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)))))
  sum(plus(cons(|0|(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
  pred(cons(s(x),nil())) -> cons(x,nil())
  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: sum#(plus(cons(|0|(),x),cons(y,l))) -> pred#(sum(cons(s(x),cons(y,l))))
p8: sum#(plus(cons(|0|(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
p9: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p10: plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
p11: plus#(s(x),s(y)) -> gt#(x,y)
p12: plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
p13: plus#(s(x),s(y)) -> not#(gt(x,y))
p14: plus#(s(x),s(y)) -> id#(x)
p15: plus#(s(x),s(y)) -> id#(y)
p16: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
p17: plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
p18: plus#(s(x),x) -> gt#(x,x)
p19: plus#(s(x),x) -> id#(x)
p20: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p21: plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
p22: plus#(id(x),s(y)) -> gt#(s(y),y)
p23: not#(x) -> if#(x,false(),true())
p24: 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: sum(plus(cons(|0|(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
r8: pred(cons(s(x),nil())) -> cons(x,nil())
r9: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r10: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r11: plus(zero(),y) -> y
r12: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r13: id(x) -> x
r14: if(true(),x,y) -> x
r15: if(false(),x,y) -> y
r16: not(x) -> if(x,false(),true())
r17: gt(s(x),zero()) -> true()
r18: gt(zero(),y) -> false()
r19: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p4}
  {p1}
  {p2}
  {p9, p16, p20}
  {p24}


-- 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: sum(plus(cons(|0|(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
r8: pred(cons(s(x),nil())) -> cons(x,nil())
r9: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r10: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r11: plus(zero(),y) -> y
r12: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r13: id(x) -> x
r14: if(true(),x,y) -> x
r15: if(false(),x,y) -> y
r16: not(x) -> if(x,false(),true())
r17: gt(s(x),zero()) -> true()
r18: gt(zero(),y) -> false()
r19: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r9, r10, r11, r12, r13, r14, r15, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        sum#_A(x1) = max{1, x1}
        app_A(x1,x2) = max{x1 + 4, x2 + 7}
        cons_A(x1,x2) = 0
        sum_A(x1) = max{0, x1 - 5}
        nil_A = 1
        plus_A(x1,x2) = x2 + 26
        s_A(x1) = 0
        if_A(x1,x2,x3) = max{x2 + 1, x3 + 42}
        gt_A(x1,x2) = max{x1 + 27, x2 - 39}
        not_A(x1) = max{0, x1 - 77}
        id_A(x1) = x1 + 41
        zero_A = 1
        true_A = 0
        false_A = 0
  
    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 + 3, x2 + 2}
          cons_A(x1,x2) = max{92, x1 - 5, x2}
          sum_A(x1) = max{9, x1 + 8}
          nil_A = 42
          plus_A(x1,x2) = max{97, x1 - 9, x2}
          s_A(x1) = 10
          if_A(x1,x2,x3) = max{44, x2 + 9, x3 + 8}
          gt_A(x1,x2) = max{x1 + 59, x2 - 17}
          not_A(x1) = 44
          id_A(x1) = max{87, x1}
          zero_A = 0
          true_A = 70
          false_A = 29
          sum#!_A(x1) = 8
          app!_A(x1,x2) = max{99, x1 + 7}
          cons!_A(x1,x2) = x2 + 14
          sum!_A(x1) = 58
          nil!_A = 57
          plus!_A(x1,x2) = 15
          s!_A(x1) = max{56, x1 - 40}
          if!_A(x1,x2,x3) = max{57, x2 - 30, x3 - 71}
          gt!_A(x1,x2) = max{57, x1 - 11, x2 + 4}
          not!_A(x1) = 56
          id!_A(x1) = max{0, x1 - 39}
          zero!_A = 6
          true!_A = 5
          false!_A = 5
  
    partial status:
  
