YES
We show the termination of the TRS R:

  minus(x,|0|()) -> x
  minus(s(x),s(y)) -> minus(x,y)
  quot(|0|(),s(y)) -> |0|()
  quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
  app(nil(),y) -> y
  app(add(n,x),y) -> add(n,app(x,y))
  reverse(nil()) -> nil()
  reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
  shuffle(nil()) -> nil()
  shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
  concat(leaf(),y) -> y
  concat(cons(u,v),y) -> cons(u,concat(v,y))
  less_leaves(x,leaf()) -> false()
  less_leaves(leaf(),cons(w,z)) -> true()
  less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))


-- SCC decomposition.

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

p1: minus#(s(x),s(y)) -> minus#(x,y)
p2: quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
p3: quot#(s(x),s(y)) -> minus#(x,y)
p4: app#(add(n,x),y) -> app#(x,y)
p5: reverse#(add(n,x)) -> app#(reverse(x),add(n,nil()))
p6: reverse#(add(n,x)) -> reverse#(x)
p7: shuffle#(add(n,x)) -> shuffle#(reverse(x))
p8: shuffle#(add(n,x)) -> reverse#(x)
p9: concat#(cons(u,v),y) -> concat#(v,y)
p10: less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z))
p11: less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v)
p12: less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: app(nil(),y) -> y
r6: app(add(n,x),y) -> add(n,app(x,y))
r7: reverse(nil()) -> nil()
r8: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r9: shuffle(nil()) -> nil()
r10: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
r11: concat(leaf(),y) -> y
r12: concat(cons(u,v),y) -> cons(u,concat(v,y))
r13: less_leaves(x,leaf()) -> false()
r14: less_leaves(leaf(),cons(w,z)) -> true()
r15: less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))

The estimated dependency graph contains the following SCCs:

  {p2}
  {p1}
  {p7}
  {p6}
  {p4}
  {p10}
  {p9}


-- Reduction pair.

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

p1: quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: app(nil(),y) -> y
r6: app(add(n,x),y) -> add(n,app(x,y))
r7: reverse(nil()) -> nil()
r8: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r9: shuffle(nil()) -> nil()
r10: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
r11: concat(leaf(),y) -> y
r12: concat(cons(u,v),y) -> cons(u,concat(v,y))
r13: less_leaves(x,leaf()) -> false()
r14: less_leaves(leaf(),cons(w,z)) -> true()
r15: less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))

The set of usable rules consists of

  r1, r2

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#_A(x1,x2) = x1
              s_A(x1) = x1 + 2
              minus_A(x1,x2) = x1 + 1
              |0|_A() = 3
              quot#!_A(x1,x2) = x1 + 1
              s!_A(x1) = x1 + 1
              minus!_A(x1,x2) = x1
              |0|!_A() = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            quot#_A(x1,x2) = max{7, x1 - 19, x2 - 21}
            s_A(x1) = x1 + 20
            minus_A(x1,x2) = max{26, x1 + 13, x2 + 13}
            |0|_A = 13
            quot#!_A(x1,x2) = max{34, x1 + 9, x2 + 15}
            s!_A(x1) = max{33, x1 + 22}
            minus!_A(x1,x2) = max{33, x1 - 21}
            |0|!_A = 23
        
  
    partial status:
  
      pi(quot#) = []
      pi(s) = [1]
      pi(minus) = [1]
      pi(|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#(s(x),s(y)) -> minus#(x,y)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: app(nil(),y) -> y
r6: app(add(n,x),y) -> add(n,app(x,y))
r7: reverse(nil()) -> nil()
r8: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r9: shuffle(nil()) -> nil()
r10: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
r11: concat(leaf(),y) -> y
r12: concat(cons(u,v),y) -> cons(u,concat(v,y))
r13: less_leaves(x,leaf()) -> false()
r14: less_leaves(leaf(),cons(w,z)) -> true()
r15: less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))

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#_A(x1,x2) = x1 + x2
              s_A(x1) = x1 + 4
              minus#!_A(x1,x2) = x1 + x2
              s!_A(x1) = 6
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus#_A(x1,x2) = 0
            s_A(x1) = max{2, x1 - 2}
            minus#!_A(x1,x2) = max{3, x1 - 3, x2 - 3}
            s!_A(x1) = 2
        
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []

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: shuffle#(add(n,x)) -> shuffle#(reverse(x))

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: app(nil(),y) -> y
r6: app(add(n,x),y) -> add(n,app(x,y))
r7: reverse(nil()) -> nil()
r8: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r9: shuffle(nil()) -> nil()
r10: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
r11: concat(leaf(),y) -> y
r12: concat(cons(u,v),y) -> cons(u,concat(v,y))
r13: less_leaves(x,leaf()) -> false()
r14: less_leaves(leaf(),cons(w,z)) -> true()
r15: less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))

The set of usable rules consists of

  r5, r6, r7, r8

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:
              shuffle#_A(x1) = x1
              add_A(x1,x2) = x1 + x2 + 4
              reverse_A(x1) = x1 + 1
              app_A(x1,x2) = x1 + x2
              nil_A() = 0
              shuffle#!_A(x1) = 0
              add!_A(x1,x2) = x2 + 3
              reverse!_A(x1) = x1 + 2
              app!_A(x1,x2) = x1 + x2
              nil!_A() = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            shuffle#_A(x1) = max{0, x1 - 7}
            add_A(x1,x2) = max{3, x1 + 2, x2}
            reverse_A(x1) = max{6, x1}
            app_A(x1,x2) = max{4, x1, x2}
            nil_A = 2
            shuffle#!_A(x1) = max{8, x1 + 3}
            add!_A(x1,x2) = max{1, x2 - 9}
            reverse!_A(x1) = max{7, x1 + 2}
            app!_A(x1,x2) = max{6, x1 - 16}
            nil!_A = 6
        
  
    partial status:
  
      pi(shuffle#) = [1]
      pi(add) = []
      pi(reverse) = []
      pi(app) = [2]
      pi(nil) = []

