YES

We show the termination of the TRS R:

  le(|0|(),Y) -> true()
  le(s(X),|0|()) -> false()
  le(s(X),s(Y)) -> le(X,Y)
  app(nil(),Y) -> Y
  app(cons(N,L),Y) -> cons(N,app(L,Y))
  low(N,nil()) -> nil()
  low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
  iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
  iflow(false(),N,cons(M,L)) -> low(N,L)
  high(N,nil()) -> nil()
  high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
  ifhigh(true(),N,cons(M,L)) -> high(N,L)
  ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
  quicksort(nil()) -> nil()
  quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))

-- SCC decomposition.

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

p1: le#(s(X),s(Y)) -> le#(X,Y)
p2: app#(cons(N,L),Y) -> app#(L,Y)
p3: low#(N,cons(M,L)) -> iflow#(le(M,N),N,cons(M,L))
p4: low#(N,cons(M,L)) -> le#(M,N)
p5: iflow#(true(),N,cons(M,L)) -> low#(N,L)
p6: iflow#(false(),N,cons(M,L)) -> low#(N,L)
p7: high#(N,cons(M,L)) -> ifhigh#(le(M,N),N,cons(M,L))
p8: high#(N,cons(M,L)) -> le#(M,N)
p9: ifhigh#(true(),N,cons(M,L)) -> high#(N,L)
p10: ifhigh#(false(),N,cons(M,L)) -> high#(N,L)
p11: quicksort#(cons(N,L)) -> app#(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))
p12: quicksort#(cons(N,L)) -> quicksort#(low(N,L))
p13: quicksort#(cons(N,L)) -> low#(N,L)
p14: quicksort#(cons(N,L)) -> quicksort#(high(N,L))
p15: quicksort#(cons(N,L)) -> high#(N,L)

and R consists of:

r1: le(|0|(),Y) -> true()
r2: le(s(X),|0|()) -> false()
r3: le(s(X),s(Y)) -> le(X,Y)
r4: app(nil(),Y) -> Y
r5: app(cons(N,L),Y) -> cons(N,app(L,Y))
r6: low(N,nil()) -> nil()
r7: low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
r8: iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
r9: iflow(false(),N,cons(M,L)) -> low(N,L)
r10: high(N,nil()) -> nil()
r11: high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
r12: ifhigh(true(),N,cons(M,L)) -> high(N,L)
r13: ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
r14: quicksort(nil()) -> nil()
r15: quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))

The estimated dependency graph contains the following SCCs:

  {p12, p14}
  {p7, p9, p10}
  {p3, p5, p6}
  {p1}
  {p2}


-- Reduction pair.

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

p1: quicksort#(cons(N,L)) -> quicksort#(high(N,L))
p2: quicksort#(cons(N,L)) -> quicksort#(low(N,L))

and R consists of:

r1: le(|0|(),Y) -> true()
r2: le(s(X),|0|()) -> false()
r3: le(s(X),s(Y)) -> le(X,Y)
r4: app(nil(),Y) -> Y
r5: app(cons(N,L),Y) -> cons(N,app(L,Y))
r6: low(N,nil()) -> nil()
r7: low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
r8: iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
r9: iflow(false(),N,cons(M,L)) -> low(N,L)
r10: high(N,nil()) -> nil()
r11: high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
r12: ifhigh(true(),N,cons(M,L)) -> high(N,L)
r13: ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
r14: quicksort(nil()) -> nil()
r15: quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))

The set of usable rules consists of

  r1, r2, r3, r6, r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            quicksort#_A(x1) = ((0,0),(1,0)) x1 + (1,1)
            cons_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (4,7)
            high_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (2,6)
            low_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (0,8)
            le_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,8)
            |0|_A() = (2,2)
            true_A() = (1,1)
            s_A(x1) = x1 + (2,1)
            false_A() = (1,3)
            iflow_A(x1,x2,x3) = x2 + x3 + (0,2)
            ifhigh_A(x1,x2,x3) = x2 + ((1,0),(1,0)) x3 + (2,4)
            nil_A() = (1,1)
    
      precedence:
      
        quicksort# = low = iflow > le > ifhigh > high = false > cons > nil > s > |0| = true
      
      partial status:
    
        pi(quicksort#) = []
        pi(cons) = []
        pi(high) = []
        pi(low) = [1]
        pi(le) = [2]
        pi(|0|) = []
        pi(true) = []
        pi(s) = []
        pi(false) = []
        pi(iflow) = []
        pi(ifhigh) = [2]
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            quicksort#_A(x1) = (0,0)
            cons_A(x1,x2) = (0,4)
            high_A(x1,x2) = (2,2)
            low_A(x1,x2) = x1 + (4,6)
            le_A(x1,x2) = x2 + (5,7)
            |0|_A() = (1,1)
            true_A() = (1,0)
            s_A(x1) = (0,0)
            false_A() = (3,3)
            iflow_A(x1,x2,x3) = (0,5)
            ifhigh_A(x1,x2,x3) = ((1,0),(1,1)) x2 + (3,1)
            nil_A() = (0,1)
    
      precedence:
      
