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

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          quot#_A(x1,x2) = x1 + 1
          s_A(x1) = x1 + 6
          minus_A(x1,x2) = x1 + 4
          |0|_A() = 1
          plus_A(x1,x2) = 3
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          quot#_A(x1,x2) = max{0, x2 - 53}
          s_A(x1) = x1 + 70
          minus_A(x1,x2) = x1 + 70
          |0|_A = 71
          plus_A(x1,x2) = max{54, x1 - 46, x2 - 8}
          quot#!_A(x1,x2) = max{x1 + 70, x2 - 52}
          s!_A(x1) = x1 + 14
          minus!_A(x1,x2) = 9
          |0|!_A = 7
          plus!_A(x1,x2) = max{268, x2 + 9}
  
    partial status:
  
      pi(quot#) = [1]
      pi(s) = []
      pi(minus) = []
      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: 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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus#_A(x1,x2) = x1
          s_A(x1) = x1 + 22
          minus_A(x1,x2) = x1 + x2 + 24
          plus_A(x1,x2) = x1 + 9
          |0|_A() = 1
          if_A(x1,x2,x3) = x2 + 11
          true_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus#_A(x1,x2) = max{0, x1 - 5}
          s_A(x1) = max{6, x1}
          minus_A(x1,x2) = max{5, x1, x2}
          plus_A(x1,x2) = 1
          |0|_A = 2
          if_A(x1,x2,x3) = max{17, x1 - 1, x2}
          true_A = 2
          minus#!_A(x1,x2) = max{3, x1 - 19}
          s!_A(x1) = 2
          minus!_A(x1,x2) = max{24, x1 + 17}
          plus!_A(x1,x2) = max{x1 - 18, x2 + 15}
          |0|!_A = 1
          if!_A(x1,x2,x3) = max{x1 + 31, x2 - 13}
          true!_A = 6
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []
      pi(minus) = [2]
      pi(plus) = [1]
      pi(|0|) = []
      pi(if) = []
      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

  r5, r6, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          sum#_A(x1) = x1
          app_A(x1,x2) = x1 + x2 + 3
          cons_A(x1,x2) = 2
          sum_A(x1) = x1
          plus_A(x1,x2) = x2 + 3
          |0|_A() = 1
          s_A(x1) = 0
          nil_A() = 2
          if_A(x1,x2,x3) = x2 + x3 + 5
          gt_A(x1,x2) = x2 + 2
          not_A(x1) = 8
          id_A(x1) = x1 + 4
          zero_A() = 2
          true_A() = 1
          false_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          sum#_A(x1) = max{0, x1 - 68}
          app_A(x1,x2) = max{x1 + 67, x2 + 32}
          cons_A(x1,x2) = max{33, x1 + 4, x2}
          sum_A(x1) = max{5, x1}
          plus_A(x1,x2) = max{29, x1 - 15, x2}
          |0|_A = 4
          s_A(x1) = max{8, x1 - 15}
          nil_A = 1
          if_A(x1,x2,x3) = max{39, x2, x3}
          gt_A(x1,x2) = max{15, x1 - 15, x2 - 15}
          not_A(x1) = max{0, x1 - 15}
          id_A(x1) = max{3, x1}
          zero_A = 0
          true_A = 16
          false_A = 16
          sum#!_A(x1) = 71
          app!_A(x1,x2) = max{x1 + 72, x2 + 37}
          cons!_A(x1,x2) = x2 + 2
          sum!_A(x1) = 5
          plus!_A(x1,x2) = max{24, x1 - 15, x2 - 15}
          |0|!_A = 2
          s!_A(x1) = max{0, x1 - 4}
          nil!_A = 4
          if!_A(x1,x2,x3) = 2
          gt!_A(x1,x2) = max{25, x1 - 3, x2 - 9}
          not!_A(x1) = max{27, x1 - 16}
          id!_A(x1) = x1 + 3
          zero!_A = 27
          true!_A = 26
          false!_A = 0
  
    partial status:
  
      pi(sum#) = [1]
      pi(app) = [2]
      pi(cons) = []
      pi(sum) = []
      pi(plus) = []
      pi(|0|) = []
      pi(s) = []
      pi(nil) = []
      pi(if) = []
      pi(gt) = [2]
      pi(not) = []
      pi(id) = [1]
      pi(zero) = []
      pi(true) = []
      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: 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

