*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {b/1,c/1,d/1,v/1} / {a/1,e/1,u/1}
      Obligation:
        Innermost
        basic terms: {b,c,d,v}/{a,e,u}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      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.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP 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()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {b#,c#,d#,v#}/{a,e,u}
    Applied Processor:
      UsableRules
    Proof:
      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()
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP 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()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {b#,c#,d#,v#}/{a,e,u}
    Applied Processor:
      Trivial
    Proof:
      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.
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {b#,c#,d#,v#}/{a,e,u}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).