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: reverse#(add(n,x)) -> reverse#(x)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: app(nil(),y) -> y
r6: app(add(n,x),y) -> add(n,app(x,y))
r7: reverse(nil()) -> nil()
r8: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r9: shuffle(nil()) -> nil()
r10: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
r11: concat(leaf(),y) -> y
r12: concat(cons(u,v),y) -> cons(u,concat(v,y))
r13: less_leaves(x,leaf()) -> false()
r14: less_leaves(leaf(),cons(w,z)) -> true()
r15: less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))

The set of usable rules consists of

  r3

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:
              reverse#_A(x1) = x1
              add_A(x1,x2) = x1 + x2 + 5
              quot_A(x1,x2) = 1
              |0|_A() = 1
              s_A(x1) = x1 + 2
              reverse#!_A(x1) = x1
              add!_A(x1,x2) = 4
              quot!_A(x1,x2) = x1 + x2
              |0|!_A() = 0
              s!_A(x1) = x1 + 5
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            reverse#_A(x1) = 0
            add_A(x1,x2) = max{25, x1 - 9, x2 - 9}
            quot_A(x1,x2) = max{3, x2 - 3}
            |0|_A = 3
            s_A(x1) = max{4, x1 - 4}
            reverse#!_A(x1) = max{3, x1 - 26}
            add!_A(x1,x2) = max{2, x1 - 10, x2 - 24}
            quot!_A(x1,x2) = max{x1 - 5, x2 + 9}
            |0|!_A = 2
            s!_A(x1) = x1 + 3
        
  
    partial status:
  
      pi(reverse#) = []
      pi(add) = [2]
      pi(quot) = []
      pi(|0|) = []
      pi(s) = [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: app#(add(n,x),y) -> app#(x,y)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: app(nil(),y) -> y
r6: app(add(n,x),y) -> add(n,app(x,y))
r7: reverse(nil()) -> nil()
r8: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r9: shuffle(nil()) -> nil()
r10: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
r11: concat(leaf(),y) -> y
r12: concat(cons(u,v),y) -> cons(u,concat(v,y))
r13: less_leaves(x,leaf()) -> false()
r14: less_leaves(leaf(),cons(w,z)) -> true()
r15: less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))

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
              add_A(x1,x2) = x1 + x2 + 2
              app#!_A(x1,x2) = 0
              add!_A(x1,x2) = 2
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = 0
            add_A(x1,x2) = max{9, x1 - 28, x2 - 28}
            app#!_A(x1,x2) = 6
            add!_A(x1,x2) = max{29, x1 + 5, x2 + 3}
        
  
    partial status:
  
      pi(app#) = [1]
      pi(add) = [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: less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z))

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: app(nil(),y) -> y
r6: app(add(n,x),y) -> add(n,app(x,y))
r7: reverse(nil()) -> nil()
r8: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r9: shuffle(nil()) -> nil()
r10: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
r11: concat(leaf(),y) -> y
r12: concat(cons(u,v),y) -> cons(u,concat(v,y))
r13: less_leaves(x,leaf()) -> false()
r14: less_leaves(leaf(),cons(w,z)) -> true()
r15: less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))

The set of usable rules consists of

  r11, r12

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:
              less_leaves#_A(x1,x2) = x1 + x2
              cons_A(x1,x2) = x1 + x2 + 2
              concat_A(x1,x2) = x1 + x2 + 1
              leaf_A() = 1
              less_leaves#!_A(x1,x2) = x1
              cons!_A(x1,x2) = 3
              concat!_A(x1,x2) = 5
              leaf!_A() = 1
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            less_leaves#_A(x1,x2) = max{0, x1 - 9, x2 - 7}
            cons_A(x1,x2) = max{6, x1 + 1, x2}
            concat_A(x1,x2) = max{5, x1, x2}
            leaf_A = 6
            less_leaves#!_A(x1,x2) = x1 + 4
            cons!_A(x1,x2) = max{1, x2 - 10}
            concat!_A(x1,x2) = max{3, x1 - 7, x2 - 8}
            leaf!_A = 2
        
  
    partial status:
  
      pi(less_leaves#) = []
      pi(cons) = [1]
      pi(concat) = [1, 2]
      pi(leaf) = []

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: concat#(cons(u,v),y) -> concat#(v,y)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: app(nil(),y) -> y
r6: app(add(n,x),y) -> add(n,app(x,y))
r7: reverse(nil()) -> nil()
r8: reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
r9: shuffle(nil()) -> nil()
r10: shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
r11: concat(leaf(),y) -> y
r12: concat(cons(u,v),y) -> cons(u,concat(v,y))
r13: less_leaves(x,leaf()) -> false()
r14: less_leaves(leaf(),cons(w,z)) -> true()
r15: less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))

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:
              concat#_A(x1,x2) = x1
              cons_A(x1,x2) = x2 + 2
              concat#!_A(x1,x2) = x1
              cons!_A(x1,x2) = x2 + 2
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            concat#_A(x1,x2) = 0
            cons_A(x1,x2) = max{x1 + 1, x2 - 5}
            concat#!_A(x1,x2) = max{1, x1 - 2, x2 - 1}
            cons!_A(x1,x2) = 0
        
  
    partial status:
  
      pi(concat#) = []
      pi(cons) = []

The following rules are removed:

  p1

Then no dependency pair remains.