      pi(sum#) = [1]
      pi(app) = [2]
      pi(cons) = []
      pi(sum) = []
      pi(nil) = []
      pi(plus) = []
      pi(s) = []
      pi(if) = []
      pi(gt) = [1]
      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: 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: sum(plus(cons(|0|(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
r8: pred(cons(s(x),nil())) -> cons(x,nil())
r9: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r10: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r11: plus(zero(),y) -> y
r12: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r13: id(x) -> x
r14: if(true(),x,y) -> x
r15: if(false(),x,y) -> y
r16: not(x) -> if(x,false(),true())
r17: gt(s(x),zero()) -> true()
r18: gt(zero(),y) -> false()
r19: 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:
      
        app#_A(x1,x2) = max{4, x2 + 1}
        cons_A(x1,x2) = max{3, x1 - 300, x2 - 108}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = max{0, x1 - 8}
          cons_A(x1,x2) = max{264, x1 - 120, x2 + 7}
          app#!_A(x1,x2) = max{4, x1 - 8}
          cons!_A(x1,x2) = max{3, x1 - 3201, x2 - 3464}
  
    partial status:
  
      pi(app#) = []
      pi(cons) = []

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: sum(plus(cons(|0|(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
r8: pred(cons(s(x),nil())) -> cons(x,nil())
r9: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r10: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r11: plus(zero(),y) -> y
r12: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r13: id(x) -> x
r14: if(true(),x,y) -> x
r15: if(false(),x,y) -> y
r16: not(x) -> if(x,false(),true())
r17: gt(s(x),zero()) -> true()
r18: gt(zero(),y) -> false()
r19: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        sum#_A(x1) = max{18, x1 + 2}
        cons_A(x1,x2) = x2 + 17
        plus_A(x1,x2) = max{x1 + 12, x2 + 12}
        s_A(x1) = x1
        if_A(x1,x2,x3) = max{x2, x3}
        gt_A(x1,x2) = max{0, x2 - 12}
        not_A(x1) = x1 + 11
        id_A(x1) = x1
        zero_A = 12
        true_A = 0
        false_A = 0
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          sum#_A(x1) = max{0, x1 - 177}
          cons_A(x1,x2) = max{x1 + 10, x2 + 88}
          plus_A(x1,x2) = max{78, x1, x2}
          s_A(x1) = max{48, x1}
          if_A(x1,x2,x3) = max{25, x1 - 50, x2, x3}
          gt_A(x1,x2) = max{1, x1 - 46, x2 - 2}
          not_A(x1) = x1 + 48
          id_A(x1) = max{48, x1}
          zero_A = 1
          true_A = 2
          false_A = 1
          sum#!_A(x1) = 93
          cons!_A(x1,x2) = max{x1 + 3, x2 + 88}
          plus!_A(x1,x2) = max{94, x1 + 22, x2 + 46}
          s!_A(x1) = 72
          if!_A(x1,x2,x3) = max{2, x1 - 2, x2 - 49, x3 - 49}
          gt!_A(x1,x2) = 71
          not!_A(x1) = max{95, x1 + 3}
          id!_A(x1) = max{1, x1 - 4}
          zero!_A = 5
          true!_A = 4
          false!_A = 4
  
    partial status:
  
