YES
We show the termination of the TRS R:

  g(x,|0|()) -> |0|()
  g(d(),s(x)) -> s(s(g(d(),x)))
  g(h(),s(|0|())) -> |0|()
  g(h(),s(s(x))) -> s(g(h(),x))
  double(x) -> g(d(),x)
  half(x) -> g(h(),x)
  f(s(x),y) -> f(half(s(x)),double(y))
  f(s(|0|()),y) -> y
  id(x) -> f(x,s(|0|()))


-- SCC decomposition.

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

p1: g#(d(),s(x)) -> g#(d(),x)
p2: g#(h(),s(s(x))) -> g#(h(),x)
p3: double#(x) -> g#(d(),x)
p4: half#(x) -> g#(h(),x)
p5: f#(s(x),y) -> f#(half(s(x)),double(y))
p6: f#(s(x),y) -> half#(s(x))
p7: f#(s(x),y) -> double#(y)
p8: id#(x) -> f#(x,s(|0|()))

and R consists of:

r1: g(x,|0|()) -> |0|()
r2: g(d(),s(x)) -> s(s(g(d(),x)))
r3: g(h(),s(|0|())) -> |0|()
r4: g(h(),s(s(x))) -> s(g(h(),x))
r5: double(x) -> g(d(),x)
r6: half(x) -> g(h(),x)
r7: f(s(x),y) -> f(half(s(x)),double(y))
r8: f(s(|0|()),y) -> y
r9: id(x) -> f(x,s(|0|()))

The estimated dependency graph contains the following SCCs:

  {p5}
  {p1}
  {p2}


-- Reduction pair.

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

p1: f#(s(x),y) -> f#(half(s(x)),double(y))

and R consists of:

r1: g(x,|0|()) -> |0|()
r2: g(d(),s(x)) -> s(s(g(d(),x)))
r3: g(h(),s(|0|())) -> |0|()
r4: g(h(),s(s(x))) -> s(g(h(),x))
r5: double(x) -> g(d(),x)
r6: half(x) -> g(h(),x)
r7: f(s(x),y) -> f(half(s(x)),double(y))
r8: f(s(|0|()),y) -> y
r9: id(x) -> f(x,s(|0|()))

The set of usable rules consists of

  r1, r3, r4, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          f#_A(x1,x2) = 6
          s_A(x1) = 5
          half_A(x1) = x1 + 3
          double_A(x1) = x1 + 2
          g_A(x1,x2) = x2 + 2
          |0|_A() = 0
          h_A() = 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          f#_A(x1,x2) = max{0, x1 - 1}
          s_A(x1) = x1 + 10
          half_A(x1) = max{2, x1 - 5}
          double_A(x1) = x1 + 20
          g_A(x1,x2) = max{0, x1 - 1, x2 - 10}
          |0|_A = 0
          h_A = 0
          f#!_A(x1,x2) = max{1, x1 - 6}
          s!_A(x1) = x1 + 11
          half!_A(x1) = 4
          double!_A(x1) = max{12, x1 - 23}
          g!_A(x1,x2) = max{32, x1 - 43, x2 - 21}
          |0|!_A = 3
          h!_A = 4
  
    partial status:
  
      pi(f#) = []
      pi(s) = []
      pi(half) = []
      pi(double) = []
      pi(g) = [2]
      pi(|0|) = []
      pi(h) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: g#(d(),s(x)) -> g#(d(),x)

and R consists of:

r1: g(x,|0|()) -> |0|()
r2: g(d(),s(x)) -> s(s(g(d(),x)))
r3: g(h(),s(|0|())) -> |0|()
r4: g(h(),s(s(x))) -> s(g(h(),x))
r5: double(x) -> g(d(),x)
r6: half(x) -> g(h(),x)
r7: f(s(x),y) -> f(half(s(x)),double(y))
r8: f(s(|0|()),y) -> y
r9: id(x) -> f(x,s(|0|()))

The set of usable rules consists of

  r1, r3

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          g#_A(x1,x2) = x2 + 2
          d_A() = 7
          s_A(x1) = x1 + 3
          g_A(x1,x2) = 0
          |0|_A() = 0
          h_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          g#_A(x1,x2) = 0
          d_A = 2
          s_A(x1) = 7
          g_A(x1,x2) = 13
          |0|_A = 3
          h_A = 4
          g#!_A(x1,x2) = max{23, x1 + 20}
          d!_A = 3
          s!_A(x1) = max{21, x1 + 5}
          g!_A(x1,x2) = max{19, x1 + 15, x2 + 10}
          |0|!_A = 7
          h!_A = 20
  
    partial status:
  
      pi(g#) = [2]
      pi(d) = []
      pi(s) = [1]
      pi(g) = []
      pi(|0|) = []
      pi(h) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: g#(h(),s(s(x))) -> g#(h(),x)

and R consists of:

r1: g(x,|0|()) -> |0|()
r2: g(d(),s(x)) -> s(s(g(d(),x)))
r3: g(h(),s(|0|())) -> |0|()
r4: g(h(),s(s(x))) -> s(g(h(),x))
r5: double(x) -> g(d(),x)
r6: half(x) -> g(h(),x)
r7: f(s(x),y) -> f(half(s(x)),double(y))
r8: f(s(|0|()),y) -> y
r9: id(x) -> f(x,s(|0|()))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          g#_A(x1,x2) = x2 + 3
          h_A() = 2
          s_A(x1) = x1 + 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          g#_A(x1,x2) = 0
          h_A = 19
          s_A(x1) = max{8, x1 - 9}
          g#!_A(x1,x2) = max{5, x2 - 34}
          h!_A = 2
          s!_A(x1) = max{4, x1 - 24}
  
    partial status:
  
      pi(g#) = [2]
      pi(h) = []
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.