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

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          times#_A(x1,x2) = x1 + x2 + 4
          plus_A(x1,x2) = x2 + 1
          |1|_A() = 5
          times_A(x1,x2) = x2 + 1
          |0|_A() = 1
          s_A(x1) = 0
          if_A(x1,x2,x3) = x2 + x3 + 3
          gt_A(x1,x2) = 3
          not_A(x1) = x1 + 5
          id_A(x1) = x1 + 2
          zero_A() = 2
          true_A() = 0
          false_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          times#_A(x1,x2) = max{0, x1 - 1, x2 - 35}
          plus_A(x1,x2) = max{97, x1 + 28, x2 + 28}
          |1|_A = 8
          times_A(x1,x2) = 68
          |0|_A = 68
          s_A(x1) = max{8, x1}
          if_A(x1,x2,x3) = max{19, x1 - 13, x2, x3}
          gt_A(x1,x2) = max{21, x1 - 11, x2 - 15}
          not_A(x1) = max{19, x1 - 2}
          id_A(x1) = max{6, x1}
          zero_A = 10
          true_A = 0
          false_A = 19
          times#!_A(x1,x2) = max{1, x2 - 3}
          plus!_A(x1,x2) = 0
          |1|!_A = 94
          times!_A(x1,x2) = max{95, x1 + 87, x2 - 28}
          |0|!_A = 88
          s!_A(x1) = max{1, x1 - 98}
          if!_A(x1,x2,x3) = max{x2 - 7, x3 + 25}
          gt!_A(x1,x2) = 13
          not!_A(x1) = max{26, x1 + 11}
          id!_A(x1) = max{0, x1 - 99}
          zero!_A = 11
          true!_A = 12
          false!_A = 14
  
    partial status:
  
      pi(times#) = [1]
      pi(plus) = []
      pi(|1|) = []
      pi(times) = [2]
      pi(|0|) = []
      pi(s) = []
      pi(if) = []
      pi(gt) = []
      pi(not) = [1]
      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

  r8, r9, r10

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = 7
          s_A(x1) = x1
          if_A(x1,x2,x3) = x2 + x3
          gt_A(x1,x2) = x1 + 6
          not_A(x1) = x1 + 2
          id_A(x1) = x1
          true_A() = 5
          false_A() = 3
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{x1, x2}
          s_A(x1) = x1 + 8
          if_A(x1,x2,x3) = max{3, x2, x3}
          gt_A(x1,x2) = max{x1, x2}
          not_A(x1) = x1
          id_A(x1) = x1 + 3
          true_A = 35
          false_A = 3
          plus#!_A(x1,x2) = max{x1 + 30, x2 + 26}
          s!_A(x1) = max{40, x1 + 16}
          if!_A(x1,x2,x3) = max{41, x2 - 4, x3 - 9}
          gt!_A(x1,x2) = max{47, x1 - 9, x2 + 39}
          not!_A(x1) = max{25, x1 - 2}
          id!_A(x1) = max{40, x1 + 39}
          true!_A = 41
          false!_A = 40
  
    partial status:
  
      pi(plus#) = []
      pi(s) = []
      pi(if) = [2, 3]
      pi(gt) = []
      pi(not) = [1]
      pi(id) = [1]
      pi(true) = []
      pi(false) = []

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,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

  r9, r10, r12, r13, r14

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = x1 + 7
          id_A(x1) = x1 + 9
          s_A(x1) = 6
          if_A(x1,x2,x3) = x2 + x3 + 2
          gt_A(x1,x2) = 2
          true_A() = 1
          false_A() = 1
          zero_A() = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 19, x2 - 9}
          id_A(x1) = x1 + 7
          s_A(x1) = max{8, x1}
          if_A(x1,x2,x3) = max{1, x1 - 1, x2, x3}
          gt_A(x1,x2) = max{1, x1 - 9, x2 - 9}
          true_A = 1
          false_A = 1
          zero_A = 8
          plus#!_A(x1,x2) = max{2, x1 - 19, x2 - 14}
          id!_A(x1) = x1 + 1
          s!_A(x1) = max{28, x1 + 19}
          if!_A(x1,x2,x3) = max{3, x1 - 16, x2 - 15, x3 - 10}
          gt!_A(x1,x2) = max{29, x1 - 13, x2 - 9}
          true!_A = 20
          false!_A = 0
          zero!_A = 21
  
    partial status:
  
      pi(plus#) = []
      pi(id) = [1]
      pi(s) = []
      pi(if) = []
      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: 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          gt#_A(x1,x2) = 8
          s_A(x1) = 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          gt#_A(x1,x2) = max{0, x1 - 2, x2 - 2}
          s_A(x1) = max{263, x1 + 1}
          gt#!_A(x1,x2) = x2 + 7
          s!_A(x1) = x1 + 7
  
    partial status:
  
      pi(gt#) = []
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.