YES
We show the termination of the TRS R:

  times(x,plus(y,|1|())) -> plus(times(x,plus(y,times(|1|(),|0|()))),x)
  times(x,|1|()) -> x
  times(x,|0|()) -> |0|()
  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,plus(y,|1|())) -> plus#(times(x,plus(y,times(|1|(),|0|()))),x)
p2: times#(x,plus(y,|1|())) -> times#(x,plus(y,times(|1|(),|0|())))
p3: times#(x,plus(y,|1|())) -> plus#(y,times(|1|(),|0|()))
p4: times#(x,plus(y,|1|())) -> times#(|1|(),|0|())
p5: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p6: plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
p7: plus#(s(x),s(y)) -> gt#(x,y)
p8: plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
p9: plus#(s(x),s(y)) -> not#(gt(x,y))
p10: plus#(s(x),s(y)) -> id#(x)
p11: plus#(s(x),s(y)) -> id#(y)
p12: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
p13: plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
p14: plus#(s(x),x) -> gt#(x,x)
p15: plus#(s(x),x) -> id#(x)
p16: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p17: plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
p18: plus#(id(x),s(y)) -> gt#(s(y),y)
p19: not#(x) -> if#(x,false(),true())
p20: gt#(s(x),s(y)) -> gt#(x,y)

and R consists of:

r1: times(x,plus(y,|1|())) -> plus(times(x,plus(y,times(|1|(),|0|()))),x)
r2: times(x,|1|()) -> x
r3: times(x,|0|()) -> |0|()
r4: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r5: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r6: plus(zero(),y) -> y
r7: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r8: id(x) -> x
r9: if(true(),x,y) -> x
r10: if(false(),x,y) -> y
r11: not(x) -> if(x,false(),true())
r12: gt(s(x),zero()) -> true()
r13: gt(zero(),y) -> false()
r14: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p2}
  {p5, p12, p16}
  {p20}


-- Reduction pair.

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

p1: times#(x,plus(y,|1|())) -> times#(x,plus(y,times(|1|(),|0|())))

and R consists of:

r1: times(x,plus(y,|1|())) -> plus(times(x,plus(y,times(|1|(),|0|()))),x)
r2: times(x,|1|()) -> x
r3: times(x,|0|()) -> |0|()
r4: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r5: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r6: plus(zero(),y) -> y
r7: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r8: id(x) -> x
r9: if(true(),x,y) -> x
r10: if(false(),x,y) -> y
r11: not(x) -> if(x,false(),true())
r12: gt(s(x),zero()) -> true()
r13: gt(zero(),y) -> false()
r14: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r3, r4, r5, r6, r7, r12, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        times#_A(x1,x2) = max{x1 + 51, x2 + 24}
        plus_A(x1,x2) = x2 + 54
        |1|_A = 233
        times_A(x1,x2) = max{101, x1 - 311, x2 - 131}
        |0|_A = 50
        s_A(x1) = 0
        if_A(x1,x2,x3) = 0
        gt_A(x1,x2) = max{57, x1 - 7, x2 - 2}
        not_A(x1) = max{57, x1 - 67}
        id_A(x1) = 0
        zero_A = 57
        true_A = 56
        false_A = 56
  
    base order 2:
  
      precedence:
      
        times# > |1| > times = not > plus = s = gt = zero > |0| = id = true = false > if
      
  
    partial status:
  
      pi(times#) = [2]
      pi(plus) = []
      pi(|1|) = []
      pi(times) = []
      pi(|0|) = []
      pi(s) = []
      pi(if) = []
      pi(gt) = []
      pi(not) = []
      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: 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,plus(y,|1|())) -> plus(times(x,plus(y,times(|1|(),|0|()))),x)
r2: times(x,|1|()) -> x
r3: times(x,|0|()) -> |0|()
r4: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r5: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r6: plus(zero(),y) -> y
r7: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r8: id(x) -> x
r9: if(true(),x,y) -> x
r10: if(false(),x,y) -> y
r11: not(x) -> if(x,false(),true())
r12: gt(s(x),zero()) -> true()
r13: gt(zero(),y) -> false()
r14: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r3, r8, r9, r10, r11, r12, r13, r14

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 + 72, x2 + 70}
        s_A(x1) = max{157, x1 + 118}
        if_A(x1,x2,x3) = max{x1 - 145, x2 + 67, x3}
        gt_A(x1,x2) = max{71, x1 - 187, x2 - 187}
        not_A(x1) = max{68, x1 - 72}
        id_A(x1) = max{62, x1 + 47}
        times_A(x1,x2) = max{6, x1 - 184, x2 - 125}
        |0|_A = 4
        true_A = 67
        false_A = 0
        zero_A = 68
  
    base order 2:
  
      precedence:
      
