YES
We show the termination of the TRS R:

  app(app(map(),f),nil()) -> nil()
  app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
  app(app(minus(),x),|0|()) -> x
  app(app(minus(),app(s(),x)),app(s(),y)) -> app(app(minus(),app(p(),app(s(),x))),app(p(),app(s(),y)))
  app(p(),app(s(),x)) -> x
  app(app(div(),|0|()),app(s(),y)) -> |0|()
  app(app(div(),app(s(),x)),app(s(),y)) -> app(s(),app(app(div(),app(app(minus(),x),app(id(),y))),app(s(),y)))
  app(id(),x) -> x
  app(id(),x) -> app(s(),app(s(),app(s(),x)))
  app(id(),app(p(),x)) -> app(id(),app(s(),app(id(),x)))

Instead we prove termination of the uncurried system:

  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(div_0(),y) -> div_1(y)
  app(div_1(x1),y) -> div_2(x1,y)
  app(id_0(),y) -> id_1(y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(minus_0(),y) -> minus_1(y)
  app(minus_1(x1),y) -> minus_2(x1,y)
  app(p_0(),y) -> p_1(y)
  app(s_0(),y) -> s_1(y)
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
  minus_2(x,|0|_0()) -> x
  minus_2(s_1(x),s_1(y)) -> minus_2(p_1(s_1(x)),p_1(s_1(y)))
  p_1(s_1(x)) -> x
  div_2(|0|_0(),s_1(y)) -> |0|_0()
  div_2(s_1(x),s_1(y)) -> s_1(div_2(minus_2(x,id_1(y)),s_1(y)))
  id_1(x) -> x
  id_1(x) -> s_1(s_1(s_1(x)))
  id_1(p_1(x)) -> id_1(s_1(id_1(x)))


-- SCC decomposition.

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

p1: app#(div_1(x1),y) -> div_2#(x1,y)
p2: app#(id_0(),y) -> id_1#(y)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: app#(minus_1(x1),y) -> minus_2#(x1,y)
p5: app#(p_0(),y) -> p_1#(y)
p6: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p7: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p8: minus_2#(s_1(x),s_1(y)) -> minus_2#(p_1(s_1(x)),p_1(s_1(y)))
p9: minus_2#(s_1(x),s_1(y)) -> p_1#(s_1(x))
p10: minus_2#(s_1(x),s_1(y)) -> p_1#(s_1(y))
p11: div_2#(s_1(x),s_1(y)) -> div_2#(minus_2(x,id_1(y)),s_1(y))
p12: div_2#(s_1(x),s_1(y)) -> minus_2#(x,id_1(y))
p13: div_2#(s_1(x),s_1(y)) -> id_1#(y)
p14: id_1#(p_1(x)) -> id_1#(s_1(id_1(x)))
p15: id_1#(p_1(x)) -> id_1#(x)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(div_0(),y) -> div_1(y)
r4: app(div_1(x1),y) -> div_2(x1,y)
r5: app(id_0(),y) -> id_1(y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(minus_0(),y) -> minus_1(y)
r9: app(minus_1(x1),y) -> minus_2(x1,y)
r10: app(p_0(),y) -> p_1(y)
r11: app(s_0(),y) -> s_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: minus_2(x,|0|_0()) -> x
r15: minus_2(s_1(x),s_1(y)) -> minus_2(p_1(s_1(x)),p_1(s_1(y)))
r16: p_1(s_1(x)) -> x
r17: div_2(|0|_0(),s_1(y)) -> |0|_0()
r18: div_2(s_1(x),s_1(y)) -> s_1(div_2(minus_2(x,id_1(y)),s_1(y)))
r19: id_1(x) -> x
r20: id_1(x) -> s_1(s_1(s_1(x)))
r21: id_1(p_1(x)) -> id_1(s_1(id_1(x)))

The estimated dependency graph contains the following SCCs:

