YES
We show the termination of the TRS R:

  app(app(minus(),x),|0|()) -> x
  app(app(minus(),app(s(),x)),app(s(),y)) -> 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(log(),app(s(),|0|())) -> |0|()
  app(log(),app(s(),app(s(),x))) -> app(s(),app(log(),app(s(),app(app(quot(),x),app(s(),app(s(),|0|()))))))
  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(log_0(),y) -> log_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(quot_0(),y) -> quot_1(y)
  app(quot_1(x1),y) -> quot_2(x1,y)
  app(s_0(),y) -> s_1(y)
  minus_2(x,|0|_0()) -> x
  minus_2(s_1(x),s_1(y)) -> 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)))
  log_1(s_1(|0|_0())) -> |0|_0()
  log_1(s_1(s_1(x))) -> s_1(log_1(s_1(quot_2(x,s_1(s_1(|0|_0()))))))
  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#(log_0(),y) -> log_1#(y)
p4: app#(map_1(x1),y) -> map_2#(x1,y)
p5: app#(minus_1(x1),y) -> minus_2#(x1,y)
p6: app#(quot_1(x1),y) -> quot_2#(x1,y)
p7: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p8: quot_2#(s_1(x),s_1(y)) -> quot_2#(minus_2(x,y),s_1(y))
p9: quot_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p10: log_1#(s_1(s_1(x))) -> log_1#(s_1(quot_2(x,s_1(s_1(|0|_0())))))
p11: log_1#(s_1(s_1(x))) -> quot_2#(x,s_1(s_1(|0|_0())))
p12: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p13: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p14: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p15: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p16: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p17: 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(log_0(),y) -> log_1(y)
r10: app(map_0(),y) -> map_1(y)
r11: app(map_1(x1),y) -> map_2(x1,y)
r12: app(minus_0(),y) -> minus_1(y)
r13: app(minus_1(x1),y) -> minus_2(x1,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: minus_2(x,|0|_0()) -> x
r18: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r19: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r20: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r21: log_1(s_1(|0|_0())) -> |0|_0()
r22: log_1(s_1(s_1(x))) -> s_1(log_1(s_1(quot_2(x,s_1(s_1(|0|_0()))))))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p4, p12, p13, p14, p15, p16, p17}
  {p10}
  {p8}
  {p7}


-- 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(log_0(),y) -> log_1(y)
r10: app(map_0(),y) -> map_1(y)
r11: app(map_1(x1),y) -> map_2(x1,y)
r12: app(minus_0(),y) -> minus_1(y)
r13: app(minus_1(x1),y) -> minus_2(x1,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: minus_2(x,|0|_0()) -> x
r18: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r19: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r20: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r21: log_1(s_1(|0|_0())) -> |0|_0()
r22: log_1(s_1(s_1(x))) -> s_1(log_1(s_1(quot_2(x,s_1(s_1(|0|_0()))))))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{3, x1 + 1}
        filter_1_A(x1) = max{64, x1 + 6}
        filter_2#_A(x1,x2) = max{5, x1 + 2}
        cons_2_A(x1,x2) = 0
        map_1_A(x1) = max{38, x1 + 23}
        map_2#_A(x1,x2) = max{25, x1 + 2}
        filter2_3_A(x1,x2,x3) = max{63, x1 - 25, x2 + 3, x3 - 24}
        filter2_4#_A(x1,x2,x3,x4) = max{5, x2 + 2}
        false_0_A = 31
        app_A(x1,x2) = max{63, x1 + 23, x2 + 23}
        true_0_A = 29
        quot_2_A(x1,x2) = max{64, x1 + 44}
        |0|_0_A = 21
        s_1_A(x1) = max{21, x1 - 22}
        filter_2_A(x1,x2) = 30
        nil_0_A = 1
  
    base order 2:
  
