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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              times#_A(x1,x2) = x2
              s_A(x1) = x1 + 5
              times#!_A(x1,x2) = x2 + 8
              s!_A(x1) = x1 + 7
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            times#_A(x1,x2) = 0
            s_A(x1) = max{30, x1 - 44}
            times#!_A(x1,x2) = max{9, x1 - 1, x2 - 31}
            s!_A(x1) = x1 + 8
        
  
    partial status:
  
      pi(times#) = [1]
      pi(s) = [1]

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, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              plus#_A(x1,x2) = 0
              s_A(x1) = x1 + 3
              if_A(x1,x2,x3) = x2 + x3
              gt_A(x1,x2) = x1 + x2
              not_A(x1) = x1
              id_A(x1) = x1 + 2
              true_A() = 0
              false_A() = 0
              zero_A() = 1
              plus#!_A(x1,x2) = 0
              s!_A(x1) = 5
              if!_A(x1,x2,x3) = x1 + 1
              gt!_A(x1,x2) = 6
              not!_A(x1) = x1 + 6
              id!_A(x1) = x1 + 7
              true!_A() = 0
              false!_A() = 0
              zero!_A() = 1
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = 0
            s_A(x1) = max{119, x1 + 25}
            if_A(x1,x2,x3) = max{118, x1 - 124, x2 + 1, x3}
            gt_A(x1,x2) = max{221, x1 + 67}
            not_A(x1) = max{149, x1 + 45}
            id_A(x1) = max{39, x1 + 16}
            true_A = 149
            false_A = 39
            zero_A = 26
            plus#!_A(x1,x2) = max{x1 + 26, x2 + 2}
            s!_A(x1) = max{0, x1 - 2}
            if!_A(x1,x2,x3) = max{146, x1 - 3, x2 - 40, x3 + 107}
            gt!_A(x1,x2) = 146
            not!_A(x1) = max{257, x1 - 2}
            id!_A(x1) = max{3, x1 - 32}
            true!_A = 0
            false!_A = 1
            zero!_A = 2
        
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(if) = [1, 2, 3]
      pi(gt) = []
      pi(not) = [1]
      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: 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#(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

  r8, r11, r12, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              plus#_A(x1,x2) = x1
              id_A(x1) = x1 + 1
              s_A(x1) = x1 + 4
              if_A(x1,x2,x3) = x2 + x3 + 1
              gt_A(x1,x2) = x1 + 1
              true_A() = 1
              zero_A() = 1
              false_A() = 1
              plus#!_A(x1,x2) = 2
              id!_A(x1) = 3
              s!_A(x1) = x1 + 10
              if!_A(x1,x2,x3) = 9
              gt!_A(x1,x2) = x2 + 1
              true!_A() = 0
              zero!_A() = 1
              false!_A() = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = max{0, x1 - 223}
            id_A(x1) = max{222, x1 + 2}
            s_A(x1) = max{121, x1 + 22}
            if_A(x1,x2,x3) = max{x1 + 124, x2 + 124}
            gt_A(x1,x2) = max{4, x1 - 14, x2 - 12}
            true_A = 9
            zero_A = 13
            false_A = 3
            plus#!_A(x1,x2) = max{1, x1 - 205}
            id!_A(x1) = max{0, x1 - 103}
            s!_A(x1) = max{18, x1 - 4}
            if!_A(x1,x2,x3) = max{199, x1 + 92, x3 - 93}
            gt!_A(x1,x2) = max{199, x1 - 144, x2 - 122}
            true!_A = 0
            zero!_A = 1
            false!_A = 0
        
  
    partial status:
  
      pi(plus#) = []
      pi(id) = []
      pi(s) = [1]
      pi(if) = [1, 2, 3]
      pi(gt) = [1, 2]
      pi(true) = []
      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: 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              gt#_A(x1,x2) = x1 + x2
              s_A(x1) = x1 + 1
              gt#!_A(x1,x2) = x1 + x2
              s!_A(x1) = 2
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            gt#_A(x1,x2) = 0
            s_A(x1) = max{8, x1 - 15}
            gt#!_A(x1,x2) = max{24, x1 + 16}
            s!_A(x1) = 5
        
  
    partial status:
  
      pi(gt#) = [2]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.