YES
We show the termination of the TRS R:

  app(app(lt(),app(s(),x)),app(s(),y)) -> app(app(lt(),x),y)
  app(app(lt(),|0|()),app(s(),y)) -> true()
  app(app(lt(),y),|0|()) -> false()
  app(app(eq(),x),x) -> true()
  app(app(eq(),app(s(),x)),|0|()) -> false()
  app(app(eq(),|0|()),app(s(),x)) -> false()
  app(app(member(),w),null()) -> false()
  app(app(member(),w),app(app(app(fork(),x),y),z)) -> app(app(app(if(),app(app(lt(),w),y)),app(app(member(),w),x)),app(app(app(if(),app(app(eq(),w),y)),true()),app(app(member(),w),z)))

Instead we prove termination of the uncurried system:

  app(eq_0(),y) -> eq_1(y)
  app(eq_1(x1),y) -> eq_2(x1,y)
  app(fork_0(),y) -> fork_1(y)
  app(fork_1(x1),y) -> fork_2(x1,y)
  app(fork_2(x1,x2),y) -> fork_3(x1,x2,y)
  app(if_0(),y) -> if_1(y)
  app(if_1(x1),y) -> if_2(x1,y)
  app(if_2(x1,x2),y) -> if_3(x1,x2,y)
  app(lt_0(),y) -> lt_1(y)
  app(lt_1(x1),y) -> lt_2(x1,y)
  app(member_0(),y) -> member_1(y)
  app(member_1(x1),y) -> member_2(x1,y)
  app(s_0(),y) -> s_1(y)
  lt_2(s_1(x),s_1(y)) -> lt_2(x,y)
  lt_2(|0|_0(),s_1(y)) -> true_0()
  lt_2(y,|0|_0()) -> false_0()
  eq_2(x,x) -> true_0()
  eq_2(s_1(x),|0|_0()) -> false_0()
  eq_2(|0|_0(),s_1(x)) -> false_0()
  member_2(w,null_0()) -> false_0()
  member_2(w,fork_3(x,y,z)) -> if_3(lt_2(w,y),member_2(w,x),if_3(eq_2(w,y),true_0(),member_2(w,z)))


-- SCC decomposition.

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

p1: app#(eq_1(x1),y) -> eq_2#(x1,y)
p2: app#(lt_1(x1),y) -> lt_2#(x1,y)
p3: app#(member_1(x1),y) -> member_2#(x1,y)
p4: lt_2#(s_1(x),s_1(y)) -> lt_2#(x,y)
p5: member_2#(w,fork_3(x,y,z)) -> lt_2#(w,y)
p6: member_2#(w,fork_3(x,y,z)) -> member_2#(w,x)
p7: member_2#(w,fork_3(x,y,z)) -> eq_2#(w,y)
p8: member_2#(w,fork_3(x,y,z)) -> member_2#(w,z)

and R consists of:

r1: app(eq_0(),y) -> eq_1(y)
r2: app(eq_1(x1),y) -> eq_2(x1,y)
r3: app(fork_0(),y) -> fork_1(y)
r4: app(fork_1(x1),y) -> fork_2(x1,y)
r5: app(fork_2(x1,x2),y) -> fork_3(x1,x2,y)
r6: app(if_0(),y) -> if_1(y)
r7: app(if_1(x1),y) -> if_2(x1,y)
r8: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r9: app(lt_0(),y) -> lt_1(y)
r10: app(lt_1(x1),y) -> lt_2(x1,y)
r11: app(member_0(),y) -> member_1(y)
r12: app(member_1(x1),y) -> member_2(x1,y)
r13: app(s_0(),y) -> s_1(y)
r14: lt_2(s_1(x),s_1(y)) -> lt_2(x,y)
r15: lt_2(|0|_0(),s_1(y)) -> true_0()
r16: lt_2(y,|0|_0()) -> false_0()
r17: eq_2(x,x) -> true_0()
r18: eq_2(s_1(x),|0|_0()) -> false_0()
r19: eq_2(|0|_0(),s_1(x)) -> false_0()
r20: member_2(w,null_0()) -> false_0()
r21: member_2(w,fork_3(x,y,z)) -> if_3(lt_2(w,y),member_2(w,x),if_3(eq_2(w,y),true_0(),member_2(w,z)))

The estimated dependency graph contains the following SCCs:

  {p6, p8}
  {p4}


-- Reduction pair.

