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, r22, r23, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        quot#_A(x1,x2) = max{20, x1 + 15, x2 - 1}
        s_A(x1) = x1 + 5
        minus_A(x1,x2) = x1 + 1
        |0|_A = 1
        plus_A(x1,x2) = max{32, x1 + 10, x2 + 15}
        zero_A = 5
        gt_A(x1,x2) = max{x1 - 6, x2 + 6}
        true_A = 2
        false_A = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          quot#_A(x1,x2) = max{0, x2 - 18}
          s_A(x1) = max{52, x1 + 19}
          minus_A(x1,x2) = max{16, x1}
          |0|_A = 72
          plus_A(x1,x2) = max{40, x1 + 6, x2}
          zero_A = 14
          gt_A(x1,x2) = max{x1 - 15, x2 + 37}
          true_A = 51
          false_A = 37
          quot#!_A(x1,x2) = max{40, x1 - 17, x2 - 12}
          s!_A(x1) = 0
          minus!_A(x1,x2) = max{41, x1 - 53}
          |0|!_A = 1
          plus!_A(x1,x2) = max{0, x2 - 27}
          zero!_A = 27
          gt!_A(x1,x2) = max{26, x1 - 53, x2 - 53}
          true!_A = 0
          false!_A = 26
  
    partial status:
  
      pi(quot#) = [1]
      pi(s) = []
      pi(minus) = [1]
      pi(|0|) = []
      pi(plus) = []
      pi(zero) = []
      pi(gt) = [2]
      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: 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

  r5, r6, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus#_A(x1,x2) = 13
        s_A(x1) = 0
        minus_A(x1,x2) = max{0, x1 - 59, x2 - 59}
        plus_A(x1,x2) = max{x1 - 105, x2 + 44}
        |0|_A = 0
        if_A(x1,x2,x3) = max{x1 - 102, x2 + 30, x3 + 15}
        gt_A(x1,x2) = 57
        not_A(x1) = max{57, x1 - 59}
        id_A(x1) = max{119, x1 + 44}
        zero_A = 2
        true_A = 0
        false_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 - 54, x2 - 57}
          s_A(x1) = max{54, x1}
          minus_A(x1,x2) = max{53, x1 + 1, x2 - 3}
          plus_A(x1,x2) = max{x1, x2}
          |0|_A = 52
          if_A(x1,x2,x3) = max{27, x1 - 35, x2, x3}
          gt_A(x1,x2) = max{35, x1 - 19, x2 + 23}
          not_A(x1) = max{35, x1 + 12}
          id_A(x1) = x1
          zero_A = 0
          true_A = 35
          false_A = 24
          minus#!_A(x1,x2) = x1 + 59
          s!_A(x1) = max{54, x1}
          minus!_A(x1,x2) = max{0, x2 - 55}
          plus!_A(x1,x2) = max{113, x2 - 26}
          |0|!_A = 1
          if!_A(x1,x2,x3) = max{114, x1 - 48, x2 - 25, x3 - 81}
          gt!_A(x1,x2) = max{114, x1 - 55, x2 - 55}
          not!_A(x1) = max{115, x1 - 47}
          id!_A(x1) = max{54, x1 - 28}
          zero!_A = 2
          true!_A = 1
          false!_A = 0
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []
      pi(minus) = []
      pi(plus) = []
      pi(|0|) = []
      pi(if) = [2, 3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(zero) = []
      pi(true) = []
      pi(false) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

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

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: 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}


-- Reduction pair.

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

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

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: 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:
      
        minus#_A(x1,x2) = 0
        s_A(x1) = max{5, x1 - 22}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus#_A(x1,x2) = max{0, x1 - 17, x2 - 17}
          s_A(x1) = x1 + 16
          minus#!_A(x1,x2) = max{1, x1 - 31, x2 - 14}
          s!_A(x1) = max{0, x1 - 32}
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        sum#_A(x1) = max{25, x1}
        app_A(x1,x2) = max{x1 + 35, x2 + 46}
        cons_A(x1,x2) = 24
        sum_A(x1) = max{24, x1 - 35}
        nil_A = 23
        plus_A(x1,x2) = max{0, x1 - 22}
        id_A(x1) = max{36, x1 + 14}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          sum#_A(x1) = max{52, x1}
          app_A(x1,x2) = max{x1 + 14, x2}
          cons_A(x1,x2) = max{39, x2}
          sum_A(x1) = max{52, x1 - 7}
          nil_A = 36
          plus_A(x1,x2) = max{12, x1}
          id_A(x1) = x1
          sum#!_A(x1) = max{30, x1 - 3}
          app!_A(x1,x2) = max{51, x1 + 12, x2 - 7}
          cons!_A(x1,x2) = max{21, x2 - 15}
          sum!_A(x1) = max{23, x1 - 19}
          nil!_A = 22
          plus!_A(x1,x2) = max{19, x1 + 1, x2}
          id!_A(x1) = max{16, x1 - 6}
  
    partial status:
  
      pi(sum#) = [1]
      pi(app) = [2]
      pi(cons) = []
      pi(sum) = []
      pi(nil) = []
      pi(plus) = []
      pi(id) = []