  r22

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          sum#_A(x1) = x1 + 1
          cons_A(x1,x2) = x2 + 2
          plus_A(x1,x2) = x1 + 6
          gt_A(x1,x2) = 2
          s_A(x1) = 3
          zero_A() = 20
          true_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          sum#_A(x1) = 0
          cons_A(x1,x2) = max{218, x2 + 73}
          plus_A(x1,x2) = max{x1 + 73, x2 + 27}
          gt_A(x1,x2) = max{81, x1 - 19, x2 - 19}
          s_A(x1) = max{18, x1 + 13}
          zero_A = 14
          true_A = 41
          sum#!_A(x1) = max{1, x1 - 144}
          cons!_A(x1,x2) = max{3, x2 - 117}
          plus!_A(x1,x2) = max{0, x1 - 77}
          gt!_A(x1,x2) = max{2, x2 - 20}
          s!_A(x1) = max{7, x1 - 19}
          zero!_A = 4
          true!_A = 1
  
    partial status:
  
      pi(sum#) = [1]
      pi(cons) = []
      pi(plus) = [1]
      pi(gt) = []
      pi(s) = []
      pi(zero) = []
      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: 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, r21, r22, r23, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = 7
          s_A(x1) = 10
          id_A(x1) = x1 + 8
          if_A(x1,x2,x3) = x2 + x3 + 1
          gt_A(x1,x2) = 2
          not_A(x1) = x1 + 4
          true_A() = 1
          false_A() = 1
          zero_A() = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 68, x2 - 101}
          s_A(x1) = x1 + 67
          id_A(x1) = max{58, x1}
          if_A(x1,x2,x3) = max{19, x1 - 196, x2, x3}
          gt_A(x1,x2) = max{188, x1 - 68, x2 - 68}
          not_A(x1) = max{19, x1 - 46}
          true_A = 1
          false_A = 19
          zero_A = 67
          plus#!_A(x1,x2) = max{x1 + 7, x2 + 3}
          s!_A(x1) = x1 + 121
          id!_A(x1) = max{0, x1 - 367}
          if!_A(x1,x2,x3) = max{122, x1 - 20, x2 - 59, x3 - 68}
          gt!_A(x1,x2) = max{73, x1 - 68, x2 - 68}
          not!_A(x1) = max{1, x1 - 189}
          true!_A = 72
          false!_A = 0
          zero!_A = 73
  
    partial status:
  
      pi(plus#) = []
      pi(s) = []
      pi(id) = [1]
      pi(if) = []
      pi(gt) = []
      pi(not) = [1]
      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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = x1 + 1
          s_A(x1) = x1 + 2
          id_A(x1) = x1 + 7
          if_A(x1,x2,x3) = x2 + x3 + 10
          gt_A(x1,x2) = x2 + 12
          true_A() = 1
          false_A() = 1
          zero_A() = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 64, x2 - 36}
          s_A(x1) = x1 + 33
          id_A(x1) = x1 + 25
          if_A(x1,x2,x3) = max{35, x1 - 1, x2 + 33, x3}
          gt_A(x1,x2) = max{34, x1 - 34, x2 - 34}
          true_A = 1
          false_A = 2
          zero_A = 33
          plus#!_A(x1,x2) = max{0, x1 - 34, x2 - 147}
          s!_A(x1) = 5
          id!_A(x1) = max{6, x1 + 2}
          if!_A(x1,x2,x3) = max{7, x1 - 44, x2 - 26, x3 - 26}
          gt!_A(x1,x2) = max{3, x1 - 113, x2 - 113}
          true!_A = 2
          false!_A = 0
          zero!_A = 3
  
    partial status:
  
      pi(plus#) = [1]
      pi(s) = []
      pi(id) = []
      pi(if) = []
      pi(gt) = [2]
      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:
  
      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 + 8
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = 0
          cons_A(x1,x2) = max{0, x1 - 17, x2 - 5}
          app#!_A(x1,x2) = max{6, x1 - 1, x2 - 1}
          cons!_A(x1,x2) = max{x1 - 69, x2 + 5}
  
    partial status:
  
      pi(app#) = []
      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: 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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          gt#_A(x1,x2) = x1 + x2 + 16
          s_A(x1) = x1 + 15
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          gt#_A(x1,x2) = max{0, x1 - 11, x2 - 11}
          s_A(x1) = x1 + 10
          gt#!_A(x1,x2) = max{1, x1 - 50, x2 - 52}
          s!_A(x1) = max{3, x1 - 51}
  
    partial status:
  
      pi(gt#) = [1]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.