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

p1: member_2#(w,fork_3(x,y,z)) -> member_2#(w,z)
p2: member_2#(w,fork_3(x,y,z)) -> member_2#(w,x)

and R consists of:

r1: app(eq_0(),y) -> eq_1(y)
r2: app(eq_1(x1),y) -> eq_2(x1,y)
r3: app(fork_0(),y) -> fork_1(y)
r4: app(fork_1(x1),y) -> fork_2(x1,y)
r5: app(fork_2(x1,x2),y) -> fork_3(x1,x2,y)
r6: app(if_0(),y) -> if_1(y)
r7: app(if_1(x1),y) -> if_2(x1,y)
r8: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r9: app(lt_0(),y) -> lt_1(y)
r10: app(lt_1(x1),y) -> lt_2(x1,y)
r11: app(member_0(),y) -> member_1(y)
r12: app(member_1(x1),y) -> member_2(x1,y)
r13: app(s_0(),y) -> s_1(y)
r14: lt_2(s_1(x),s_1(y)) -> lt_2(x,y)
r15: lt_2(|0|_0(),s_1(y)) -> true_0()
r16: lt_2(y,|0|_0()) -> false_0()
r17: eq_2(x,x) -> true_0()
r18: eq_2(s_1(x),|0|_0()) -> false_0()
r19: eq_2(|0|_0(),s_1(x)) -> false_0()
r20: member_2(w,null_0()) -> false_0()
r21: member_2(w,fork_3(x,y,z)) -> if_3(lt_2(w,y),member_2(w,x),if_3(eq_2(w,y),true_0(),member_2(w,z)))

The set of usable rules consists of

  r1, r3, r4, r6, r7, r8, r11, r13, r15, r16, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        member_2#_A(x1,x2) = x2
        fork_3_A(x1,x2,x3) = max{x1, x2 - 26, x3}
        app_A(x1,x2) = max{8, x1 - 10, x2 - 1}
        eq_0_A = 9
        eq_1_A(x1) = max{8, x1 - 16}
        fork_0_A = 9
        fork_1_A(x1) = max{8, x1 - 1}
        fork_2_A(x1,x2) = max{8, x1 - 11, x2 - 27}
        if_0_A = 1
        if_1_A(x1) = max{2, x1 - 2}
        if_2_A(x1,x2) = max{3, x1 - 13, x2 - 2}
        if_3_A(x1,x2,x3) = max{4, x1 - 24, x2 - 13, x3 - 6}
        member_0_A = 3
        member_1_A(x1) = max{4, x1 - 2}
        s_0_A = 6
        s_1_A(x1) = max{7, x1 - 53}
        lt_2_A(x1,x2) = max{8, x1 - 8, x2 - 52}
        |0|_0_A = 7
        true_0_A = 6
        false_0_A = 0
        eq_2_A(x1,x2) = max{5, x1 - 26, x2 - 52}
  
    base order 2:
  
      precedence:
      
        app > s_1 = |0|_0 > true_0 > eq_0 = if_0 = s_0 > member_1 = lt_2 = eq_2 > fork_3 = eq_1 = if_3 = member_0 = false_0 > member_2# = fork_2 = if_2 > fork_1 = if_1 > fork_0
      
  
    partial status:
  
      pi(member_2#) = [2]
      pi(fork_3) = [1, 3]
      pi(app) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(fork_0) = []
      pi(fork_1) = []
      pi(fork_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(if_3) = []
      pi(member_0) = []
      pi(member_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(lt_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(eq_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: member_2#(w,fork_3(x,y,z)) -> member_2#(w,x)

and R consists of:

