Input TRS:
    1: eq(0(),0()) -> true()
    2: eq(0(),s(x)) -> false()
    3: eq(s(x),0()) -> false()
    4: eq(s(x),s(y)) -> eq(x,y)
    5: le(0(),y) -> true()
    6: le(s(x),0()) -> false()
    7: le(s(x),s(y)) -> le(x,y)
    8: app(nil(),y) -> y
    9: app(add(n,x),y) -> add(n,app(x,y))
    10: min(add(n,nil())) -> n
    11: min(add(n,add(m,x))) -> if_min(le(n,m),add(n,add(m,x)))
    12: if_min(true(),add(n,add(m,x))) -> min(add(n,x))
    13: if_min(false(),add(n,add(m,x))) -> min(add(m,x))
    14: rm(n,nil()) -> nil()
    15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
    16: if_rm(true(),n,add(m,x)) -> rm(n,x)
    17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
    18: minsort(nil(),nil()) -> nil()
    19: minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
    20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
    21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))
Number of strict rules: 21
Direct Order(PosReal,>,Poly) ... failed.
Freezing min
1: eq(0(),0()) -> true()
2: eq(0(),s(x)) -> false()
3: eq(s(x),0()) -> false()
4: eq(s(x),s(y)) -> eq(x,y)
5: le(0(),y) -> true()
6: le(s(x),0()) -> false()
7: le(s(x),s(y)) -> le(x,y)
8: app(nil(),y) -> y
9: app(add(n,x),y) -> add(n,app(x,y))
10: min❆1_add(n,nil()) -> n
11: min❆1_add(n,add(m,x)) -> if_min(le(n,m),add(n,add(m,x)))
12: if_min(true(),add(n,add(m,x))) -> min❆1_add(n,x)
13: if_min(false(),add(n,add(m,x))) -> min❆1_add(m,x)
14: rm(n,nil()) -> nil()
15: rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
16: if_rm(true(),n,add(m,x)) -> rm(n,x)
17: if_rm(false(),n,add(m,x)) -> add(m,rm(n,x))
18: minsort(nil(),nil()) -> nil()
19: minsort(add(n,x),y) -> if_minsort(eq(n,min❆1_add(n,x)),add(n,x),y)
20: if_minsort(true(),add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil()))
21: if_minsort(false(),add(n,x),y) -> minsort(x,add(n,y))
22: min(add(_1,_2)) ->= min❆1_add(_1,_2)
Number of strict rules: 21
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #if_min(false(),add(n,add(m,x))) -> #min❆1_add(m,x)
   #2: #app(add(n,x),y) -> #app(x,y)
   #3: #min❆1_add(n,add(m,x)) -> #if_min(le(n,m),add(n,add(m,x)))
   #4: #min❆1_add(n,add(m,x)) -> #le(n,m)
   #5: #if_min(true(),add(n,add(m,x))) -> #min❆1_add(n,x)
   #6: #if_minsort(true(),add(n,x),y) -> #minsort(app(rm(n,x),y),nil())
   #7: #if_minsort(true(),add(n,x),y) -> #app(rm(n,x),y)
   #8: #if_minsort(true(),add(n,x),y) -> #rm(n,x)
   #9: #le(s(x),s(y)) -> #le(x,y)
   #10: #min(add(_1,_2)) ->? #min❆1_add(_1,_2)
   #11: #if_rm(false(),n,add(m,x)) -> #rm(n,x)
   #12: #minsort(add(n,x),y) -> #if_minsort(eq(n,min❆1_add(n,x)),add(n,x),y)
   #13: #minsort(add(n,x),y) -> #eq(n,min❆1_add(n,x))
   #14: #minsort(add(n,x),y) -> #min❆1_add(n,x)
   #15: #if_minsort(false(),add(n,x),y) -> #minsort(x,add(n,y))
   #16: #if_rm(true(),n,add(m,x)) -> #rm(n,x)
   #17: #rm(n,add(m,x)) -> #if_rm(eq(n,m),n,add(m,x))
   #18: #rm(n,add(m,x)) -> #eq(n,m)
   #19: #eq(s(x),s(y)) -> #eq(x,y)
Number of SCCs: 6, DPs: 12, edges: 15
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
    le(x1,x2)	weight: 0
 if_rm(x1,x2,x3)	weight: 0
     s(x1)	weight: 0
  #le(x1,x2)	weight: 0
#if_rm(x1,x2,x3)	weight: 0
#if_min(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #min(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
min❆1_add(x1,x2)	weight: 0
     0()	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: x1
#if_minsort(x1,x2,x3)	weight: 0
   min(x1)	weight: 0
#minsort(x1,x2)	weight: 0
   add(x1,x2)	weight: (/ 1 2) + x2
if_min(x1,x2)	weight: 0
if_minsort(x1,x2,x3)	weight: 0
minsort(x1,x2)	weight: 0
#min❆1_add(x1,x2)	weight: 0
    rm(x1,x2)	weight: 0
  #rm(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #2
Number of SCCs: 5, DPs: 11, edges: 14
	SCC { #9 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
    le(x1,x2)	weight: 0
 if_rm(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
  #le(x1,x2)	weight: x2
#if_rm(x1,x2,x3)	weight: 0
#if_min(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #min(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
min❆1_add(x1,x2)	weight: 0
     0()	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
#if_minsort(x1,x2,x3)	weight: 0
   min(x1)	weight: 0
#minsort(x1,x2)	weight: 0
   add(x1,x2)	weight: (/ 1 2)
if_min(x1,x2)	weight: 0
if_minsort(x1,x2,x3)	weight: 0
minsort(x1,x2)	weight: 0
