YES
We show the termination of the TRS R:

  app(app(plus(),|0|()),y) -> y
  app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
  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(app(curry(),g),x),y) -> app(app(g,x),y)
  inc() -> app(map(),app(app(curry(),plus()),app(s(),|0|())))

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(curry_0(),y) -> curry_1(y)
  app(curry_1(x1),y) -> curry_2(x1,y)
  app(curry_2(x1,x2),y) -> curry_3(x1,x2,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(plus_0(),y) -> plus_1(y)
  app(plus_1(x1),y) -> plus_2(x1,y)
  app(s_0(),y) -> s_1(y)
  plus_2(|0|_0(),y) -> y
  plus_2(s_1(x),y) -> s_1(plus_2(x,y))
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
  curry_3(g,x,y) -> app(app(g,x),y)
  inc_0() -> map_1(curry_2(plus_0(),s_1(|0|_0())))


-- SCC decomposition.

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

p1: app#(curry_2(x1,x2),y) -> curry_3#(x1,x2,y)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: app#(plus_1(x1),y) -> plus_2#(x1,y)
p4: plus_2#(s_1(x),y) -> plus_2#(x,y)
p5: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p7: curry_3#(g,x,y) -> app#(app(g,x),y)
p8: curry_3#(g,x,y) -> app#(g,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(curry_0(),y) -> curry_1(y)
r4: app(curry_1(x1),y) -> curry_2(x1,y)
r5: app(curry_2(x1,x2),y) -> curry_3(x1,x2,y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(plus_0(),y) -> plus_1(y)
r9: app(plus_1(x1),y) -> plus_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: plus_2(|0|_0(),y) -> y
r12: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r13: map_2(f,nil_0()) -> nil_0()
r14: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r15: curry_3(g,x,y) -> app(app(g,x),y)
r16: inc_0() -> map_1(curry_2(plus_0(),s_1(|0|_0())))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p5, p6, p7, p8}
  {p4}


-- Reduction pair.

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

p1: app#(curry_2(x1,x2),y) -> curry_3#(x1,x2,y)
p2: curry_3#(g,x,y) -> app#(g,x)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p5: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: curry_3#(g,x,y) -> app#(app(g,x),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(curry_0(),y) -> curry_1(y)
r4: app(curry_1(x1),y) -> curry_2(x1,y)
r5: app(curry_2(x1,x2),y) -> curry_3(x1,x2,y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(plus_0(),y) -> plus_1(y)
r9: app(plus_1(x1),y) -> plus_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: plus_2(|0|_0(),y) -> y
r12: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r13: map_2(f,nil_0()) -> nil_0()
r14: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r15: curry_3(g,x,y) -> app(app(g,x),y)
r16: inc_0() -> map_1(curry_2(plus_0(),s_1(|0|_0())))

The set of usable rules consists of

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

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1 + 2
          curry_2_A(x1,x2) = x1 + x2 + 6
          curry_3#_A(x1,x2,x3) = x1 + x2 + 5
          map_1_A(x1) = x1 + 4
          map_2#_A(x1,x2) = x1 + 3
          cons_2_A(x1,x2) = x1 + 1
          app_A(x1,x2) = x1 + x2 + 2
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          curry_0_A() = 4
          curry_1_A(x1) = x1 + 5
          curry_3_A(x1,x2,x3) = x1 + x2 + x3 + 7
          map_0_A() = 3
          map_2_A(x1,x2) = x1 + x2 + 5
          plus_0_A() = 1
          plus_1_A(x1) = x1 + 2
          plus_2_A(x1,x2) = x2 + 3
          s_0_A() = 2
          s_1_A(x1) = 1
          |0|_0_A() = 1
          nil_0_A() = 1
          inc_0_A() = 13
  
    base order 2:
  
      precedence:
      
        curry_3# > app# = curry_3 = map_2 = plus_2 > map_2# = app = s_0 = nil_0 = inc_0 > curry_2 = map_1 = cons_2 = plus_1 = s_1 = |0|_0 > cons_1 = curry_1 = map_0 = plus_0 > cons_0 = curry_0
      
