YES

We show the termination of the TRS R:

  app(app(neq(),|0|()),|0|()) -> false()
  app(app(neq(),|0|()),app(s(),y)) -> true()
  app(app(neq(),app(s(),x)),|0|()) -> true()
  app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
  app(app(filter(),f),nil()) -> nil()
  app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
  app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
  app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
  nonzero() -> app(filter(),app(neq(),|0|()))

-- SCC decomposition.

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

p1: app#(app(neq(),app(s(),x)),app(s(),y)) -> app#(app(neq(),x),y)
p2: app#(app(neq(),app(s(),x)),app(s(),y)) -> app#(neq(),x)
p3: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
p4: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(app(filtersub(),app(f,y)),f)
p5: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(filtersub(),app(f,y))
p6: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(f,y)
p7: app#(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app#(app(cons(),y),app(app(filter(),f),ys))
p8: app#(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)
p9: app#(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app#(filter(),f)
p10: app#(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)
p11: app#(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app#(filter(),f)
p12: nonzero#() -> app#(filter(),app(neq(),|0|()))
p13: nonzero#() -> app#(neq(),|0|())

and R consists of:

r1: app(app(neq(),|0|()),|0|()) -> false()
r2: app(app(neq(),|0|()),app(s(),y)) -> true()
r3: app(app(neq(),app(s(),x)),|0|()) -> true()
r4: app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
r5: app(app(filter(),f),nil()) -> nil()
r6: app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
r7: app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
r8: app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
r9: nonzero() -> app(filter(),app(neq(),|0|()))

The estimated dependency graph contains the following SCCs:

  {p3, p6, p8, p10}
  {p1}


-- Reduction pair.

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

p1: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(f,y)
p2: app#(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)
p3: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
p4: app#(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)

and R consists of:

r1: app(app(neq(),|0|()),|0|()) -> false()
r2: app(app(neq(),|0|()),app(s(),y)) -> true()
r3: app(app(neq(),app(s(),x)),|0|()) -> true()
r4: app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
r5: app(app(filter(),f),nil()) -> nil()
r6: app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
r7: app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
r8: app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
r9: nonzero() -> app(filter(),app(neq(),|0|()))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            app#_A(x1,x2) = ((1,0),(0,0)) x1 + (4,6)
            app_A(x1,x2) = ((1,0),(0,0)) x2 + (5,1)
            filter_A() = (13,4)
            cons_A() = (1,2)
            filtersub_A() = (2,0)
            false_A() = (14,5)
            true_A() = (3,3)
            neq_A() = (1,0)
            |0|_A() = (15,6)
            s_A() = (2,0)
            nil_A() = (6,2)
    
      precedence:
      
        app# = filter > app = neq > |0| = s > false > cons = filtersub = nil > true
      
      partial status:
    
        pi(app#) = []
        pi(app) = []
        pi(filter) = []
        pi(cons) = []
        pi(filtersub) = []
        pi(false) = []
        pi(true) = []
        pi(neq) = []
        pi(|0|) = []
        pi(s) = []
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            app#_A(x1,x2) = (0,0)
            app_A(x1,x2) = (3,3)
            filter_A() = (0,0)
            cons_A() = (3,3)
            filtersub_A() = (4,4)
            false_A() = (4,4)
            true_A() = (4,4)
            neq_A() = (1,1)
            |0|_A() = (5,5)
            s_A() = (2,2)
            nil_A() = (1,1)
    
      precedence:
      
        true > neq > filtersub = |0| = s > nil > app# = app = filter = cons = false
      
      partial status:
    
        pi(app#) = []
        pi(app) = []
        pi(filter) = []
        pi(cons) = []
        pi(filtersub) = []
        pi(false) = []
        pi(true) = []
        pi(neq) = []
        pi(|0|) = []
        pi(s) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: app#(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)
p2: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
p3: app#(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)

and R consists of:

r1: app(app(neq(),|0|()),|0|()) -> false()
r2: app(app(neq(),|0|()),app(s(),y)) -> true()
r3: app(app(neq(),app(s(),x)),|0|()) -> true()
r4: app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
r5: app(app(filter(),f),nil()) -> nil()
r6: app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
r7: app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
r8: app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
r9: nonzero() -> app(filter(),app(neq(),|0|()))

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: app#(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)
p2: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
p3: app#(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)

and R consists of:

r1: app(app(neq(),|0|()),|0|()) -> false()
r2: app(app(neq(),|0|()),app(s(),y)) -> true()
r3: app(app(neq(),app(s(),x)),|0|()) -> true()
r4: app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
r5: app(app(filter(),f),nil()) -> nil()
r6: app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
r7: app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
r8: app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
r9: nonzero() -> app(filter(),app(neq(),|0|()))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            app#_A(x1,x2) = x2
            app_A(x1,x2) = ((1,0),(1,1)) x2
            filtersub_A() = (1,1)
            false_A() = (3,0)
            cons_A() = (0,3)
            filter_A() = (5,2)
            true_A() = (0,0)
            neq_A() = (4,1)
            |0|_A() = (4,1)
            s_A() = (5,2)
            nil_A() = (1,1)
    
      precedence:
      
        neq > app# = app = cons = filter = |0| > false > filtersub = true > s = nil
      
      partial status:
    
