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!):
        
          max/plus interpretation on natural numbers:
          
            times#_A(x1,x2) = max{0, x2 - 26}
            s_A(x1) = x1 + 25
            times#!_A(x1,x2) = max{2, x1 - 1, x2 - 171}
            s!_A(x1) = max{1, x1 - 170}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            times#_A(x1,x2) = max{0, x2 - 11}
            s_A(x1) = max{17, x1 + 10}
            times#!_A(x1,x2) = max{1, x1 - 1, x2 - 49}
            s!_A(x1) = max{32, x1 - 50}
        
  
    partial status:
  
      pi(times#) = [2]
      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!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = max{0, x1 - 99, x2 - 99}
            s_A(x1) = max{23, x1 + 1}
            if_A(x1,x2,x3) = max{27, x1 - 7, x2, x3}
            gt_A(x1,x2) = max{105, x1 - 24, x2 - 2}
            not_A(x1) = max{32, x1 - 7}
            id_A(x1) = max{68, x1}
            true_A = 32
            false_A = 32
            zero_A = 1
            plus#!_A(x1,x2) = max{8, x1 - 141, x2 - 142}
            s!_A(x1) = x1 + 7
            if!_A(x1,x2,x3) = max{0, x1 - 119, x2 - 139, x3 - 146}
            gt!_A(x1,x2) = max{3, x1 - 139, x2 - 131}
            not!_A(x1) = max{9, x1 - 115}
            id!_A(x1) = max{4, x1 - 139}
            true!_A = 2
            false!_A = 0
            zero!_A = 3
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = max{0, x1 - 155, x2 - 155}
            s_A(x1) = max{154, x1 + 43}
            if_A(x1,x2,x3) = max{83, x1 - 2, x2, x3}
            gt_A(x1,x2) = max{101, x1 - 155, x2 - 44}
            not_A(x1) = max{99, x1 - 2}
            id_A(x1) = max{44, x1}
            true_A = 1
            false_A = 99
            zero_A = 0
            plus#!_A(x1,x2) = max{1, x1 - 152, x2 - 190}
            s!_A(x1) = max{54, x1 + 45}
            if!_A(x1,x2,x3) = max{55, x1 - 304, x2 - 360, x3 - 56}
            gt!_A(x1,x2) = max{54, x1 - 315, x2 - 189}
            not!_A(x1) = max{56, x1 - 102}
            id!_A(x1) = max{53, x1 - 100}
            true!_A = 46
            false!_A = 0
            zero!_A = 47
        
  
    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, r9, 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!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = max{0, x1 - 51, x2 - 42}
            id_A(x1) = max{6, x1}
            s_A(x1) = x1 + 20
            if_A(x1,x2,x3) = max{41, x1 - 1, x2 + 5, x3}
            gt_A(x1,x2) = max{21, x1 - 21, x2 - 21}
            true_A = 10
            false_A = 17
            zero_A = 20
            plus#!_A(x1,x2) = max{2, x1 - 8, x2 - 42}
            id!_A(x1) = max{4, x1 - 10}
            s!_A(x1) = max{3, x1 - 131}
            if!_A(x1,x2,x3) = max{5, x2 - 18, x3 - 31}
            gt!_A(x1,x2) = max{5, x1 - 29, x2 - 21}
            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) = max{0, x1 - 35, x2 - 27}
            id_A(x1) = max{1, x1}
            s_A(x1) = max{26, x1}
            if_A(x1,x2,x3) = max{14, x1 - 5, x2, x3}
            gt_A(x1,x2) = max{4, x1 - 27, x2 - 18}
            true_A = 3
            false_A = 3
            zero_A = 17
            plus#!_A(x1,x2) = max{15, x1 - 58, x2 - 50}
            id!_A(x1) = max{10, x1 - 59}
            s!_A(x1) = max{14, x1 + 12}
            if!_A(x1,x2,x3) = max{x1 + 17, x2 - 24, x3 - 27}
            gt!_A(x1,x2) = max{0, x2 - 49}
            true!_A = 13
            false!_A = 0
            zero!_A = 14
        
  
    partial status:
  
      pi(plus#) = [1]
      pi(id) = [1]
      pi(s) = [1]
      pi(if) = [1, 2, 3]
      pi(gt) = [2]
      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

  (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!):
        
          max/plus interpretation on natural numbers:
          
            gt#_A(x1,x2) = max{0, x1 - 63, x2 - 63}
            s_A(x1) = x1 + 62
            gt#!_A(x1,x2) = max{7, x1 - 63, x2 - 102}
            s!_A(x1) = max{6, x1 - 103}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            gt#_A(x1,x2) = max{0, x1 - 10, x2 - 10}
            s_A(x1) = x1 + 9
            gt#!_A(x1,x2) = max{1, x1 - 10, x2 - 10}
            s!_A(x1) = 0
        
  
    partial status:
  
      pi(gt#) = [2]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.