  
    partial status:
  
      pi(app#) = []
      pi(curry_2) = [2]
      pi(curry_3#) = [2]
      pi(map_1) = [1]
      pi(map_2#) = [1]
      pi(cons_2) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(curry_0) = []
      pi(curry_1) = [1]
      pi(curry_3) = [3]
      pi(map_0) = []
      pi(map_2) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(plus_2) = [2]
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(nil_0) = []
      pi(inc_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: curry_3#(g,x,y) -> app#(g,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p4: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: curry_3#(g,x,y) -> app#(app(g,x),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(curry_0(),y) -> curry_1(y)
r4: app(curry_1(x1),y) -> curry_2(x1,y)
r5: app(curry_2(x1,x2),y) -> curry_3(x1,x2,y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(plus_0(),y) -> plus_1(y)
r9: app(plus_1(x1),y) -> plus_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: plus_2(|0|_0(),y) -> y
r12: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r13: map_2(f,nil_0()) -> nil_0()
r14: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r15: curry_3(g,x,y) -> app(app(g,x),y)
r16: inc_0() -> map_1(curry_2(plus_0(),s_1(|0|_0())))

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4}


-- Reduction pair.

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)) -> app#(f,x)
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(curry_0(),y) -> curry_1(y)
r4: app(curry_1(x1),y) -> curry_2(x1,y)
r5: app(curry_2(x1,x2),y) -> curry_3(x1,x2,y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(plus_0(),y) -> plus_1(y)
r9: app(plus_1(x1),y) -> plus_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: plus_2(|0|_0(),y) -> y
r12: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r13: map_2(f,nil_0()) -> nil_0()
r14: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r15: curry_3(g,x,y) -> app(app(g,x),y)
r16: inc_0() -> map_1(curry_2(plus_0(),s_1(|0|_0())))

The set of usable rules consists of

  r3, r12, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x2 + 3
          map_1_A(x1) = 3
          map_2#_A(x1,x2) = x2 + 2
          cons_2_A(x1,x2) = x1 + x2 + 2
          app_A(x1,x2) = 0
          curry_0_A() = 1
          curry_1_A(x1) = 0
          plus_2_A(x1,x2) = 2
          s_1_A(x1) = 1
          inc_0_A() = 5
          curry_2_A(x1,x2) = 2
          plus_0_A() = 4
          |0|_0_A() = 1
  
    base order 2:
  
      precedence:
      
        curry_0 = inc_0 > curry_2 = plus_0 = |0|_0 > app# = plus_2 > cons_2 = s_1 > map_2# = app > map_1 = curry_1
      
  
    partial status:
  
      pi(app#) = [2]
      pi(map_1) = []
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(curry_0) = []
      pi(curry_1) = []
      pi(plus_2) = []
      pi(s_1) = []
      pi(inc_0) = []
      pi(curry_2) = []
      pi(plus_0) = []
      pi(|0|_0) = []

The following rules are removed:

  p3



-- 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)) -> app#(f,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(curry_0(),y) -> curry_1(y)
r4: app(curry_1(x1),y) -> curry_2(x1,y)
r5: app(curry_2(x1,x2),y) -> curry_3(x1,x2,y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(plus_0(),y) -> plus_1(y)
r9: app(plus_1(x1),y) -> plus_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: plus_2(|0|_0(),y) -> y
r12: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r13: map_2(f,nil_0()) -> nil_0()
r14: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r15: curry_3(g,x,y) -> app(app(g,x),y)
r16: inc_0() -> map_1(curry_2(plus_0(),s_1(|0|_0())))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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)) -> app#(f,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(curry_0(),y) -> curry_1(y)
r4: app(curry_1(x1),y) -> curry_2(x1,y)
r5: app(curry_2(x1,x2),y) -> curry_3(x1,x2,y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(plus_0(),y) -> plus_1(y)
r9: app(plus_1(x1),y) -> plus_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: plus_2(|0|_0(),y) -> y
r12: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r13: map_2(f,nil_0()) -> nil_0()
r14: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r15: curry_3(g,x,y) -> app(app(g,x),y)
r16: inc_0() -> map_1(curry_2(plus_0(),s_1(|0|_0())))

