/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:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
             runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
             runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
             runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          implies(x,z){z -> or(y,z)} =
            implies(x,or(y,z)) ->^+ or(y,implies(x,z))
              = C[implies(x,z) = implies(x,z){}]

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
             runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + 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: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          implies#(x,or(y,z)) -> c_1(implies#(x,z))
          implies#(not(x),y) -> c_2()
          implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:3: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),y) -> c_2()
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Strict TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          implies#(x,or(y,z)) -> c_1(implies#(x,z))
          implies#(not(x),y) -> c_2()
          implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
** Step 1.b:4: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),y) -> c_2()
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1,3}.
        Here rules are labelled as follows:
          1: implies#(x,or(y,z)) -> c_1(implies#(x,z))
          2: implies#(not(x),y) -> c_2()
          3: implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
** Step 1.b:5: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Weak DPs:
            implies#(not(x),y) -> c_2()
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:implies#(x,or(y,z)) -> c_1(implies#(x,z))
             -->_1 implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z))):2
             -->_1 implies#(not(x),y) -> c_2():3
             -->_1 implies#(x,or(y,z)) -> c_1(implies#(x,z)):1
          
          2:S:implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
             -->_1 implies#(not(x),y) -> c_2():3
             -->_1 implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z))):2
             -->_1 implies#(x,or(y,z)) -> c_1(implies#(x,z)):1
          
          3:W:implies#(not(x),y) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: implies#(not(x),y) -> c_2()
** Step 1.b:6: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          2: implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:6.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {implies#}
        TcT has computed the following interpretation:
           p(implies) = [1] x2 + [0]         
               p(not) = [1] x1 + [2]         
                p(or) = [1] x1 + [1] x2 + [0]
          p(implies#) = [2] x1 + [2] x2 + [0]
               p(c_1) = [1] x1 + [0]         
               p(c_2) = [1]                  
               p(c_3) = [1] x1 + [3]         
        
        Following rules are strictly oriented:
        implies#(not(x),or(y,z)) = [2] x + [2] y + [2] z + [4]
                                 > [2] x + [2] y + [2] z + [3]
                                 = c_3(implies#(y,or(x,z)))   
        
        
        Following rules are (at-least) weakly oriented:
        implies#(x,or(y,z)) =  [2] x + [2] y + [2] z + [0]
                            >= [2] x + [2] z + [0]        
                            =  c_1(implies#(x,z))         
        
*** Step 1.b:6.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
        - Weak DPs:
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 1.b:6.b:1: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
        - Weak DPs:
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: implies#(x,or(y,z)) -> c_1(implies#(x,z))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.b:1.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
        - Weak DPs:
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {implies#}
        TcT has computed the following interpretation:
           p(implies) = [0]         
               p(not) = [1]         
                p(or) = [1] x2 + [2]
          p(implies#) = [4] x2 + [0]
               p(c_1) = [1] x1 + [6]
               p(c_2) = [1]         
               p(c_3) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        implies#(x,or(y,z)) = [4] z + [8]       
                            > [4] z + [6]       
                            = c_1(implies#(x,z))
        
        
        Following rules are (at-least) weakly oriented:
        implies#(not(x),or(y,z)) =  [4] z + [8]             
                                 >= [4] z + [8]             
                                 =  c_3(implies#(y,or(x,z)))
        
**** Step 1.b:6.b:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

**** Step 1.b:6.b:1.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            implies#(x,or(y,z)) -> c_1(implies#(x,z))
            implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:implies#(x,or(y,z)) -> c_1(implies#(x,z))
             -->_1 implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z))):2
             -->_1 implies#(x,or(y,z)) -> c_1(implies#(x,z)):1
          
          2:W:implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
             -->_1 implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z))):2
             -->_1 implies#(x,or(y,z)) -> c_1(implies#(x,z)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: implies#(x,or(y,z)) -> c_1(implies#(x,z))
          2: implies#(not(x),or(y,z)) -> c_3(implies#(y,or(x,z)))
**** Step 1.b:6.b:1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {implies/2,implies#/2} / {not/1,or/2,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies#} and constructors {not,or}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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