      pi(sum#) = [1]
      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: sum(plus(cons(|0|(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
r8: pred(cons(s(x),nil())) -> cons(x,nil())
r9: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r10: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r11: plus(zero(),y) -> y
r12: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r13: id(x) -> x
r14: if(true(),x,y) -> x
r15: if(false(),x,y) -> y
r16: not(x) -> if(x,false(),true())
r17: gt(s(x),zero()) -> true()
r18: gt(zero(),y) -> false()
r19: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r13, r14, r15, r16, r17, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = max{11, x1 - 111, x2 - 130}
        s_A(x1) = x1 + 123
        if_A(x1,x2,x3) = max{109, x1 - 5, x2, x3}
        gt_A(x1,x2) = max{2, x1 - 119, x2 - 131}
        not_A(x1) = max{129, x1 - 3}
        id_A(x1) = x1 + 111
        true_A = 3
        false_A = 1
        zero_A = 118
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 590, x2 - 590}
          s_A(x1) = x1 + 34
          if_A(x1,x2,x3) = max{18, x1 - 102, x2, x3}
          gt_A(x1,x2) = max{65, x1 - 66, x2 - 35}
          not_A(x1) = max{691, x1 - 102}
          id_A(x1) = max{25, x1}
          true_A = 65
          false_A = 1
          zero_A = 0
          plus#!_A(x1,x2) = max{728, x1 - 1187, x2 - 1179}
          s!_A(x1) = max{728, x1 - 587}
          if!_A(x1,x2,x3) = max{164, x1 - 1117, x2 - 586, x3 - 1180}
          gt!_A(x1,x2) = max{167, x1 - 1152, x2 - 1153}
          not!_A(x1) = max{165, x1 - 1116}
          id!_A(x1) = max{727, x1 - 1481}
          true!_A = 0
          false!_A = 0
          zero!_A = 1
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(if) = [2, 3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      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: sum(plus(cons(|0|(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
r8: pred(cons(s(x),nil())) -> cons(x,nil())
r9: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r10: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r11: plus(zero(),y) -> y
r12: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r13: id(x) -> x
r14: if(true(),x,y) -> x
r15: if(false(),x,y) -> y
r16: not(x) -> if(x,false(),true())
r17: gt(s(x),zero()) -> true()
r18: gt(zero(),y) -> false()
r19: 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: sum(plus(cons(|0|(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
r8: pred(cons(s(x),nil())) -> cons(x,nil())
r9: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r10: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r11: plus(zero(),y) -> y
r12: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r13: id(x) -> x
r14: if(true(),x,y) -> x
r15: if(false(),x,y) -> y
r16: not(x) -> if(x,false(),true())
r17: gt(s(x),zero()) -> true()
r18: gt(zero(),y) -> false()
r19: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r14, r15, r17, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = max{0, x2 - 84}
        id_A(x1) = max{18, x1 - 87}
        s_A(x1) = x1 + 28
        if_A(x1,x2,x3) = max{45, x1 - 6, x2 + 14, x3}
        gt_A(x1,x2) = max{4, x1 - 77, x2 - 29}
        true_A = 3
        false_A = 0
        zero_A = 28
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 4153, x2 - 3685}
          id_A(x1) = max{465, x1 + 172}
          s_A(x1) = max{557, x1 + 370}
          if_A(x1,x2,x3) = max{3684, x1 - 1, x2 + 365, x3}
          gt_A(x1,x2) = max{194, x1 - 558, x2 - 371}
          true_A = 194
          false_A = 194
          zero_A = 0
          plus#!_A(x1,x2) = max{3, x1 - 168}
          id!_A(x1) = max{1, x1 - 1018}
          s!_A(x1) = max{2, x1 - 554}
          if!_A(x1,x2,x3) = max{300, x1 - 274, x2 - 466, x3 - 925}
          gt!_A(x1,x2) = max{298, x1 - 4975, x2 - 4975}
          true!_A = 0
          false!_A = 0
          zero!_A = 1
  
    partial status:
  
      pi(plus#) = []
      pi(id) = []
      pi(s) = [1]
      pi(if) = [3]
      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: sum(plus(cons(|0|(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
r8: pred(cons(s(x),nil())) -> cons(x,nil())
r9: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r10: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r11: plus(zero(),y) -> y
r12: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r13: id(x) -> x
r14: if(true(),x,y) -> x
r15: if(false(),x,y) -> y
r16: not(x) -> if(x,false(),true())
r17: gt(s(x),zero()) -> true()
r18: gt(zero(),y) -> false()
r19: 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) = 1
        s_A(x1) = max{9, x1 - 154}
  
    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{64, x1 + 55}
          gt#!_A(x1,x2) = max{7, x1 - 56, x2 - 65}
          s!_A(x1) = max{6, x1 - 304}
  
    partial status:
  
      pi(gt#) = []
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.