The set of usable rules consists of

  r1, r2, r3, r4, r6, r8, r10, r12, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x2 + 3
          map_1_A(x1) = 1
          map_2#_A(x1,x2) = x2 + 2
          cons_2_A(x1,x2) = x1 + x2 + 4
          app_A(x1,x2) = x1 + x2 + 5
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          curry_0_A() = 4
          curry_1_A(x1) = x1 + 3
          curry_2_A(x1,x2) = 2
          map_0_A() = 2
          plus_0_A() = 1
          plus_1_A(x1) = x1 + 3
          s_0_A() = 2
          s_1_A(x1) = 1
          plus_2_A(x1,x2) = 2
          inc_0_A() = 3
          |0|_0_A() = 4
  
    base order 2:
  
      precedence:
      
        map_2# = map_0 = inc_0 > map_1 = |0|_0 > app = s_0 > cons_2 = curry_1 = s_1 > app# = cons_1 = curry_0 = curry_2 = plus_1 = plus_2 > cons_0 = plus_0
      
  
    partial status:
  
      pi(app#) = [2]
      pi(map_1) = []
      pi(map_2#) = [2]
      pi(cons_2) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(curry_0) = []
      pi(curry_1) = [1]
      pi(curry_2) = []
      pi(map_0) = []
      pi(plus_0) = []
      pi(plus_1) = [1]
      pi(s_0) = []
      pi(s_1) = []
      pi(plus_2) = []
      pi(inc_0) = []
      pi(|0|_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: map_2#(f,cons_2(x,xs)) -> app#(f,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(curry_0(),y) -> curry_1(y)
r4: app(curry_1(x1),y) -> curry_2(x1,y)
r5: app(curry_2(x1,x2),y) -> curry_3(x1,x2,y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(plus_0(),y) -> plus_1(y)
r9: app(plus_1(x1),y) -> plus_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: plus_2(|0|_0(),y) -> y
r12: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r13: map_2(f,nil_0()) -> nil_0()
r14: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r15: curry_3(g,x,y) -> app(app(g,x),y)
r16: inc_0() -> map_1(curry_2(plus_0(),s_1(|0|_0())))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: plus_2#(s_1(x),y) -> plus_2#(x,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(curry_0(),y) -> curry_1(y)
r4: app(curry_1(x1),y) -> curry_2(x1,y)
r5: app(curry_2(x1,x2),y) -> curry_3(x1,x2,y)
r6: app(map_0(),y) -> map_1(y)
r7: app(map_1(x1),y) -> map_2(x1,y)
r8: app(plus_0(),y) -> plus_1(y)
r9: app(plus_1(x1),y) -> plus_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: plus_2(|0|_0(),y) -> y
r12: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r13: map_2(f,nil_0()) -> nil_0()
r14: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r15: curry_3(g,x,y) -> app(app(g,x),y)
r16: inc_0() -> map_1(curry_2(plus_0(),s_1(|0|_0())))

The set of usable rules consists of

  r1, r3, r4, r8, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus_2#_A(x1,x2) = x1
          s_1_A(x1) = x1
          app_A(x1,x2) = 4
          cons_0_A() = 2
          cons_1_A(x1) = 1
          curry_0_A() = 4
          curry_1_A(x1) = 3
          curry_2_A(x1,x2) = 2
          plus_0_A() = 2
          plus_1_A(x1) = 1
          inc_0_A() = 8
          map_1_A(x1) = x1 + 5
          |0|_0_A() = 1
  
    base order 2:
  
      precedence:
      
        inc_0 > plus_0 = |0|_0 > s_1 = plus_1 > plus_2# > app = cons_0 = curry_0 > cons_1 = curry_1 = map_1 > curry_2
      
  
    partial status:
  
      pi(plus_2#) = [1]
      pi(s_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(curry_0) = []
      pi(curry_1) = []
      pi(curry_2) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(inc_0) = []
      pi(map_1) = []
      pi(|0|_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.