MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            f(true(),x,y,z) -> g(gt(x,y),x,y,z)
            g(true(),x,y,z) -> f(gt(x,z),x,y,s(z))
            g(true(),x,y,z) -> f(gt(x,z),x,s(y),z)
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
        - Signature:
            {f/4,g/4,gt/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,gt} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
          g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
          g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
          gt#(0(),v) -> c_4()
          gt#(s(u),0()) -> c_5()
          gt#(s(u),s(v)) -> c_6(gt#(u,v))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
            g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
            g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
            gt#(0(),v) -> c_4()
            gt#(s(u),0()) -> c_5()
            gt#(s(u),s(v)) -> c_6(gt#(u,v))
        - Weak TRS:
            f(true(),x,y,z) -> g(gt(x,y),x,y,z)
            g(true(),x,y,z) -> f(gt(x,z),x,y,s(z))
            g(true(),x,y,z) -> f(gt(x,z),x,s(y),z)
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
        - Signature:
            {f/4,g/4,gt/2,f#/4,g#/4,gt#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,gt#} and constructors {0,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          gt(0(),v) -> false()
          gt(s(u),0()) -> true()
          gt(s(u),s(v)) -> gt(u,v)
          f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
          g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
          g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
          gt#(0(),v) -> c_4()
          gt#(s(u),0()) -> c_5()
          gt#(s(u),s(v)) -> c_6(gt#(u,v))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
            g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
            g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
            gt#(0(),v) -> c_4()
            gt#(s(u),0()) -> c_5()
            gt#(s(u),s(v)) -> c_6(gt#(u,v))
        - Weak TRS:
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
        - Signature:
            {f/4,g/4,gt/2,f#/4,g#/4,gt#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,gt#} and constructors {0,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4,5}
        by application of
          Pre({4,5}) = {1,2,3,6}.
        Here rules are labelled as follows:
          1: f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
          2: g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
          3: g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
          4: gt#(0(),v) -> c_4()
          5: gt#(s(u),0()) -> c_5()
          6: gt#(s(u),s(v)) -> c_6(gt#(u,v))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
            g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
            g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
            gt#(s(u),s(v)) -> c_6(gt#(u,v))
        - Weak DPs:
            gt#(0(),v) -> c_4()
            gt#(s(u),0()) -> c_5()
        - Weak TRS:
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
        - Signature:
            {f/4,g/4,gt/2,f#/4,g#/4,gt#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,gt#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
             -->_2 gt#(s(u),s(v)) -> c_6(gt#(u,v)):4
             -->_1 g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z)):3
             -->_1 g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z)):2
             -->_2 gt#(s(u),0()) -> c_5():6
             -->_2 gt#(0(),v) -> c_4():5
          
          2:S:g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
             -->_2 gt#(s(u),s(v)) -> c_6(gt#(u,v)):4
             -->_2 gt#(s(u),0()) -> c_5():6
             -->_2 gt#(0(),v) -> c_4():5
             -->_1 f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y)):1
          
          3:S:g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
             -->_2 gt#(s(u),s(v)) -> c_6(gt#(u,v)):4
             -->_2 gt#(s(u),0()) -> c_5():6
             -->_2 gt#(0(),v) -> c_4():5
             -->_1 f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y)):1
          
          4:S:gt#(s(u),s(v)) -> c_6(gt#(u,v))
             -->_1 gt#(s(u),0()) -> c_5():6
             -->_1 gt#(0(),v) -> c_4():5
             -->_1 gt#(s(u),s(v)) -> c_6(gt#(u,v)):4
          
          5:W:gt#(0(),v) -> c_4()
             
          
          6:W:gt#(s(u),0()) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: gt#(0(),v) -> c_4()
          6: gt#(s(u),0()) -> c_5()
* Step 5: WeightGap MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
            g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
            g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
            gt#(s(u),s(v)) -> c_6(gt#(u,v))
        - Weak TRS:
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
        - Signature:
            {f/4,g/4,gt/2,f#/4,g#/4,gt#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,gt#} and constructors {0,false,s,true}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(f#) = {1},
            uargs(g#) = {1},
            uargs(c_1) = {1,2},
            uargs(c_2) = {1,2},
            uargs(c_3) = {1,2},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [0]                  
                p(f) = [0]                  
            p(false) = [0]                  
                p(g) = [0]                  
               p(gt) = [1]                  
                p(s) = [1]                  
             p(true) = [1]                  
               p(f#) = [1] x1 + [0]         
               p(g#) = [1] x1 + [6]         
              p(gt#) = [5]                  
              p(c_1) = [1] x1 + [1] x2 + [0]
              p(c_2) = [1] x1 + [1] x2 + [1]
              p(c_3) = [1] x1 + [1] x2 + [0]
              p(c_4) = [0]                  
              p(c_5) = [0]                  
              p(c_6) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          g#(true(),x,y,z) = [7]                               
                           > [6]                               
                           = c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
          
