YES
We show the termination of the TRS R:

  app(pred(),app(s(),x)) -> x
  app(app(minus(),x),|0|()) -> x
  app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
  app(app(quot(),|0|()),app(s(),y)) -> |0|()
  app(app(quot(),app(s(),x)),app(s(),y)) -> app(s(),app(app(quot(),app(app(minus(),x),y)),app(s(),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(filter(),f),nil()) -> nil()
  app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
  app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
  app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

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(filter_0(),y) -> filter_1(y)
  app(filter_1(x1),y) -> filter_2(x1,y)
  app(filter2_0(),y) -> filter2_1(y)
  app(filter2_1(x1),y) -> filter2_2(x1,y)
  app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
  app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,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(pred_0(),y) -> pred_1(y)
  app(quot_0(),y) -> quot_1(y)
  app(quot_1(x1),y) -> quot_2(x1,y)
  app(s_0(),y) -> s_1(y)
  pred_1(s_1(x)) -> x
  minus_2(x,|0|_0()) -> x
  minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
  quot_2(|0|_0(),s_1(y)) -> |0|_0()
  quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,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))
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
  filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
  filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)


-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: app#(minus_1(x1),y) -> minus_2#(x1,y)
p5: app#(pred_0(),y) -> pred_1#(y)
p6: app#(quot_1(x1),y) -> quot_2#(x1,y)
p7: minus_2#(x,s_1(y)) -> pred_1#(minus_2(x,y))
p8: minus_2#(x,s_1(y)) -> minus_2#(x,y)
p9: quot_2#(s_1(x),s_1(y)) -> quot_2#(minus_2(x,y),s_1(y))
p10: quot_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p11: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p12: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p13: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p14: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p15: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p16: filter2_4#(false_0(),f,x,xs) -> filter_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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(minus_0(),y) -> minus_1(y)
r12: app(minus_1(x1),y) -> minus_2(x1,y)
r13: app(pred_0(),y) -> pred_1(y)
r14: app(quot_0(),y) -> quot_1(y)
r15: app(quot_1(x1),y) -> quot_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: pred_1(s_1(x)) -> x
r18: minus_2(x,|0|_0()) -> x
r19: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r20: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r21: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r22: map_2(f,nil_0()) -> nil_0()
r23: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r24: filter_2(f,nil_0()) -> nil_0()
r25: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r26: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r27: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p11, p12, p13, p14, p15, p16}
  {p9}
  {p8}


-- Reduction pair.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> app#(f,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: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p7: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p8: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p9: filter2_4#(true_0(),f,x,xs) -> filter_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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(minus_0(),y) -> minus_1(y)
r12: app(minus_1(x1),y) -> minus_2(x1,y)
r13: app(pred_0(),y) -> pred_1(y)
r14: app(quot_0(),y) -> quot_1(y)
r15: app(quot_1(x1),y) -> quot_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: pred_1(s_1(x)) -> x
r18: minus_2(x,|0|_0()) -> x
r19: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r20: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r21: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r22: map_2(f,nil_0()) -> nil_0()
r23: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r24: filter_2(f,nil_0()) -> nil_0()
r25: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r26: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r27: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r15, r20, r21

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              app#_A(x1,x2) = 0
              filter_1_A(x1) = 5
              filter_2#_A(x1,x2) = 0
              cons_2_A(x1,x2) = x2 + 2
              map_1_A(x1) = 2
              map_2#_A(x1,x2) = 0
              filter2_3_A(x1,x2,x3) = 6
              filter2_4#_A(x1,x2,x3,x4) = 0
              false_0_A() = 1
              app_A(x1,x2) = 1
              true_0_A() = 1
              quot_1_A(x1) = 4
              quot_2_A(x1,x2) = 1
              |0|_0_A() = 1
              s_1_A(x1) = 1
              minus_2_A(x1,x2) = 2
              app#!_A(x1,x2) = x2 + 3
              filter_1!_A(x1) = 1
              filter_2#!_A(x1,x2) = x2
              cons_2!_A(x1,x2) = x1 + x2 + 4
              map_1!_A(x1) = 6
              map_2#!_A(x1,x2) = 2
              filter2_3!_A(x1,x2,x3) = 8
              filter2_4#!_A(x1,x2,x3,x4) = x4 + 1
              false_0!_A() = 1
              app!_A(x1,x2) = x1 + 5
              true_0!_A() = 8
              quot_1!_A(x1) = 9
              quot_2!_A(x1,x2) = 8
              |0|_0!_A() = 0
              s_1!_A(x1) = 7
              minus_2!_A(x1,x2) = 9
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{1, x1 - 36, x2 - 7}
            filter_1_A(x1) = x1 + 35
            filter_2#_A(x1,x2) = max{1, x1 - 36, x2 - 37}
            cons_2_A(x1,x2) = max{x1 + 30, x2 + 36}
            map_1_A(x1) = x1 + 18
            map_2#_A(x1,x2) = max{1, x1 - 36, x2 - 37}
            filter2_3_A(x1,x2,x3) = max{x1 - 9, x2 + 35, x3 - 9}
            filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 36, x3 - 45, x4 - 37}
            false_0_A = 65
            app_A(x1,x2) = max{x1 - 62, x2 + 17}
            true_0_A = 45
            quot_1_A(x1) = max{59, x1 + 18}
            quot_2_A(x1,x2) = max{17, x2 - 18}
            |0|_0_A = 17
            s_1_A(x1) = 17
            minus_2_A(x1,x2) = max{x1 - 2, x2 + 17}
            app#!_A(x1,x2) = max{14, x2 + 6}
            filter_1!_A(x1) = max{3, x1 - 110}
            filter_2#!_A(x1,x2) = max{4, x2 - 30}
            cons_2!_A(x1,x2) = max{x1 + 15, x2 + 17}
            map_1!_A(x1) = max{2, x1 - 1}
            map_2#!_A(x1,x2) = max{x1 + 8, x2 - 23}
            filter2_3!_A(x1,x2,x3) = max{19, x1 - 47, x2 - 3}
            filter2_4#!_A(x1,x2,x3,x4) = x4 + 5
            false_0!_A = 1
            app!_A(x1,x2) = max{20, x1 - 36, x2 + 16}
            true_0!_A = 5
            quot_1!_A(x1) = 15
            quot_2!_A(x1,x2) = 22
            |0|_0!_A = 1
            s_1!_A(x1) = 21
            minus_2!_A(x1,x2) = max{23, x1 - 1, x2 - 17}
        