        high = ifhigh > cons > low > nil > le > s = false > |0| = true > quicksort# = iflow
      
      partial status:
    
        pi(quicksort#) = []
        pi(cons) = []
        pi(high) = []
        pi(low) = [1]
        pi(le) = [2]
        pi(|0|) = []
        pi(true) = []
        pi(s) = []
        pi(false) = []
        pi(iflow) = []
        pi(ifhigh) = []
        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: quicksort#(cons(N,L)) -> quicksort#(low(N,L))

and R consists of:

r1: le(|0|(),Y) -> true()
r2: le(s(X),|0|()) -> false()
r3: le(s(X),s(Y)) -> le(X,Y)
r4: app(nil(),Y) -> Y
r5: app(cons(N,L),Y) -> cons(N,app(L,Y))
r6: low(N,nil()) -> nil()
r7: low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
r8: iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
r9: iflow(false(),N,cons(M,L)) -> low(N,L)
r10: high(N,nil()) -> nil()
r11: high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
r12: ifhigh(true(),N,cons(M,L)) -> high(N,L)
r13: ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
r14: quicksort(nil()) -> nil()
r15: quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: quicksort#(cons(N,L)) -> quicksort#(low(N,L))

and R consists of:

r1: le(|0|(),Y) -> true()
r2: le(s(X),|0|()) -> false()
r3: le(s(X),s(Y)) -> le(X,Y)
r4: app(nil(),Y) -> Y
r5: app(cons(N,L),Y) -> cons(N,app(L,Y))
r6: low(N,nil()) -> nil()
r7: low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
r8: iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
r9: iflow(false(),N,cons(M,L)) -> low(N,L)
r10: high(N,nil()) -> nil()
r11: high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
r12: ifhigh(true(),N,cons(M,L)) -> high(N,L)
r13: ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
r14: quicksort(nil()) -> nil()
r15: quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))

The set of usable rules consists of

  r1, r2, r3, r6, r7, r8, r9

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            quicksort#_A(x1) = ((0,0),(1,0)) x1 + (1,1)
            cons_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (4,7)
            low_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (0,6)
            le_A(x1,x2) = x1 + (2,0)
            |0|_A() = (2,7)
            true_A() = (3,5)
            s_A(x1) = x1
            false_A() = (1,7)
            iflow_A(x1,x2,x3) = x3 + (0,1)
            nil_A() = (0,0)
    
      precedence:
      
        le = |0| > true > cons = s = false > low > nil > quicksort# = iflow
      
      partial status:
    
        pi(quicksort#) = []
        pi(cons) = []
        pi(low) = []
        pi(le) = [1]
        pi(|0|) = []
        pi(true) = []
        pi(s) = [1]
        pi(false) = []
        pi(iflow) = [3]
        pi(nil) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            quicksort#_A(x1) = (0,0)
            cons_A(x1,x2) = (2,2)
            low_A(x1,x2) = (2,2)
            le_A(x1,x2) = ((1,0),(1,1)) x1 + (0,3)
            |0|_A() = (1,2)
            true_A() = (0,1)
            s_A(x1) = (0,0)
            false_A() = (0,0)
            iflow_A(x1,x2,x3) = x3
            nil_A() = (1,3)
    
      precedence:
      
        cons = low = iflow > nil > le > |0| = s = false > quicksort# = true
      
      partial status:
    
        pi(quicksort#) = []
        pi(cons) = []
        pi(low) = []
        pi(le) = []
        pi(|0|) = []
        pi(true) = []
        pi(s) = []
        pi(false) = []
        pi(iflow) = []
        pi(nil) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: ifhigh#(false(),N,cons(M,L)) -> high#(N,L)
p2: high#(N,cons(M,L)) -> ifhigh#(le(M,N),N,cons(M,L))
p3: ifhigh#(true(),N,cons(M,L)) -> high#(N,L)

and R consists of:

r1: le(|0|(),Y) -> true()
r2: le(s(X),|0|()) -> false()
r3: le(s(X),s(Y)) -> le(X,Y)
r4: app(nil(),Y) -> Y
r5: app(cons(N,L),Y) -> cons(N,app(L,Y))
r6: low(N,nil()) -> nil()
r7: low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
r8: iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
r9: iflow(false(),N,cons(M,L)) -> low(N,L)
r10: high(N,nil()) -> nil()
r11: high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
r12: ifhigh(true(),N,cons(M,L)) -> high(N,L)
r13: ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
r14: quicksort(nil()) -> nil()
r15: quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            ifhigh#_A(x1,x2,x3) = x2 + ((1,0),(0,0)) x3 + (1,2)
            false_A() = (3,1)
            cons_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (5,1)
            high#_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (4,2)
            le_A(x1,x2) = x1 + (1,3)
            true_A() = (1,1)
            |0|_A() = (2,2)
            s_A(x1) = x1 + (4,4)
    
      precedence:
      
        ifhigh# > false = cons = high# = s > le = true = |0|
      
      partial status:
    
        pi(ifhigh#) = [2]
        pi(false) = []
        pi(cons) = [2]
        pi(high#) = []
        pi(le) = [1]
        pi(true) = []
        pi(|0|) = []
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            ifhigh#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + (1,1)
            false_A() = (1,1)
            cons_A(x1,x2) = x2
            high#_A(x1,x2) = (0,0)
            le_A(x1,x2) = (0,0)
            true_A() = (1,1)
            |0|_A() = (2,2)
            s_A(x1) = (2,2)
    
      precedence:
      
        ifhigh# = false = cons = high# = le = true = |0| = s
      
      partial status:
    
        pi(ifhigh#) = []
        pi(false) = []
        pi(cons) = []
        pi(high#) = []
        pi(le) = []
        pi(true) = []
        pi(|0|) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1, p2, p3

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

-- Reduction pair.