        pi(app#) = [2]
        pi(app) = [2]
        pi(filtersub) = []
        pi(false) = []
        pi(cons) = []
        pi(filter) = []
        pi(true) = []
        pi(neq) = []
        pi(|0|) = []
        pi(s) = []
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            app#_A(x1,x2) = ((0,0),(1,0)) x2 + (12,5)
            app_A(x1,x2) = ((1,0),(0,0)) x2 + (4,3)
            filtersub_A() = (9,4)
            false_A() = (5,4)
            cons_A() = (3,0)
            filter_A() = (2,2)
            true_A() = (7,4)
            neq_A() = (1,1)
            |0|_A() = (8,5)
            s_A() = (2,2)
            nil_A() = (1,1)
    
      precedence:
      
        app# = true > false > s > |0| > filtersub > app = nil > neq > cons = filter
      
      partial status:
    
        pi(app#) = []
        pi(app) = []
        pi(filtersub) = []
        pi(false) = []
        pi(cons) = []
        pi(filter) = []
        pi(true) = []
        pi(neq) = []
        pi(|0|) = []
        pi(s) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: app#(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)
p2: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))

and R consists of:

r1: app(app(neq(),|0|()),|0|()) -> false()
r2: app(app(neq(),|0|()),app(s(),y)) -> true()
r3: app(app(neq(),app(s(),x)),|0|()) -> true()
r4: app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
r5: app(app(filter(),f),nil()) -> nil()
r6: app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
r7: app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
r8: app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
r9: nonzero() -> app(filter(),app(neq(),|0|()))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: app#(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app#(app(filter(),f),ys)
p2: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))

and R consists of:

r1: app(app(neq(),|0|()),|0|()) -> false()
r2: app(app(neq(),|0|()),app(s(),y)) -> true()
r3: app(app(neq(),app(s(),x)),|0|()) -> true()
r4: app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
r5: app(app(filter(),f),nil()) -> nil()
r6: app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
r7: app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
r8: app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
r9: nonzero() -> app(filter(),app(neq(),|0|()))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            app#_A(x1,x2) = ((1,0),(1,1)) x2 + (27,25)
            app_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (4,2)
            filtersub_A() = (0,24)
            false_A() = (22,0)
            cons_A() = (3,7)
            filter_A() = (15,23)
            neq_A() = (3,1)
            |0|_A() = (19,19)
            s_A() = (0,19)
            true_A() = (2,18)
            nil_A() = (1,1)
    
      precedence:
      
        filter > app# = app = neq = s > filtersub = |0| = true = nil > false = cons
      
      partial status:
    
        pi(app#) = []
        pi(app) = []
        pi(filtersub) = []
        pi(false) = []
        pi(cons) = []
        pi(filter) = []
        pi(neq) = []
        pi(|0|) = []
        pi(s) = []
        pi(true) = []
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            app#_A(x1,x2) = (4,4)
            app_A(x1,x2) = (3,3)
            filtersub_A() = (4,4)
            false_A() = (5,5)
            cons_A() = (8,5)
            filter_A() = (0,0)
            neq_A() = (1,1)
            |0|_A() = (10,7)
            s_A() = (2,2)
            true_A() = (9,6)
            nil_A() = (1,1)
    
      precedence:
      
        neq > cons > |0| > s > false = true > filtersub > app# = app = nil > filter
      
      partial status:
    
        pi(app#) = []
        pi(app) = []
        pi(filtersub) = []
        pi(false) = []
        pi(cons) = []
        pi(filter) = []
        pi(neq) = []
        pi(|0|) = []
        pi(s) = []
        pi(true) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: app#(app(filter(),f),app(app(cons(),y),ys)) -> app#(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))

and R consists of:

r1: app(app(neq(),|0|()),|0|()) -> false()
r2: app(app(neq(),|0|()),app(s(),y)) -> true()
r3: app(app(neq(),app(s(),x)),|0|()) -> true()
r4: app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
r5: app(app(filter(),f),nil()) -> nil()
r6: app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
r7: app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
r8: app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
r9: nonzero() -> app(filter(),app(neq(),|0|()))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: app#(app(neq(),app(s(),x)),app(s(),y)) -> app#(app(neq(),x),y)

and R consists of:

r1: app(app(neq(),|0|()),|0|()) -> false()
r2: app(app(neq(),|0|()),app(s(),y)) -> true()
r3: app(app(neq(),app(s(),x)),|0|()) -> true()
r4: app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
r5: app(app(filter(),f),nil()) -> nil()
r6: app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
r7: app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
r8: app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
r9: nonzero() -> app(filter(),app(neq(),|0|()))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            app#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (4,11)
            app_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (3,1)
            neq_A() = (5,5)
            s_A() = (6,4)
    
      precedence:
      
        app# > neq > app = s
      
      partial status:
    
        pi(app#) = [2]
        pi(app) = [2]
        pi(neq) = []
        pi(s) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            app#_A(x1,x2) = ((1,0),(1,1)) x2
            app_A(x1,x2) = x2 + (3,0)
            neq_A() = (1,1)
            s_A() = (2,2)
    
      precedence:
      
        app# > app = neq = s
      
      partial status:
    
        pi(app#) = []
        pi(app) = [2]
        pi(neq) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.