YES(O(1),O(n^2))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x))
  , sqr(x) -> *(x, x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We use the processor 'custom shape polynomial interpretation' to
orient following rules strictly.

Trs:
  { sum(s(x)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x)) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  TcT has computed the following constructor-restricted polynomial
  interpretation.
    [sum](x1) = x1 + x1^2
                         
        [0]() = 0        
                         
      [s](x1) = 2 + x1   
                         
  [+](x1, x2) = x1 + x2  
                         
    [sqr](x1) = 2*x1     
                         
  [*](x1, x2) = x1 + x2  
                         
  
  This order satisfies the following ordering constraints.
  
     [sum(0())] =                            
                >=                           
                =  [0()]                     
                                             
    [sum(s(x))] =  6 + 5*x + x^2             
                >  4 + 3*x + x^2             
                =  [+(sqr(s(x)), sum(x))]    
                                             
    [sum(s(x))] =  6 + 5*x + x^2             
                >  4 + 3*x + x^2             
                =  [+(*(s(x), s(x)), sum(x))]
                                             
       [sqr(x)] =  2*x                       
                >= 2*x                       
                =  [*(x, x)]                 
                                             

We return to the main proof.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs:
  { sum(0()) -> 0()
  , sqr(x) -> *(x, x) }
Weak Trs:
  { sum(s(x)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We use the processor 'custom shape polynomial interpretation' to
orient following rules strictly.

Trs: { sqr(x) -> *(x, x) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  TcT has computed the following constructor-restricted polynomial
  interpretation.
    [sum](x1) = x1 + x1^2
                         
        [0]() = 0        
                         
      [s](x1) = 2 + x1   
                         
  [+](x1, x2) = x1 + x2  
                         
    [sqr](x1) = 1 + 2*x1 
                         
  [*](x1, x2) = x1 + x2  
                         
  
  This order satisfies the following ordering constraints.
  
     [sum(0())] =                            
                >=                           
                =  [0()]                     
                                             
    [sum(s(x))] =  6 + 5*x + x^2             
                >  5 + 3*x + x^2             
                =  [+(sqr(s(x)), sum(x))]    
                                             
    [sum(s(x))] =  6 + 5*x + x^2             
                >  4 + 3*x + x^2             
                =  [+(*(s(x), s(x)), sum(x))]
                                             
       [sqr(x)] =  1 + 2*x                   
                >  2*x                       
                =  [*(x, x)]                 
                                             

We return to the main proof.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs: { sum(0()) -> 0() }
Weak Trs:
  { sum(s(x)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x))
  , sqr(x) -> *(x, x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We use the processor 'custom shape polynomial interpretation' to
orient following rules strictly.

Trs: { sum(0()) -> 0() }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  TcT has computed the following constructor-restricted polynomial
  interpretation.
    [sum](x1) = 1 + x1 + x1^2
                             
        [0]() = 0            
                             
      [s](x1) = 2 + x1       
                             
  [+](x1, x2) = x1 + x2      
                             
    [sqr](x1) = 2*x1         
                             
  [*](x1, x2) = x1 + x2      
                             
  
  This order satisfies the following ordering constraints.
  
     [sum(0())] =  1                         
                >                            
                =  [0()]                     
                                             
    [sum(s(x))] =  7 + 5*x + x^2             
                >  5 + 3*x + x^2             
                =  [+(sqr(s(x)), sum(x))]    
                                             
    [sum(s(x))] =  7 + 5*x + x^2             
                >  5 + 3*x + x^2             
                =  [+(*(s(x), s(x)), sum(x))]
                                             
       [sqr(x)] =  2*x                       
                >= 2*x                       
                =  [*(x, x)]                 
                                             

We return to the main proof.

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

Weak Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x))
  , sqr(x) -> *(x, x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(n^2))