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)))
  plus(|0|(),y) -> y
  plus(s(x),y) -> s(plus(x,y))
  minus(minus(x,y),z) -> minus(x,plus(y,z))
  app(nil(),k) -> k
  app(l,nil()) -> l
  app(cons(x,l),k) -> cons(x,app(l,k))
  sum(cons(x,nil())) -> cons(x,nil())
  sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
  sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
  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: plus#(s(x),y) -> plus#(x,y)
p5: minus#(minus(x,y),z) -> minus#(x,plus(y,z))
p6: minus#(minus(x,y),z) -> plus#(y,z)
p7: app#(cons(x,l),k) -> app#(l,k)
p8: sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
p9: sum#(cons(x,cons(y,l))) -> plus#(x,y)
p10: sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
p11: sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
p12: sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
p13: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p14: plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
p15: plus#(s(x),s(y)) -> gt#(x,y)
p16: plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
p17: plus#(s(x),s(y)) -> not#(gt(x,y))
p18: plus#(s(x),s(y)) -> id#(x)
p19: plus#(s(x),s(y)) -> id#(y)
p20: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
p21: plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
p22: plus#(s(x),x) -> gt#(x,x)
p23: plus#(s(x),x) -> id#(x)
p24: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p25: plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
p26: plus#(id(x),s(y)) -> gt#(s(y),y)
p27: not#(x) -> if#(x,false(),true())
p28: 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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p2}
  {p1, p5}
  {p10}
  {p8}
  {p4, p13, p20, p24}
  {p7}
  {p28}


-- 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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r1, r2, r5, r7, r16, r19

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!):
        
          max/plus interpretation on natural numbers:
          
            quot#_A(x1,x2) = max{0, x1 - 294, x2 - 166}
            s_A(x1) = x1 + 166
            minus_A(x1,x2) = max{293, x1 + 83}
            |0|_A = 208
            plus_A(x1,x2) = max{40, x1 - 85, x2 + 39}
            zero_A = 10
            if_A(x1,x2,x3) = max{1, x1 - 206, x2}
            true_A = 372
            quot#!_A(x1,x2) = max{0, x1 - 292}
            s!_A(x1) = max{452, x1 - 124}
            minus!_A(x1,x2) = max{0, x1 - 125}
            |0|!_A = 1
            plus!_A(x1,x2) = x1 + 286
            zero!_A = 7
            if!_A(x1,x2,x3) = max{451, x1 - 207}
            true!_A = 8
        
        2. 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 + 5
              minus_A(x1,x2) = x1 + 1
              |0|_A() = 2
              plus_A(x1,x2) = x2 + 5
              zero_A() = 1
              if_A(x1,x2,x3) = x2 + 2
              true_A() = 3
              quot#!_A(x1,x2) = 14
              s!_A(x1) = 15
              minus!_A(x1,x2) = x1 + 16
              |0|!_A() = 0
              plus!_A(x1,x2) = x1 + x2 + 11
              zero!_A() = 15
              if!_A(x1,x2,x3) = 12
              true!_A() = 14
        
  
    partial status:
  
      pi(quot#) = [1]
      pi(s) = [1]
      pi(minus) = [1]
      pi(|0|) = []
      pi(plus) = [2]
      pi(zero) = []
      pi(if) = [1, 2]
      pi(true) = []

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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r1, r19

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!):
        
          max/plus interpretation on natural numbers:
          
            minus#_A(x1,x2) = max{0, x1 - 97}
            s_A(x1) = max{72, x1 + 28}
            minus_A(x1,x2) = max{x1 + 96, x2 + 96}
            plus_A(x1,x2) = max{x1 - 97, x2 + 28}
            |0|_A = 96
            if_A(x1,x2,x3) = max{24, x1 - 85, x2 + 17}
            true_A = 29
            minus#!_A(x1,x2) = max{139, x1 - 73}
            s!_A(x1) = 137
            minus!_A(x1,x2) = max{91, x2 + 64}
            plus!_A(x1,x2) = max{138, x1 - 25, x2 + 65}
            |0|!_A = 93
            if!_A(x1,x2,x3) = max{143, x1 - 45, x2 - 73}
            true!_A = 137
        
        2. 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
              s_A(x1) = x1 + 4
              minus_A(x1,x2) = x1 + 6
              plus_A(x1,x2) = x1 + x2 + 1
              |0|_A() = 2
              if_A(x1,x2,x3) = x2 + 3
              true_A() = 3
              minus#!_A(x1,x2) = x1
              s!_A(x1) = 9
              minus!_A(x1,x2) = x1 + 1
              plus!_A(x1,x2) = x1 + 8
              |0|!_A() = 0
              if!_A(x1,x2,x3) = 13
              true!_A() = 8
        