      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) = x1 + x2
            filter_1_A(x1) = x1 + 3
            filter_2#_A(x1,x2) = x1 + x2 + 3
            cons_2_A(x1,x2) = x1 + x2 + 14
            map_1_A(x1) = x1 + 13
            map_2#_A(x1,x2) = x1 + x2
            filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 11
            filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 3
            false_0_A() = 1
            app_A(x1,x2) = x1 + x2 + 1
            true_0_A() = 1
            quot_2_A(x1,x2) = x1 + 7
            |0|_0_A() = 7
            s_1_A(x1) = 6
            filter_2_A(x1,x2) = x1 + x2 + 14
            nil_0_A() = 16
            app#!_A(x1,x2) = x1 + x2 + 8
            filter_1!_A(x1) = 7
            filter_2#!_A(x1,x2) = x2 + 11
            cons_2!_A(x1,x2) = 13
            map_1!_A(x1) = 15
            map_2#!_A(x1,x2) = 14
            filter2_3!_A(x1,x2,x3) = x1 + x2 + 14
            filter2_4#!_A(x1,x2,x3,x4) = x4 + 12
            false_0!_A() = 10
            app!_A(x1,x2) = x1 + x2 + 5
            true_0!_A() = 0
            quot_2!_A(x1,x2) = x1
            |0|_0!_A() = 6
            s_1!_A(x1) = 8
            filter_2!_A(x1,x2) = x2 + 9
            nil_0!_A() = 0
  
    partial status:
  
      pi(app#) = [1]
      pi(filter_1) = []
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(map_1) = []
      pi(map_2#) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(quot_2) = [1]
      pi(|0|_0) = []
      pi(s_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

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



-- SCC decomposition.

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(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(log_0(),y) -> log_1(y)
r10: app(map_0(),y) -> map_1(y)
r11: app(map_1(x1),y) -> map_2(x1,y)
r12: app(minus_0(),y) -> minus_1(y)
r13: app(minus_1(x1),y) -> minus_2(x1,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: minus_2(x,|0|_0()) -> x
r18: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r19: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r20: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r21: log_1(s_1(|0|_0())) -> |0|_0()
r22: log_1(s_1(s_1(x))) -> s_1(log_1(s_1(quot_2(x,s_1(s_1(|0|_0()))))))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

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,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(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(log_0(),y) -> log_1(y)
r10: app(map_0(),y) -> map_1(y)
r11: app(map_1(x1),y) -> map_2(x1,y)
r12: app(minus_0(),y) -> minus_1(y)
r13: app(minus_1(x1),y) -> minus_2(x1,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: minus_2(x,|0|_0()) -> x
r18: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r19: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r20: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r21: log_1(s_1(|0|_0())) -> |0|_0()
r22: log_1(s_1(s_1(x))) -> s_1(log_1(s_1(quot_2(x,s_1(s_1(|0|_0()))))))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: 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:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{7, x1 + 1}
        cons_2_A(x1,x2) = max{6, x1 + 4, x2 + 4}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            map_2#_A(x1,x2) = x2
            cons_2_A(x1,x2) = x1 + x2 + 5
            map_2#!_A(x1,x2) = x2
            cons_2!_A(x1,x2) = x1 + x2 + 7
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [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: log_1#(s_1(s_1(x))) -> log_1#(s_1(quot_2(x,s_1(s_1(|0|_0())))))

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(log_0(),y) -> log_1(y)
r10: app(map_0(),y) -> map_1(y)
r11: app(map_1(x1),y) -> map_2(x1,y)
r12: app(minus_0(),y) -> minus_1(y)
r13: app(minus_1(x1),y) -> minus_2(x1,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: minus_2(x,|0|_0()) -> x
r18: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r19: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r20: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r21: log_1(s_1(|0|_0())) -> |0|_0()
r22: log_1(s_1(s_1(x))) -> s_1(log_1(s_1(quot_2(x,s_1(s_1(|0|_0()))))))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r13, r17, r18, r19, r20, r21