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

p1: iflow#(false(),N,cons(M,L)) -> low#(N,L)
p2: low#(N,cons(M,L)) -> iflow#(le(M,N),N,cons(M,L))
p3: iflow#(true(),N,cons(M,L)) -> low#(N,L)

and R consists of:

r1: le(|0|(),Y) -> true()
r2: le(s(X),|0|()) -> false()
r3: le(s(X),s(Y)) -> le(X,Y)
r4: app(nil(),Y) -> Y
r5: app(cons(N,L),Y) -> cons(N,app(L,Y))
r6: low(N,nil()) -> nil()
r7: low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
r8: iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
r9: iflow(false(),N,cons(M,L)) -> low(N,L)
r10: high(N,nil()) -> nil()
r11: high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
r12: ifhigh(true(),N,cons(M,L)) -> high(N,L)
r13: ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
r14: quicksort(nil()) -> nil()
r15: quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            iflow#_A(x1,x2,x3) = x2 + x3 + (1,3)
            false_A() = (1,1)
            cons_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,1)
            low#_A(x1,x2) = x1 + x2 + (2,2)
            le_A(x1,x2) = x1 + (6,4)
            true_A() = (1,1)
            |0|_A() = (2,2)
            s_A(x1) = ((1,0),(1,1)) x1 + (7,5)
    
      precedence:
      
        iflow# = cons = low# > false = le = true = |0| = s
      
      partial status:
    
        pi(iflow#) = [2, 3]
        pi(false) = []
        pi(cons) = [2]
        pi(low#) = [1, 2]
        pi(le) = [1]
        pi(true) = []
        pi(|0|) = []
        pi(s) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            iflow#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,1)) x3 + (3,1)
            false_A() = (1,1)
            cons_A(x1,x2) = x2
            low#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,2)
            le_A(x1,x2) = (0,0)
            true_A() = (1,1)
            |0|_A() = (2,2)
            s_A(x1) = (2,2)
    
      precedence:
      
        iflow# = false = cons = le = true = s > low# = |0|
      
      partial status:
    
        pi(iflow#) = []
        pi(false) = []
        pi(cons) = []
        pi(low#) = [1]
        pi(le) = []
        pi(true) = []
        pi(|0|) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1, p2, p3

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

-- Reduction pair.

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

p1: le#(s(X),s(Y)) -> le#(X,Y)

and R consists of:

r1: le(|0|(),Y) -> true()
r2: le(s(X),|0|()) -> false()
r3: le(s(X),s(Y)) -> le(X,Y)
r4: app(nil(),Y) -> Y
r5: app(cons(N,L),Y) -> cons(N,app(L,Y))
r6: low(N,nil()) -> nil()
r7: low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
r8: iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
r9: iflow(false(),N,cons(M,L)) -> low(N,L)
r10: high(N,nil()) -> nil()
r11: high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
r12: ifhigh(true(),N,cons(M,L)) -> high(N,L)
r13: ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
r14: quicksort(nil()) -> nil()
r15: quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))

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:
            le#_A(x1,x2) = ((1,0),(0,0)) x1
            s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        le# > s
      
      partial status:
    
        pi(le#) = []
        pi(s) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            le#_A(x1,x2) = (0,0)
            s_A(x1) = (1,1)
    
      precedence:
      
        le# = s
      
      partial status:
    
        pi(le#) = []
        pi(s) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: app#(cons(N,L),Y) -> app#(L,Y)

and R consists of:

r1: le(|0|(),Y) -> true()
r2: le(s(X),|0|()) -> false()
r3: le(s(X),s(Y)) -> le(X,Y)
r4: app(nil(),Y) -> Y
r5: app(cons(N,L),Y) -> cons(N,app(L,Y))
r6: low(N,nil()) -> nil()
r7: low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
r8: iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
r9: iflow(false(),N,cons(M,L)) -> low(N,L)
r10: high(N,nil()) -> nil()
r11: high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
r12: ifhigh(true(),N,cons(M,L)) -> high(N,L)
r13: ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
r14: quicksort(nil()) -> nil()
r15: quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))

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) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (2,2)
            cons_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (1,1)
    
      precedence:
      
        app# = cons
      
      partial status:
    
        pi(app#) = []
        pi(cons) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            app#_A(x1,x2) = (1,1)
            cons_A(x1,x2) = x1 + (2,2)
    
      precedence:
      
        app# = cons
      
      partial status:
    
        pi(app#) = []
        pi(cons) = []
    

The next rules are strictly ordered:

  p1

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