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: or(true(),y) -> true()
    6: or(false(),y) -> y
    7: union(empty(),h) -> h
    8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
    9: reach(x,y,empty(),h) -> false()
    10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
    11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
    12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
    13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
    14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
Number of strict rules: 14
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #if_reach_2(false(),x,y,edge(u,v,i),h) -> #or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
   #2: #if_reach_2(false(),x,y,edge(u,v,i),h) -> #reach(x,y,i,h)
   #3: #if_reach_2(false(),x,y,edge(u,v,i),h) -> #reach(v,y,union(i,h),empty())
   #4: #if_reach_2(false(),x,y,edge(u,v,i),h) -> #union(i,h)
   #5: #if_reach_1(true(),x,y,edge(u,v,i),h) -> #if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
   #6: #if_reach_1(true(),x,y,edge(u,v,i),h) -> #eq(y,v)
   #7: #if_reach_1(false(),x,y,edge(u,v,i),h) -> #reach(x,y,i,edge(u,v,h))
   #8: #reach(x,y,edge(u,v,i),h) -> #if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
   #9: #reach(x,y,edge(u,v,i),h) -> #eq(x,u)
   #10: #union(edge(x,y,i),h) -> #union(i,h)
   #11: #eq(s(x),s(y)) -> #eq(x,y)
Number of SCCs: 3, DPs: 7, edges: 9
	SCC { #11 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
#if_reach_1(x1,x2,x3,x4,x5)	weight: 0
  edge(x1,x2,x3)	weight: 0
#if_reach_2(x1,x2,x3,x4,x5)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 reach(x1,x2,x3,x4)	weight: 0
if_reach_2(x1,x2,x3,x4,x5)	weight: 0
  true()	weight: 0
#reach(x1,x2,x3,x4)	weight: 0
  #eq(x1,x2)	weight: x2
     0()	weight: 0
 union(x1,x2)	weight: 0
    or(x1,x2)	weight: 0
 empty()	weight: 0
if_reach_1(x1,x2,x3,x4,x5)	weight: 0
  #or(x1,x2)	weight: 0
#union(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #11
Number of SCCs: 2, DPs: 6, edges: 8
	SCC { #10 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2)
#if_reach_1(x1,x2,x3,x4,x5)	weight: 0
  edge(x1,x2,x3)	weight: (/ 1 2) + x3
#if_reach_2(x1,x2,x3,x4,x5)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 reach(x1,x2,x3,x4)	weight: 0
if_reach_2(x1,x2,x3,x4,x5)	weight: 0
  true()	weight: 0
#reach(x1,x2,x3,x4)	weight: 0
  #eq(x1,x2)	weight: 0
     0()	weight: 0
 union(x1,x2)	weight: 0
    or(x1,x2)	weight: 0
 empty()	weight: 0
if_reach_1(x1,x2,x3,x4,x5)	weight: 0
  #or(x1,x2)	weight: 0
#union(x1,x2)	weight: x1
    Usable rules: { }
    Removed DPs: #10
Number of SCCs: 1, DPs: 5, edges: 7
	SCC { #2 #3 #5 #7 #8 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 4)
#if_reach_1(x1,x2,x3,x4,x5)	weight: (/ 1 4) + x4 + x5
  edge(x1,x2,x3)	weight: (/ 3 4) + x1 + x2 + x3
#if_reach_2(x1,x2,x3,x4,x5)	weight: x4 + x5
    eq(x1,x2)	weight: (/ 1 4) + x1
 false()	weight: 0
 reach(x1,x2,x3,x4)	weight: 0
if_reach_2(x1,x2,x3,x4,x5)	weight: 0
  true()	weight: 0
#reach(x1,x2,x3,x4)	weight: (/ 1 4) + x3 + x4
  #eq(x1,x2)	weight: 0
     0()	weight: 0
 union(x1,x2)	weight: (/ 1 4) + x1 + x2
    or(x1,x2)	weight: 0
 empty()	weight: 0
if_reach_1(x1,x2,x3,x4,x5)	weight: 0
  #or(x1,x2)	weight: 0
#union(x1,x2)	weight: 0
    Usable rules: { 7 8 }
    Removed DPs: #2 #3 #5
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #7 #8 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 4)
#if_reach_1(x1,x2,x3,x4,x5)	weight: (/ 1 4) + x4
  edge(x1,x2,x3)	weight: (/ 1 2) + x1 + x2 + x3
#if_reach_2(x1,x2,x3,x4,x5)	weight: 0
    eq(x1,x2)	weight: (/ 1 4)
 false()	weight: 0
 reach(x1,x2,x3,x4)	weight: 0
if_reach_2(x1,x2,x3,x4,x5)	weight: 0
  true()	weight: 0
#reach(x1,x2,x3,x4)	weight: (/ 1 2) + x3
  #eq(x1,x2)	weight: 0
     0()	weight: 0
 union(x1,x2)	weight: (/ 1 4) + x1 + x2
    or(x1,x2)	weight: 0
 empty()	weight: 0
if_reach_1(x1,x2,x3,x4,x5)	weight: 0
  #or(x1,x2)	weight: 0
#union(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #7 #8
Number of SCCs: 0, DPs: 0, edges: 0
YES