YES
We show the termination of the TRS R:

  app(app(append(),nil()),l) -> l
  app(app(append(),app(app(cons(),h),t)),l) -> app(app(cons(),h),app(app(append(),t),l))
  app(app(map(),f),nil()) -> nil()
  app(app(map(),f),app(app(cons(),h),t)) -> app(app(cons(),app(f,h)),app(app(map(),f),t))
  app(app(append(),app(app(append(),l1),l2)),l3) -> app(app(append(),l1),app(app(append(),l2),l3))
  app(app(map(),f),app(app(append(),l1),l2)) -> app(app(append(),app(app(map(),f),l1)),app(app(map(),f),l2))

Instead we prove termination of the uncurried system:

  app(append_0(),y) -> append_1(y)
  app(append_1(x1),y) -> append_2(x1,y)
  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  append_2(nil_0(),l) -> l
  append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
  append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
  map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))


-- SCC decomposition.

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

p1: app#(append_1(x1),y) -> append_2#(x1,y)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: append_2#(cons_2(h,t),l) -> append_2#(t,l)
p4: map_2#(f,cons_2(h,t)) -> app#(f,h)
p5: map_2#(f,cons_2(h,t)) -> map_2#(f,t)
p6: append_2#(append_2(l1,l2),l3) -> append_2#(l1,append_2(l2,l3))
p7: append_2#(append_2(l1,l2),l3) -> append_2#(l2,l3)
p8: map_2#(f,append_2(l1,l2)) -> append_2#(map_2(f,l1),map_2(f,l2))
p9: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1)
p10: map_2#(f,append_2(l1,l2)) -> map_2#(f,l2)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The estimated dependency graph contains the following SCCs:

  {p2, p4, p5, p9, p10}
  {p3, p6, p7}


-- Reduction pair.

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

p1: map_2#(f,cons_2(h,t)) -> app#(f,h)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,append_2(l1,l2)) -> map_2#(f,l2)
p4: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1)
p5: map_2#(f,cons_2(h,t)) -> map_2#(f,t)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The set of usable rules consists of

  r1, r3, r4, r9

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 + 5
        cons_2_A(x1,x2) = max{8, x1 - 11, x2 - 11}
        app#_A(x1,x2) = x1 + 4
        map_1_A(x1) = max{9, x1 + 6}
        append_2_A(x1,x2) = max{3, x1 - 18, x2 - 37}
        app_A(x1,x2) = max{9, x1 - 10, x2 - 1}
        append_0_A = 1
        append_1_A(x1) = max{2, x1 - 7}
        cons_0_A = 6
        cons_1_A(x1) = max{7, x1 - 1}
        map_2_A(x1,x2) = max{7, x1 - 31, x2 - 13}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        map_1 = map_2 > app# = app = nil_0 > cons_2 = append_2 > map_2# = append_1 = cons_1 > append_0 = cons_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = [1]
      pi(map_1) = [1]
      pi(append_2) = []
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_2) = []
      pi(nil_0) = []

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,append_2(l1,l2)) -> map_2#(f,l2)
p3: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1)
p4: map_2#(f,cons_2(h,t)) -> map_2#(f,t)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

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: map_2#(f,append_2(l1,l2)) -> map_2#(f,l2)
p2: map_2#(f,cons_2(h,t)) -> map_2#(f,t)
p3: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The set of usable rules consists of

  r1, r3, r5, r7, r9

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 + 6
        append_2_A(x1,x2) = max{19, x1 + 9, x2 + 10}
        cons_2_A(x1,x2) = max{18, x1 - 4, x2 + 7}
        app_A(x1,x2) = max{3, x1 - 28, x2 - 25}
        append_0_A = 1
        append_1_A(x1) = max{2, x1 - 26}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 26}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 26}
        nil_0_A = 0
        map_2_A(x1,x2) = max{20, x1 - 27, x2 - 20}
  
    base order 2:
  
      precedence:
      
        map_2 > append_2 = cons_2 = app = nil_0 > map_2# = append_1 = cons_1 = map_1 > append_0 = cons_0 = map_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(append_2) = [2]
      pi(cons_2) = []
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(nil_0) = []
      pi(map_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: map_2#(f,append_2(l1,l2)) -> map_2#(f,l2)
p2: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: map_2#(f,append_2(l1,l2)) -> map_2#(f,l2)
p2: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The set of usable rules consists of

  r1, r3, r4, r5, r7, r9

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 + 6
        append_2_A(x1,x2) = max{x1 + 7, x2 + 9}
        app_A(x1,x2) = max{10, x1 - 11, x2 - 8}
        append_0_A = 10
        append_1_A(x1) = max{3, x1 - 9}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 9}
        cons_2_A(x1,x2) = max{3, x1 - 21, x2 - 32}
        map_0_A = 9
        map_1_A(x1) = max{10, x1 - 33}
        nil_0_A = 0
        map_2_A(x1,x2) = max{10, x1 - 32, x2 - 8}
  
    base order 2:
  
      precedence:
      
