MAYBE
* Step 1: WeightGap MAYBE
    + Considered Problem:
        - Strict TRS:
            diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y)))
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            leq(0(),Y) -> true()
            leq(s(X),0()) -> false()
            leq(s(X),s(Y)) -> leq(X,Y)
            p(0()) -> 0()
            p(s(X)) -> X
        - Signature:
            {diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(diff) = {1},
            uargs(if) = {1,3},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [0]                           
             p(diff) = [1] x1 + [0]                  
            p(false) = [0]                           
               p(if) = [1] x1 + [2] x2 + [1] x3 + [0]
              p(leq) = [0]                           
                p(p) = [1] x1 + [9]                  
                p(s) = [1] x1 + [12]                 
             p(true) = [0]                           
          
          Following rules are strictly oriented:
           p(0()) = [9]         
                  > [0]         
                  = 0()         
          
          p(s(X)) = [1] X + [21]
                  > [1] X + [0] 
                  = X           
          
          
          Following rules are (at-least) weakly oriented:
                diff(X,Y) =  [1] X + [0]                     
                          >= [1] X + [21]                    
                          =  if(leq(X,Y),0(),s(diff(p(X),Y)))
          
          if(false(),X,Y) =  [2] X + [1] Y + [0]             
                          >= [1] Y + [0]                     
                          =  Y                               
          
           if(true(),X,Y) =  [2] X + [1] Y + [0]             
                          >= [1] X + [0]                     
                          =  X                               
          
               leq(0(),Y) =  [0]                             
                          >= [0]                             
                          =  true()                          
          
            leq(s(X),0()) =  [0]                             
                          >= [0]                             
                          =  false()                         
          
           leq(s(X),s(Y)) =  [0]                             
                          >= [0]                             
                          =  leq(X,Y)                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap MAYBE
    + Considered Problem:
        - Strict TRS:
            diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y)))
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            leq(0(),Y) -> true()
            leq(s(X),0()) -> false()
            leq(s(X),s(Y)) -> leq(X,Y)
        - Weak TRS:
            p(0()) -> 0()
            p(s(X)) -> X
        - Signature:
            {diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(diff) = {1},
            uargs(if) = {1,3},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [0]                           
             p(diff) = [1] x1 + [4] x2 + [0]         
            p(false) = [5]                           
               p(if) = [1] x1 + [4] x2 + [1] x3 + [0]
              p(leq) = [3]                           
                p(p) = [1] x1 + [8]                  
                p(s) = [1] x1 + [8]                  
             p(true) = [0]                           
          
          Following rules are strictly oriented:
          if(false(),X,Y) = [4] X + [1] Y + [5]
                          > [1] Y + [0]        
                          = Y                  
          
               leq(0(),Y) = [3]                
                          > [0]                
                          = true()             
          
          
          Following rules are (at-least) weakly oriented:
               diff(X,Y) =  [1] X + [4] Y + [0]             
                         >= [1] X + [4] Y + [19]            
                         =  if(leq(X,Y),0(),s(diff(p(X),Y)))
          
          if(true(),X,Y) =  [4] X + [1] Y + [0]             
                         >= [1] X + [0]                     
                         =  X                               
          
           leq(s(X),0()) =  [3]                             
                         >= [5]                             
                         =  false()                         
          
          leq(s(X),s(Y)) =  [3]                             
                         >= [3]                             
                         =  leq(X,Y)                        
          
                  p(0()) =  [8]                             
                         >= [0]                             
                         =  0()                             
          
                 p(s(X)) =  [1] X + [16]                    
                         >= [1] X + [0]                     
                         =  X                               
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap MAYBE
    + Considered Problem:
        - Strict TRS:
            diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y)))
            if(true(),X,Y) -> X
            leq(s(X),0()) -> false()
            leq(s(X),s(Y)) -> leq(X,Y)
        - Weak TRS:
            if(false(),X,Y) -> Y
            leq(0(),Y) -> true()
            p(0()) -> 0()
            p(s(X)) -> X
        - Signature:
            {diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(diff) = {1},
            uargs(if) = {1,3},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [2]                           
             p(diff) = [1] x1 + [1] x2 + [2]         
            p(false) = [1]                           
               p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
              p(leq) = [4]                           
                p(p) = [1] x1 + [8]                  
                p(s) = [1] x1 + [0]                  
             p(true) = [3]                           
          
          Following rules are strictly oriented:
          if(true(),X,Y) = [1] X + [1] Y + [3]
                         > [1] X + [0]        
                         = X                  
          
           leq(s(X),0()) = [4]                
                         > [1]                
                         = false()            
          
          
          Following rules are (at-least) weakly oriented:
                diff(X,Y) =  [1] X + [1] Y + [2]             
                          >= [1] X + [1] Y + [16]            
                          =  if(leq(X,Y),0(),s(diff(p(X),Y)))
          
          if(false(),X,Y) =  [1] X + [1] Y + [1]             
                          >= [1] Y + [0]                     
                          =  Y                               
          
               leq(0(),Y) =  [4]                             
                          >= [3]                             
                          =  true()                          
          
           leq(s(X),s(Y)) =  [4]                             
                          >= [4]                             
                          =  leq(X,Y)                        
          
                   p(0()) =  [10]                            
                          >= [2]                             
                          =  0()                             
          
                  p(s(X)) =  [1] X + [8]                     
                          >= [1] X + [0]                     
                          =  X                               
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: Failure MAYBE
  + Considered Problem:
      - Strict TRS:
          diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y)))
          leq(s(X),s(Y)) -> leq(X,Y)
      - Weak TRS:
          if(false(),X,Y) -> Y
          if(true(),X,Y) -> X
          leq(0(),Y) -> true()
          leq(s(X),0()) -> false()
          p(0()) -> 0()
          p(s(X)) -> X
      - Signature:
          {diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE