Input TRS:
    1: le(0(),y) -> true()
    2: le(s(x),0()) -> false()
    3: le(s(x),s(y)) -> le(x,y)
    4: minus(0(),y) -> 0()
    5: minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
    6: if_minus(true(),s(x),y) -> 0()
    7: if_minus(false(),s(x),y) -> s(minus(x,y))
    8: gcd(0(),y) -> y
    9: gcd(s(x),0()) -> s(x)
    10: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
    11: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
    12: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
    13: rand(x) ->= x
    14: rand(x) ->= rand(s(x))
Number of strict rules: 12
Direct Order(PosReal,>,Poly) ... failed.
Freezing le
1: le❆1_0(y) -> true()
2: le❆1_s(x,0()) -> false()
3: le❆1_s(x,s(y)) -> le(x,y)
4: minus(0(),y) -> 0()
5: minus(s(x),y) -> if_minus(le❆1_s(x,y),s(x),y)
6: if_minus(true(),s(x),y) -> 0()
7: if_minus(false(),s(x),y) -> s(minus(x,y))
8: gcd(0(),y) -> y
9: gcd(s(x),0()) -> s(x)
10: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
11: if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
12: if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
13: rand(x) ->= x
14: rand(x) ->= rand(s(x))
15: le(0(),_1) ->= le❆1_0(_1)
16: le(s(_1),_2) ->= le❆1_s(_1,_2)
Number of strict rules: 12
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #if_gcd(true(),s(x),s(y)) -> #gcd(minus(x,y),s(y))
   #2: #if_gcd(true(),s(x),s(y)) -> #minus(x,y)
   #3: #if_gcd(false(),s(x),s(y)) -> #gcd(minus(y,x),s(x))
   #4: #if_gcd(false(),s(x),s(y)) -> #minus(y,x)
   #5: #if_minus(false(),s(x),y) -> #minus(x,y)
   #6: #gcd(s(x),s(y)) -> #if_gcd(le(y,x),s(x),s(y))
   #7: #gcd(s(x),s(y)) -> #le(y,x)
   #8: #minus(s(x),y) -> #if_minus(le❆1_s(x,y),s(x),y)
   #9: #minus(s(x),y) -> #le❆1_s(x,y)
   #10: #le(s(_1),_2) ->? #le❆1_s(_1,_2)
   #11: #le❆1_s(x,s(y)) -> #le(x,y)
   #12: #le(0(),_1) ->? #le❆1_0(_1)
Number of SCCs: 3, DPs: 7, edges: 8
	SCC { #10 #11 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
    le(x1,x2)	weight: 0	status: []	precedence above: s #le minus false true le❆1_s 0 #le❆1_s le❆1_0
     s(x1)	weight: x1	status: [x1]	precedence above: le #le minus false true le❆1_s 0 #le❆1_s le❆1_0
  #le(x1,x2)	weight: (/ 1 2) + x1	status: [x1]	precedence above: #le❆1_s
 minus(x1,x2)	weight: max{0, x1}	status: x1
   gcd(x1,x2)	weight: max{(/ 1 2) + x2, (/ 1 2) + x1}	status: []	precedence above: le s #le minus false true le❆1_s 0 #le❆1_s if_gcd le❆1_0
 false()	weight: 0	status: []	precedence above: le s #le minus true le❆1_s 0 #le❆1_s le❆1_0
  true()	weight: 0	status: []	precedence above: minus
  rand(x1)	weight: (/ 1 2) + x1	status: []	precedence above:
#le❆1_0(x1)	weight: x1	status: []	precedence above:
le❆1_s(x1,x2)	weight: 0	status: []	precedence above: le s #le minus false true 0 #le❆1_s le❆1_0
     0()	weight: 0	status: []	precedence above: minus true le❆1_0
#if_minus(x1,x2,x3)	weight: max{(/ 1 2) + x3, (/ 1 2) + x2, (/ 1 2) + x1}	status: [x2,x3,x1]	precedence above:
#le❆1_s(x1,x2)	weight: (/ 1 2) + x1	status: [x1]	precedence above: #le
#minus(x1,x2)	weight: max{0, (/ 1 2) + x1}	status: [x1]	precedence above:
if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
if_gcd(x1,x2,x3)	weight: max{0, (/ 1 2) + x3, (/ 1 2) + x2}	status: []	precedence above: le s #le minus gcd false true le❆1_s 0 #le❆1_s le❆1_0
#if_gcd(x1,x2,x3)	weight: max{(/ 1 2) + x3, (/ 1 2) + x2, (/ 1 2) + x1}	status: [x2,x3,x1]	precedence above:
le❆1_0(x1)	weight: 0	status: []	precedence above: minus true
 #gcd(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above:
    Removed DPs: #10
Number of SCCs: 2, DPs: 5, edges: 6
	SCC { #5 #8 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
    le(x1,x2)	weight: 0	status: []	precedence above: s #le minus false true le❆1_s 0 #le❆1_s le❆1_0
     s(x1)	weight: x1	status: [x1]	precedence above: le #le minus false true le❆1_s 0 #le❆1_s le❆1_0
  #le(x1,x2)	weight: (/ 1 2) + x1	status: [x1]	precedence above: #le❆1_s
 minus(x1,x2)	weight: max{0, x1}	status: x1
   gcd(x1,x2)	weight: max{(/ 1 2) + x2, (/ 1 2) + x1}	status: []	precedence above: le s #le minus false true le❆1_s 0 #le❆1_s if_gcd le❆1_0
 false()	weight: 0	status: []	precedence above: le s #le minus true le❆1_s 0 #le❆1_s le❆1_0
  true()	weight: 0	status: []	precedence above: minus
  rand(x1)	weight: (/ 1 2) + x1	status: []	precedence above:
#le❆1_0(x1)	weight: x1	status: []	precedence above:
le❆1_s(x1,x2)	weight: 0	status: []	precedence above: le s #le minus false true 0 #le❆1_s le❆1_0
     0()	weight: 0	status: []	precedence above: minus true le❆1_0
#if_minus(x1,x2,x3)	weight: max{0, (/ 1 2) + x3, x2}	status: [x3,x2]	precedence above: minus true #minus le❆1_0
#le❆1_s(x1,x2)	weight: (/ 1 2) + x1	status: [x1]	precedence above: #le
#minus(x1,x2)	weight: max{(/ 1 2) + x2, x1}	status: [x2,x1]	precedence above: minus true #if_minus le❆1_0
if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
if_gcd(x1,x2,x3)	weight: max{0, (/ 1 2) + x3, (/ 1 2) + x2}	status: []	precedence above: le s #le minus gcd false true le❆1_s 0 #le❆1_s le❆1_0
#if_gcd(x1,x2,x3)	weight: max{(/ 1 2) + x3, (/ 1 2) + x2, (/ 1 2) + x1}	status: [x2,x3,x1]	precedence above:
le❆1_0(x1)	weight: 0	status: []	precedence above: minus true
 #gcd(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above:
    Removed DPs: #5
Number of SCCs: 1, DPs: 3, edges: 4
	SCC { #1 #3 #6 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... Order(PosReal,>,Sum-Sum; NegReal,≥,Sum)... Order(PosReal,>,MaxSum-Sum; NegReal,≥,Sum)... failed.
Removing edges: failed.
Finding a loop...  failed.
MAYBE