  {p3, p6, p7}
  {p11}
  {p15}
  {p8}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(div_0(),y) -> div_1(y)
r4: app(div_1(x1),y) -> div_2(x1,y)
r5: app(id_0(),y) -> id_1(y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(minus_0(),y) -> minus_1(y)
r9: app(minus_1(x1),y) -> minus_2(x1,y)
r10: app(p_0(),y) -> p_1(y)
r11: app(s_0(),y) -> s_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: minus_2(x,|0|_0()) -> x
r15: minus_2(s_1(x),s_1(y)) -> minus_2(p_1(s_1(x)),p_1(s_1(y)))
r16: p_1(s_1(x)) -> x
r17: div_2(|0|_0(),s_1(y)) -> |0|_0()
r18: div_2(s_1(x),s_1(y)) -> s_1(div_2(minus_2(x,id_1(y)),s_1(y)))
r19: id_1(x) -> x
r20: id_1(x) -> s_1(s_1(s_1(x)))
r21: id_1(p_1(x)) -> id_1(s_1(id_1(x)))

The set of usable rules consists of

  r1, r2, r3, r8, r11, r17, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x1 + 3
        cons_2_A(x1,x2) = max{7, x1 - 30, x2 - 30}
        app#_A(x1,x2) = max{2, x1 - 1}
        map_1_A(x1) = max{14, x1 + 5}
        app_A(x1,x2) = max{26, x1 - 15, x2 - 14}
        cons_0_A = 16
        cons_1_A(x1) = max{23, x1 - 15}
        div_0_A = 16
        div_1_A(x1) = max{14, x1 - 16}
        minus_0_A = 16
        minus_1_A(x1) = max{14, x1 - 15}
        s_0_A = 14
        s_1_A(x1) = max{20, x1 - 32}
        div_2_A(x1,x2) = max{21, x1 - 32, x2 - 33}
        |0|_0_A = 0
        id_1_A(x1) = max{22, x1 - 45}
  
    base order 2:
  
      precedence:
      
        map_1 > app = cons_0 > cons_2 > map_2# = cons_1 = id_1 > app# = minus_1 = s_1 = div_2 > div_1 = minus_0 = s_0 = |0|_0 > div_0
      
  
    partial status:
  
      pi(map_2#) = [1]
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(div_2) = []
      pi(|0|_0) = []
      pi(id_1) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: app#(map_1(x1),y) -> map_2#(x1,y)
p2: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(div_0(),y) -> div_1(y)
r4: app(div_1(x1),y) -> div_2(x1,y)
r5: app(id_0(),y) -> id_1(y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(minus_0(),y) -> minus_1(y)
r9: app(minus_1(x1),y) -> minus_2(x1,y)
r10: app(p_0(),y) -> p_1(y)
r11: app(s_0(),y) -> s_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: minus_2(x,|0|_0()) -> x
r15: minus_2(s_1(x),s_1(y)) -> minus_2(p_1(s_1(x)),p_1(s_1(y)))
r16: p_1(s_1(x)) -> x
r17: div_2(|0|_0(),s_1(y)) -> |0|_0()
r18: div_2(s_1(x),s_1(y)) -> s_1(div_2(minus_2(x,id_1(y)),s_1(y)))
r19: id_1(x) -> x
r20: id_1(x) -> s_1(s_1(s_1(x)))
r21: id_1(p_1(x)) -> id_1(s_1(id_1(x)))

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(div_0(),y) -> div_1(y)
r4: app(div_1(x1),y) -> div_2(x1,y)
r5: app(id_0(),y) -> id_1(y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(minus_0(),y) -> minus_1(y)
r9: app(minus_1(x1),y) -> minus_2(x1,y)
r10: app(p_0(),y) -> p_1(y)
r11: app(s_0(),y) -> s_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: minus_2(x,|0|_0()) -> x
r15: minus_2(s_1(x),s_1(y)) -> minus_2(p_1(s_1(x)),p_1(s_1(y)))
r16: p_1(s_1(x)) -> x
r17: div_2(|0|_0(),s_1(y)) -> |0|_0()
r18: div_2(s_1(x),s_1(y)) -> s_1(div_2(minus_2(x,id_1(y)),s_1(y)))
r19: id_1(x) -> x
r20: id_1(x) -> s_1(s_1(s_1(x)))
r21: id_1(p_1(x)) -> id_1(s_1(id_1(x)))

