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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          c#_A(x1) = ((1,0),(0,0)) x1 + (71,0)
          c_A(x1) = ((1,1),(1,0)) x1 + (4,0)
          a_A(x1,x2) = x1 + ((1,1),(1,0)) x2 + (29,0)
          |0|_A() = (3,2)
          b_A(x1,x2) = x1 + (75,0)
  
    precedence:
    
      c# = c = |0| = b > a
    
    partial status:
  
      pi(c#) = []
      pi(c) = []
      pi(a) = []
      pi(|0|) = []
      pi(b) = []

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

  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) = ((1,1),(1,1)) x1 + (12,1)
          a_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (8,13)
          |0|_A() = (1,2)
          b_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,1),(0,0)) x2 + (27,21)
  
    precedence:
    
      c = |0| = b > c# = a
    
    partial status:
  
      pi(c#) = []
      pi(c) = [1]
      pi(a) = [2]
      pi(|0|) = []
      pi(b) = []

The next rules are strictly ordered:

  p2

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(y))
p3: c#(c(c(a(x,y)))) -> c#(y)
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(c(a(x,y)))) -> c#(y)
p4: c#(c(c(a(x,y)))) -> 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          c#_A(x1) = ((0,1),(0,1)) x1 + (9,0)
          c_A(x1) = ((0,1),(1,1)) x1 + (13,4)
          a_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,1),(1,1)) x2 + (56,1)
          |0|_A() = (8,1)
          b_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (36,76)
  
    precedence:
    
      c# = c = a = |0| > b
    
    partial status:
  
      pi(c#) = []
      pi(c) = []
      pi(a) = []
      pi(|0|) = []
      pi(b) = [2]

The next rules are strictly ordered:

  p4

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(c(a(x,y)))) -> 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}


-- 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#(y)
p3: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          c#_A(x1) = x1 + (1,0)
          c_A(x1) = ((1,1),(1,1)) x1
          a_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (0,11)
          |0|_A() = (2,3)
          b_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,0),(1,1)) x2 + (0,44)
  
    precedence:
    
      |0| > c > b > c# = a
    
    partial status:
  
      pi(c#) = [1]
      pi(c) = [1]
      pi(a) = [2]
      pi(|0|) = []
      pi(b) = []

The next rules are strictly ordered:

  p2

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)

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          c#_A(x1) = ((1,1),(0,1)) x1 + (4,38)
          c_A(x1) = ((0,1),(1,1)) x1 + (5,3)
          a_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,1),(1,1)) x2 + (15,0)
          |0|_A() = (0,2)
          b_A(x1,x2) = ((0,0),(0,1)) x1 + ((0,1),(0,1)) x2 + (24,11)
  
    precedence:
    
      a > c > 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(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}


-- Reduction pair.

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

p1: 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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          c#_A(x1) = x1 + (4,3)
          c_A(x1) = ((1,0),(1,1)) x1 + (1,0)
          a_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (2,1)
          |0|_A() = (1,0)
  
    precedence:
    
      c = a > c# = |0|
    
    partial status:
  
      pi(c#) = [1]
      pi(c) = []
      pi(a) = []
      pi(|0|) = []

The next rules are strictly ordered:

  p1

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