YES
We show the termination of the TRS R:

  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: app#(cons(x,l),k) -> app#(l,k)
p2: sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
p3: sum#(cons(x,cons(y,l))) -> plus#(x,y)
p4: sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
p5: sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
p6: sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
p7: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p8: plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
p9: plus#(s(x),s(y)) -> gt#(x,y)
p10: plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
p11: plus#(s(x),s(y)) -> not#(gt(x,y))
p12: plus#(s(x),s(y)) -> id#(x)
p13: plus#(s(x),s(y)) -> id#(y)
p14: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
p15: plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
p16: plus#(s(x),x) -> gt#(x,x)
p17: plus#(s(x),x) -> id#(x)
p18: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p19: plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
p20: plus#(id(x),s(y)) -> gt#(s(y),y)
p21: not#(x) -> if#(x,false(),true())
p22: gt#(s(x),s(y)) -> gt#(x,y)

and R consists of:

r1: app(nil(),k) -> k
r2: app(l,nil()) -> l
r3: app(cons(x,l),k) -> cons(x,app(l,k))
r4: sum(cons(x,nil())) -> cons(x,nil())
r5: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r6: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r7: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r8: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r9: plus(zero(),y) -> y
r10: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r11: id(x) -> x
r12: if(true(),x,y) -> x
r13: if(false(),x,y) -> y
r14: not(x) -> if(x,false(),true())
r15: gt(s(x),zero()) -> true()
r16: gt(zero(),y) -> false()
r17: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p4}
  {p1}
  {p2}
  {p7, p14, p18}
  {p22}


-- 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: app(nil(),k) -> k
r2: app(l,nil()) -> l
r3: app(cons(x,l),k) -> cons(x,app(l,k))
r4: sum(cons(x,nil())) -> cons(x,nil())
r5: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r6: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r7: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r8: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r9: plus(zero(),y) -> y
r10: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r11: id(x) -> x
r12: if(true(),x,y) -> x
r13: if(false(),x,y) -> y
r14: not(x) -> if(x,false(),true())
r15: gt(s(x),zero()) -> true()
r16: gt(zero(),y) -> false()
r17: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17

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 + 6
          cons_A(x1,x2) = 5
          sum_A(x1) = 5
          nil_A() = 5
          plus_A(x1,x2) = x2 + 4
          s_A(x1) = 0
          if_A(x1,x2,x3) = x2 + x3 + 5
          gt_A(x1,x2) = 2
          not_A(x1) = x1 + 8
          id_A(x1) = x1 + 3
          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) = 0
          app_A(x1,x2) = max{x1 + 76, x2 + 4}
          cons_A(x1,x2) = max{25, x2}
          sum_A(x1) = max{46, x1 - 14}
          nil_A = 20
          plus_A(x1,x2) = max{49, x1, x2}
          s_A(x1) = max{42, x1}
          if_A(x1,x2,x3) = max{49, x1 - 12, x2, x3}
          gt_A(x1,x2) = max{1, x1 - 43, x2 - 43}
          not_A(x1) = max{54, x1}
          id_A(x1) = max{48, x1}
          zero_A = 21
          true_A = 1
          false_A = 1
          sum#!_A(x1) = 57
          app!_A(x1,x2) = max{x1 + 58, x2 + 9}
          cons!_A(x1,x2) = max{2, x2 - 14}
          sum!_A(x1) = max{1, x1 - 17}
          nil!_A = 7
          plus!_A(x1,x2) = max{59, x1 + 10, x2 + 17}
          s!_A(x1) = 2
          if!_A(x1,x2,x3) = 1
          gt!_A(x1,x2) = max{x1 - 1, x2 + 18}
          not!_A(x1) = max{41, x1 - 2}
          id!_A(x1) = x1 + 3
          zero!_A = 41
          true!_A = 40
          false!_A = 19
  
    partial status:
  
      pi(sum#) = [1]
      pi(app) = [1, 2]
      pi(cons) = []
      pi(sum) = []
      pi(nil) = []
      pi(plus) = []
      pi(s) = []
      pi(if) = []
      pi(gt) = []
      pi(not) = [1]
      pi(id) = []
      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: app#(cons(x,l),k) -> app#(l,k)