The set of usable rules consists of

  r1, r3, r6, r8, r11, r17, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x2 + 160
        cons_2_A(x1,x2) = max{x1 - 91, x2 + 161}
        app_A(x1,x2) = max{68, x2 - 91}
        cons_0_A = 1
        cons_1_A(x1) = max{66, x1 - 91}
        div_0_A = 66
        div_1_A(x1) = max{65, x1 - 91}
        map_0_A = 1
        map_1_A(x1) = 68
        minus_0_A = 1
        minus_1_A(x1) = max{68, x1 - 91}
        s_0_A = 1
        s_1_A(x1) = max{56, x1 - 92}
        div_2_A(x1,x2) = max{66, x1 - 83, x2 - 113}
        |0|_0_A = 55
        id_1_A(x1) = max{67, x1 - 129}
  
    base order 2:
  
      precedence:
      
        cons_2 > map_2# = app = cons_0 > cons_1 = map_1 = id_1 > map_0 = s_1 = div_2 > div_1 = minus_1 = s_0 = |0|_0 > div_0 = minus_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(div_2) = []
      pi(|0|_0) = []
      pi(id_1) = []

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: div_2#(s_1(x),s_1(y)) -> div_2#(minus_2(x,id_1(y)),s_1(y))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(div_0(),y) -> div_1(y)
r4: app(div_1(x1),y) -> div_2(x1,y)
r5: app(id_0(),y) -> id_1(y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(minus_0(),y) -> minus_1(y)
r9: app(minus_1(x1),y) -> minus_2(x1,y)
r10: app(p_0(),y) -> p_1(y)
r11: app(s_0(),y) -> s_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: minus_2(x,|0|_0()) -> x
r15: minus_2(s_1(x),s_1(y)) -> minus_2(p_1(s_1(x)),p_1(s_1(y)))
r16: p_1(s_1(x)) -> x
r17: div_2(|0|_0(),s_1(y)) -> |0|_0()
r18: div_2(s_1(x),s_1(y)) -> s_1(div_2(minus_2(x,id_1(y)),s_1(y)))
r19: id_1(x) -> x
r20: id_1(x) -> s_1(s_1(s_1(x)))
r21: id_1(p_1(x)) -> id_1(s_1(id_1(x)))

The set of usable rules consists of

  r1, r3, r6, r8, r14, r15, r16, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        div_2#_A(x1,x2) = max{1, x1 - 6}
        s_1_A(x1) = max{33, x1 + 8}
        minus_2_A(x1,x2) = x1 + 3
        id_1_A(x1) = max{15, x1 - 10}
        app_A(x1,x2) = max{34, x1 - 4, x2 - 9}
        cons_0_A = 3
        cons_1_A(x1) = 2
        div_0_A = 40
        div_1_A(x1) = 35
        map_0_A = 3
        map_1_A(x1) = max{2, x1 - 10}
        minus_0_A = 3
        minus_1_A(x1) = max{0, x1 - 9}
        |0|_0_A = 32
        p_1_A(x1) = max{33, x1}
        div_2_A(x1,x2) = max{34, x2 - 42}
  
    base order 2:
  
      precedence:
      
        p_1 > div_2# = s_1 = minus_2 > app = cons_0 = div_0 = map_0 = minus_0 > id_1 = map_1 = minus_1 > cons_1 = div_2 > div_1 = |0|_0
      
  
    partial status:
  
      pi(div_2#) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(id_1) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(|0|_0) = []
      pi(p_1) = [1]
      pi(div_2) = []

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: id_1#(p_1(x)) -> id_1#(x)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(div_0(),y) -> div_1(y)
r4: app(div_1(x1),y) -> div_2(x1,y)
r5: app(id_0(),y) -> id_1(y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(minus_0(),y) -> minus_1(y)
r9: app(minus_1(x1),y) -> minus_2(x1,y)
r10: app(p_0(),y) -> p_1(y)
r11: app(s_0(),y) -> s_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: minus_2(x,|0|_0()) -> x
r15: minus_2(s_1(x),s_1(y)) -> minus_2(p_1(s_1(x)),p_1(s_1(y)))
r16: p_1(s_1(x)) -> x
r17: div_2(|0|_0(),s_1(y)) -> |0|_0()
r18: div_2(s_1(x),s_1(y)) -> s_1(div_2(minus_2(x,id_1(y)),s_1(y)))
r19: id_1(x) -> x
r20: id_1(x) -> s_1(s_1(s_1(x)))
r21: id_1(p_1(x)) -> id_1(s_1(id_1(x)))

