YES
We show the termination of the TRS R:

  plus(|0|(),x) -> x
  plus(s(x),y) -> s(plus(p(s(x)),y))
  times(|0|(),y) -> |0|()
  times(s(x),y) -> plus(y,times(p(s(x)),y))
  exp(x,|0|()) -> s(|0|())
  exp(x,s(y)) -> times(x,exp(x,y))
  p(s(|0|())) -> |0|()
  p(s(s(x))) -> s(p(s(x)))
  tower(x,y) -> towerIter(x,y,s(|0|()))
  towerIter(|0|(),y,z) -> z
  towerIter(s(x),y,z) -> towerIter(p(s(x)),y,exp(y,z))


-- SCC decomposition.

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

p1: plus#(s(x),y) -> plus#(p(s(x)),y)
p2: plus#(s(x),y) -> p#(s(x))
p3: times#(s(x),y) -> plus#(y,times(p(s(x)),y))
p4: times#(s(x),y) -> times#(p(s(x)),y)
p5: times#(s(x),y) -> p#(s(x))
p6: exp#(x,s(y)) -> times#(x,exp(x,y))
p7: exp#(x,s(y)) -> exp#(x,y)
p8: p#(s(s(x))) -> p#(s(x))
p9: tower#(x,y) -> towerIter#(x,y,s(|0|()))
p10: towerIter#(s(x),y,z) -> towerIter#(p(s(x)),y,exp(y,z))
p11: towerIter#(s(x),y,z) -> p#(s(x))
p12: towerIter#(s(x),y,z) -> exp#(y,z)

and R consists of:

r1: plus(|0|(),x) -> x
r2: plus(s(x),y) -> s(plus(p(s(x)),y))
r3: times(|0|(),y) -> |0|()
r4: times(s(x),y) -> plus(y,times(p(s(x)),y))
r5: exp(x,|0|()) -> s(|0|())
r6: exp(x,s(y)) -> times(x,exp(x,y))
r7: p(s(|0|())) -> |0|()
r8: p(s(s(x))) -> s(p(s(x)))
r9: tower(x,y) -> towerIter(x,y,s(|0|()))
r10: towerIter(|0|(),y,z) -> z
r11: towerIter(s(x),y,z) -> towerIter(p(s(x)),y,exp(y,z))

The estimated dependency graph contains the following SCCs:

  {p10}
  {p7}
  {p4}
  {p1}
  {p8}


-- Reduction pair.

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

p1: towerIter#(s(x),y,z) -> towerIter#(p(s(x)),y,exp(y,z))

and R consists of:

r1: plus(|0|(),x) -> x
r2: plus(s(x),y) -> s(plus(p(s(x)),y))
r3: times(|0|(),y) -> |0|()
r4: times(s(x),y) -> plus(y,times(p(s(x)),y))
r5: exp(x,|0|()) -> s(|0|())
r6: exp(x,s(y)) -> times(x,exp(x,y))
r7: p(s(|0|())) -> |0|()
r8: p(s(s(x))) -> s(p(s(x)))
r9: tower(x,y) -> towerIter(x,y,s(|0|()))
r10: towerIter(|0|(),y,z) -> z
r11: towerIter(s(x),y,z) -> towerIter(p(s(x)),y,exp(y,z))

The set of usable rules consists of

  r1, r7, r8

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:
          towerIter#_A(x1,x2,x3) = 1
          s_A(x1) = 4
          p_A(x1) = 6
          exp_A(x1,x2) = x2 + 3
          plus_A(x1,x2) = x1 + x2
          |0|_A() = 5
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          towerIter#_A(x1,x2,x3) = max{0, x2 - 1}
          s_A(x1) = x1 + 6
          p_A(x1) = max{3, x1 - 3}
          exp_A(x1,x2) = x1 + 10
          plus_A(x1,x2) = max{13, x2}
          |0|_A = 13
          towerIter#!_A(x1,x2,x3) = max{4, x1}
          s!_A(x1) = 5
          p!_A(x1) = 5
          exp!_A(x1,x2) = max{x1 + 23, x2 - 14}
          plus!_A(x1,x2) = max{23, x1 - 15, x2 - 13}
          |0|!_A = 22
  
    partial status:
  
      pi(towerIter#) = []
      pi(s) = []
      pi(p) = []
      pi(exp) = []
      pi(plus) = []
      pi(|0|) = []

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: exp#(x,s(y)) -> exp#(x,y)

and R consists of:

r1: plus(|0|(),x) -> x
r2: plus(s(x),y) -> s(plus(p(s(x)),y))
r3: times(|0|(),y) -> |0|()
r4: times(s(x),y) -> plus(y,times(p(s(x)),y))
r5: exp(x,|0|()) -> s(|0|())
r6: exp(x,s(y)) -> times(x,exp(x,y))
r7: p(s(|0|())) -> |0|()
r8: p(s(s(x))) -> s(p(s(x)))
r9: tower(x,y) -> towerIter(x,y,s(|0|()))
r10: towerIter(|0|(),y,z) -> z
r11: towerIter(s(x),y,z) -> towerIter(p(s(x)),y,exp(y,z))