and R consists of:

r1: app(nil(),k) -> k
r2: app(l,nil()) -> l
r3: app(cons(x,l),k) -> cons(x,app(l,k))
r4: sum(cons(x,nil())) -> cons(x,nil())
r5: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r6: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r7: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r8: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r9: plus(zero(),y) -> y
r10: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r11: id(x) -> x
r12: if(true(),x,y) -> x
r13: if(false(),x,y) -> y
r14: not(x) -> if(x,false(),true())
r15: gt(s(x),zero()) -> true()
r16: gt(zero(),y) -> false()
r17: 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 + 3
  
    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{5, x1 - 239, x2 - 9}
          app#!_A(x1,x2) = max{109, x1 - 142, x2 - 1}
          cons!_A(x1,x2) = max{108, x1 - 384, x2 - 143}
  
    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: sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))

and R consists of:

r1: app(nil(),k) -> k
r2: app(l,nil()) -> l
r3: app(cons(x,l),k) -> cons(x,app(l,k))
r4: sum(cons(x,nil())) -> cons(x,nil())
r5: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r6: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r7: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r8: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r9: plus(zero(),y) -> y
r10: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r11: id(x) -> x
r12: if(true(),x,y) -> x
r13: if(false(),x,y) -> y
r14: not(x) -> if(x,false(),true())
r15: gt(s(x),zero()) -> true()
r16: gt(zero(),y) -> false()
r17: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17

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) = 1
          cons_A(x1,x2) = 2
          plus_A(x1,x2) = x2 + 5
          s_A(x1) = 0
          if_A(x1,x2,x3) = x2 + x3 + 6
          gt_A(x1,x2) = 2
          not_A(x1) = x1 + 9
          id_A(x1) = x1 + 1
          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 - 143}
          cons_A(x1,x2) = max{84, x2 + 58}
          plus_A(x1,x2) = max{x1, x2}
          s_A(x1) = max{11, x1}
          if_A(x1,x2,x3) = max{8, x1 - 30, x2, x3}
          gt_A(x1,x2) = max{12, x1 - 13, x2 - 18}
          not_A(x1) = max{26, x1 - 30}
          id_A(x1) = max{6, x1}
          zero_A = 0
          true_A = 1
          false_A = 1
          sum#!_A(x1) = max{54, x1 - 29}
          cons!_A(x1,x2) = max{0, x2 - 29}
          plus!_A(x1,x2) = max{24, x1 + 17, x2 + 2}
          s!_A(x1) = x1 + 3
          if!_A(x1,x2,x3) = max{30, x1 + 17, x2 - 2, x3 - 7}
          gt!_A(x1,x2) = max{27, x1 + 14, x2 - 12}
          not!_A(x1) = max{29, x1 - 13}
          id!_A(x1) = max{16, x1 - 31}
          zero!_A = 16
          true!_A = 4
          false!_A = 15
  
    partial status:
  
      pi(sum#) = []
      pi(cons) = []
      pi(plus) = []
      pi(s) = []
      pi(if) = [2, 3]
      pi(gt) = []
      pi(not) = [1]
      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),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p2: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p3: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))

and R consists of:

r1: app(nil(),k) -> k
r2: app(l,nil()) -> l
r3: app(cons(x,l),k) -> cons(x,app(l,k))
r4: sum(cons(x,nil())) -> cons(x,nil())
r5: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r6: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r7: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r8: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r9: plus(zero(),y) -> y
r10: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r11: id(x) -> x
r12: if(true(),x,y) -> x
r13: if(false(),x,y) -> y
r14: not(x) -> if(x,false(),true())
r15: gt(s(x),zero()) -> true()
r16: gt(zero(),y) -> false()
r17: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r11, r12, r13, r14, r15, r16, r17

