YES

Problem:
 eq(0(),0()) -> true()
 eq(0(),s(x)) -> false()
 eq(s(x),0()) -> false()
 eq(s(x),s(y)) -> eq(x,y)
 or(true(),y) -> true()
 or(false(),y) -> y
 union(empty(),h) -> h
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 reach(x,y,empty(),h) -> false()
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
 if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
 if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

Proof:
 DP Processor:
  DPs:
   eq#(s(x),s(y)) -> eq#(x,y)
   union#(edge(x,y,i),h) -> union#(i,h)
   reach#(x,y,edge(u,v,i),h) -> eq#(x,u)
   reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
   if_reach_1#(true(),x,y,edge(u,v,i),h) -> eq#(y,v)
   if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
   if_reach_2#(false(),x,y,edge(u,v,i),h) -> union#(i,h)
   if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(v,y,union(i,h),empty())
   if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h)
   if_reach_2#(false(),x,y,edge(u,v,i),h) -> or#(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
   if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))
  TRS:
   eq(0(),0()) -> true()
   eq(0(),s(x)) -> false()
   eq(s(x),0()) -> false()
   eq(s(x),s(y)) -> eq(x,y)
   or(true(),y) -> true()
   or(false(),y) -> y
   union(empty(),h) -> h
   union(edge(x,y,i),h) -> edge(x,y,union(i,h))
   reach(x,y,empty(),h) -> false()
   reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
   if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
   if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
   if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
   if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
  TDG Processor:
   DPs:
    eq#(s(x),s(y)) -> eq#(x,y)
    union#(edge(x,y,i),h) -> union#(i,h)
    reach#(x,y,edge(u,v,i),h) -> eq#(x,u)
    reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
    if_reach_1#(true(),x,y,edge(u,v,i),h) -> eq#(y,v)
    if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> union#(i,h)
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(v,y,union(i,h),empty())
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h)
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> or#(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
    if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))
   TRS:
    eq(0(),0()) -> true()
    eq(0(),s(x)) -> false()
    eq(s(x),0()) -> false()
    eq(s(x),s(y)) -> eq(x,y)
    or(true(),y) -> true()
    or(false(),y) -> y
    union(empty(),h) -> h
    union(edge(x,y,i),h) -> edge(x,y,union(i,h))
    reach(x,y,empty(),h) -> false()
    reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
    if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
    if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
    if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
    if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
   graph:
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h) ->
    reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h) ->
    reach#(x,y,edge(u,v,i),h) -> eq#(x,u)
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(v,y,union(i,h),empty()) ->
    reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(v,y,union(i,h),empty()) ->
    reach#(x,y,edge(u,v,i),h) -> eq#(x,u)
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> union#(i,h) ->
    union#(edge(x,y,i),h) -> union#(i,h)
    if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h)) ->
    reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
    if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h)) ->
    reach#(x,y,edge(u,v,i),h) -> eq#(x,u)
    if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h) ->
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> or#(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
    if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h) ->
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h)
    if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h) ->
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(v,y,union(i,h),empty())
    if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h) ->
    if_reach_2#(false(),x,y,edge(u,v,i),h) -> union#(i,h)
    if_reach_1#(true(),x,y,edge(u,v,i),h) -> eq#(y,v) ->
    eq#(s(x),s(y)) -> eq#(x,y)
    reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h) ->
    if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))
    reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h) ->
    if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
    reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h) ->
    if_reach_1#(true(),x,y,edge(u,v,i),h) -> eq#(y,v)
    reach#(x,y,edge(u,v,i),h) -> eq#(x,u) ->
    eq#(s(x),s(y)) -> eq#(x,y)
    union#(edge(x,y,i),h) -> union#(i,h) ->
    union#(edge(x,y,i),h) -> union#(i,h)
    eq#(s(x),s(y)) -> eq#(x,y) -> eq#(s(x),s(y)) -> eq#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 7
    #arcs: 18/121
    DPs:
     if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h)
     reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
     if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
     if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(v,y,union(i,h),empty())
     if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))
    TRS:
     eq(0(),0()) -> true()
     eq(0(),s(x)) -> false()
     eq(s(x),0()) -> false()
     eq(s(x),s(y)) -> eq(x,y)
     or(true(),y) -> true()
     or(false(),y) -> y
     union(empty(),h) -> h
     union(edge(x,y,i),h) -> edge(x,y,union(i,h))
     reach(x,y,empty(),h) -> false()
     reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
     if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
     if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
     if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
     if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [if_reach_2#](x0, x1, x2, x3, x4) = 7x1 + 4x2 + x3 + x4,
      