The set of usable rules consists of

  r1, r2, r3, r6, r8, r11, r17, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        id_1#_A(x1) = max{23, x1 + 12}
        p_1_A(x1) = max{23, x1 + 12}
        app_A(x1,x2) = max{17, x1 - 21, x2 - 19}
        cons_0_A = 13
        cons_1_A(x1) = max{14, x1 - 20}
        cons_2_A(x1,x2) = max{15, x1 - 42, x2 - 24}
        div_0_A = 9
        div_1_A(x1) = max{10, x1 - 20}
        map_0_A = 20
        map_1_A(x1) = max{1, x1 - 20}
        minus_0_A = 20
        minus_1_A(x1) = max{0, x1 - 20}
        s_0_A = 12
        s_1_A(x1) = max{13, x1 - 54}
        div_2_A(x1,x2) = max{11, x1 - 42, x2 - 66}
        |0|_0_A = 0
        id_1_A(x1) = max{16, x1 - 52}
  
    base order 2:
  
      precedence:
      
        p_1 > id_1# = app = map_0 > cons_2 = map_1 = id_1 > cons_1 = s_1 > cons_0 = minus_1 = s_0 = div_2 > div_1 = minus_0 = |0|_0 > div_0
      
  
    partial status:
  
      pi(id_1#) = [1]
      pi(p_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(div_2) = []
      pi(|0|_0) = []
      pi(id_1) = []

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: minus_2#(s_1(x),s_1(y)) -> minus_2#(p_1(s_1(x)),p_1(s_1(y)))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(div_0(),y) -> div_1(y)
r4: app(div_1(x1),y) -> div_2(x1,y)
r5: app(id_0(),y) -> id_1(y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(minus_0(),y) -> minus_1(y)
r9: app(minus_1(x1),y) -> minus_2(x1,y)
r10: app(p_0(),y) -> p_1(y)
r11: app(s_0(),y) -> s_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: minus_2(x,|0|_0()) -> x
r15: minus_2(s_1(x),s_1(y)) -> minus_2(p_1(s_1(x)),p_1(s_1(y)))
r16: p_1(s_1(x)) -> x
r17: div_2(|0|_0(),s_1(y)) -> |0|_0()
r18: div_2(s_1(x),s_1(y)) -> s_1(div_2(minus_2(x,id_1(y)),s_1(y)))
r19: id_1(x) -> x
r20: id_1(x) -> s_1(s_1(s_1(x)))
r21: id_1(p_1(x)) -> id_1(s_1(id_1(x)))

The set of usable rules consists of

  r1, r2, r3, r6, r8, r10, r16, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus_2#_A(x1,x2) = max{6, x1 + 3}
        s_1_A(x1) = x1 + 9
        p_1_A(x1) = max{5, x1 - 8}
        app_A(x1,x2) = max{19, x1 - 13, x2 - 7}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 8}
        cons_2_A(x1,x2) = max{3, x1 - 22, x2 - 36}
        div_0_A = 12
        div_1_A(x1) = max{1, x1 - 34}
        map_0_A = 11
        map_1_A(x1) = max{12, x1 - 36}
        minus_0_A = 11
        minus_1_A(x1) = max{12, x1 - 8}
        p_0_A = 4
        div_2_A(x1,x2) = max{10, x1 - 33, x2 - 67}
        |0|_0_A = 5
  
    base order 2:
  
      precedence:
      
        div_2 > minus_2# = s_1 > app = cons_0 = |0|_0 > p_1 = cons_2 = div_1 = map_1 = minus_1 > cons_1 = div_0 = map_0 = minus_0 = p_0
      
  
    partial status:
  
      pi(minus_2#) = [1]
      pi(s_1) = [1]
      pi(p_1) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(p_0) = []
      pi(div_2) = []
      pi(|0|_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.