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

  r5, r6, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        sum#_A(x1) = x1 + 46
        cons_A(x1,x2) = max{28, x1 - 40, x2 + 12}
        plus_A(x1,x2) = max{19, x2 + 11}
        |0|_A = 0
        s_A(x1) = 0
        if_A(x1,x2,x3) = max{10, x1 - 29, x2, x3}
        gt_A(x1,x2) = 12
        not_A(x1) = max{10, x1 - 29}
        id_A(x1) = max{9, x1 + 6}
        zero_A = 10
        true_A = 9
        false_A = 9
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          sum#_A(x1) = max{0, x1 - 5}
          cons_A(x1,x2) = max{2, x1 - 13, x2}
          plus_A(x1,x2) = max{17, x1, x2}
          |0|_A = 2
          s_A(x1) = max{41, x1}
          if_A(x1,x2,x3) = max{20, x1 - 41, x2, x3}
          gt_A(x1,x2) = max{1, x1 - 42, x2 - 42}
          not_A(x1) = max{20, x1}
          id_A(x1) = max{18, x1}
          zero_A = 0
          true_A = 1
          false_A = 1
          sum#!_A(x1) = max{3, x1 - 5}
          cons!_A(x1,x2) = max{2, x1 - 23, x2 - 9}
          plus!_A(x1,x2) = max{82, x1 + 39}
          |0|!_A = 0
          s!_A(x1) = max{58, x1 + 18}
          if!_A(x1,x2,x3) = max{81, x1 - 43, x2 - 19, x3 - 42}
          gt!_A(x1,x2) = max{60, x1 - 42, x2 - 42}
          not!_A(x1) = max{83, x1 - 42}
          id!_A(x1) = max{58, x1 - 45}
          zero!_A = 60
          true!_A = 59
          false!_A = 0
  
    partial status:
  
      pi(sum#) = [1]
      pi(cons) = [2]
      pi(plus) = []
      pi(|0|) = []
      pi(s) = []
      pi(if) = [2, 3]
      pi(gt) = []
      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: 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:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = 1
        s_A(x1) = max{8, x1}
        id_A(x1) = max{8, x1}
        if_A(x1,x2,x3) = max{21, x1 - 11, x2 + 10, x3 + 2}
        gt_A(x1,x2) = max{21, x1 - 13, x2 - 13}
        not_A(x1) = max{21, x1 - 11}
        true_A = 7
        false_A = 11
        zero_A = 12
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 280, x2 - 340}
          s_A(x1) = max{279, x1 + 152}
          id_A(x1) = max{159, x1 + 42}
          if_A(x1,x2,x3) = max{220, x1 - 11, x2 + 110, x3}
          gt_A(x1,x2) = max{287, x1 - 280, x2 - 153}
          not_A(x1) = max{278, x1 - 11}
          true_A = 1
          false_A = 0
          zero_A = 0
          plus#!_A(x1,x2) = max{1, x1 - 277, x2 - 391}
          s!_A(x1) = max{109, x1 - 169}
          id!_A(x1) = max{108, x1 - 170}
          if!_A(x1,x2,x3) = max{112, x1 - 446, x2 - 160, x3 - 280}
          gt!_A(x1,x2) = max{112, x1 - 1186, x2 - 1302}
          not!_A(x1) = max{113, x1 - 445}
          true!_A = 0
          false!_A = 0
          zero!_A = 1
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(id) = [1]
      pi(if) = []
      pi(gt) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  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)))
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: 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, p3}


-- Reduction pair.

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

p1: plus#(s(x),y) -> plus#(x,y)
p2: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
p3: 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

  r18, r19, r20, r22, r23, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = x1 + 7
        s_A(x1) = max{48, x1 + 37}
        if_A(x1,x2,x3) = max{40, x1 - 64, x2 + 2, x3 + 6}
        gt_A(x1,x2) = max{56, x1 - 49, x2 - 49}
        id_A(x1) = max{34, x1 + 8}
        true_A = 38
        false_A = 38
        zero_A = 48
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 69, x2 - 64}
          s_A(x1) = max{27, x1}
          if_A(x1,x2,x3) = max{63, x1 - 2, x2, x3}
          gt_A(x1,x2) = max{1, x1 - 29, x2 - 1}
          id_A(x1) = x1
          true_A = 1
          false_A = 1
          zero_A = 0
          plus#!_A(x1,x2) = max{2, x2 - 61}
          s!_A(x1) = max{6, x1 - 139}
          if!_A(x1,x2,x3) = max{2, x1 - 60, x2 - 62, x3 - 87}
          gt!_A(x1,x2) = max{4, x1 - 268, x2 - 268}
          id!_A(x1) = max{3, x1 + 1}
          true!_A = 0
          false!_A = 0
          zero!_A = 1
  
    partial status:
  
      pi(plus#) = [1]
      pi(s) = [1]
      pi(if) = []
      pi(gt) = []
      pi(id) = [1]
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p1, p2, p3

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

  r23

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{59, x1 + 1}
        cons_A(x1,x2) = max{58, x2}
        gt_A(x1,x2) = max{20, x1 - 34, x2 - 34}
        zero_A = 33
        false_A = 19
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = max{0, x1 - 4}
          cons_A(x1,x2) = max{5, x1 - 37, x2 + 3}
          gt_A(x1,x2) = max{7, x1 - 208, x2 - 45}
          zero_A = 44
          false_A = 7
          app#!_A(x1,x2) = max{4, x1 - 6, x2 - 1}
          cons!_A(x1,x2) = max{3, x1 - 4083, x2 - 2667}
          gt!_A(x1,x2) = max{5, x1 - 2814, x2 - 2816}
          zero!_A = 1
          false!_A = 0
  
    partial status:
  
      pi(app#) = [1]
      pi(cons) = [2]
      pi(gt) = []
      pi(zero) = []
      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: 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) = 0
        s_A(x1) = max{31, x1 - 101}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          gt#_A(x1,x2) = max{0, x1 - 47, x2 - 47}
          s_A(x1) = max{54, x1 + 46}
          gt#!_A(x1,x2) = max{6, x1 - 47, x2 - 55}
          s!_A(x1) = max{6, x1 - 322}
  
    partial status:
  
      pi(gt#) = []
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.