  
    partial status:
  
      pi(app#) = [2]
      pi(filter_1) = [1]
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(map_1) = [1]
      pi(map_2#) = [2]
      pi(filter2_3) = [1, 2, 3]
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(quot_1) = [1]
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(s_1) = []
      pi(minus_2) = [1]

The following rules are removed:

  p1, p2, p3, p4, p5, p6, p7, p8, p9

Then no dependency pair remains.

-- Reduction pair.

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

p1: quot_2#(s_1(x),s_1(y)) -> quot_2#(minus_2(x,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(minus_0(),y) -> minus_1(y)
r12: app(minus_1(x1),y) -> minus_2(x1,y)
r13: app(pred_0(),y) -> pred_1(y)
r14: app(quot_0(),y) -> quot_1(y)
r15: app(quot_1(x1),y) -> quot_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: pred_1(s_1(x)) -> x
r18: minus_2(x,|0|_0()) -> x
r19: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r20: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r21: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r22: map_2(f,nil_0()) -> nil_0()
r23: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r24: filter_2(f,nil_0()) -> nil_0()
r25: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r26: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r27: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r17, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              quot_2#_A(x1,x2) = x1
              s_1_A(x1) = x1 + 6
              minus_2_A(x1,x2) = x1 + 5
              pred_1_A(x1) = x1
              |0|_0_A() = 6
              quot_2#!_A(x1,x2) = x1 + x2
              s_1!_A(x1) = 15
              minus_2!_A(x1,x2) = x2 + 13
              pred_1!_A(x1) = 10
              |0|_0!_A() = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            quot_2#_A(x1,x2) = max{0, x1 - 4, x2 - 6}
            s_1_A(x1) = x1 + 3
            minus_2_A(x1,x2) = max{2, x1 + 1, x2 - 8}
            pred_1_A(x1) = max{2, x1 - 3}
            |0|_0_A = 27
            quot_2#!_A(x1,x2) = max{2, x1 - 4, x2 - 2}
            s_1!_A(x1) = 8
            minus_2!_A(x1,x2) = max{x1 + 9, x2 - 2}
            pred_1!_A(x1) = 5
            |0|_0!_A = 1
        
  
    partial status:
  
      pi(quot_2#) = []
      pi(s_1) = [1]
      pi(minus_2) = [1]
      pi(pred_1) = [1]
      pi(|0|_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: minus_2#(x,s_1(y)) -> minus_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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(minus_0(),y) -> minus_1(y)
r12: app(minus_1(x1),y) -> minus_2(x1,y)
r13: app(pred_0(),y) -> pred_1(y)
r14: app(quot_0(),y) -> quot_1(y)
r15: app(quot_1(x1),y) -> quot_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: pred_1(s_1(x)) -> x
r18: minus_2(x,|0|_0()) -> x
r19: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r20: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r21: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r22: map_2(f,nil_0()) -> nil_0()
r23: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r24: filter_2(f,nil_0()) -> nil_0()
r25: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r26: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r27: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              minus_2#_A(x1,x2) = x2
              s_1_A(x1) = x1 + 6
              minus_2#!_A(x1,x2) = x2 + 3
              s_1!_A(x1) = 2
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus_2#_A(x1,x2) = x2
            s_1_A(x1) = x1 + 25
            minus_2#!_A(x1,x2) = max{1, x1 - 1, x2 - 26}
            s_1!_A(x1) = max{0, x1 - 4}
        
  
    partial status:
  
      pi(minus_2#) = []
      pi(s_1) = []

The following rules are removed:

  p1

Then no dependency pair remains.