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []
      pi(minus) = [1]
      pi(plus) = []
      pi(|0|) = []
      pi(if) = [2]
      pi(true) = []

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: sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))

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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r8, r9, r10, r11, r12, r13

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!):
        
          max/plus interpretation on natural numbers:
          
            sum#_A(x1) = 0
            app_A(x1,x2) = max{x1 + 3, x2 + 13}
            cons_A(x1,x2) = max{8, x2}
            sum_A(x1) = 8
            nil_A = 7
            plus_A(x1,x2) = max{26, x2}
            sum#!_A(x1) = max{10, x1 - 2}
            app!_A(x1,x2) = max{x1 + 20, x2 + 14}
            cons!_A(x1,x2) = max{4, x2}
            sum!_A(x1) = max{6, x1}
            nil!_A = 5
            plus!_A(x1,x2) = max{43, x1 + 26}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              sum#_A(x1) = x1
              app_A(x1,x2) = x1 + x2 + 1
              cons_A(x1,x2) = x2 + 4
              sum_A(x1) = 5
              nil_A() = 1
              plus_A(x1,x2) = x1 + 3
              sum#!_A(x1) = 10
              app!_A(x1,x2) = x1 + x2
              cons!_A(x1,x2) = x2 + 2
              sum!_A(x1) = x1 + 1
              nil!_A() = 0
              plus!_A(x1,x2) = x1 + 10
        
  
    partial status:
  
      pi(sum#) = [1]
      pi(app) = [1, 2]
      pi(cons) = [2]
      pi(sum) = [1]
      pi(nil) = []
      pi(plus) = [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: sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))

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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r18, r20

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!):
        
          max/plus interpretation on natural numbers:
          
            sum#_A(x1) = max{0, x1 - 69}
            cons_A(x1,x2) = max{31, x2 + 6}
            plus_A(x1,x2) = max{6, x1 - 19, x2 - 19}
            id_A(x1) = x1
            if_A(x1,x2,x3) = max{x1 + 1, x2, x3 + 12}
            false_A = 32
            sum#!_A(x1) = max{30, x1 - 7}
            cons!_A(x1,x2) = max{4, x2 + 1}
            plus!_A(x1,x2) = 0
            id!_A(x1) = x1 + 2
            if!_A(x1,x2,x3) = max{x1 + 30, x2 - 1, x3 + 18}
            false!_A = 33
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              sum#_A(x1) = 0
              cons_A(x1,x2) = 2
              plus_A(x1,x2) = x1 + x2 + 3
              id_A(x1) = x1 + 1
              if_A(x1,x2,x3) = x3 + 1
              false_A() = 2
              sum#!_A(x1) = 1
              cons!_A(x1,x2) = 2
              plus!_A(x1,x2) = 0
              id!_A(x1) = 8
              if!_A(x1,x2,x3) = 9
              false!_A() = 10
        
  
    partial status:
  
      pi(sum#) = [1]
      pi(cons) = [2]
      pi(plus) = [1, 2]
      pi(id) = [1]
      pi(if) = [1, 2, 3]
      pi(false) = []

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: plus#(s(x),y) -> plus#(x,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))
p4: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r11, r18, r19, r20, r22, r23, r24

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!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = max{0, x1 - 316, x2 - 372}
            s_A(x1) = max{86, x1 + 62}
            id_A(x1) = x1 + 5
            if_A(x1,x2,x3) = max{62, x2, x3}
            gt_A(x1,x2) = max{31, x1 - 336, x2 - 93}
            not_A(x1) = max{62, x1 - 186}
            sum_A(x1) = max{181, x1 + 51}
            cons_A(x1,x2) = x1 + 102
            nil_A = 91
            true_A = 2
            false_A = 31
            zero_A = 92
            plus#!_A(x1,x2) = max{247, x1 + 243, x2 + 240}
            s!_A(x1) = max{325, x1 + 243}
            id!_A(x1) = max{24, x1 + 2}
            if!_A(x1,x2,x3) = max{331, x2 + 300, x3 + 329}
            gt!_A(x1,x2) = max{1, x2 - 87}
            not!_A(x1) = max{330, x1 - 32}
            sum!_A(x1) = x1 + 129
            cons!_A(x1,x2) = 23
            nil!_A = 22
            true!_A = 4
            false!_A = 2
            zero!_A = 3
        
        2. 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) = 3
              id_A(x1) = x1 + 1
              if_A(x1,x2,x3) = x2 + x3 + 1
              gt_A(x1,x2) = 1
              not_A(x1) = 3
              sum_A(x1) = x1 + 4
              cons_A(x1,x2) = x1 + x2 + 1
              nil_A() = 0
              true_A() = 1
              false_A() = 1
              zero_A() = 1
              plus#!_A(x1,x2) = 10
              s!_A(x1) = x1 + 7
              id!_A(x1) = 4
              if!_A(x1,x2,x3) = 9
              gt!_A(x1,x2) = 5
              not!_A(x1) = 8
              sum!_A(x1) = 7
              cons!_A(x1,x2) = 5
              nil!_A() = 0
              true!_A() = 4
              false!_A() = 6
              zero!_A() = 3
        
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(id) = [1]
      pi(if) = [2, 3]
      pi(gt) = [1, 2]
      pi(not) = [1]
      pi(sum) = [1]
      pi(cons) = [1]
      pi(nil) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p3, p4



-- SCC decomposition.

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

p1: plus#(s(x),y) -> plus#(x,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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: 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),y) -> plus#(x,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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r19, r20, r22, r23, r24

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!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = max{0, x1 - 255, x2 - 142}
            s_A(x1) = max{131, x1 + 91}
            id_A(x1) = max{110, x1 + 29}
            if_A(x1,x2,x3) = max{141, x1 - 1, x2 + 63, x3}
            gt_A(x1,x2) = max{64, x1 - 132, x2 - 92}
            true_A = 36
            false_A = 64
            zero_A = 0
            plus#!_A(x1,x2) = max{77, x1 - 132, x2 - 64}
            s!_A(x1) = max{77, x1 - 2}
            id!_A(x1) = max{0, x1 - 388}
            if!_A(x1,x2,x3) = max{14, x1 + 2, x3 - 115}
            gt!_A(x1,x2) = max{84, x1 - 5}
            true!_A = 85
            false!_A = 1
            zero!_A = 2
        
        2. 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
              s_A(x1) = x1 + 6
              id_A(x1) = x1 + 1
              if_A(x1,x2,x3) = x2 + x3 + 7
              gt_A(x1,x2) = x1 + 1
              true_A() = 1
              false_A() = 1
              zero_A() = 2
              plus#!_A(x1,x2) = x1 + 17
              s!_A(x1) = x1 + 14
              id!_A(x1) = 11
              if!_A(x1,x2,x3) = x2 + 12
              gt!_A(x1,x2) = x2 + 17
              true!_A() = 18
              false!_A() = 18
              zero!_A() = 19
        
  
    partial status:
  
      pi(plus#) = [2]
      pi(s) = [1]
      pi(id) = []
      pi(if) = [2, 3]
      pi(gt) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

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: app#(cons(x,l),k) -> app#(l,k)

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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: 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!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{0, x1 - 11}
            cons_A(x1,x2) = max{x1 - 23, x2 + 10}
            app#!_A(x1,x2) = x1 + 27
            cons!_A(x1,x2) = x2 + 26
        
        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
              cons_A(x1,x2) = x1 + x2 + 1
              app#!_A(x1,x2) = x1 + 3
              cons!_A(x1,x2) = x2 + 2
        
  
    partial status:
  
      pi(app#) = []
      pi(cons) = [1, 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: 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: plus(|0|(),y) -> y
r6: plus(s(x),y) -> s(plus(x,y))
r7: minus(minus(x,y),z) -> minus(x,plus(y,z))
r8: app(nil(),k) -> k
r9: app(l,nil()) -> l
r10: app(cons(x,l),k) -> cons(x,app(l,k))
r11: sum(cons(x,nil())) -> cons(x,nil())
r12: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r13: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r14: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r15: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r16: plus(zero(),y) -> y
r17: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r18: id(x) -> x
r19: if(true(),x,y) -> x
r20: if(false(),x,y) -> y
r21: not(x) -> if(x,false(),true())
r22: gt(s(x),zero()) -> true()
r23: gt(zero(),y) -> false()
r24: 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!):
        
          max/plus interpretation on natural numbers:
          
            gt#_A(x1,x2) = 0
            s_A(x1) = 102
            gt#!_A(x1,x2) = 234
            s!_A(x1) = 233
        
        2. 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 + 8
              gt#!_A(x1,x2) = x1 + x2 + 17
              s!_A(x1) = x1 + 16
        
  
    partial status:
  
      pi(gt#) = []
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.