r1: app(eq_0(),y) -> eq_1(y)
r2: app(eq_1(x1),y) -> eq_2(x1,y)
r3: app(fork_0(),y) -> fork_1(y)
r4: app(fork_1(x1),y) -> fork_2(x1,y)
r5: app(fork_2(x1,x2),y) -> fork_3(x1,x2,y)
r6: app(if_0(),y) -> if_1(y)
r7: app(if_1(x1),y) -> if_2(x1,y)
r8: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r9: app(lt_0(),y) -> lt_1(y)
r10: app(lt_1(x1),y) -> lt_2(x1,y)
r11: app(member_0(),y) -> member_1(y)
r12: app(member_1(x1),y) -> member_2(x1,y)
r13: app(s_0(),y) -> s_1(y)
r14: lt_2(s_1(x),s_1(y)) -> lt_2(x,y)
r15: lt_2(|0|_0(),s_1(y)) -> true_0()
r16: lt_2(y,|0|_0()) -> false_0()
r17: eq_2(x,x) -> true_0()
r18: eq_2(s_1(x),|0|_0()) -> false_0()
r19: eq_2(|0|_0(),s_1(x)) -> false_0()
r20: member_2(w,null_0()) -> false_0()
r21: member_2(w,fork_3(x,y,z)) -> if_3(lt_2(w,y),member_2(w,x),if_3(eq_2(w,y),true_0(),member_2(w,z)))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: member_2#(w,fork_3(x,y,z)) -> member_2#(w,x)

and R consists of:

r1: app(eq_0(),y) -> eq_1(y)
r2: app(eq_1(x1),y) -> eq_2(x1,y)
r3: app(fork_0(),y) -> fork_1(y)
r4: app(fork_1(x1),y) -> fork_2(x1,y)
r5: app(fork_2(x1,x2),y) -> fork_3(x1,x2,y)
r6: app(if_0(),y) -> if_1(y)
r7: app(if_1(x1),y) -> if_2(x1,y)
r8: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r9: app(lt_0(),y) -> lt_1(y)
r10: app(lt_1(x1),y) -> lt_2(x1,y)
r11: app(member_0(),y) -> member_1(y)
r12: app(member_1(x1),y) -> member_2(x1,y)
r13: app(s_0(),y) -> s_1(y)
r14: lt_2(s_1(x),s_1(y)) -> lt_2(x,y)
r15: lt_2(|0|_0(),s_1(y)) -> true_0()
r16: lt_2(y,|0|_0()) -> false_0()
r17: eq_2(x,x) -> true_0()
r18: eq_2(s_1(x),|0|_0()) -> false_0()
r19: eq_2(|0|_0(),s_1(x)) -> false_0()
r20: member_2(w,null_0()) -> false_0()
r21: member_2(w,fork_3(x,y,z)) -> if_3(lt_2(w,y),member_2(w,x),if_3(eq_2(w,y),true_0(),member_2(w,z)))

The set of usable rules consists of

  r1, r3, r4, r6, r7, r8, r9, r11, r13, r15, r16, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        member_2#_A(x1,x2) = max{3, x2 - 1}
        fork_3_A(x1,x2,x3) = max{5, x1 + 2, x2 - 39, x3 - 16}
        app_A(x1,x2) = max{9, x1 - 10, x2 - 1}
        eq_0_A = 9
        eq_1_A(x1) = max{6, x1 - 29}
        fork_0_A = 9
        fork_1_A(x1) = max{8, x1 - 2}
        fork_2_A(x1,x2) = max{9, x1 - 12, x2 - 30}
        if_0_A = 3
        if_1_A(x1) = max{4, x1 - 2}
        if_2_A(x1,x2) = max{5, x1 - 12, x2 - 1}
        if_3_A(x1,x2,x3) = max{6, x1 - 22, x2 - 11, x3 - 23}
        lt_0_A = 5
        lt_1_A(x1) = max{6, x1 - 29}
        member_0_A = 6
        member_1_A(x1) = max{7, x1 - 2}
        s_0_A = 5
        s_1_A(x1) = max{6, x1 - 85}
        lt_2_A(x1,x2) = max{7, x1 - 40, x2 - 84}
        |0|_0_A = 39
        true_0_A = 3
        false_0_A = 0
        eq_2_A(x1,x2) = max{7, x1 - 40, x2 - 84}
  
    base order 2:
  
      precedence:
      
