YES(O(1),O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict Trs:
  { prime(0()) -> false()
  , prime(s(0())) -> false()
  , prime(s(s(x))) -> prime1(s(s(x)), s(x))
  , prime1(x, 0()) -> false()
  , prime1(x, s(0())) -> true()
  , prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
  , divp(x, y) -> =(rem(x, y), 0()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { prime^#(0()) -> c_1()
  , prime^#(s(0())) -> c_2()
  , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
  , prime1^#(x, 0()) -> c_4()
  , prime1^#(x, s(0())) -> c_5()
  , prime1^#(x, s(s(y))) ->
    c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
  , divp^#(x, y) -> c_7() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { prime^#(0()) -> c_1()
  , prime^#(s(0())) -> c_2()
  , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
  , prime1^#(x, 0()) -> c_4()
  , prime1^#(x, s(0())) -> c_5()
  , prime1^#(x, s(s(y))) ->
    c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
  , divp^#(x, y) -> c_7() }
Strict Trs:
  { prime(0()) -> false()
  , prime(s(0())) -> false()
  , prime(s(s(x))) -> prime1(s(s(x)), s(x))
  , prime1(x, 0()) -> false()
  , prime1(x, s(0())) -> true()
  , prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
  , divp(x, y) -> =(rem(x, y), 0()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { prime^#(0()) -> c_1()
  , prime^#(s(0())) -> c_2()
  , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
  , prime1^#(x, 0()) -> c_4()
  , prime1^#(x, s(0())) -> c_5()
  , prime1^#(x, s(s(y))) ->
    c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
  , divp^#(x, y) -> c_7() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

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

The following argument positions are usable:
  Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}

TcT has computed the following constructor-restricted matrix
interpretation.

                 [0] = [0]                      
                       [0]                      
                                                
             [s](x1) = [0]                      
                       [0]                      
                                                
       [prime^#](x1) = [0]                      
                       [0]                      
                                                
               [c_1] = [1]                      
                       [0]                      
                                                
               [c_2] = [1]                      
                       [0]                      
                                                
           [c_3](x1) = [1 0] x1 + [1]           
                       [0 1]      [0]           
                                                
  [prime1^#](x1, x2) = [0]                      
                       [0]                      
                                                
               [c_4] = [1]                      
                       [0]                      
                                                
               [c_5] = [2]                      
                       [0]                      
                                                
       [c_6](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                       [0 1]      [0 1]      [0]
                                                
    [divp^#](x1, x2) = [0 0] x2 + [1]           
                       [2 2]      [0]           
                                                
               [c_7] = [0]                      
                       [0]                      

The following symbols are considered usable

  {prime^#, prime1^#, divp^#}

The order satisfies the following ordering constraints:

          [prime^#(0())] = [0]                                         
                           [0]                                         
                         ? [1]                                         
                           [0]                                         
                         = [c_1()]                                     
                                                                       
       [prime^#(s(0()))] = [0]                                         
                           [0]                                         
                         ? [1]                                         
                           [0]                                         
                         = [c_2()]                                     
                                                                       
      [prime^#(s(s(x)))] = [0]                                         
                           [0]                                         
                         ? [1]                                         
                           [0]                                         
                         = [c_3(prime1^#(s(s(x)), s(x)))]              
                                                                       
      [prime1^#(x, 0())] = [0]                                         
                           [0]                                         
                         ? [1]                                         
                           [0]                                         
                         = [c_4()]                                     
                                                                       
   [prime1^#(x, s(0()))] = [0]                                         
                           [0]                                         
                         ? [2]                                         
                           [0]                                         
                         = [c_5()]                                     
                                                                       
  [prime1^#(x, s(s(y)))] = [0]                                         
                           [0]                                         
                         ? [0 0] x + [1]                               
                           [2 2]     [0]                               
                         = [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
                                                                       
          [divp^#(x, y)] = [0 0] y + [1]                               
                           [2 2]     [0]                               
                         > [0]                                         
                           [0]                                         
                         = [c_7()]                                     
                                                                       

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 YES(O(1),O(n^1)).

