YES
We show the termination of the TRS R:

  times(x,|0|()) -> |0|()
  times(x,s(y)) -> plus(times(x,y),x)
  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: times#(x,s(y)) -> plus#(times(x,y),x)
p2: times#(x,s(y)) -> times#(x,y)
p3: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p4: plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
p5: plus#(s(x),s(y)) -> gt#(x,y)
p6: plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
p7: plus#(s(x),s(y)) -> not#(gt(x,y))
p8: plus#(s(x),s(y)) -> id#(x)
p9: plus#(s(x),s(y)) -> id#(y)
p10: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
p11: plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
p12: plus#(s(x),x) -> gt#(x,x)
p13: plus#(s(x),x) -> id#(x)
p14: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p15: plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
p16: plus#(id(x),s(y)) -> gt#(s(y),y)
p17: not#(x) -> if#(x,false(),true())
p18: gt#(s(x),s(y)) -> gt#(x,y)

and R consists of:

r1: times(x,|0|()) -> |0|()
r2: times(x,s(y)) -> plus(times(x,y),x)
r3: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r4: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r5: plus(zero(),y) -> y
r6: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r7: id(x) -> x
r8: if(true(),x,y) -> x
r9: if(false(),x,y) -> y
r10: not(x) -> if(x,false(),true())
r11: gt(s(x),zero()) -> true()
r12: gt(zero(),y) -> false()
r13: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p2}
  {p3, p10, p14}
  {p18}


-- Reduction pair.

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

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

and R consists of:

r1: times(x,|0|()) -> |0|()
r2: times(x,s(y)) -> plus(times(x,y),x)
r3: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r4: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r5: plus(zero(),y) -> y
r6: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r7: id(x) -> x
r8: if(true(),x,y) -> x
r9: if(false(),x,y) -> y
r10: not(x) -> if(x,false(),true())
r11: gt(s(x),zero()) -> true()
r12: gt(zero(),y) -> false()
r13: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r11, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        times#_A(x1,x2) = max{25, x2 - 1}
        s_A(x1) = max{27, x1 + 24}
        gt_A(x1,x2) = max{111, x1 - 40, x2 - 28}
        zero_A = 27
        true_A = 26
        false_A = 8
  
    base order 2:
  
      precedence:
      
        s = gt = zero > times# = true = false
      
  
    partial status:
  
      pi(times#) = []
      pi(s) = [1]
      pi(gt) = []
      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: times(x,|0|()) -> |0|()
r2: times(x,s(y)) -> plus(times(x,y),x)
r3: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r4: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r5: plus(zero(),y) -> y
r6: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r7: id(x) -> x
r8: if(true(),x,y) -> x
r9: if(false(),x,y) -> y
r10: not(x) -> if(x,false(),true())
r11: gt(s(x),zero()) -> true()
r12: gt(zero(),y) -> false()
r13: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = max{x1 + 17, x2 + 16}
        s_A(x1) = max{315, x1 + 29}
        if_A(x1,x2,x3) = max{39, x1 - 64, x2 + 1, x3}
        gt_A(x1,x2) = max{36, x1 - 107, x2 - 353}
        not_A(x1) = max{47, x1 - 37}
        id_A(x1) = max{37, x1 + 14}
        true_A = 35
        false_A = 0
        zero_A = 36
  
    base order 2:
  
      precedence:
      
        plus# > not > if > s = gt = zero > id = true = false
      
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(if) = [3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p3



-- SCC decomposition.

