Input TRS:
    1: tower(x) -> f(a(),x,s(0()))
    2: f(a(),0(),y) -> y
    3: f(a(),s(x),y) -> f(b(),y,s(x))
    4: f(b(),y,x) -> f(a(),half(x),exp(y))
    5: exp(0()) -> s(0())
    6: exp(s(x)) -> double(exp(x))
    7: double(0()) -> 0()
    8: double(s(x)) -> s(s(double(x)))
    9: half(0()) -> double(0())
    10: half(s(0())) -> half(0())
    11: half(s(s(x))) -> s(half(x))
Number of strict rules: 11
Direct Order(PosReal,>,Poly) ... removes: 1 2
     a()	weight: 0
 tower(x1)	weight: (/ 1 2) + x1
     s(x1)	weight: x1
     b()	weight: 0
     f(x1,x2,x3)	weight: (/ 1 4) + x1 + x2 + x3
  half(x1)	weight: x1
     0()	weight: 0
double(x1)	weight: x1
   exp(x1)	weight: x1
Number of strict rules: 9
Direct Order(PosReal,>,Poly) ... failed.
Freezing half f double
3: f❆1_a(s(x),y) -> f❆1_b(y,s(x))
4: f❆1_b(y,x) -> f❆1_a(half(x),exp(y))
5: exp(0()) -> s(0())
6: exp(s(x)) -> double(exp(x))
7: double❆1_0() -> 0()
8: double❆1_s(x) -> s(s(double(x)))
9: half❆1_0() -> double❆1_0()
10: half❆1_s(0()) -> half❆1_0()
11: half❆1_s(s(x)) -> s(half(x))
12: double(0()) ->= double❆1_0()
13: double(s(_1)) ->= double❆1_s(_1)
14: f(a(),_2,_3) ->= f❆1_a(_2,_3)
15: f(b(),_2,_3) ->= f❆1_b(_2,_3)
16: half(0()) ->= half❆1_0()
17: half(s(_1)) ->= half❆1_s(_1)
Number of strict rules: 9
Direct Order(PosReal,>,Poly) ... removes: 15 14
     a()	weight: 0
 tower(x1)	weight: x1
     s(x1)	weight: x1
     b()	weight: 0
     f(x1,x2,x3)	weight: (/ 1 2) + x1 + 2 * x2 + 2 * x3
  half(x1)	weight: x1
half❆1_s(x1)	weight: x1
     0()	weight: 0
double(x1)	weight: x1
double❆1_0()	weight: 0
f❆1_b(x1,x2)	weight: 2 * x1 + x2
   exp(x1)	weight: x1
double❆1_s(x1)	weight: x1
f❆1_a(x1,x2)	weight: x1 + 2 * x2
half❆1_0()	weight: 0
Number of strict rules: 9
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #exp(s(x)) -> #double(exp(x))
   #2: #exp(s(x)) -> #exp(x)
   #3: #double(s(_1)) ->? #double❆1_s(_1)
   #4: #half❆1_0() -> #double❆1_0()
   #5: #half❆1_s(s(x)) -> #half(x)
   #6: #double(0()) ->? #double❆1_0()
   #7: #half❆1_s(0()) -> #half❆1_0()
   #8: #half(s(_1)) ->? #half❆1_s(_1)
   #9: #half(0()) ->? #half❆1_0()
   #10: #f❆1_a(s(x),y) -> #f❆1_b(y,s(x))
   #11: #double❆1_s(x) -> #double(x)
   #12: #f❆1_b(y,x) -> #f❆1_a(half(x),exp(y))
   #13: #f❆1_b(y,x) -> #half(x)
   #14: #f❆1_b(y,x) -> #exp(y)
Number of SCCs: 4, DPs: 7, edges: 7
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a()	weight: 0
#double❆1_0()	weight: 0
 tower(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
     b()	weight: 0
#double❆1_s(x1)	weight: 0
#half❆1_s(x1)	weight: 0
 #exp(x1)	weight: x1
#half(x1)	weight: 0
#f❆1_b(x1,x2)	weight: 0
     f(x1,x2,x3)	weight: 0
