(x)) -> c_2()
            eq0#(S(x),0()) -> c_3()
            eq0#(S(x'),S(x)) -> c_4(eq0#(x',x))
        - Weak TRS:
            eq0(0(),0()) -> S(0())
            eq0(0(),S(x)) -> 0()
            eq0(S(x),0()) -> 0()
            eq0(S(x'),S(x)) -> eq0(x',x)
        - Signature:
            {eq0/2,eq0#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq0#} and constructors {0,S}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3}
        by application of
          Pre({1,2,3}) = {4}.
        Here rules are labelled as follows:
          1: eq0#(0(),0()) -> c_1()
          2: eq0#(0(),S(x)) -> c_2()
          3: eq0#(S(x),0()) -> c_3()
          4: eq0#(S(x'),S(x)) -> c_4(eq0#(x',x))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            eq0#(S(x'),S(x)) -> c_4(eq0#(x',x))
        - Weak DPs:
            eq0#(0(),0()) -> c_1()
            eq0#(0(),S(x)) -> c_2()
            eq0#(S(x),0()) -> c_3()
        - Weak TRS:
            eq0(0(),0()) -> S(0())
            eq0(0(),S(x)) -> 0()
            eq0(S(x),0()) -> 0()
            eq0(S(x'),S(x)) -> eq0(x',x)
        - Signature:
            {eq0/2,eq0#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq0#} and constructors {0,S}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:eq0#(S(x'),S(x)) -> c_4(eq0#(x',x))
             -->_1 eq0#(S(x),0()) -> c_3():4
             -->_1 eq0#(0(),S(x)) -> c_2():3
             -->_1 eq0#(0(),0()) -> c_1():2
             -->_1 eq0#(S(x'),S(x)) -> c_4(eq0#(x',x)):1
          
          2:W:eq0#(0(),0()) -> c_1()
             
          
          3:W:eq0#(0(),S(x)) -> c_2()
             
          
          4:W:eq0#(S(x),0()) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: eq0#(0(),0()) -> c_1()
          3: eq0#(0(),S(x)) -> c_2()
          4: eq0#(S(x),0()) -> c_3()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            eq0#(S(x'),S(x)) -> c_4(eq0#(x',x))
        - Weak TRS:
            eq0(0(),0()) -> S(0())
            eq0(0(),S(x)) -> 0()
            eq0(S(x),0()) -> 0()
            eq0(S(x'),S(x)) -> eq0(x',x)
        - Signature:
            {eq0/2,eq0#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq0#} and constructors {0,S}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          eq0#(S(x'),S(x)) -> c_4(eq0#(x',x))
** Step 1.b:5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            eq0#(S(x'),S(x)) -> c_4(eq0#(x',x))
        - Signature:
            {eq0/2,eq0#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq0#} and constructors {0,S}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {eq0#}
        TcT has computed the following interpretation:
             p(0) = [0]         
             p(S) = [1] x1 + [8]
           p(eq0) = [4] x1 + [1]
          p(eq0#) = [2] x1 + [4]
           p(c_1) = [2]         
           p(c_2) = [1]         
           p(c_3) = [1]         
           p(c_4) = [1] x1 + [8]
        
        Following rules are strictly oriented:
        eq0#(S(x'),S(x)) = [2] x' + [20]  
                         > [2] x' + [12]  
                         = c_4(eq0#(x',x))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:6: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            eq0#(S(x'),S(x)) -> c_4(eq0#(x',x))
        - Signature:
            {eq0/2,eq0#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq0#} and constructors {0,S}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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