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)))

and R consists of:

r1: times(x,|0|()) -> |0|()
r2: times(x,s(y)) -> plus(times(x,y),x)
r3: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r4: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r5: plus(zero(),y) -> y
r6: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r7: id(x) -> x
r8: if(true(),x,y) -> x
r9: if(false(),x,y) -> y
r10: not(x) -> if(x,false(),true())
r11: gt(s(x),zero()) -> true()
r12: gt(zero(),y) -> false()
r13: 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),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)))

and R consists of:

r1: times(x,|0|()) -> |0|()
r2: times(x,s(y)) -> plus(times(x,y),x)
r3: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r4: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r5: plus(zero(),y) -> y
r6: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r7: id(x) -> x
r8: if(true(),x,y) -> x
r9: if(false(),x,y) -> y
r10: not(x) -> if(x,false(),true())
r11: gt(s(x),zero()) -> true()
r12: gt(zero(),y) -> false()
r13: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = max{27, x1 + 2, x2 - 32}
        s_A(x1) = max{36, x1 + 35}
        if_A(x1,x2,x3) = max{x1 - 68, x2 + 28, x3}
        gt_A(x1,x2) = max{26, x1 - 65, x2 - 76}
        not_A(x1) = max{67, x1 - 67}
        id_A(x1) = max{18, x1 + 1}
        true_A = 25
        false_A = 0
        zero_A = 26
  
    base order 2:
  
      precedence:
      
        not > if > plus# = s = gt = zero > id = true = false
      
  
    partial status:
  
      pi(plus#) = [1]
      pi(s) = [1]
      pi(if) = [3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p2



-- SCC decomposition.

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)))

and R consists of:

r1: times(x,|0|()) -> |0|()
r2: times(x,s(y)) -> plus(times(x,y),x)
r3: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r4: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r5: plus(zero(),y) -> y
r6: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r7: id(x) -> x
r8: if(true(),x,y) -> x
r9: if(false(),x,y) -> y
r10: not(x) -> if(x,false(),true())
r11: gt(s(x),zero()) -> true()
r12: gt(zero(),y) -> false()
r13: 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#(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: times(x,|0|()) -> |0|()
r2: times(x,s(y)) -> plus(times(x,y),x)
r3: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r4: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r5: plus(zero(),y) -> y
r6: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r7: id(x) -> x
r8: if(true(),x,y) -> x
r9: if(false(),x,y) -> y
r10: not(x) -> if(x,false(),true())
r11: gt(s(x),zero()) -> true()
r12: gt(zero(),y) -> false()
r13: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = max{1, x1, x2 - 55}
        s_A(x1) = max{108, x1 + 58}
        if_A(x1,x2,x3) = max{54, x1 - 102, x2 + 39, x3 + 1}
        gt_A(x1,x2) = max{4, x1 - 103, x2 - 42}
        not_A(x1) = max{84, x1 - 5}
        id_A(x1) = max{39, x1 + 1}
        true_A = 59
        false_A = 0
        zero_A = 102
  
    base order 2:
  
      precedence:
      
        plus# = not > if > s = gt = zero > id = true = false
      
  
    partial status:
  
      pi(plus#) = [1]
      pi(s) = [1]
      pi(if) = []
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      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: times(x,|0|()) -> |0|()
r2: times(x,s(y)) -> plus(times(x,y),x)
r3: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r4: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r5: plus(zero(),y) -> y
r6: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r7: id(x) -> x
r8: if(true(),x,y) -> x
r9: if(false(),x,y) -> y
r10: not(x) -> if(x,false(),true())
r11: gt(s(x),zero()) -> true()
r12: gt(zero(),y) -> false()
r13: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r9, r11, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gt#_A(x1,x2) = max{22, x2 + 20}
        s_A(x1) = max{70, x1 + 21}
        if_A(x1,x2,x3) = max{64, x1 - 2, x2 - 8, x3 + 34}
        false_A = 0
        gt_A(x1,x2) = max{61, x1 - 71, x2 - 2}
        zero_A = 1
        true_A = 0
  
    base order 2:
  
      precedence:
      
        if > s = gt = zero > gt# = false = true
      
  
    partial status:
  
      pi(gt#) = [2]
      pi(s) = [1]
      pi(if) = []
      pi(false) = []
      pi(gt) = []
      pi(zero) = []
      pi(true) = []

The following rules are removed:

  p1

Then no dependency pair remains.