* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            circ(s,id()) -> s
            circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
            circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
            circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
            circ(id(),s) -> s
            msubst(a,id()) -> a
            subst(a,id()) -> a
        - Signature:
            {circ/2,msubst/2,subst/2} / {cons/2,id/0,lift/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ,msubst,subst} and constructors {cons,id,lift}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          circ#(s,id()) -> c_1()
          circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
          circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
          circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
          circ#(id(),s) -> c_5()
          msubst#(a,id()) -> c_6()
          subst#(a,id()) -> c_7()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            circ#(s,id()) -> c_1()
            circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
            circ#(id(),s) -> c_5()
            msubst#(a,id()) -> c_6()
            subst#(a,id()) -> c_7()
        - Strict TRS:
            circ(s,id()) -> s
            circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
            circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
            circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
            circ(id(),s) -> s
            msubst(a,id()) -> a
            subst(a,id()) -> a
        - Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0
            ,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          circ#(s,id()) -> c_1()
          circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
          circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
          circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
          circ#(id(),s) -> c_5()
          msubst#(a,id()) -> c_6()
          subst#(a,id()) -> c_7()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            circ#(s,id()) -> c_1()
            circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
            circ#(id(),s) -> c_5()
            msubst#(a,id()) -> c_6()
            subst#(a,id()) -> c_7()
        - Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0
            ,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,5,6,7}
        by application of
          Pre({1,5,6,7}) = {2,3,4}.
        Here rules are labelled as follows:
          1: circ#(s,id()) -> c_1()
          2: circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
          3: circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
          4: circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
          5: circ#(id(),s) -> c_5()
          6: msubst#(a,id()) -> c_6()
          7: subst#(a,id()) -> c_7()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
        - Weak DPs:
            circ#(s,id()) -> c_1()
            circ#(id(),s) -> c_5()
            msubst#(a,id()) -> c_6()
            subst#(a,id()) -> c_7()
        - Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0
            ,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
             -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
             -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
             -->_1 msubst#(a,id()) -> c_6():6
             -->_2 circ#(id(),s) -> c_5():5
             -->_2 circ#(s,id()) -> c_1():4
             -->_2 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1
          
          2:S:circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
             -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
             -->_1 circ#(id(),s) -> c_5():5
             -->_1 circ#(s,id()) -> c_1():4
             -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
             -->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1
          
          3:S:circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
             -->_1 circ#(id(),s) -> c_5():5
             -->_1 circ#(s,id()) -> c_1():4
             -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
             -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
             -->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1
          
          4:W:circ#(s,id()) -> c_1()
             
          
          5:W:circ#(id(),s) -> c_5()
             
          
          6:W:msubst#(a,id()) -> c_6()
             
          
          7:W:subst#(a,id()) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: subst#(a,id()) -> c_7()
          6: msubst#(a,id()) -> c_6()
          4: circ#(s,id()) -> c_1()
          5: circ#(id(),s) -> c_5()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
        - Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0
            ,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t))
             -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
             -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
             -->_2 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1
          
          2:S:circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
             -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
             -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
             -->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1
          
          3:S:circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
             -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
             -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
             -->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          circ#(cons(a,s),t) -> c_2(circ#(s,t))
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            circ#(cons(a,s),t) -> c_2(circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
        - Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
    + 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: circ#(cons(a,s),t) -> c_2(circ#(s,t))
          2: circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
          3: circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
          
        The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            circ#(cons(a,s),t) -> c_2(circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
        - Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
    + 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_2) = {1},
          uargs(c_3) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {circ#,msubst#,subst#}
        TcT has computed the following interpretation:
             p(circ) = [1] x2 + [1]         
             p(cons) = [1] x1 + [1] x2 + [1]
               p(id) = [1]                  
             p(lift) = [3]                  
           p(msubst) = [8] x1 + [1] x2 + [1]
            p(subst) = [2] x1 + [4]         
            p(circ#) = [1] x1 + [1] x2 + [1]
          p(msubst#) = [1] x1 + [0]         
           p(subst#) = [1] x2 + [0]         
              p(c_1) = [8]                  
              p(c_2) = [1] x1 + [0]         
              p(c_3) = [1] x1 + [0]         
              p(c_4) = [1] x1 + [5]         
              p(c_5) = [1]                  
              p(c_6) = [1]                  
              p(c_7) = [0]                  
        
        Following rules are strictly oriented:
                          circ#(cons(a,s),t) = [1] a + [1] s + [1] t + [2]
                                             > [1] s + [1] t + [1]        
                                             = c_2(circ#(s,t))            
        
             circ#(cons(lift(),s),cons(a,t)) = [1] a + [1] s + [1] t + [6]
                                             > [1] s + [1] t + [1]        
                                             = c_3(circ#(s,t))            
        
        circ#(cons(lift(),s),cons(lift(),t)) = [1] s + [1] t + [9]        
                                             > [1] s + [1] t + [6]        
                                             = c_4(circ#(s,t))            
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            circ#(cons(a,s),t) -> c_2(circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
        - Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            circ#(cons(a,s),t) -> c_2(circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
        - Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:circ#(cons(a,s),t) -> c_2(circ#(s,t))
             -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
             -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
             -->_1 circ#(cons(a,s),t) -> c_2(circ#(s,t)):1
          
          2:W:circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
             -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
             -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
             -->_1 circ#(cons(a,s),t) -> c_2(circ#(s,t)):1
          
          3:W:circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
             -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3
             -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2
             -->_1 circ#(cons(a,s),t) -> c_2(circ#(s,t)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: circ#(cons(a,s),t) -> c_2(circ#(s,t))
          3: circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t))
          2: circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
            ,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))