          
          Following rules are (at-least) weakly oriented:
          f#(true(),x,y,z) =  [1]                               
                           >= [12]                              
                           =  c_1(g#(gt(x,y),x,y,z),gt#(x,y))   
          
          g#(true(),x,y,z) =  [7]                               
                           >= [7]                               
                           =  c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
          
            gt#(s(u),s(v)) =  [5]                               
                           >= [5]                               
                           =  c_6(gt#(u,v))                     
          
                 gt(0(),v) =  [1]                               
                           >= [0]                               
                           =  false()                           
          
              gt(s(u),0()) =  [1]                               
                           >= [1]                               
                           =  true()                            
          
             gt(s(u),s(v)) =  [1]                               
                           >= [1]                               
                           =  gt(u,v)                           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
            g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
            gt#(s(u),s(v)) -> c_6(gt#(u,v))
        - Weak DPs:
            g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
        - Weak TRS:
            gt(0(),v) -> false()
            gt(s(u),0()) -> true()
            gt(s(u),s(v)) -> gt(u,v)
        - Signature:
            {f/4,g/4,gt/2,f#/4,g#/4,gt#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,gt#} and constructors {0,false,s,true}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(f#) = {1},
            uargs(g#) = {1},
            uargs(c_1) = {1,2},
            uargs(c_2) = {1,2},
            uargs(c_3) = {1,2},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [0]                  
                p(f) = [1] x2 + [1] x3 + [1]
            p(false) = [0]                  
                p(g) = [2] x2 + [1] x3 + [0]
               p(gt) = [1]                  
                p(s) = [1]                  
             p(true) = [0]                  
               p(f#) = [1] x1 + [1] x4 + [0]
               p(g#) = [1] x1 + [1] x4 + [3]
              p(gt#) = [0]                  
              p(c_1) = [1] x1 + [1] x2 + [4]
              p(c_2) = [1] x1 + [1] x2 + [0]
              p(c_3) = [1] x1 + [1] x2 + [2]
              p(c_4) = [0]                  
              p(c_5) = [1]                  
              p(c_6) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          g#(true(),x,y,z) = [1] z + [3]                       
                           > [2]                               
                           = c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
          
          
          Following rules are (at-least) weakly oriented:
          f#(true(),x,y,z) =  [1] z + [0]                       
                           >= [1] z + [8]                       
                           =  c_1(g#(gt(x,y),x,y,z),gt#(x,y))   
          
          g#(true(),x,y,z) =  [1] z + [3]                       
                           >= [1] z + [3]                       
                           =  c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
          
            gt#(s(u),s(v)) =  [0]                               
                           >= [0]                               
                           =  c_6(gt#(u,v))                     
          
                 gt(0(),v) =  [1]                               
                           >= [0]                               
                           =  false()                           
          
              gt(s(u),0()) =  [1]                               
                           >= [0]                               
                           =  true()                            
          
             gt(s(u),s(v)) =  [1]                               
                           >= [1]                               
                           =  gt(u,v)                           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          f#(true(),x,y,z) -> c_1(g#(gt(x,y),x,y,z),gt#(x,y))
          gt#(s(u),s(v)) -> c_6(gt#(u,v))
      - Weak DPs:
          g#(true(),x,y,z) -> c_2(f#(gt(x,z),x,y,s(z)),gt#(x,z))
          g#(true(),x,y,z) -> c_3(f#(gt(x,z),x,s(y),z),gt#(x,z))
      - Weak TRS:
          gt(0(),v) -> false()
          gt(s(u),0()) -> true()
          gt(s(u),s(v)) -> gt(u,v)
      - Signature:
          {f/4,g/4,gt/2,f#/4,g#/4,gt#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0,c_6/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {f#,g#,gt#} and constructors {0,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE