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, r7, r8, r9, r11

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        quot#_A(x1,x2) = max{x1 - 28, x2 + 6}
        s_A(x1) = x1 + 30
        minus_A(x1,x2) = x1 + 25
        |0|_A = 24
        plus_A(x1,x2) = max{x1 - 12, x2 + 28}
        app_A(x1,x2) = max{x1 + 74, x2 + 10}
        nil_A = 69
        sum_A(x1) = max{73, x1 + 11}
        cons_A(x1,x2) = max{110, x1 - 40, x2 + 41}
  
    base order 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 + 3
            minus_A(x1,x2) = x1 + 1
            |0|_A() = 2
            plus_A(x1,x2) = 1
            app_A(x1,x2) = x1 + x2
            nil_A() = 1
            sum_A(x1) = x1 + 1
            cons_A(x1,x2) = x1 + 1
            quot#!_A(x1,x2) = x1 + x2
            s!_A(x1) = 5
            minus!_A(x1,x2) = x1 + 7
            |0|!_A() = 0
            plus!_A(x1,x2) = x1 + 6
            app!_A(x1,x2) = x1
            nil!_A() = 0
            sum!_A(x1) = 3
            cons!_A(x1,x2) = 2
  
    partial status:
  
      pi(quot#) = [2]
      pi(s) = [1]
      pi(minus) = [1]
      pi(|0|) = []
      pi(plus) = [2]
      pi(app) = [2]
      pi(nil) = []
      pi(sum) = []
      pi(cons) = [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: 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:
  
      max/plus interpretation on natural numbers:
      
        minus#_A(x1,x2) = 0
        s_A(x1) = 27
        minus_A(x1,x2) = max{25, x1 + 24, x2 + 24}
        plus_A(x1,x2) = max{x1 + 9, x2 + 23}
        |0|_A = 1
        if_A(x1,x2,x3) = max{x1 - 86, x2 + 56, x3 - 32}
        true_A = 26
  
    base order 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 + 2
            minus_A(x1,x2) = x1 + x2 + 3
            plus_A(x1,x2) = x1 + x2 + 6
            |0|_A() = 2
            if_A(x1,x2,x3) = x2 + 2
            true_A() = 3
            minus#!_A(x1,x2) = x1
            s!_A(x1) = x1 + 2
            minus!_A(x1,x2) = x1 + x2 + 5
            plus!_A(x1,x2) = x1 + x2 + 8
            |0|!_A() = 0
            if!_A(x1,x2,x3) = 13
            true!_A() = 1
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []
      pi(minus) = [1, 2]
      pi(plus) = [1, 2]
      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

  r4, r5, r6, r8, r9, r10, r11, r12, r14, r16, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        sum#_A(x1) = 4
        app_A(x1,x2) = max{5, x1 + 3, x2}
        cons_A(x1,x2) = 2
        sum_A(x1) = 3
        quot_A(x1,x2) = max{13, x2 - 32}
        s_A(x1) = 12
        minus_A(x1,x2) = max{0, x1 - 22, x2 - 22}
        plus_A(x1,x2) = x2 + 12
        |0|_A = 11
        nil_A = 0
        if_A(x1,x2,x3) = max{0, x3 - 22}
        gt_A(x1,x2) = max{12, x1 - 16, x2 - 16}
        not_A(x1) = max{13, x1 - 16}
        id_A(x1) = max{0, x1 - 22}
        zero_A = 15
  
    base order 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
            app_A(x1,x2) = x1 + x2 + 1
            cons_A(x1,x2) = x2 + 4
            sum_A(x1) = 5
            quot_A(x1,x2) = x2
            s_A(x1) = 2
            minus_A(x1,x2) = x1 + x2 + 3
            plus_A(x1,x2) = x1 + x2 + 4
            |0|_A() = 3
            nil_A() = 1
            if_A(x1,x2,x3) = 1
            gt_A(x1,x2) = x2 + 1
            not_A(x1) = 1
            id_A(x1) = x1 + 3
            zero_A() = 1
            sum#!_A(x1) = x1
            app!_A(x1,x2) = x1 + 10
            cons!_A(x1,x2) = x1 + 7
            sum!_A(x1) = 8
            quot!_A(x1,x2) = x1 + 5
            s!_A(x1) = 6
            minus!_A(x1,x2) = x1 + x2
            plus!_A(x1,x2) = x1 + 4
            |0|!_A() = 0
            nil!_A() = 0
            if!_A(x1,x2,x3) = 5
            gt!_A(x1,x2) = 6
            not!_A(x1) = 6
            id!_A(x1) = 6
            zero!_A() = 8
  
    partial status:
  
      pi(sum#) = []
      pi(app) = [2]
      pi(cons) = []
      pi(sum) = []
      pi(quot) = []
      pi(s) = []
      pi(minus) = []
      pi(plus) = [2]
      pi(|0|) = []
      pi(nil) = []
      pi(if) = []
      pi(gt) = []
      pi(not) = []
      pi(id) = []
      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: 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