Strict DPs:
  { prime^#(0()) -> c_1()
  , prime^#(s(0())) -> c_2()
  , prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
  , prime1^#(x, 0()) -> c_4()
  , prime1^#(x, s(0())) -> c_5()
  , prime1^#(x, s(s(y))) ->
    c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs: { divp^#(x, y) -> c_7() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We estimate the number of application of {1,2,4,5} by applications
of Pre({1,2,4,5}) = {3,6}. Here rules are labeled as follows:

  DPs:
    { 1: prime^#(0()) -> c_1()
    , 2: prime^#(s(0())) -> c_2()
    , 3: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
    , 4: prime1^#(x, 0()) -> c_4()
    , 5: prime1^#(x, s(0())) -> c_5()
    , 6: prime1^#(x, s(s(y))) ->
         c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
    , 7: divp^#(x, y) -> c_7() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
  , prime1^#(x, s(s(y))) ->
    c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
  { prime^#(0()) -> c_1()
  , prime^#(s(0())) -> c_2()
  , prime1^#(x, 0()) -> c_4()
  , prime1^#(x, s(0())) -> c_5()
  , divp^#(x, y) -> c_7() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

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

{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, divp^#(x, y) -> c_7() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
  , prime1^#(x, s(s(y))) ->
    c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { prime1^#(x, s(s(y))) ->
    c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { prime^#(s(s(x))) -> c_1(prime1^#(s(s(x)), s(x)))
  , prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

Consider the dependency graph

  1: prime^#(s(s(x))) -> c_1(prime1^#(s(s(x)), s(x)))
     -->_1 prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) :2
  
  2: prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y)))
     -->_1 prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) :2
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { prime^#(s(s(x))) -> c_1(prime1^#(s(s(x)), s(x))) }


We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs: { prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.

DPs:
  { 1: prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }

Sub-proof:
----------
  The input was oriented with the instance of 'Small Polynomial Path
  Order (PS,1-bounded)' as induced by the safe mapping
  
   safe(prime) = {}, safe(0) = {}, safe(false) = {}, safe(s) = {1},
   safe(prime1) = {}, safe(true) = {}, safe(and) = {1, 2},
   safe(not) = {1}, safe(divp) = {}, safe(=) = {1, 2},
   safe(rem) = {1, 2}, safe(prime^#) = {}, safe(c_1) = {},
   safe(c_2) = {}, safe(c_3) = {}, safe(prime1^#) = {1},
   safe(c_4) = {}, safe(c_5) = {}, safe(c_6) = {}, safe(divp^#) = {},
   safe(c_7) = {}, safe(c) = {}, safe(c_1) = {}, safe(c_2) = {}
  
  and precedence
  
   empty .
  
  Following symbols are considered recursive:
  
   {prime1^#}
  
  The recursion depth is 1.
  
  Further, following argument filtering is employed:
  
   pi(prime) = [], pi(0) = [], pi(false) = [], pi(s) = [1],
   pi(prime1) = [], pi(true) = [], pi(and) = [], pi(not) = [],
   pi(divp) = [], pi(=) = [], pi(rem) = [], pi(prime^#) = [],
   pi(c_1) = [], pi(c_2) = [], pi(c_3) = [], pi(prime1^#) = [1, 2],
   pi(c_4) = [], pi(c_5) = [], pi(c_6) = [], pi(divp^#) = [],
   pi(c_7) = [], pi(c) = [], pi(c_1) = [], pi(c_2) = [1]
  
  Usable defined function symbols are a subset of:
  
   {prime^#, prime1^#, divp^#}
  
  For your convenience, here are the satisfied ordering constraints:
  
    pi(prime1^#(x, s(s(y)))) = prime1^#(s(; s(; y)); x)  
                             > c_2(prime1^#(s(; y); x);) 
                             = pi(c_2(prime1^#(x, s(y))))
                                                         

The strictly oriented rules are moved into the weak component.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak DPs: { prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

{ prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(n^1))