YES

We show the termination of the TRS R:

  t(N) -> cs(r(q(N)),nt(ns(N)))
  q(|0|()) -> |0|()
  q(s(X)) -> s(p(q(X),d(X)))
  d(|0|()) -> |0|()
  d(s(X)) -> s(s(d(X)))
  p(|0|(),X) -> X
  p(X,|0|()) -> X
  p(s(X),s(Y)) -> s(s(p(X,Y)))
  f(|0|(),X) -> nil()
  f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
  t(X) -> nt(X)
  s(X) -> ns(X)
  f(X1,X2) -> nf(X1,X2)
  a(nt(X)) -> t(a(X))
  a(ns(X)) -> s(a(X))
  a(nf(X1,X2)) -> f(a(X1),a(X2))
  a(X) -> X

-- SCC decomposition.

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

p1: t#(N) -> q#(N)
p2: q#(s(X)) -> s#(p(q(X),d(X)))
p3: q#(s(X)) -> p#(q(X),d(X))
p4: q#(s(X)) -> q#(X)
p5: q#(s(X)) -> d#(X)
p6: d#(s(X)) -> s#(s(d(X)))
p7: d#(s(X)) -> s#(d(X))
p8: d#(s(X)) -> d#(X)
p9: p#(s(X),s(Y)) -> s#(s(p(X,Y)))
p10: p#(s(X),s(Y)) -> s#(p(X,Y))
p11: p#(s(X),s(Y)) -> p#(X,Y)
p12: f#(s(X),cs(Y,Z)) -> a#(Z)
p13: a#(nt(X)) -> t#(a(X))
p14: a#(nt(X)) -> a#(X)
p15: a#(ns(X)) -> s#(a(X))
p16: a#(ns(X)) -> a#(X)
p17: a#(nf(X1,X2)) -> f#(a(X1),a(X2))
p18: a#(nf(X1,X2)) -> a#(X1)
p19: a#(nf(X1,X2)) -> a#(X2)

and R consists of:

r1: t(N) -> cs(r(q(N)),nt(ns(N)))
r2: q(|0|()) -> |0|()
r3: q(s(X)) -> s(p(q(X),d(X)))
r4: d(|0|()) -> |0|()
r5: d(s(X)) -> s(s(d(X)))
r6: p(|0|(),X) -> X
r7: p(X,|0|()) -> X
r8: p(s(X),s(Y)) -> s(s(p(X,Y)))
r9: f(|0|(),X) -> nil()
r10: f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
r11: t(X) -> nt(X)
r12: s(X) -> ns(X)
r13: f(X1,X2) -> nf(X1,X2)
r14: a(nt(X)) -> t(a(X))
r15: a(ns(X)) -> s(a(X))
r16: a(nf(X1,X2)) -> f(a(X1),a(X2))
r17: a(X) -> X

The estimated dependency graph contains the following SCCs:

  {p12, p14, p16, p17, p18, p19}
  {p4}
  {p11}
  {p8}


-- Reduction pair.

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

p1: a#(nf(X1,X2)) -> a#(X2)
p2: a#(nf(X1,X2)) -> a#(X1)
p3: a#(nf(X1,X2)) -> f#(a(X1),a(X2))
p4: f#(s(X),cs(Y,Z)) -> a#(Z)
p5: a#(ns(X)) -> a#(X)
p6: a#(nt(X)) -> a#(X)

and R consists of:

r1: t(N) -> cs(r(q(N)),nt(ns(N)))
r2: q(|0|()) -> |0|()
r3: q(s(X)) -> s(p(q(X),d(X)))
r4: d(|0|()) -> |0|()
r5: d(s(X)) -> s(s(d(X)))
r6: p(|0|(),X) -> X
r7: p(X,|0|()) -> X
r8: p(s(X),s(Y)) -> s(s(p(X,Y)))
r9: f(|0|(),X) -> nil()
r10: f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
r11: t(X) -> nt(X)
r12: s(X) -> ns(X)
r13: f(X1,X2) -> nf(X1,X2)
r14: a(nt(X)) -> t(a(X))
r15: a(ns(X)) -> s(a(X))
r16: a(nf(X1,X2)) -> f(a(X1),a(X2))
r17: a(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a#_A(x1) = x1
          nf_A(x1,x2) = max{10, x1 + 6, x2 + 3}
          f#_A(x1,x2) = max{2, x1 + 1, x2 + 1}
          a_A(x1) = max{4, x1}
          s_A(x1) = max{3, x1}
          cs_A(x1,x2) = max{x1 + 7, x2}
          ns_A(x1) = max{2, x1}
          nt_A(x1) = max{21, x1 + 17}
          d_A(x1) = x1 + 6
          |0|_A = 6
          p_A(x1,x2) = max{13, x1, x2 + 4}
          q_A(x1) = x1 + 10
          t_A(x1) = max{21, x1 + 17}
          r_A(x1) = max{9, x1}
          f_A(x1,x2) = max{10, x1 + 6, x2 + 3}
          nil_A = 11
    
      precedence:
      
        f# = d > nf = a = |0| = q = t = f = nil > nt > cs = p > a# = s = ns > r
      
      partial status:
    