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) = 22
          s_A(x1) = 24
          if_A(x1,x2,x3) = x2 + x3 + 9
          gt_A(x1,x2) = 9
          not_A(x1) = x1 + 12
          id_A(x1) = x1 + 7
          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) = 0
          s_A(x1) = x1 + 30
          if_A(x1,x2,x3) = max{x1 + 24, x2 + 4, x3}
          gt_A(x1,x2) = max{3, x1 - 36, x2 - 36}
          not_A(x1) = x1 + 30
          id_A(x1) = x1 + 22
          true_A = 3
          false_A = 3
          zero_A = 35
          plus#!_A(x1,x2) = max{x1 + 40, x2 + 11}
          s!_A(x1) = 24
          if!_A(x1,x2,x3) = max{70, x1 - 34, x2 + 25, x3 - 34}
          gt!_A(x1,x2) = max{x1 - 34, x2 + 41}
          not!_A(x1) = max{37, x1 - 4}
          id!_A(x1) = max{9, x1 - 6}
          true!_A = 23
          false!_A = 0
          zero!_A = 24
  
    partial status:
  
      pi(plus#) = []
      pi(s) = []
      pi(if) = [3]
      pi(gt) = []
      pi(not) = [1]
      pi(id) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p1, p3



-- SCC decomposition.

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

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

and R consists of:

r1: app(nil(),k) -> k
r2: app(l,nil()) -> l
r3: app(cons(x,l),k) -> cons(x,app(l,k))
r4: sum(cons(x,nil())) -> cons(x,nil())
r5: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r6: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r7: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r8: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r9: plus(zero(),y) -> y
r10: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r11: id(x) -> x
r12: if(true(),x,y) -> x
r13: if(false(),x,y) -> y
r14: not(x) -> if(x,false(),true())
r15: gt(s(x),zero()) -> true()
r16: gt(zero(),y) -> false()
r17: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

r1: app(nil(),k) -> k
r2: app(l,nil()) -> l
r3: app(cons(x,l),k) -> cons(x,app(l,k))
r4: sum(cons(x,nil())) -> cons(x,nil())
r5: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r6: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r7: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r8: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r9: plus(zero(),y) -> y
r10: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r11: id(x) -> x
r12: if(true(),x,y) -> x
r13: if(false(),x,y) -> y
r14: not(x) -> if(x,false(),true())
r15: gt(s(x),zero()) -> true()
r16: gt(zero(),y) -> false()
r17: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r12, r13, r15, r16, r17

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 + 4
          id_A(x1) = x1 + 8
          s_A(x1) = x1 + 3
          if_A(x1,x2,x3) = x2 + x3 + 10
          gt_A(x1,x2) = x2 + 2
          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 - 386, x2 - 167}
          id_A(x1) = x1 + 82
          s_A(x1) = x1 + 110
          if_A(x1,x2,x3) = max{154, x1 - 21, x2 + 84, x3}
          gt_A(x1,x2) = max{187, x1 - 170, x2 - 170}
          true_A = 110
          false_A = 172
          zero_A = 169
          plus#!_A(x1,x2) = max{390, x1 - 193, x2 - 284}
          id!_A(x1) = 0
          s!_A(x1) = x1 + 388
          if!_A(x1,x2,x3) = max{389, x2 - 173, x3 - 111}
          gt!_A(x1,x2) = max{0, x1 - 167, x2 - 167}
          true!_A = 1
          false!_A = 1
          zero!_A = 2
  
    partial status:
  
      pi(plus#) = [1]
      pi(id) = [1]
      pi(s) = []
      pi(if) = []
      pi(gt) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: app(nil(),k) -> k
r2: app(l,nil()) -> l
r3: app(cons(x,l),k) -> cons(x,app(l,k))
r4: sum(cons(x,nil())) -> cons(x,nil())
r5: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
r6: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
r7: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r8: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r9: plus(zero(),y) -> y
r10: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r11: id(x) -> x
r12: if(true(),x,y) -> x
r13: if(false(),x,y) -> y
r14: not(x) -> if(x,false(),true())
r15: gt(s(x),zero()) -> true()
r16: gt(zero(),y) -> false()
r17: 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
          s_A(x1) = x1 + 3
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          gt#_A(x1,x2) = max{0, x1 - 56, x2 - 56}
          s_A(x1) = max{138, x1 + 55}
          gt#!_A(x1,x2) = max{12, x1 - 139, x2 - 139}
          s!_A(x1) = max{11, x1 + 1}
  
    partial status:
  
      pi(gt#) = [1]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.