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

  (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:
              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 + 5
              map_1_A(x1) = x1 + 3
              map_2#_A(x1,x2) = x1 + x2
              filter2_3_A(x1,x2,x3) = x2 + x3 + 7
              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
              app#!_A(x1,x2) = 6
              filter_1!_A(x1) = 10
              filter_2#!_A(x1,x2) = 7
              cons_2!_A(x1,x2) = x1 + 5
              map_1!_A(x1) = 10
              map_2#!_A(x1,x2) = 7
              filter2_3!_A(x1,x2,x3) = 8
              filter2_4#!_A(x1,x2,x3,x4) = 7
              false_0!_A() = 8
              app!_A(x1,x2) = x2 + 9
              true_0!_A() = 0
        
        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 + 5
              filter_2#_A(x1,x2) = x1 + x2 + 5
              cons_2_A(x1,x2) = x1 + x2 + 5
              map_1_A(x1) = x1 + 3
              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 + 5
              false_0_A() = 1
              app_A(x1,x2) = x1 + x2 + 1
              true_0_A() = 1
              app#!_A(x1,x2) = x2 + 4
              filter_1!_A(x1) = 4
              filter_2#!_A(x1,x2) = 3
              cons_2!_A(x1,x2) = x2 + 15
              map_1!_A(x1) = x1 + 1
              map_2#!_A(x1,x2) = x2 + 5
              filter2_3!_A(x1,x2,x3) = 8
              filter2_4#!_A(x1,x2,x3,x4) = 3
              false_0!_A() = 0
              app!_A(x1,x2) = x1 + x2 + 2
              true_0!_A() = 0
        
  
    partial status:
  
      pi(app#) = [1]
      pi(filter_1) = [1]
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(map_1) = [1]
      pi(map_2#) = [1]
      pi(filter2_3) = [2, 3]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(app) = [1]
      pi(true_0) = []

The following rules are removed:

  p1, p2, p3, p4, p5, p6



-- SCC decomposition.

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

p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p3: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p3: 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

  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:
              filter2_4#_A(x1,x2,x3,x4) = x4
              false_0_A() = 1
              filter_2#_A(x1,x2) = x2
              cons_2_A(x1,x2) = x1 + x2 + 11
              app_A(x1,x2) = x1 + 1
              true_0_A() = 1
              log_1_A(x1) = 6
              s_1_A(x1) = 7
              |0|_0_A() = 1
              filter2_4#!_A(x1,x2,x3,x4) = 4
              false_0!_A() = 5
              filter_2#!_A(x1,x2) = 4
              cons_2!_A(x1,x2) = x1 + 12
              app!_A(x1,x2) = x1 + x2 + 7
              true_0!_A() = 7
              log_1!_A(x1) = 10
              s_1!_A(x1) = x1 + 5
              |0|_0!_A() = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              filter2_4#_A(x1,x2,x3,x4) = x4
              false_0_A() = 1
              filter_2#_A(x1,x2) = x2
              cons_2_A(x1,x2) = x1 + x2 + 5
              app_A(x1,x2) = x1 + x2 + 1
              true_0_A() = 1
              log_1_A(x1) = 6
              s_1_A(x1) = 7
              |0|_0_A() = 6
              filter2_4#!_A(x1,x2,x3,x4) = 2
              false_0!_A() = 0
              filter_2#!_A(x1,x2) = 1
              cons_2!_A(x1,x2) = 14
              app!_A(x1,x2) = x1 + 9
              true_0!_A() = 2
              log_1!_A(x1) = 2
              s_1!_A(x1) = 1
              |0|_0!_A() = 0
        
  
    partial status:
  
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(true_0) = []
      pi(log_1) = []
      pi(s_1) = []
      pi(|0|_0) = []

The following rules are removed:

  p1, p2, p3

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

  r17, r18, r19, 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:
              log_1#_A(x1) = 0
              s_1_A(x1) = x1 + 8
              quot_2_A(x1,x2) = x1 + 3
              |0|_0_A() = 2
              minus_2_A(x1,x2) = x1
              log_1_A(x1) = 8
              log_1#!_A(x1) = x1
              s_1!_A(x1) = x1 + 12
              quot_2!_A(x1,x2) = x1 + 7
              |0|_0!_A() = 15
              minus_2!_A(x1,x2) = 10
              log_1!_A(x1) = x1 + 10
        
        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 + 11
              quot_2_A(x1,x2) = x1 + 5
              |0|_0_A() = 1
              minus_2_A(x1,x2) = x1
              log_1_A(x1) = 10
              log_1#!_A(x1) = x1 + 1
              s_1!_A(x1) = x1 + 20
              quot_2!_A(x1,x2) = x1 + 14
              |0|_0!_A() = 22
              minus_2!_A(x1,x2) = x1 + x2 + 19
              log_1!_A(x1) = x1 + 17
        
  
    partial status:
  
      pi(log_1#) = []
      pi(s_1) = [1]
      pi(quot_2) = [1]
      pi(|0|_0) = []
      pi(minus_2) = [1]
      pi(log_1) = [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

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 + 4
              |0|_0_A() = 5
              quot_2#!_A(x1,x2) = x2 + 16
              s_1!_A(x1) = x1 + 21
              minus_2!_A(x1,x2) = x2 + 8
              |0|_0!_A() = 0
        
        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 + 4
              |0|_0_A() = 8
              quot_2#!_A(x1,x2) = x1 + x2 + 9
              s_1!_A(x1) = 18
              minus_2!_A(x1,x2) = x1 + x2 + 17
              |0|_0!_A() = 13
        
  
    partial status:
  
      pi(quot_2#) = [1]
      pi(s_1) = [1]
      pi(minus_2) = [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#(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

  r6

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) = x1 + x2
              s_1_A(x1) = x1 + 3
              app_A(x1,x2) = 1
              filter2_1_A(x1) = 2
              filter2_2_A(x1,x2) = 1
              minus_2#!_A(x1,x2) = 0
              s_1!_A(x1) = 5
              app!_A(x1,x2) = x1 + 2
              filter2_1!_A(x1) = 1
              filter2_2!_A(x1,x2) = 0
        
        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 + 6
              app_A(x1,x2) = x1 + x2 + 1
              filter2_1_A(x1) = x1 + 3
              filter2_2_A(x1,x2) = x1 + x2 + 1
              minus_2#!_A(x1,x2) = x2 + 8
              s_1!_A(x1) = x1 + 9
              app!_A(x1,x2) = x1 + x2 + 14
              filter2_1!_A(x1) = 1
              filter2_2!_A(x1,x2) = x1
        
  
    partial status:
  
      pi(minus_2#) = [2]
      pi(s_1) = [1]
      pi(app) = [1]
      pi(filter2_1) = []
      pi(filter2_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.