*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a(y,x) -> y
        a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
        b(x,y) -> c(a(c(y),a(0(),x)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a/2,b/2} / {0/0,c/1}
      Obligation:
        Innermost
        basic terms: {a,b}/{0,c}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      Arguments of following rules are not normal-forms.
        a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
      All above mentioned rules can be savely removed.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        a(y,x) -> y
        b(x,y) -> c(a(c(y),a(0(),x)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {a/2,b/2} / {0/0,c/1}
      Obligation:
        Innermost
        basic terms: {a,b}/{0,c}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        a#(y,x) -> c_1()
        b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        a#(y,x) -> c_1()
        b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a(y,x) -> y
        b(x,y) -> c(a(c(y),a(0(),x)))
      Signature:
        {a/2,b/2,a#/2,b#/2} / {0/0,c/1,c_1/0,c_2/2}
      Obligation:
        Innermost
        basic terms: {a#,b#}/{0,c}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        a(y,x) -> y
        a#(y,x) -> c_1()
        b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        a#(y,x) -> c_1()
        b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        a(y,x) -> y
      Signature:
        {a/2,b/2,a#/2,b#/2} / {0/0,c/1,c_1/0,c_2/2}
      Obligation:
        Innermost
        basic terms: {a#,b#}/{0,c}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:a#(y,x) -> c_1()
           
        
        2:S:b#(x,y) -> c_2(a#(c(y),a(0(),x)),a#(0(),x))
           -->_2 a#(y,x) -> c_1():1
           -->_1 a#(y,x) -> 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:
        a(y,x) -> y
      Signature:
        {a/2,b/2,a#/2,b#/2} / {0/0,c/1,c_1/0,c_2/2}
      Obligation:
        Innermost
        basic terms: {a#,b#}/{0,c}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).