#min❆1_add(x1,x2)	weight: 0
    rm(x1,x2)	weight: 0
  #rm(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #9
Number of SCCs: 4, DPs: 10, edges: 13
	SCC { #19 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
    le(x1,x2)	weight: 0
 if_rm(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
  #le(x1,x2)	weight: 0
#if_rm(x1,x2,x3)	weight: 0
#if_min(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #min(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: x2
min❆1_add(x1,x2)	weight: 0
     0()	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
#if_minsort(x1,x2,x3)	weight: 0
   min(x1)	weight: 0
#minsort(x1,x2)	weight: 0
   add(x1,x2)	weight: (/ 1 2)
if_min(x1,x2)	weight: 0
if_minsort(x1,x2,x3)	weight: 0
minsort(x1,x2)	weight: 0
#min❆1_add(x1,x2)	weight: 0
    rm(x1,x2)	weight: 0
  #rm(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #19
Number of SCCs: 3, DPs: 9, edges: 12
	SCC { #11 #16 #17 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
    le(x1,x2)	weight: 0
 if_rm(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 4)
  #le(x1,x2)	weight: 0
#if_rm(x1,x2,x3)	weight: x3
#if_min(x1,x2)	weight: 0
    eq(x1,x2)	weight: (/ 1 4) + x1
 false()	weight: 0
 #min(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
min❆1_add(x1,x2)	weight: 0
     0()	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
#if_minsort(x1,x2,x3)	weight: 0
   min(x1)	weight: 0
#minsort(x1,x2)	weight: 0
   add(x1,x2)	weight: (/ 1 2) + x1 + x2
if_min(x1,x2)	weight: 0
if_minsort(x1,x2,x3)	weight: 0
minsort(x1,x2)	weight: 0
#min❆1_add(x1,x2)	weight: 0
    rm(x1,x2)	weight: 0
  #rm(x1,x2)	weight: (/ 1 4) + x2
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #11 #16 #17
Number of SCCs: 2, DPs: 6, edges: 8
	SCC { #1 #3 #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
    le(x1,x2)	weight: (/ 1 8)
 if_rm(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 8)
  #le(x1,x2)	weight: 0
#if_rm(x1,x2,x3)	weight: 0
#if_min(x1,x2)	weight: x2
    eq(x1,x2)	weight: (/ 1 8) + x1
 false()	weight: 0
 #min(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
min❆1_add(x1,x2)	weight: 0
     0()	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
#if_minsort(x1,x2,x3)	weight: 0
   min(x1)	weight: 0
#minsort(x1,x2)	weight: 0
   add(x1,x2)	weight: (/ 1 4) + x2
if_min(x1,x2)	weight: 0
if_minsort(x1,x2,x3)	weight: 0
minsort(x1,x2)	weight: 0
#min❆1_add(x1,x2)	weight: (/ 3 8) + x2
    rm(x1,x2)	weight: 0
  #rm(x1,x2)	weight: (/ 1 8)
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #1 #3 #5
Number of SCCs: 1, DPs: 3, edges: 4
	SCC { #6 #12 #15 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
    le(x1,x2)	weight: (/ 1 8)
 if_rm(x1,x2,x3)	weight: x3
     s(x1)	weight: (/ 1 8)
  #le(x1,x2)	weight: 0
#if_rm(x1,x2,x3)	weight: 0
#if_min(x1,x2)	weight: 0
    eq(x1,x2)	weight: (/ 1 8) + x1
 false()	weight: 0
 #min(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
min❆1_add(x1,x2)	weight: (/ 1 8)
     0()	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
#if_minsort(x1,x2,x3)	weight: x2 + x3
   min(x1)	weight: 0
#minsort(x1,x2)	weight: x1 + x2
   add(x1,x2)	weight: (/ 1 4) + x1 + x2
if_min(x1,x2)	weight: (/ 1 4)
if_minsort(x1,x2,x3)	weight: 0
minsort(x1,x2)	weight: 0
#min❆1_add(x1,x2)	weight: (/ 3 8)
    rm(x1,x2)	weight: x2
  #rm(x1,x2)	weight: (/ 1 8)
   app(x1,x2)	weight: (/ 1 8) + x1 + x2
    Usable rules: { 8 9 14..17 }
    Removed DPs: #6
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #12 #15 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
    le(x1,x2)	weight: (/ 1 8)
 if_rm(x1,x2,x3)	weight: x3
     s(x1)	weight: (/ 1 8)
  #le(x1,x2)	weight: 0
#if_rm(x1,x2,x3)	weight: 0
#if_min(x1,x2)	weight: 0
    eq(x1,x2)	weight: (/ 1 8) + x1
 false()	weight: 0
 #min(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
min❆1_add(x1,x2)	weight: (/ 1 8)
     0()	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
#if_minsort(x1,x2,x3)	weight: x2
   min(x1)	weight: 0
#minsort(x1,x2)	weight: (/ 1 8) + x1
   add(x1,x2)	weight: (/ 1 4) + x1 + x2
if_min(x1,x2)	weight: (/ 1 4)
if_minsort(x1,x2,x3)	weight: 0
minsort(x1,x2)	weight: 0
#min❆1_add(x1,x2)	weight: (/ 3 8)
    rm(x1,x2)	weight: x2
  #rm(x1,x2)	weight: (/ 1 8)
   app(x1,x2)	weight: (/ 1 8) + x1 + x2
    Usable rules: { }
    Removed DPs: #12 #15
Number of SCCs: 0, DPs: 0, edges: 0
YES