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        log_1#_A(x1) = max{22, x1 + 8}
        s_1_A(x1) = max{15, x1 + 8}
        quot_2_A(x1,x2) = max{5, x1}
        |0|_0_A = 12
        app_A(x1,x2) = max{x1, x2 + 4}
        minus_1_A(x1) = x1 + 7
        minus_2_A(x1,x2) = max{7, x1}
        log_1_A(x1) = max{17, x1 - 20}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            log_1#_A(x1) = 0
            s_1_A(x1) = x1 + 4
            quot_2_A(x1,x2) = x1
            |0|_0_A() = 1
            app_A(x1,x2) = x1 + x2 + 1
            minus_1_A(x1) = x1 + 3
            minus_2_A(x1,x2) = x1
            log_1_A(x1) = 3
            log_1#!_A(x1) = x1 + 16
            s_1!_A(x1) = x1 + 18
            quot_2!_A(x1,x2) = x2 + 14
            |0|_0!_A() = 25
            app!_A(x1,x2) = 13
            minus_1!_A(x1) = 15
            minus_2!_A(x1,x2) = 14
            log_1!_A(x1) = 12
  
    partial status:
  
      pi(log_1#) = [1]
      pi(s_1) = []
      pi(quot_2) = [1]
      pi(|0|_0) = []
      pi(app) = [1]
      pi(minus_1) = []
      pi(minus_2) = [1]
      pi(log_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: 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(log_0(),y) -> log_1(y)
r10: app(map_0(),y) -> map_1(y)
r11: app(map_1(x1),y) -> map_2(x1,y)
r12: app(minus_0(),y) -> minus_1(y)
r13: app(minus_1(x1),y) -> minus_2(x1,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: minus_2(x,|0|_0()) -> x
r18: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r19: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r20: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r21: log_1(s_1(|0|_0())) -> |0|_0()
r22: log_1(s_1(s_1(x))) -> s_1(log_1(s_1(quot_2(x,s_1(s_1(|0|_0()))))))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r17, r18, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        quot_2#_A(x1,x2) = max{1, x1 - 29}
        s_1_A(x1) = x1 + 33
        minus_2_A(x1,x2) = max{20, x1 + 5}
        |0|_0_A = 40
        filter_2_A(x1,x2) = x1 + 22
        nil_0_A = 21
  
    base order 2:
  
      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
            |0|_0_A() = 8
            filter_2_A(x1,x2) = x2 + 10
            nil_0_A() = 25
            quot_2#!_A(x1,x2) = 16
            s_1!_A(x1) = 17
            minus_2!_A(x1,x2) = x2 + 18
            |0|_0!_A() = 18
            filter_2!_A(x1,x2) = 14
            nil_0!_A() = 0
  
    partial status:
  
      pi(quot_2#) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(filter_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: minus_2#(s_1(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(log_0(),y) -> log_1(y)
r10: app(map_0(),y) -> map_1(y)
r11: app(map_1(x1),y) -> map_2(x1,y)
r12: app(minus_0(),y) -> minus_1(y)
r13: app(minus_1(x1),y) -> minus_2(x1,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: minus_2(x,|0|_0()) -> x
r18: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r19: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r20: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r21: log_1(s_1(|0|_0())) -> |0|_0()
r22: log_1(s_1(s_1(x))) -> s_1(log_1(s_1(quot_2(x,s_1(s_1(|0|_0()))))))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r25

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus_2#_A(x1,x2) = max{1, x1 - 22}
        s_1_A(x1) = x1 + 28
        filter_2_A(x1,x2) = max{x1 + 4, x2 + 11}
        nil_0_A = 8
  
    base order 2:
  
      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) = x1 + x2
            s_1_A(x1) = x1 + 10
            filter_2_A(x1,x2) = x2 + 10
            nil_0_A() = 6
            minus_2#!_A(x1,x2) = x1 + x2 + 14
            s_1!_A(x1) = 13
            filter_2!_A(x1,x2) = x2 + 5
            nil_0!_A() = 0
  
    partial status:
  
      pi(minus_2#) = []
      pi(s_1) = [1]
      pi(filter_2) = [1, 2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.