  r1, r5, r9, r16, r19, r20, r22, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        sum#_A(x1) = x1 + 10
        cons_A(x1,x2) = max{8, x2}
        plus_A(x1,x2) = max{x1 + 80, x2 + 13}
        minus_A(x1,x2) = max{44, x1 + 24, x2 + 13}
        |0|_A = 19
        app_A(x1,x2) = max{7, x1 + 1, x2 - 1}
        nil_A = 13
        zero_A = 16
        if_A(x1,x2,x3) = max{x1 - 142, x2 + 18, x3 + 18}
        true_A = 108
        false_A = 15
        gt_A(x1,x2) = max{125, x1 + 61, x2 - 17}
        s_A(x1) = max{20, x1 - 31}
  
    base order 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) = x2 + 2
            plus_A(x1,x2) = x2 + 1
            minus_A(x1,x2) = x1 + x2 + 6
            |0|_A() = 6
            app_A(x1,x2) = x1 + x2 + 1
            nil_A() = 1
            zero_A() = 1
            if_A(x1,x2,x3) = x2 + x3 + 1
            true_A() = 1
            false_A() = 1
            gt_A(x1,x2) = 1
            s_A(x1) = 5
            sum#!_A(x1) = x1
            cons!_A(x1,x2) = 3
            plus!_A(x1,x2) = x1 + 6
            minus!_A(x1,x2) = x1 + x2 + 5
            |0|!_A() = 9
            app!_A(x1,x2) = x2 + 2
            nil!_A() = 0
            zero!_A() = 1
            if!_A(x1,x2,x3) = 6
            true!_A() = 0
            false!_A() = 0
            gt!_A(x1,x2) = 7
            s!_A(x1) = 10
  
    partial status:
  
      pi(sum#) = [1]
      pi(cons) = [2]
      pi(plus) = []
      pi(minus) = []
      pi(|0|) = []
      pi(app) = []
      pi(nil) = []
      pi(zero) = []
      pi(if) = [2, 3]
      pi(true) = []
      pi(false) = []
      pi(gt) = [1]
      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: 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

  r18, r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = max{44, x1 + 18, x2 + 13}
        s_A(x1) = max{33, x1 + 18}
        id_A(x1) = max{32, x1 + 6}
        if_A(x1,x2,x3) = max{25, x2, x3}
        gt_A(x1,x2) = max{x1 - 18, x2}
        not_A(x1) = max{51, x1 - 18}
        true_A = 0
        false_A = 0
  
    base order 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) = 7
            id_A(x1) = x1 + 4
            if_A(x1,x2,x3) = x2 + x3 + 8
            gt_A(x1,x2) = x1 + 1
            not_A(x1) = 1
            true_A() = 9
            false_A() = 3
            plus#!_A(x1,x2) = 10
            s!_A(x1) = 11
            id!_A(x1) = x1 + 9
            if!_A(x1,x2,x3) = 8
            gt!_A(x1,x2) = 16
            not!_A(x1) = 9
            true!_A() = 0
            false!_A() = 2
  
    partial status:
  
      pi(plus#) = [1]
      pi(s) = [1]
      pi(id) = [1]
      pi(if) = [2, 3]
      pi(gt) = [2]
      pi(not) = []
      pi(true) = []
      pi(false) = []

The following rules are removed:

  p1, p2, p3, p4

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

  r5, r11, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x2 + 82
        cons_A(x1,x2) = max{80, x1 - 77, x2 - 41}
        plus_A(x1,x2) = max{x1 + 40, x2 + 20}
        |0|_A = 0
        sum_A(x1) = max{197, x1 + 160}
        nil_A = 1
        zero_A = 1
  
    base order 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
            plus_A(x1,x2) = x2 + 3
            |0|_A() = 3
            sum_A(x1) = x1 + 4
            nil_A() = 3
            zero_A() = 1
            app#!_A(x1,x2) = x1 + x2
            cons!_A(x1,x2) = x1 + x2 + 3
            plus!_A(x1,x2) = x2 + 10
            |0|!_A() = 2
            sum!_A(x1) = 5
            nil!_A() = 5
            zero!_A() = 3
  
    partial status:
  
      pi(app#) = [2]
      pi(cons) = []
      pi(plus) = [1]
      pi(|0|) = []
      pi(sum) = [1]
      pi(nil) = []
      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: 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:
  
      max/plus interpretation on natural numbers:
      
        gt#_A(x1,x2) = max{108, x1 + 54, x2 - 108}
        s_A(x1) = max{107, x1 + 55}
  
    base order 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 + 1
            gt#!_A(x1,x2) = x1 + x2
            s!_A(x1) = 4
  
    partial status:
  
      pi(gt#) = [1]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.