Input TRS:
    1: app(app(and(),true()),true()) -> true()
    2: app(app(and(),true()),false()) -> false()
    3: app(app(and(),false()),true()) -> false()
    4: app(app(and(),false()),false()) -> false()
    5: app(app(or(),true()),true()) -> true()
    6: app(app(or(),true()),false()) -> true()
    7: app(app(or(),false()),true()) -> true()
    8: app(app(or(),false()),false()) -> false()
    9: app(app(forall(),p),nil()) -> true()
    10: app(app(forall(),p),app(app(cons(),x),xs)) -> app(app(and(),app(p,x)),app(app(forall(),p),xs))
    11: app(app(forsome(),p),nil()) -> false()
    12: app(app(forsome(),p),app(app(cons(),x),xs)) -> app(app(or(),app(p,x)),app(app(forsome(),p),xs))
Number of strict rules: 12
Direct Order(PosReal,>,Poly) ... failed.
Freezing app
1: app❆2_and(true(),true()) -> true()
2: app❆2_and(true(),false()) -> false()
3: app❆2_and(false(),true()) -> false()
4: app❆2_and(false(),false()) -> false()
5: app❆2_or(true(),true()) -> true()
6: app❆2_or(true(),false()) -> true()
7: app❆2_or(false(),true()) -> true()
8: app❆2_or(false(),false()) -> false()
9: app❆2_forall(p,nil()) -> true()
10: app❆2_forall(p,app❆2_cons(x,xs)) -> app❆2_and(app(p,x),app❆2_forall(p,xs))
11: app❆2_forsome(p,nil()) -> false()
12: app❆2_forsome(p,app❆2_cons(x,xs)) -> app❆2_or(app(p,x),app❆2_forsome(p,xs))
13: app(cons(),_1) ->= app❆1_cons(_1)
14: app(app❆1_cons(_1),_2) ->= app❆2_cons(_1,_2)
15: app(or(),_1) ->= app❆1_or(_1)
16: app(app❆1_or(_1),_2) ->= app❆2_or(_1,_2)
17: app(and(),_1) ->= app❆1_and(_1)
18: app(app❆1_and(_1),_2) ->= app❆2_and(_1,_2)
19: app(forsome(),_1) ->= app❆1_forsome(_1)
20: app(app❆1_forsome(_1),_2) ->= app❆2_forsome(_1,_2)
21: app(forall(),_1) ->= app❆1_forall(_1)
22: app(app❆1_forall(_1),_2) ->= app❆2_forall(_1,_2)
Number of strict rules: 12
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #app❆2_forsome(p,app❆2_cons(x,xs)) -> #app❆2_or(app(p,x),app❆2_forsome(p,xs))
   #2: #app❆2_forsome(p,app❆2_cons(x,xs)) -> #app(p,x)
   #3: #app❆2_forsome(p,app❆2_cons(x,xs)) -> #app❆2_forsome(p,xs)
   #4: #app(app❆1_forsome(_1),_2) ->? #app❆2_forsome(_1,_2)
   #5: #app❆2_forall(p,app❆2_cons(x,xs)) -> #app❆2_and(app(p,x),app❆2_forall(p,xs))
   #6: #app❆2_forall(p,app❆2_cons(x,xs)) -> #app(p,x)
   #7: #app❆2_forall(p,app❆2_cons(x,xs)) -> #app❆2_forall(p,xs)
   #8: #app(app❆1_forall(_1),_2) ->? #app❆2_forall(_1,_2)
   #9: #app(app❆1_or(_1),_2) ->? #app❆2_or(_1,_2)
   #10: #app(app❆1_and(_1),_2) ->? #app❆2_and(_1,_2)
Number of SCCs: 1, DPs: 6, edges: 12
	SCC { #2..4 #6..8 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
app❆2_forall(x1,x2)	weight: 0
app❆1_and(x1)	weight: 0
   and()	weight: 0
 false()	weight: 0
app❆2_and(x1,x2)	weight: 0
app❆1_forall(x1)	weight: (/ 1 4)
#app❆2_forall(x1,x2)	weight: x2
  true()	weight: 0
app❆1_or(x1)	weight: 0
app❆1_forsome(x1)	weight: (/ 1 4)
forsome()	weight: 0
#app❆2_or(x1,x2)	weight: 0
forall()	weight: 0
#app❆2_and(x1,x2)	weight: 0
#app❆2_forsome(x1,x2)	weight: x2
app❆2_or(x1,x2)	weight: 0
   nil()	weight: 0
    or()	weight: 0
 #app(x1,x2)	weight: (/ 1 4) + x2
app❆2_forsome(x1,x2)	weight: 0
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 1 2) + x1 + x2
app❆1_cons(x1)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #2..4 #6..8
Number of SCCs: 0, DPs: 0, edges: 0
YES