*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        b(x,b(z,y)) -> f(b(f(f(z)),c(x,z,y)))
        b(y,z) -> z
        f(c(a(),z,x)) -> b(a(),z)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {b/2,f/1} / {a/0,c/3}
      Obligation:
        Innermost
        basic terms: {b,f}/{a,c}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      Arguments of following rules are not normal-forms.
        b(x,b(z,y)) -> f(b(f(f(z)),c(x,z,y)))
      All above mentioned rules can be savely removed.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        b(y,z) -> z
        f(c(a(),z,x)) -> b(a(),z)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {b/2,f/1} / {a/0,c/3}
      Obligation:
        Innermost
        basic terms: {b,f}/{a,c}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        b#(y,z) -> c_1()
        f#(c(a(),z,x)) -> c_2(b#(a(),z))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        b#(y,z) -> c_1()
        f#(c(a(),z,x)) -> c_2(b#(a(),z))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        b(y,z) -> z
        f(c(a(),z,x)) -> b(a(),z)
      Signature:
        {b/2,f/1,b#/2,f#/1} / {a/0,c/3,c_1/0,c_2/1}
      Obligation:
        Innermost
        basic terms: {b#,f#}/{a,c}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        b#(y,z) -> c_1()
        f#(c(a(),z,x)) -> c_2(b#(a(),z))
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        b#(y,z) -> c_1()
        f#(c(a(),z,x)) -> c_2(b#(a(),z))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {b/2,f/1,b#/2,f#/1} / {a/0,c/3,c_1/0,c_2/1}
      Obligation:
        Innermost
        basic terms: {b#,f#}/{a,c}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:b#(y,z) -> c_1()
           
        
        2:S:f#(c(a(),z,x)) -> c_2(b#(a(),z))
           -->_1 b#(y,z) -> c_1():1
        
      The dependency graph contains no loops, we remove all dependency pairs.
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {b/2,f/1,b#/2,f#/1} / {a/0,c/3,c_1/0,c_2/1}
      Obligation:
        Innermost
        basic terms: {b#,f#}/{a,c}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).