The set of usable rules consists of

  r3

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:
          exp#_A(x1,x2) = 1
          s_A(x1) = 4
          times_A(x1,x2) = 3
          |0|_A() = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          exp#_A(x1,x2) = x2
          s_A(x1) = x1 + 17
          times_A(x1,x2) = max{45, x1 - 27, x2 - 92}
          |0|_A = 26
          exp#!_A(x1,x2) = max{6, x2 - 8}
          s!_A(x1) = max{19, x1 + 5}
          times!_A(x1,x2) = max{32, x1 - 176, x2 - 150}
          |0|!_A = 18
  
    partial status:
  
      pi(exp#) = []
      pi(s) = []
      pi(times) = []
      pi(|0|) = []

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

and R consists of:

r1: plus(|0|(),x) -> x
r2: plus(s(x),y) -> s(plus(p(s(x)),y))
r3: times(|0|(),y) -> |0|()
r4: times(s(x),y) -> plus(y,times(p(s(x)),y))
r5: exp(x,|0|()) -> s(|0|())
r6: exp(x,s(y)) -> times(x,exp(x,y))
r7: p(s(|0|())) -> |0|()
r8: p(s(s(x))) -> s(p(s(x)))
r9: tower(x,y) -> towerIter(x,y,s(|0|()))
r10: towerIter(|0|(),y,z) -> z
r11: towerIter(s(x),y,z) -> towerIter(p(s(x)),y,exp(y,z))

The set of usable rules consists of

  r7, r8

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) = 2
          s_A(x1) = x1 + 3
          p_A(x1) = x1
          |0|_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 - 32, x2 - 2}
          s_A(x1) = max{40, x1 + 22}
          p_A(x1) = max{31, x1 - 7}
          |0|_A = 20
          times#!_A(x1,x2) = max{13, x1 - 21}
          s!_A(x1) = x1 + 16
          p!_A(x1) = max{15, x1 - 23}
          |0|!_A = 11
  
    partial status:
  
      pi(times#) = []
      pi(s) = [1]
      pi(p) = [1]
      pi(|0|) = []

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),y) -> plus#(p(s(x)),y)

and R consists of:

r1: plus(|0|(),x) -> x
r2: plus(s(x),y) -> s(plus(p(s(x)),y))
r3: times(|0|(),y) -> |0|()
r4: times(s(x),y) -> plus(y,times(p(s(x)),y))
r5: exp(x,|0|()) -> s(|0|())
r6: exp(x,s(y)) -> times(x,exp(x,y))
r7: p(s(|0|())) -> |0|()
r8: p(s(s(x))) -> s(p(s(x)))
r9: tower(x,y) -> towerIter(x,y,s(|0|()))
r10: towerIter(|0|(),y,z) -> z
r11: towerIter(s(x),y,z) -> towerIter(p(s(x)),y,exp(y,z))

The set of usable rules consists of

  r7, r8

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 + x2 + 1
          s_A(x1) = x1
          p_A(x1) = 0
          |0|_A() = 0
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 22, x2 - 5}
          s_A(x1) = max{50, x1 + 21}
          p_A(x1) = max{18, x1 - 12}
          |0|_A = 27
          plus#!_A(x1,x2) = max{1, x1 - 19}
          s!_A(x1) = max{57, x1 + 7}
          p!_A(x1) = max{105, x1 - 23}
          |0|!_A = 105
  
    partial status:
  
      pi(plus#) = [2]
      pi(s) = []
      pi(p) = []
      pi(|0|) = []

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: p#(s(s(x))) -> p#(s(x))

and R consists of:

r1: plus(|0|(),x) -> x
r2: plus(s(x),y) -> s(plus(p(s(x)),y))
r3: times(|0|(),y) -> |0|()
r4: times(s(x),y) -> plus(y,times(p(s(x)),y))
r5: exp(x,|0|()) -> s(|0|())
r6: exp(x,s(y)) -> times(x,exp(x,y))
r7: p(s(|0|())) -> |0|()
r8: p(s(s(x))) -> s(p(s(x)))
r9: tower(x,y) -> towerIter(x,y,s(|0|()))
r10: towerIter(|0|(),y,z) -> z
r11: towerIter(s(x),y,z) -> towerIter(p(s(x)),y,exp(y,z))

The set of usable rules consists of

  r3

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:
          p#_A(x1) = x1
          s_A(x1) = x1 + 3
          times_A(x1,x2) = x2 + 2
          |0|_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          p#_A(x1) = 0
          s_A(x1) = max{8, x1 - 9}
          times_A(x1,x2) = max{14, x1 - 10, x2 - 29}
          |0|_A = 8
          p#!_A(x1) = max{0, x1 - 88}
          s!_A(x1) = max{0, x1 - 70}
          times!_A(x1,x2) = max{1, x1 - 78, x2 - 44}
          |0|!_A = 0
  
    partial status:
  
      pi(p#) = []
      pi(s) = [1]
      pi(times) = []
      pi(|0|) = []

The following rules are removed:

  p1

Then no dependency pair remains.