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.
   |0|()	weight: (/ 3 4)	status: []	precedence above:
    le(x1,x2)	weight: max{0, (/ 3 4) + x2}	status: [x2]	precedence above: le❆1_s le❆1_|0|
     s(x1)	weight: x1	status: [x1]	precedence above:
  #le(x1,x2)	weight: x1	status: x1
 minus(x1,x2)	weight: max{(/ 5 4) + x2, x1}	status: [x2,x1]	precedence above: |0| s #le #le❆1_s if_minus
   mod(x1,x2)	weight: max{(/ 3 2) + x2, (/ 1 8) + x1}	status: []	precedence above: |0| s #le if_mod #le❆1_s
 false()	weight: (/ 11 8)	status: []	precedence above:
  true()	weight: (/ 1 2)	status: []	precedence above:
  rand(x1)	weight: (/ 1 8) + x1	status: []	precedence above:
le❆1_s(x1,x2)	weight: max{0, (/ 3 4) + x2}	status: [x2]	precedence above: le le❆1_|0|
if_mod(x1,x2,x3)	weight: max{0, (/ 3 2) + x3, (/ 1 8) + x2}	status: []	precedence above: |0| s #le mod #le❆1_s
#if_minus(x1,x2,x3)	weight: max{0, (/ 1 8) + x3}	status: [x3]	precedence above:
le❆1_|0|(x1)	weight: (/ 5 8) + x1	status: []	precedence above:
#le❆1_|0|(x1)	weight: x1	status: []	precedence above:
#le❆1_s(x1,x2)	weight: x1	status: x1
#minus(x1,x2)	weight: max{0, x2}	status: []	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{(/ 1 8) + x3, (/ 1 8) + x2, (/ 1 8) + x1}	status: [x1,x2,x3]	precedence above:
if_minus(x1,x2,x3)	weight: max{(/ 5 4) + x3, x2, (/ 3 8) + x1}	status: [x3,x2]	precedence above: |0| s #le minus #le❆1_s
    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.
   |0|()	weight: 0	status: []	precedence above:
    le(x1,x2)	weight: max{0, (/ 3 8) + x2}	status: []	precedence above: false le❆1_s le❆1_|0| #minus
     s(x1)	weight: x1	status: [x1]	precedence above: |0| #minus
  #le(x1,x2)	weight: x1	status: x1
 minus(x1,x2)	weight: max{0, x1}	status: x1
   mod(x1,x2)	weight: max{(/ 1 2) + x2, x1}	status: [x1,x2]	precedence above: |0| s #le minus if_mod #le❆1_s #minus if_minus
 false()	weight: (/ 1 8)	status: []	precedence above: #minus
  true()	weight: (/ 1 8)	status: []	precedence above:
  rand(x1)	weight: (/ 1 8) + x1	status: []	precedence above:
le❆1_s(x1,x2)	weight: max{0, (/ 3 8) + x2}	status: []	precedence above: le false le❆1_|0| #minus
if_mod(x1,x2,x3)	weight: max{0, (/ 1 2) + x3, x2}	status: [x2,x3]	precedence above: |0| s #le minus mod #le❆1_s #minus if_minus
#if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
le❆1_|0|(x1)	weight: (/ 1 4)	status: []	precedence above:
#le❆1_|0|(x1)	weight: x1	status: []	precedence above:
#le❆1_s(x1,x2)	weight: x1	status: x1
#minus(x1,x2)	weight: max{0, x1}	status: [x1]	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{(/ 1 8) + x3, (/ 1 8) + x2, (/ 1 8) + x1}	status: [x1,x2,x3]	precedence above:
if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
    Removed DPs: #3 #6
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.
   |0|()	weight: 0	status: []	precedence above:
    le(x1,x2)	weight: max{0, (/ 1 2) + x2}	status: []	precedence above: |0| s #le minus le❆1_s le❆1_|0| #le❆1_s #minus #mod #if_mod
     s(x1)	weight: x1	status: [x1]	precedence above: |0| le #le minus le❆1_s le❆1_|0| #le❆1_s #minus #mod #if_mod
  #le(x1,x2)	weight: x1	status: x1
 minus(x1,x2)	weight: max{0, x1}	status: x1
   mod(x1,x2)	weight: max{(/ 5 8) + x2, (/ 1 8) + x1}	status: []	precedence above: |0| le s #le minus false le❆1_s if_mod le❆1_|0| #le❆1_s #minus #mod #if_mod if_minus
 false()	weight: (/ 1 8)	status: []	precedence above: #minus
  true()	weight: (/ 1 4)	status: []	precedence above: |0| le s #le minus le❆1_s le❆1_|0| #le❆1_s #minus #mod #if_mod
  rand(x1)	weight: (/ 1 8) + x1	status: []	precedence above:
le❆1_s(x1,x2)	weight: max{0, (/ 1 2) + x2}	status: []	precedence above: |0| le s #le minus le❆1_|0| #le❆1_s #minus #mod #if_mod
if_mod(x1,x2,x3)	weight: max{0, (/ 5 8) + x3, (/ 1 8) + x2}	status: []	precedence above: |0| le s #le minus mod false le❆1_s le❆1_|0| #le❆1_s #minus #mod #if_mod if_minus
#if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
le❆1_|0|(x1)	weight: (/ 3 8)	status: []	precedence above:
#le❆1_|0|(x1)	weight: x1	status: []	precedence above:
#le❆1_s(x1,x2)	weight: x1	status: x1
#minus(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above:
 #mod(x1,x2)	weight: max{0, (/ 5 8) + x1}	status: [x1]	precedence above: |0| le s #le minus le❆1_s le❆1_|0| #le❆1_s #minus #if_mod
#if_mod(x1,x2,x3)	weight: max{0, (/ 5 8) + x2, (/ 1 8) + x1}	status: [x2]	precedence above: |0| le s #le minus le❆1_s le❆1_|0| #le❆1_s #minus #mod
if_minus(x1,x2,x3)	weight: max{0, x2}	status: x2
    Removed DPs: #1
Number of SCCs: 0, DPs: 0, edges: 0
YES