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: mod(0(),y) -> 0()
    9: mod(s(x),0()) -> 0()
    10: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
    11: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
    12: if_mod(false(),s(x),s(y)) -> 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: mod(0(),y) -> 0()
9: mod(s(x),0()) -> 0()
10: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
11: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
12: if_mod(false(),s(x),s(y)) -> 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_mod(true(),s(x),s(y)) -> #mod(minus(x,y),s(y))
   #2: #if_mod(true(),s(x),s(y)) -> #minus(x,y)
   #3: #if_minus(false(),s(x),y) -> #minus(x,y)
   #4: #mod(s(x),s(y)) -> #if_mod(le(y,x),s(x),s(y))
   #5: #mod(s(x),s(y)) -> #le(y,x)
   #6: #minus(s(x),y) -> #if_minus(le❆1_s(x,y),s(x),y)
   #7: #minus(s(x),y) -> #le❆1_s(x,y)
   #8: #le(s(_1),_2) ->? #le❆1_s(_1,_2)
   #9: #le❆1_s(x,s(y)) -> #le(x,y)
   #10: #le(0(),_1) ->? #le❆1_0(_1)
Number of SCCs: 3, DPs: 6, edges: 6
	SCC { #8 #9 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
    le(x1,x2)	weight: max{0, (/ 3 8) + x2}	status: []	precedence above: minus false le❆1_s 0 if_minus le❆1_0
     s(x1)	weight: x1	status: [x1]	precedence above: le #le minus false le❆1_s 0 #le❆1_s if_minus le❆1_0
  #le(x1,x2)	weight: x1	status: [x1]	precedence above: le s minus false le❆1_s 0 #le❆1_s if_minus le❆1_0
 minus(x1,x2)	weight: max{0, x1}	status: x1
   mod(x1,x2)	weight: max{0, (/ 1 2) + x1}	status: []	precedence above: le s #le minus false le❆1_s if_mod 0 #le❆1_s if_minus le❆1_0
 false()	weight: 0	status: []	precedence above:
  true()	weight: (/ 1 8)	status: []	precedence above: 0
  rand(x1)	weight: (/ 1 8) + x1	status: []	precedence above:
#le❆1_0(x1)	weight: x1	status: []	precedence above:
le❆1_s(x1,x2)	weight: max{0, (/ 3 8) + x2}	status: []	precedence above: le minus false 0 if_minus le❆1_0
if_mod(x1,x2,x3)	weight: max{0, (/ 1 2) + x2, x1}	status: []	precedence above: le s #le minus mod false le❆1_s 0 #le❆1_s if_minus le❆1_0
     0()	weight: 0	status: []	precedence above:
#if_minus(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 1 8) + x2, (/ 1 8) + x1}	status: [x3,x1,x2]	precedence above:
#le❆1_s(x1,x2)	weight: x1	status: [x1]	precedence above: le s #le minus false le❆1_s 0 if_minus le❆1_0
#minus(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}	status: [x1,x2]	precedence above:
 #mod(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}	status: [x1,x2]	precedence above:
#if_mod(x1,x2,x3)	weight: max{0, (/ 1 8) + x1}	status: [x1]	precedence above:
if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
le❆1_0(x1)	weight: (/ 1 4) + x1	status: [x1]	precedence above:
    Removed DPs: #8
Number of SCCs: 2, DPs: 4, edges: 4
	SCC { #3 #6 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
    le(x1,x2)	weight: max{0, (/ 3 8) + x2}	status: []	precedence above: minus false le❆1_s 0 if_minus le❆1_0
     s(x1)	weight: x1	status: [x1]	precedence above: le #le minus false le❆1_s 0 #le❆1_s if_minus le❆1_0
  #le(x1,x2)	weight: x1	status: [x1]	precedence above: le s minus false le❆1_s 0 #le❆1_s if_minus le❆1_0
 minus(x1,x2)	weight: max{0, x1}	status: x1
   mod(x1,x2)	weight: max{0, (/ 1 2) + x1}	status: []	precedence above: le s #le minus false le❆1_s if_mod 0 #le❆1_s if_minus le❆1_0
 false()	weight: 0	status: []	precedence above:
  true()	weight: (/ 1 8)	status: []	precedence above: 0
  rand(x1)	weight: (/ 1 8) + x1	status: []	precedence above:
#le❆1_0(x1)	weight: x1	status: []	precedence above:
le❆1_s(x1,x2)	weight: max{0, (/ 3 8) + x2}	status: []	precedence above: le minus false 0 if_minus le❆1_0
if_mod(x1,x2,x3)	weight: max{0, (/ 1 2) + x2, x1}	status: []	precedence above: le s #le minus mod false le❆1_s 0 #le❆1_s if_minus le❆1_0
     0()	weight: 0	status: []	precedence above:
#if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
#le❆1_s(x1,x2)	weight: x1	status: [x1]	precedence above: le s #le minus false le❆1_s 0 if_minus le❆1_0
#minus(x1,x2)	weight: max{0, x1}	status: x1
 #mod(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}	status: [x1,x2]	precedence above:
#if_mod(x1,x2,x3)	weight: max{0, (/ 1 8) + x1}	status: [x1]	precedence above:
if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
le❆1_0(x1)	weight: (/ 1 4) + x1	status: [x1]	precedence above:
    Removed DPs: #3
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #1 #4 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
    le(x1,x2)	weight: max{(/ 1 16) + x2, (/ 5 16) + x1}	status: []	precedence above: s false le❆1_s 0 le❆1_0
     s(x1)	weight: x1	status: [x1]	precedence above: le false le❆1_s 0 le❆1_0
  #le(x1,x2)	weight: x1	status: [x1]	precedence above: le s minus false le❆1_s 0 #le❆1_s if_minus le❆1_0
 minus(x1,x2)	weight: max{0, x1}	status: x1
   mod(x1,x2)	weight: max{0, (/ 3 8) + x1}	status: []	precedence above: le s #le minus false le❆1_s if_mod 0 #le❆1_s #mod #if_mod if_minus le❆1_0
 false()	weight: 0	status: []	precedence above:
  true()	weight: (/ 3 16)	status: []	precedence above: minus 0 if_minus
  rand(x1)	weight: (/ 1 16) + x1	status: []	precedence above:
#le❆1_0(x1)	weight: x1	status: []	precedence above:
le❆1_s(x1,x2)	weight: max{(/ 1 16) + x2, (/ 5 16) + x1}	status: []	precedence above: le s false 0 le❆1_0
if_mod(x1,x2,x3)	weight: max{0, (/ 3 8) + x2}	status: []	precedence above: le s #le minus mod false le❆1_s 0 #le❆1_s #mod #if_mod if_minus le❆1_0
     0()	weight: 0	status: []	precedence above:
#if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
#le❆1_s(x1,x2)	weight: x1	status: [x1]	precedence above: le s #le minus false le❆1_s 0 if_minus le❆1_0
#minus(x1,x2)	weight: max{0, x1}	status: x1
 #mod(x1,x2)	weight: max{(/ 9 16) + x2, (/ 3 8) + x1}	status: [x1]	precedence above: le s false le❆1_s 0 #if_mod le❆1_0
#if_mod(x1,x2,x3)	weight: max{(/ 9 16) + x3, (/ 3 8) + x2, (/ 1 4) + x1}	status: [x2]	precedence above: le s false le❆1_s 0 #mod le❆1_0
if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
le❆1_0(x1)	weight: (/ 1 4)	status: []	precedence above:
    Removed DPs: #1
Number of SCCs: 0, DPs: 0, edges: 0
YES