        app > eq_1 = if_3 = lt_2 = eq_2 > member_2# = fork_3 = eq_0 = if_2 = lt_1 = member_1 = s_1 = |0|_0 > fork_2 = if_1 = lt_0 = member_0 = s_0 = true_0 = false_0 > fork_1 = if_0 > fork_0
      
  
    partial status:
  
      pi(member_2#) = []
      pi(fork_3) = [1]
      pi(app) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(fork_0) = []
      pi(fork_1) = []
      pi(fork_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(if_3) = []
      pi(lt_0) = []
      pi(lt_1) = []
      pi(member_0) = []
      pi(member_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(lt_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(eq_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: lt_2#(s_1(x),s_1(y)) -> lt_2#(x,y)

and R consists of:

r1: app(eq_0(),y) -> eq_1(y)
r2: app(eq_1(x1),y) -> eq_2(x1,y)
r3: app(fork_0(),y) -> fork_1(y)
r4: app(fork_1(x1),y) -> fork_2(x1,y)
r5: app(fork_2(x1,x2),y) -> fork_3(x1,x2,y)
r6: app(if_0(),y) -> if_1(y)
r7: app(if_1(x1),y) -> if_2(x1,y)
r8: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r9: app(lt_0(),y) -> lt_1(y)
r10: app(lt_1(x1),y) -> lt_2(x1,y)
r11: app(member_0(),y) -> member_1(y)
r12: app(member_1(x1),y) -> member_2(x1,y)
r13: app(s_0(),y) -> s_1(y)
r14: lt_2(s_1(x),s_1(y)) -> lt_2(x,y)
r15: lt_2(|0|_0(),s_1(y)) -> true_0()
r16: lt_2(y,|0|_0()) -> false_0()
r17: eq_2(x,x) -> true_0()
r18: eq_2(s_1(x),|0|_0()) -> false_0()
r19: eq_2(|0|_0(),s_1(x)) -> false_0()
r20: member_2(w,null_0()) -> false_0()
r21: member_2(w,fork_3(x,y,z)) -> if_3(lt_2(w,y),member_2(w,x),if_3(eq_2(w,y),true_0(),member_2(w,z)))

The set of usable rules consists of

  r1, r3, r4, r6, r7, r8, r9, r11, r15, r16, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        lt_2#_A(x1,x2) = max{x1 - 1, x2 + 32}
        s_1_A(x1) = x1 + 33
        app_A(x1,x2) = max{36, x1 - 37, x2 - 35}
        eq_0_A = 36
        eq_1_A(x1) = max{35, x1 - 36}
        fork_0_A = 36
        fork_1_A(x1) = max{31, x1 - 35}
        fork_2_A(x1,x2) = max{32, x1 - 72, x2 - 35}
        if_0_A = 1
        if_1_A(x1) = max{2, x1 - 36}
        if_2_A(x1,x2) = max{3, x1 - 73, x2 - 41}
        if_3_A(x1,x2,x3) = max{4, x1 - 111, x2 - 79, x3 - 54}
        lt_0_A = 34
        lt_1_A(x1) = max{35, x1 - 36}
        member_0_A = 29
        member_1_A(x1) = max{30, x1 - 94}
        lt_2_A(x1,x2) = max{37, x1 - 34, x2 - 34}
        |0|_0_A = 33
        true_0_A = 32
        false_0_A = 30
        eq_2_A(x1,x2) = max{34, x1 - 74, x2 - 35}
  
    base order 2:
  
      precedence:
      
        lt_2# = lt_2 > s_1 > app > eq_0 = eq_2 > eq_1 = lt_1 = |0|_0 > if_3 = lt_0 = member_1 = false_0 > if_2 = member_0 > fork_2 = if_1 = true_0 > fork_1 = if_0 > fork_0
      
  
    partial status:
  
      pi(lt_2#) = [2]
      pi(s_1) = [1]
      pi(app) = []
      pi(eq_0) = []
      pi(eq_1) = []
      pi(fork_0) = []
      pi(fork_1) = []
      pi(fork_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(if_3) = []
      pi(lt_0) = []
      pi(lt_1) = []
      pi(member_0) = []
      pi(member_1) = []
      pi(lt_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(eq_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.