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)))
  minus(minus(x,y),z) -> minus(x,plus(y,z))
  plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
  plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
  plus(zero(),y) -> y
  plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
  id(x) -> x
  if(true(),x,y) -> x
  if(false(),x,y) -> y
  not(x) -> if(x,false(),true())
  gt(s(x),zero()) -> true()
  gt(zero(),y) -> false()
  gt(s(x),s(y)) -> gt(x,y)


-- 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: minus#(minus(x,y),z) -> minus#(x,plus(y,z))
p5: minus#(minus(x,y),z) -> plus#(y,z)
p6: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p7: plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
p8: plus#(s(x),s(y)) -> gt#(x,y)
p9: plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
p10: plus#(s(x),s(y)) -> not#(gt(x,y))
p11: plus#(s(x),s(y)) -> id#(x)
p12: plus#(s(x),s(y)) -> id#(y)
p13: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
p14: plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
p15: plus#(s(x),x) -> gt#(x,x)
p16: plus#(s(x),x) -> id#(x)
p17: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p18: plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
p19: plus#(id(x),s(y)) -> gt#(s(y),y)
p20: not#(x) -> if#(x,false(),true())
p21: gt#(s(x),s(y)) -> gt#(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: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p2}
  {p1, p4}
  {p6, p13, p17}
  {p21}


-- 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: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r1, r2, r5

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) = 0
              s_A(x1) = x1 + 3
              minus_A(x1,x2) = x1 + 1
              |0|_A() = 0
              plus_A(x1,x2) = x1 + 1
              quot#!_A(x1,x2) = 3
              s!_A(x1) = 5
              minus!_A(x1,x2) = 4
              |0|!_A() = 0
              plus!_A(x1,x2) = 3
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            quot#_A(x1,x2) = max{0, x2 - 98}
            s_A(x1) = max{33, x1 + 8}
            minus_A(x1,x2) = max{30, x1 + 4}
            |0|_A = 49
            plus_A(x1,x2) = max{80, x1 - 80}
            quot#!_A(x1,x2) = max{0, x1 - 6}
            s!_A(x1) = max{42, x1 - 2}
            minus!_A(x1,x2) = max{0, x1 - 3}
            |0|!_A = 41
            plus!_A(x1,x2) = max{26, x1 - 80, x2 + 17}
        
  
    partial status:
  
      pi(quot#) = []
      pi(s) = [1]
      pi(minus) = [1]
      pi(|0|) = []
      pi(plus) = []

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)
p2: minus#(minus(x,y),z) -> minus#(x,plus(y,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: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

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) = 0
              s_A(x1) = x1 + 1
              minus_A(x1,x2) = x1 + 2
              plus_A(x1,x2) = 0
              minus#!_A(x1,x2) = x1 + 2
              s!_A(x1) = x1 + 3
              minus!_A(x1,x2) = 6
              plus!_A(x1,x2) = 5
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus#_A(x1,x2) = 0
            s_A(x1) = 5
            minus_A(x1,x2) = 0
            plus_A(x1,x2) = 7
            minus#!_A(x1,x2) = 0
            s!_A(x1) = 0
            minus!_A(x1,x2) = max{2, x2 - 11}
            plus!_A(x1,x2) = max{3, x1 - 67}
        
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []
      pi(minus) = [1, 2]
      pi(plus) = [1]

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

p1: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p2: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p3: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(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: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r10, r11, r12, r14, r15, r16

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:
              plus#_A(x1,x2) = 0
              s_A(x1) = x1 + 2
              if_A(x1,x2,x3) = x2 + x3 + 1
              gt_A(x1,x2) = x1 + 1
              not_A(x1) = x1 + 2
              id_A(x1) = x1 + 1
              true_A() = 1
              false_A() = 1
              zero_A() = 1
              plus#!_A(x1,x2) = 4
              s!_A(x1) = x1 + 6
              if!_A(x1,x2,x3) = 3
              gt!_A(x1,x2) = x2 + 5
              not!_A(x1) = 7
              id!_A(x1) = 4
              true!_A() = 0
              false!_A() = 0
              zero!_A() = 1
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = 0
            s_A(x1) = max{43, x1 + 38}
            if_A(x1,x2,x3) = max{x2 + 6, x3}
            gt_A(x1,x2) = max{77, x1 - 43}
            not_A(x1) = 77
            id_A(x1) = x1 + 31
            true_A = 63
            false_A = 1
            zero_A = 43
            plus#!_A(x1,x2) = max{15, x1 - 5, x2 - 43}
            s!_A(x1) = 72
            if!_A(x1,x2,x3) = max{74, x2 - 22, x3 - 63}
            gt!_A(x1,x2) = max{74, x1 + 37, x2 - 151}
            not!_A(x1) = x1 + 76
            id!_A(x1) = 72
            true!_A = 15
            false!_A = 75
            zero!_A = 76
        
