MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
  , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
  , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , or(x, true()) -> true()
  , or(true(), x) -> true()
  , or(false(), false()) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x
  , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
  , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following weak dependency pairs:

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, 0()), x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y))
  , cond2^#(false(), x, y) -> c_3(cond3^#(gr(y, 0()), x, y))
  , cond3^#(true(), x, y) ->
    c_12(cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y)))
  , cond3^#(false(), x, y) ->
    c_13(cond1^#(or(gr(x, 0()), gr(y, 0())), x, y))
  , gr^#(0(), x) -> c_4()
  , gr^#(s(x), 0()) -> c_5()
  , gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
  , or^#(x, true()) -> c_7()
  , or^#(true(), x) -> c_8()
  , or^#(false(), false()) -> c_9()
  , p^#(0()) -> c_10()
  , p^#(s(x)) -> c_11() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, 0()), x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y))
  , cond2^#(false(), x, y) -> c_3(cond3^#(gr(y, 0()), x, y))
  , cond3^#(true(), x, y) ->
    c_12(cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y)))
  , cond3^#(false(), x, y) ->
    c_13(cond1^#(or(gr(x, 0()), gr(y, 0())), x, y))
  , gr^#(0(), x) -> c_4()
  , gr^#(s(x), 0()) -> c_5()
  , gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
  , or^#(x, true()) -> c_7()
  , or^#(true(), x) -> c_8()
  , or^#(false(), false()) -> c_9()
  , p^#(0()) -> c_10()
  , p^#(s(x)) -> c_11() }
Strict Trs:
  { cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
  , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
  , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , or(x, true()) -> true()
  , or(true(), x) -> true()
  , or(false(), false()) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x
  , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
  , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { gr(0(), x) -> false()
    , gr(s(x), 0()) -> true()
    , gr(s(x), s(y)) -> gr(x, y)
    , or(x, true()) -> true()
    , or(true(), x) -> true()
    , or(false(), false()) -> false()
    , p(0()) -> 0()
    , p(s(x)) -> x }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, 0()), x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y))
  , cond2^#(false(), x, y) -> c_3(cond3^#(gr(y, 0()), x, y))
  , cond3^#(true(), x, y) ->
    c_12(cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y)))
  , cond3^#(false(), x, y) ->
    c_13(cond1^#(or(gr(x, 0()), gr(y, 0())), x, y))
  , gr^#(0(), x) -> c_4()
  , gr^#(s(x), 0()) -> c_5()
  , gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
  , or^#(x, true()) -> c_7()
  , or^#(true(), x) -> c_8()
  , or^#(false(), false()) -> c_9()
  , p^#(0()) -> c_10()
  , p^#(s(x)) -> c_11() }
Strict Trs:
  { gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , or(x, true()) -> true()
  , or(true(), x) -> true()
  , or(false(), false()) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(or) = {1, 2}, Uargs(cond1^#) = {1, 2, 3}, Uargs(c_1) = {1},
  Uargs(cond2^#) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1},
  Uargs(cond3^#) = {1}, Uargs(c_6) = {1}, Uargs(c_12) = {1},
  Uargs(c_13) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

                 [true] = [0]                           
                                                        
           [gr](x1, x2) = [2] x2 + [1]                  
                                                        
                    [0] = [0]                           
                                                        
           [or](x1, x2) = [1] x1 + [1] x2 + [1]         
                                                        
                [p](x1) = [1] x1 + [1]                  
                                                        
                [false] = [0]                           
                                                        
                [s](x1) = [1] x1 + [2]                  
                                                        
  [cond1^#](x1, x2, x3) = [2] x1 + [1] x2 + [1] x3 + [0]
                                                        
              [c_1](x1) = [1] x1 + [2]                  
                                                        
  [cond2^#](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
                                                        
              [c_2](x1) = [1] x1 + [0]                  
                                                        
              [c_3](x1) = [1] x1 + [2]                  
                                                        
  [cond3^#](x1, x2, x3) = [2] x1 + [1] x2 + [1] x3 + [2]
                                                        
         [gr^#](x1, x2) = [1] x1 + [2] x2 + [1]         
                                                        
                  [c_4] = [0]                           
                                                        
                  [c_5] = [2]                           
                                                        
              [c_6](x1) = [1] x1 + [1]                  
                                                        
         [or^#](x1, x2) = [1] x1 + [1] x2 + [2]         
                                                        
                  [c_7] = [1]                           
                                                        
                  [c_8] = [1]                           
                                                        
                  [c_9] = [1]                           
                                                        
              [p^#](x1) = [2] x1 + [2]                  
                                                        
                 [c_10] = [1]                           
                                                        
                 [c_11] = [1]                           
                                                        
             [c_12](x1) = [1] x1 + [0]                  
                                                        
             [c_13](x1) = [1] x1 + [1]                  

This order satisfies following ordering constraints:

            [gr(0(), x)] = [2] x + [1]
                         > [0]        
                         = [false()]  
                                      
         [gr(s(x), 0())] = [1]        
                         > [0]        
                         = [true()]   
                                      
        [gr(s(x), s(y))] = [2] y + [5]
                         > [2] y + [1]
                         = [gr(x, y)] 
                                      
         [or(x, true())] = [1] x + [1]
                         > [0]        
                         = [true()]   
                                      
         [or(true(), x)] = [1] x + [1]
                         > [0]        
                         = [true()]   
                                      
  [or(false(), false())] = [1]        
                         > [0]        
                         = [false()]  
                                      
                [p(0())] = [1]        
                         > [0]        
                         = [0()]      
                                      
               [p(s(x))] = [1] x + [3]
                         > [1] x + [0]
                         = [x]        
                                      

Further, it can be verified that all rules not oriented are covered by the weightgap condition.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, 0()), x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y))
  , cond2^#(false(), x, y) -> c_3(cond3^#(gr(y, 0()), x, y))
  , cond3^#(true(), x, y) ->
    c_12(cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y)))
  , cond3^#(false(), x, y) ->
    c_13(cond1^#(or(gr(x, 0()), gr(y, 0())), x, y)) }
Weak DPs:
  { gr^#(0(), x) -> c_4()
  , gr^#(s(x), 0()) -> c_5()
  , gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
  , or^#(x, true()) -> c_7()
  , or^#(true(), x) -> c_8()
  , or^#(false(), false()) -> c_9()
  , p^#(0()) -> c_10()
  , p^#(s(x)) -> c_11() }
Weak Trs:
  { gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , or(x, true()) -> true()
  , or(true(), x) -> true()
  , or(false(), false()) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ gr^#(0(), x) -> c_4()
, gr^#(s(x), 0()) -> c_5()
, gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
, or^#(x, true()) -> c_7()
, or^#(true(), x) -> c_8()
, or^#(false(), false()) -> c_9()
, p^#(0()) -> c_10()
, p^#(s(x)) -> c_11() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, 0()), x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y))
  , cond2^#(false(), x, y) -> c_3(cond3^#(gr(y, 0()), x, y))
  , cond3^#(true(), x, y) ->
    c_12(cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y)))
  , cond3^#(false(), x, y) ->
    c_13(cond1^#(or(gr(x, 0()), gr(y, 0())), x, y)) }
Weak Trs:
  { gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , or(x, true()) -> true()
  , or(true(), x) -> true()
  , or(false(), false()) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..