        pi(a#) = [1]
        pi(nf) = [1, 2]
        pi(f#) = [1, 2]
        pi(a) = [1]
        pi(s) = [1]
        pi(cs) = [1, 2]
        pi(ns) = [1]
        pi(nt) = [1]
        pi(d) = [1]
        pi(|0|) = []
        pi(p) = [1, 2]
        pi(q) = [1]
        pi(t) = [1]
        pi(r) = [1]
        pi(f) = [1, 2]
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a#_A(x1) = x1 + 30
          nf_A(x1,x2) = max{x1 + 2, x2 + 25}
          f#_A(x1,x2) = max{29, x1 - 5, x2}
          a_A(x1) = x1 + 33
          s_A(x1) = max{34, x1}
          cs_A(x1,x2) = max{x1 + 1, x2 + 17}
          ns_A(x1) = x1
          nt_A(x1) = x1 + 1
          d_A(x1) = 34
          |0|_A = 38
          p_A(x1,x2) = max{x1 + 74, x2 + 74}
          q_A(x1) = x1 + 38
          t_A(x1) = max{35, x1 + 17}
          r_A(x1) = max{36, x1 - 1}
          f_A(x1,x2) = max{x1 + 74, x2 + 40}
          nil_A = 39
    
      precedence:
      
        r = f = nil > a# = nf = f# = a = cs = nt = d = t > s > ns > |0| = p = q
      
      partial status:
    
        pi(a#) = []
        pi(nf) = [2]
        pi(f#) = [2]
        pi(a) = [1]
        pi(s) = []
        pi(cs) = [2]
        pi(ns) = []
        pi(nt) = [1]
        pi(d) = []
        pi(|0|) = []
        pi(p) = [1]
        pi(q) = [1]
        pi(t) = []
        pi(r) = []
        pi(f) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6

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

-- Reduction pair.

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

p1: q#(s(X)) -> q#(X)

and R consists of:

r1: t(N) -> cs(r(q(N)),nt(ns(N)))
r2: q(|0|()) -> |0|()
r3: q(s(X)) -> s(p(q(X),d(X)))
r4: d(|0|()) -> |0|()
r5: d(s(X)) -> s(s(d(X)))
r6: p(|0|(),X) -> X
r7: p(X,|0|()) -> X
r8: p(s(X),s(Y)) -> s(s(p(X,Y)))
r9: f(|0|(),X) -> nil()
r10: f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
r11: t(X) -> nt(X)
r12: s(X) -> ns(X)
r13: f(X1,X2) -> nf(X1,X2)
r14: a(nt(X)) -> t(a(X))
r15: a(ns(X)) -> s(a(X))
r16: a(nf(X1,X2)) -> f(a(X1),a(X2))
r17: a(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          q#_A(x1) = max{4, x1 + 3}
          s_A(x1) = max{3, x1 + 2}
    
      precedence:
      
        q# = s
      
      partial status:
    
        pi(q#) = [1]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          q#_A(x1) = max{1, x1 - 1}
          s_A(x1) = x1
    
      precedence:
      
        q# = s
      
      partial status:
    
        pi(q#) = []
        pi(s) = [1]
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: p#(s(X),s(Y)) -> p#(X,Y)

and R consists of:

r1: t(N) -> cs(r(q(N)),nt(ns(N)))
r2: q(|0|()) -> |0|()
r3: q(s(X)) -> s(p(q(X),d(X)))
r4: d(|0|()) -> |0|()
r5: d(s(X)) -> s(s(d(X)))
r6: p(|0|(),X) -> X
r7: p(X,|0|()) -> X
r8: p(s(X),s(Y)) -> s(s(p(X,Y)))
r9: f(|0|(),X) -> nil()
r10: f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
r11: t(X) -> nt(X)
r12: s(X) -> ns(X)
r13: f(X1,X2) -> nf(X1,X2)
r14: a(nt(X)) -> t(a(X))
r15: a(ns(X)) -> s(a(X))
r16: a(nf(X1,X2)) -> f(a(X1),a(X2))
r17: a(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          p#_A(x1,x2) = max{2, x1 - 1, x2 + 1}
          s_A(x1) = max{1, x1}
    
      precedence:
      
        p# = s
      
      partial status:
    
        pi(p#) = [2]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          p#_A(x1,x2) = 0
          s_A(x1) = max{2, x1}
    
      precedence:
      
        p# = s
      
      partial status:
    
        pi(p#) = []
        pi(s) = [1]
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: d#(s(X)) -> d#(X)

and R consists of:

r1: t(N) -> cs(r(q(N)),nt(ns(N)))
r2: q(|0|()) -> |0|()
r3: q(s(X)) -> s(p(q(X),d(X)))
r4: d(|0|()) -> |0|()
r5: d(s(X)) -> s(s(d(X)))
r6: p(|0|(),X) -> X
r7: p(X,|0|()) -> X
r8: p(s(X),s(Y)) -> s(s(p(X,Y)))
r9: f(|0|(),X) -> nil()
r10: f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
r11: t(X) -> nt(X)
r12: s(X) -> ns(X)
r13: f(X1,X2) -> nf(X1,X2)
r14: a(nt(X)) -> t(a(X))
r15: a(ns(X)) -> s(a(X))
r16: a(nf(X1,X2)) -> f(a(X1),a(X2))
r17: a(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          d#_A(x1) = max{4, x1 + 3}
          s_A(x1) = max{3, x1 + 2}
    
      precedence:
      
        d# = s
      
      partial status:
    
        pi(d#) = [1]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          d#_A(x1) = max{1, x1 - 1}
          s_A(x1) = x1
    
      precedence:
      
        d# = s
      
      partial status:
    
        pi(d#) = []
        pi(s) = [1]
    

The next rules are strictly ordered:

  p1

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