YES

We show the termination of the TRS R:

  admit(x,nil()) -> nil()
  admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
  cond(true(),y) -> y

-- SCC decomposition.

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

p1: admit#(x,.(u,.(v,.(w(),z)))) -> cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
p2: admit#(x,.(u,.(v,.(w(),z)))) -> admit#(carry(x,u,v),z)

and R consists of:

r1: admit(x,nil()) -> nil()
r2: admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
r3: cond(true(),y) -> y

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

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

p1: admit#(x,.(u,.(v,.(w(),z)))) -> admit#(carry(x,u,v),z)

and R consists of:

r1: admit(x,nil()) -> nil()
r2: admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
r3: cond(true(),y) -> y

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        admit#_A(x1,x2) = x1 + x2
        ._A(x1,x2) = x1 + x2 + 1
        w_A() = 3
        carry_A(x1,x2,x3) = x1 + x2 + x3 + 3
    
    2. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        admit#_A(x1,x2) = x1 + x2
        ._A(x1,x2) = x1 + x2
        w_A() = 2
        carry_A(x1,x2,x3) = x1 + x2 + x3 + 3
    

The next rules are strictly ordered:

  p1
  r1, r2, r3

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