        append_0 = map_2 > append_2 = nil_0 > map_2# = app > append_1 = cons_2 = map_1 > cons_1 = map_0 > cons_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(append_2) = [2]
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(nil_0) = []
      pi(map_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: map_2#(f,append_2(l1,l2)) -> map_2#(f,l1)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The set of usable rules consists of

  r1, r3, r4, r5, r9

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 + 7
        append_2_A(x1,x2) = max{x1 + 7, x2 - 5}
        app_A(x1,x2) = max{9, x1 - 3, x2 - 8}
        append_0_A = 1
        append_1_A(x1) = max{2, x1 - 9}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 9}
        cons_2_A(x1,x2) = max{3, x1 - 13, x2 - 22}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 9}
        map_2_A(x1,x2) = max{8, x1 - 16, x2 - 8}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app = append_0 = cons_0 = cons_2 = map_2 > append_2 = cons_1 = map_1 = nil_0 > map_2# = append_1 = map_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(append_2) = [1]
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(map_2) = []
      pi(nil_0) = []

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: append_2#(cons_2(h,t),l) -> append_2#(t,l)
p2: append_2#(append_2(l1,l2),l3) -> append_2#(l2,l3)
p3: append_2#(append_2(l1,l2),l3) -> append_2#(l1,append_2(l2,l3))

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The set of usable rules consists of

  r1, r3, r5, r7, r9

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        append_2#_A(x1,x2) = x1 + 9
        cons_2_A(x1,x2) = max{x1 - 7, x2 + 9}
        append_2_A(x1,x2) = max{25, x1 + 13, x2 + 13}
        app_A(x1,x2) = max{15, x1 - 3, x2 - 1}
        append_0_A = 1
        append_1_A(x1) = max{2, x1 - 2}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 4}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 2}
        nil_0_A = 0
        map_2_A(x1,x2) = max{14, x1 - 12, x2 - 10}
  
    base order 2:
  
      precedence:
      
        map_2 > app > cons_2 = append_2 = map_1 = nil_0 > append_2# = append_1 = cons_1 = map_0 > append_0 = cons_0
      
  
    partial status:
  
      pi(append_2#) = [1]
      pi(cons_2) = [2]
      pi(append_2) = [1, 2]
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(nil_0) = []
      pi(map_2) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: append_2#(cons_2(h,t),l) -> append_2#(t,l)
p2: append_2#(append_2(l1,l2),l3) -> append_2#(l2,l3)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: append_2#(cons_2(h,t),l) -> append_2#(t,l)
p2: append_2#(append_2(l1,l2),l3) -> append_2#(l2,l3)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The set of usable rules consists of

  r1, r3, r5, r7, r9

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        append_2#_A(x1,x2) = x1 + 2
        cons_2_A(x1,x2) = max{8, x1, x2 + 2}
        append_2_A(x1,x2) = max{x1 + 5, x2 + 9}
        app_A(x1,x2) = max{0, x1 - 2, x2 - 8}
        append_0_A = 1
        append_1_A(x1) = max{0, x1 - 8}
        cons_0_A = 1
        cons_1_A(x1) = max{0, x1 - 9}
        map_0_A = 1
        map_1_A(x1) = max{0, x1 - 8}
        nil_0_A = 0
        map_2_A(x1,x2) = max{0, x1 - 10}
  
    base order 2:
  
      precedence:
      
        app > cons_2 = map_2 > append_2 = cons_1 = map_1 = nil_0 > append_2# = append_1 = cons_0 = map_0 > append_0
      
  
    partial status:
  
      pi(append_2#) = [1]
      pi(cons_2) = []
      pi(append_2) = [2]
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(nil_0) = []
      pi(map_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: append_2#(cons_2(h,t),l) -> append_2#(t,l)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: append_2#(cons_2(h,t),l) -> append_2#(t,l)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: append_2(nil_0(),l) -> l
r8: append_2(cons_2(h,t),l) -> cons_2(h,append_2(t,l))
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r11: append_2(append_2(l1,l2),l3) -> append_2(l1,append_2(l2,l3))
r12: map_2(f,append_2(l1,l2)) -> append_2(map_2(f,l1),map_2(f,l2))

The set of usable rules consists of

  r1, r3, r5, r9

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        append_2#_A(x1,x2) = max{3, x1 - 1}
        cons_2_A(x1,x2) = max{5, x1 - 8, x2 + 2}
        app_A(x1,x2) = max{7, x1 - 6, x2 - 18}
        append_0_A = 4
        append_1_A(x1) = max{5, x1 - 19}
        cons_0_A = 3
        cons_1_A(x1) = max{4, x1 - 19}
        map_0_A = 4
        map_1_A(x1) = max{5, x1 - 19}
        map_2_A(x1,x2) = max{25, x1 - 3, x2 - 9}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        map_2 > cons_2 = app = nil_0 > append_2# = append_1 = cons_1 = map_1 > append_0 = cons_0 = map_0
      
  
    partial status:
  
      pi(append_2#) = []
      pi(cons_2) = [2]
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(map_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.