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(if) = [2, 3]
      pi(gt) = [2]
      pi(not) = [1]
      pi(id) = [1]
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p2: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),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: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p2: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),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: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r10, r11, r12, r13, r14, r15, r16

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:
              plus#_A(x1,x2) = 0
              s_A(x1) = 7
              if_A(x1,x2,x3) = x2 + x3
              gt_A(x1,x2) = 1
              not_A(x1) = x1 + 2
              id_A(x1) = x1 + 3
              true_A() = 1
              false_A() = 1
              zero_A() = 1
              plus#!_A(x1,x2) = 6
              s!_A(x1) = 8
              if!_A(x1,x2,x3) = 14
              gt!_A(x1,x2) = 1
              not!_A(x1) = 15
              id!_A(x1) = x1 + 7
              true!_A() = 0
              false!_A() = 0
              zero!_A() = 1
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = max{0, x1 - 66, x2 - 49}
            s_A(x1) = x1 + 44
            if_A(x1,x2,x3) = max{32, x1 - 61, x2, x3}
            gt_A(x1,x2) = max{79, x2 - 1}
            not_A(x1) = max{35, x1 - 44}
            id_A(x1) = max{48, x1 + 27}
            true_A = 27
            false_A = 27
            zero_A = 0
            plus#!_A(x1,x2) = max{17, x1 - 19, x2 - 25}
            s!_A(x1) = 5
            if!_A(x1,x2,x3) = max{16, x1 - 46, x2 - 28, x3 - 28}
            gt!_A(x1,x2) = max{3, x1 - 37, x2 - 45}
            not!_A(x1) = max{26, x1 - 2}
            id!_A(x1) = max{6, x1 - 20}
            true!_A = 4
            false!_A = 0
            zero!_A = 5
        
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(if) = [1, 2, 3]
      pi(gt) = []
      pi(not) = [1]
      pi(id) = [1]
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),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: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),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: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r11, r12, r14, r15, r16

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:
              plus#_A(x1,x2) = x1
              id_A(x1) = x1
              s_A(x1) = x1 + 4
              if_A(x1,x2,x3) = x2 + x3 + 5
              gt_A(x1,x2) = x1 + 1
              true_A() = 1
              false_A() = 1
              zero_A() = 1
              plus#!_A(x1,x2) = 4
              id!_A(x1) = 15
              s!_A(x1) = x1 + 3
              if!_A(x1,x2,x3) = 16
              gt!_A(x1,x2) = x2 + 2
              true!_A() = 0
              false!_A() = 0
              zero!_A() = 1
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = max{0, x2 - 179}
            id_A(x1) = max{66, x1 - 25}
            s_A(x1) = x1 + 150
            if_A(x1,x2,x3) = max{178, x1 + 130, x2 + 28, x3}
            gt_A(x1,x2) = max{46, x1 - 227, x2 - 123}
            true_A = 46
            false_A = 46
            zero_A = 0
            plus#!_A(x1,x2) = max{33, x1 - 31}
            id!_A(x1) = max{59, x1 + 24}
            s!_A(x1) = max{3, x1 - 25}
            if!_A(x1,x2,x3) = max{x1 + 122, x2 - 92, x3 - 92}
            gt!_A(x1,x2) = max{34, x1 - 151, x2 - 151}
            true!_A = 0
            false!_A = 0
            zero!_A = 1
        
  
    partial status:
  
      pi(plus#) = [1]
      pi(id) = [1]
      pi(s) = []
      pi(if) = [1, 2, 3]
      pi(gt) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

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: gt#(s(x),s(y)) -> gt#(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: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

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:
              gt#_A(x1,x2) = x1 + x2
              s_A(x1) = x1 + 4
              gt#!_A(x1,x2) = x1
              s!_A(x1) = x1 + 9
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            gt#_A(x1,x2) = 0
            s_A(x1) = max{13, x1 - 21}
            gt#!_A(x1,x2) = max{16, x1 - 14, x2 - 14}
            s!_A(x1) = max{15, x1 + 11}
        
  
    partial status:
  
      pi(gt#) = []
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.