YES

We show the termination of the TRS R:

  c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
  c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
  c(c(a(a(y,|0|()),x))) -> c(y)

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(c(a(x,y)))) -> c#(c(c(c(y))))
p2: c#(c(c(a(x,y)))) -> c#(c(c(y)))
p3: c#(c(c(a(x,y)))) -> c#(c(y))
p4: c#(c(c(a(x,y)))) -> c#(y)
p5: c#(c(b(c(y),|0|()))) -> c#(c(a(y,|0|())))
p6: c#(c(b(c(y),|0|()))) -> c#(a(y,|0|()))
p7: c#(c(a(a(y,|0|()),x))) -> c#(y)

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(c(a(x,y)))) -> c#(c(c(c(y))))
p2: c#(c(a(a(y,|0|()),x))) -> c#(y)
p3: c#(c(b(c(y),|0|()))) -> c#(c(a(y,|0|())))
p4: c#(c(c(a(x,y)))) -> c#(y)
p5: c#(c(c(a(x,y)))) -> c#(c(y))
p6: c#(c(c(a(x,y)))) -> c#(c(c(y)))

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = ((0,1),(1,0)) x1
            c_A(x1) = ((1,1),(1,1)) x1 + (2,13)
            a_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + (108,2)
            |0|_A() = (1,9)
            b_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,1),(1,1)) x2 + (380,276)
    
      precedence:
      
        c# = c = a = |0| = b
      
      partial status:
    
        pi(c#) = []
        pi(c) = [1]
        pi(a) = []
        pi(|0|) = []
        pi(b) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = (2,0)
            c_A(x1) = (3,0)
            a_A(x1,x2) = (0,0)
            |0|_A() = (1,0)
            b_A(x1,x2) = (0,0)
    
      precedence:
      
        c# = c = |0| > a = b
      
      partial status:
    
        pi(c#) = []
        pi(c) = []
        pi(a) = []
        pi(|0|) = []
        pi(b) = []
    

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(c(a(x,y)))) -> c#(c(c(c(y))))
p2: c#(c(a(a(y,|0|()),x))) -> c#(y)
p3: c#(c(b(c(y),|0|()))) -> c#(c(a(y,|0|())))
p4: c#(c(c(a(x,y)))) -> c#(y)
p5: c#(c(c(a(x,y)))) -> c#(c(c(y)))

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(c(a(x,y)))) -> c#(c(c(c(y))))
p2: c#(c(c(a(x,y)))) -> c#(c(c(y)))
p3: c#(c(c(a(x,y)))) -> c#(y)
p4: c#(c(b(c(y),|0|()))) -> c#(c(a(y,|0|())))
p5: c#(c(a(a(y,|0|()),x))) -> c#(y)

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = ((1,0),(1,1)) x1 + (39,0)
            c_A(x1) = ((1,1),(1,0)) x1 + (16,1)
            a_A(x1,x2) = x1 + ((1,1),(1,0)) x2 + (12,49)
            b_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (84,40)
            |0|_A() = (1,2)
    
      precedence:
      
        c > b = |0| > c# = a
      
      partial status:
    
        pi(c#) = [1]
        pi(c) = []
        pi(a) = [1]
        pi(b) = [1]
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = (0,0)
            c_A(x1) = (1,0)
            a_A(x1,x2) = ((1,0),(1,1)) x1
            b_A(x1,x2) = ((0,0),(1,0)) x1
            |0|_A() = (0,2)
    
      precedence:
      
        c# = c > |0| > a = b
      
      partial status:
    
        pi(c#) = []
        pi(c) = []
        pi(a) = []
        pi(b) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(c(a(x,y)))) -> c#(c(c(c(y))))
p2: c#(c(c(a(x,y)))) -> c#(c(c(y)))
p3: c#(c(b(c(y),|0|()))) -> c#(c(a(y,|0|())))
p4: c#(c(a(a(y,|0|()),x))) -> c#(y)

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(c(a(x,y)))) -> c#(c(c(c(y))))
p2: c#(c(a(a(y,|0|()),x))) -> c#(y)
p3: c#(c(b(c(y),|0|()))) -> c#(c(a(y,|0|())))
p4: c#(c(c(a(x,y)))) -> c#(c(c(y)))

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = ((1,0),(0,0)) x1
            c_A(x1) = ((1,0),(1,0)) x1
            a_A(x1,x2) = ((1,0),(1,0)) x1 + x2
            |0|_A() = (0,1)
            b_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2
    
      precedence:
      
        c# = c = a = |0| = b
      
      partial status:
    
        pi(c#) = []
        pi(c) = []
        pi(a) = []
        pi(|0|) = []
        pi(b) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = x1 + (0,7)
            c_A(x1) = ((1,1),(1,1)) x1
            a_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (3,4)
            |0|_A() = (1,1)
            b_A(x1,x2) = ((0,1),(1,0)) x1 + (20,28)
    
      precedence:
      
        a > c > |0| = b > c#
      
      partial status:
    
        pi(c#) = [1]
        pi(c) = [1]
        pi(a) = [2]
        pi(|0|) = []
        pi(b) = []
    

The next rules are strictly ordered:

  p1, p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(a(a(y,|0|()),x))) -> c#(y)
p2: c#(c(c(a(x,y)))) -> c#(c(c(y)))

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(a(a(y,|0|()),x))) -> c#(y)
p2: c#(c(c(a(x,y)))) -> c#(c(c(y)))

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = ((1,0),(0,0)) x1
            c_A(x1) = ((1,0),(1,0)) x1 + (2,0)
            a_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (0,1)
            |0|_A() = (0,3)
            b_A(x1,x2) = ((0,1),(0,0)) x1 + (0,3)
    
      precedence:
      
        c# = c = a = |0| = b
      
      partial status:
    
        pi(c#) = []
        pi(c) = []
        pi(a) = []
        pi(|0|) = []
        pi(b) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = x1
            c_A(x1) = ((0,1),(1,0)) x1 + (2,3)
            a_A(x1,x2) = ((1,0),(1,1)) x2 + (1,1)
            |0|_A() = (0,1)
            b_A(x1,x2) = (8,9)
    
      precedence:
      
        c > a = b > c# > |0|
      
      partial status:
    
        pi(c#) = [1]
        pi(c) = []
        pi(a) = []
        pi(|0|) = []
        pi(b) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(c(a(x,y)))) -> c#(c(c(y)))

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: c#(c(c(a(x,y)))) -> c#(c(c(y)))

and R consists of:

r1: c(c(c(a(x,y)))) -> b(c(c(c(c(y)))),x)
r2: c(c(b(c(y),|0|()))) -> a(|0|(),c(c(a(y,|0|()))))
r3: c(c(a(a(y,|0|()),x))) -> c(y)

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = x1 + (1,12)
            c_A(x1) = ((1,1),(1,1)) x1
            a_A(x1,x2) = ((0,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (0,12)
            b_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (23,13)
            |0|_A() = (1,1)
    
      precedence:
      
        c = a > c# = |0| > b
      
      partial status:
    
        pi(c#) = [1]
        pi(c) = []
        pi(a) = []
        pi(b) = [2]
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            c#_A(x1) = ((1,0),(0,0)) x1 + (1,4)
            c_A(x1) = (3,3)
            a_A(x1,x2) = (0,1)
            b_A(x1,x2) = (2,2)
            |0|_A() = (1,0)
    
      precedence:
      
        c > a = b = |0| > c#
      
      partial status:
    
        pi(c#) = []
        pi(c) = []
        pi(a) = []
        pi(b) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.