#half❆1_0()	weight: 0
  half(x1)	weight: 0
half❆1_s(x1)	weight: 0
#f❆1_a(x1,x2)	weight: 0
     0()	weight: 0
#double(x1)	weight: 0
double(x1)	weight: 0
double❆1_0()	weight: 0
f❆1_b(x1,x2)	weight: 0
   exp(x1)	weight: 0
double❆1_s(x1)	weight: 0
f❆1_a(x1,x2)	weight: 0
half❆1_0()	weight: 0
    Usable rules: { }
    Removed DPs: #2
Number of SCCs: 3, DPs: 6, edges: 6
	SCC { #3 #11 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a()	weight: 0
#double❆1_0()	weight: 0
 tower(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
     b()	weight: 0
#double❆1_s(x1)	weight: (/ 1 4) + x1
#half❆1_s(x1)	weight: 0
 #exp(x1)	weight: 0
#half(x1)	weight: 0
#f❆1_b(x1,x2)	weight: 0
     f(x1,x2,x3)	weight: 0
#half❆1_0()	weight: 0
  half(x1)	weight: 0
half❆1_s(x1)	weight: 0
#f❆1_a(x1,x2)	weight: 0
     0()	weight: 0
#double(x1)	weight: x1
double(x1)	weight: 0
double❆1_0()	weight: 0
f❆1_b(x1,x2)	weight: 0
   exp(x1)	weight: 0
double❆1_s(x1)	weight: 0
f❆1_a(x1,x2)	weight: 0
half❆1_0()	weight: 0
    Usable rules: { }
    Removed DPs: #3 #11
Number of SCCs: 2, DPs: 4, edges: 4
	SCC { #5 #8 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a()	weight: 0
#double❆1_0()	weight: 0
 tower(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
     b()	weight: 0
#double❆1_s(x1)	weight: (/ 1 2)
#half❆1_s(x1)	weight: x1
 #exp(x1)	weight: 0
#half(x1)	weight: x1
#f❆1_b(x1,x2)	weight: 0
     f(x1,x2,x3)	weight: 0
#half❆1_0()	weight: 0
  half(x1)	weight: 0
half❆1_s(x1)	weight: 0
#f❆1_a(x1,x2)	weight: 0
     0()	weight: 0
#double(x1)	weight: 0
double(x1)	weight: 0
double❆1_0()	weight: 0
f❆1_b(x1,x2)	weight: 0
   exp(x1)	weight: 0
double❆1_s(x1)	weight: 0
f❆1_a(x1,x2)	weight: 0
half❆1_0()	weight: 0
    Usable rules: { }
    Removed DPs: #5 #8
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #10 #12 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... succeeded.
     a()	weight: 0
#double❆1_0()	weight: 0
 tower(x1)	weight: 0
     s(x1)	weight: max{0, (/ 1 4) + x1}
     b()	weight: 0
#double❆1_s(x1)	weight: 0
#half❆1_s(x1)	weight: 0
 #exp(x1)	weight: 0
#half(x1)	weight: 0
#f❆1_b(x1,x2)	weight: max{0, (- (/ 1 8)) + x2}
     f(x1,x2,x3)	weight: 0
#half❆1_0()	weight: 0
  half(x1)	weight: max{0, (- (/ 1 8)) + x1}
half❆1_s(x1)	weight: max{0, (/ 1 16) + x1}
#f❆1_a(x1,x2)	weight: max{0, (- (/ 1 16)) + x1}
     0()	weight: 0
#double(x1)	weight: 0
double(x1)	weight: max{0, (/ 1 8) + x1}
double❆1_0()	weight: 0
f❆1_b(x1,x2)	weight: 0
   exp(x1)	weight: (max (/ 1 4) 0)
double❆1_s(x1)	weight: (max (/ 3 16) 0)
f❆1_a(x1,x2)	weight: 0
half❆1_0()	weight: 0
    Usable rules: { 5 7 9..12 16 17 }
    Removed DPs: #10
Number of SCCs: 0, DPs: 0, edges: 0
YES