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:
  
      max/plus interpretations on natural numbers:
      
        c#_A(x1) = x1 + 2
        c_A(x1) = x1 + 21
        a_A(x1,x2) = max{x1 - 9, x2 + 24}
        |0|_A = 22
        b_A(x1,x2) = max{70, x1 + 3, x2 - 2}
  
    precedence:
    
      c = a = |0| = b > c#
    
    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)
p2: c#(c(b(c(y),|0|()))) -> c#(c(a(y,|0|())))
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(a(a(y,|0|()),x))) -> 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|())))

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:
  
      max/plus interpretations on natural numbers:
      
        c#_A(x1) = max{19, x1 - 96}
        c_A(x1) = max{78, x1 + 36}
        a_A(x1,x2) = max{71, x1 - 8, x2 + 30}
        |0|_A = 9
        b_A(x1,x2) = max{x1 - 8, x2 + 98}
  
    precedence:
    
      c# = c = a = |0| = b
    
    partial status:
  
      pi(c#) = []
      pi(c) = []
      pi(a) = [2]
      pi(|0|) = []
      pi(b) = [2]

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

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

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        c#_A(x1) = max{13, x1 + 6}
        c_A(x1) = max{62, x1 + 45}
        a_A(x1,x2) = max{x1 - 19, x2 + 15}
        |0|_A = 0
        b_A(x1,x2) = max{151, x1 - 48, x2 + 1}
  
    precedence:
    
      a = b > c# > c = |0|
    
    partial status:
  
      pi(c#) = [1]
      pi(c) = []
      pi(a) = []
      pi(|0|) = []
      pi(b) = [2]

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


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

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:
  
      max/plus interpretations on natural numbers:
      
        c#_A(x1) = max{1, x1 - 79}
        c_A(x1) = max{31, x1 + 22}
        a_A(x1,x2) = max{64, x1 - 1, x2 + 55}
        |0|_A = 5
        b_A(x1,x2) = max{120, x1 + 32, x2 + 54}
  
    precedence:
    
      c# = c = a = |0| = b
    
    partial status:
  
      pi(c#) = []
      pi(c) = []
      pi(a) = []
      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(a(a(y,|0|()),x))) -> c#(y)
p2: c#(c(b(c(y),|0|()))) -> c#(c(a(y,|0|())))

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

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:
  
      max/plus interpretations on natural numbers:
      
        c#_A(x1) = max{33, x1 + 17}
        c_A(x1) = x1 + 7
        a_A(x1,x2) = max{x1 + 8, x2 - 2}
        |0|_A = 12
        b_A(x1,x2) = max{11, x1 + 7, x2}
  
    precedence:
    
      c# = c = a = |0| = b
    
    partial status:
  
      pi(c#) = []
      pi(c) = [1]
      pi(a) = [1]
      pi(|0|) = []
      pi(b) = [2]

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

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:

  (no SCCs)