        plus# > not > if > id > s = gt = zero > times = true = false > |0|
      
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(if) = [2, 3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(times) = []
      pi(|0|) = []
      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,plus(y,|1|())) -> plus(times(x,plus(y,times(|1|(),|0|()))),x)
r2: times(x,|1|()) -> x
r3: times(x,|0|()) -> |0|()
r4: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r5: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r6: plus(zero(),y) -> y
r7: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r8: id(x) -> x
r9: if(true(),x,y) -> x
r10: if(false(),x,y) -> y
r11: not(x) -> if(x,false(),true())
r12: gt(s(x),zero()) -> true()
r13: gt(zero(),y) -> false()
r14: 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,plus(y,|1|())) -> plus(times(x,plus(y,times(|1|(),|0|()))),x)
r2: times(x,|1|()) -> x
r3: times(x,|0|()) -> |0|()
r4: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r5: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r6: plus(zero(),y) -> y
r7: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r8: id(x) -> x
r9: if(true(),x,y) -> x
r10: if(false(),x,y) -> y
r11: not(x) -> if(x,false(),true())
r12: gt(s(x),zero()) -> true()
r13: gt(zero(),y) -> false()
r14: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r3, r8, r9, r10, r11, r12, r13, r14

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 - 13, x2 - 20}
        s_A(x1) = x1 + 18
        if_A(x1,x2,x3) = max{12, x2 + 2, x3}
        gt_A(x1,x2) = max{57, x1 - 69, x2 - 51}
        not_A(x1) = max{56, x1 - 108}
        id_A(x1) = x1 + 4
        times_A(x1,x2) = max{38, x1 - 40, x2 - 79}
        |0|_A = 6
        true_A = 17
        false_A = 0
        zero_A = 18
  
    base order 2:
  
      precedence:
      
        gt > plus# > s = not = id = zero > if = true = false > times > |0|
      
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(if) = [3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(times) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p1



-- 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: times(x,plus(y,|1|())) -> plus(times(x,plus(y,times(|1|(),|0|()))),x)
r2: times(x,|1|()) -> x
r3: times(x,|0|()) -> |0|()
r4: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r5: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r6: plus(zero(),y) -> y
r7: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r8: id(x) -> x
r9: if(true(),x,y) -> x
r10: if(false(),x,y) -> y
r11: not(x) -> if(x,false(),true())
r12: gt(s(x),zero()) -> true()
r13: gt(zero(),y) -> false()
r14: 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: times(x,plus(y,|1|())) -> plus(times(x,plus(y,times(|1|(),|0|()))),x)
r2: times(x,|1|()) -> x
r3: times(x,|0|()) -> |0|()
r4: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r5: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r6: plus(zero(),y) -> y
r7: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r8: id(x) -> x
r9: if(true(),x,y) -> x
r10: if(false(),x,y) -> y
r11: not(x) -> if(x,false(),true())
r12: gt(s(x),zero()) -> true()
r13: gt(zero(),y) -> false()
r14: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r3, 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{4, x1 + 2}
        id_A(x1) = max{18, x1 + 3}
        s_A(x1) = max{7, x1 - 144}
        if_A(x1,x2,x3) = max{6, x1 - 9, x2 - 217, x3 - 73}
        gt_A(x1,x2) = max{6, x1 - 64, x2 - 208}
        times_A(x1,x2) = max{41, x1 - 44, x2 - 20}
        |0|_A = 40
        zero_A = 63
        true_A = 5
        false_A = 5
  
    base order 2:
  
      precedence:
      
        times > |0| > plus# = s > if = gt = zero > id = true = false
      
  
    partial status:
  
      pi(plus#) = [1]
      pi(id) = [1]
      pi(s) = []
      pi(if) = []
      pi(gt) = []
      pi(times) = []
      pi(|0|) = []
      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: gt#(s(x),s(y)) -> gt#(x,y)

and R consists of:

r1: times(x,plus(y,|1|())) -> plus(times(x,plus(y,times(|1|(),|0|()))),x)
r2: times(x,|1|()) -> x
r3: times(x,|0|()) -> |0|()
r4: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r5: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r6: plus(zero(),y) -> y
r7: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r8: id(x) -> x
r9: if(true(),x,y) -> x
r10: if(false(),x,y) -> y
r11: not(x) -> if(x,false(),true())
r12: gt(s(x),zero()) -> true()
r13: gt(zero(),y) -> false()
r14: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r3, r6, r12, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gt#_A(x1,x2) = max{1, x1 - 7, x2 - 10}
        s_A(x1) = max{9, x1 + 6}
        times_A(x1,x2) = max{36, x1 - 71}
        |0|_A = 2
        plus_A(x1,x2) = max{35, x2}
        zero_A = 8
        gt_A(x1,x2) = max{8, x1 - 10, x2 - 10}
        true_A = 7
        false_A = 7
  
    base order 2:
  
      precedence:
      
        gt# > s = times = plus = zero = gt > |0| = true = false
      
  
    partial status:
  
      pi(gt#) = []
      pi(s) = [1]
      pi(times) = []
      pi(|0|) = []
      pi(plus) = [2]
      pi(zero) = []
      pi(gt) = []
      pi(true) = []
      pi(false) = []

The following rules are removed:

  p1

Then no dependency pair remains.