      [if_reach_1#](x0, x1, x2, x3, x4) = 7x1 + 4x2 + x3 + x4,
      
      [reach#](x0, x1, x2, x3) = 7x0 + 4x1 + x2 + x3,
      
      [if_reach_2](x0, x1, x2, x3, x4) = 0,
      
      [if_reach_1](x0, x1, x2, x3, x4) = 0,
      
      [reach](x0, x1, x2, x3) = 0,
      
      [edge](x0, x1, x2) = 2x0 + 7x1 + x2 + 1,
      
      [union](x0, x1) = x0 + x1,
      
      [empty] = 1,
      
      [or](x0, x1) = 4x1,
      
      [false] = 0,
      
      [s](x0) = 0,
      
      [true] = 0,
      
      [eq](x0, x1) = 0,
      
      [0] = 0
     orientation:
      if_reach_2#(false(),x,y,edge(u,v,i),h) = h + i + 2u + 7v + 7x + 4y + 1 >= h + i + 7x + 4y = reach#(x,y,i,h)
      
      reach#(x,y,edge(u,v,i),h) = h + i + 2u + 7v + 7x + 4y + 1 >= h + i + 2u + 7v + 7x + 4y + 1 = if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
      
      if_reach_1#(true(),x,y,edge(u,v,i),h) = h + i + 2u + 7v + 7x + 4y + 1 >= h + i + 2u + 7v + 7x + 4y + 1 = if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
      
      if_reach_2#(false(),x,y,edge(u,v,i),h) = h + i + 2u + 7v + 7x + 4y + 1 >= h + i + 7v + 4y + 1 = reach#(v,y,union(i,h),empty())
      
      if_reach_1#(false(),x,y,edge(u,v,i),h) = h + i + 2u + 7v + 7x + 4y + 1 >= h + i + 2u + 7v + 7x + 4y + 1 = reach#(x,y,i,edge(u,v,h))
      
      eq(0(),0()) = 0 >= 0 = true()
      
      eq(0(),s(x)) = 0 >= 0 = false()
      
      eq(s(x),0()) = 0 >= 0 = false()
      
      eq(s(x),s(y)) = 0 >= 0 = eq(x,y)
      
      or(true(),y) = 4y >= 0 = true()
      
      or(false(),y) = 4y >= y = y
      
      union(empty(),h) = h + 1 >= h = h
      
      union(edge(x,y,i),h) = h + i + 2x + 7y + 1 >= h + i + 2x + 7y + 1 = edge(x,y,union(i,h))
      
      reach(x,y,empty(),h) = 0 >= 0 = false()
      
      reach(x,y,edge(u,v,i),h) = 0 >= 0 = if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
      
      if_reach_1(true(),x,y,edge(u,v,i),h) = 0 >= 0 = if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
      
      if_reach_2(true(),x,y,edge(u,v,i),h) = 0 >= 0 = true()
      
      if_reach_2(false(),x,y,edge(u,v,i),h) = 0 >= 0 = or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
      
      if_reach_1(false(),x,y,edge(u,v,i),h) = 0 >= 0 = reach(x,y,i,edge(u,v,h))
     problem:
      DPs:
       reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
       if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
       if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(v,y,union(i,h),empty())
       if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))
      TRS:
       eq(0(),0()) -> true()
       eq(0(),s(x)) -> false()
       eq(s(x),0()) -> false()
       eq(s(x),s(y)) -> eq(x,y)
       or(true(),y) -> true()
       or(false(),y) -> y
       union(empty(),h) -> h
       union(edge(x,y,i),h) -> edge(x,y,union(i,h))
       reach(x,y,empty(),h) -> false()
       reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
       if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
       if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
       if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
       if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [if_reach_2#](x0, x1, x2, x3, x4) = x2 + 3x3 + 3x4,
       
       [if_reach_1#](x0, x1, x2, x3, x4) = 2x1 + x2 + 3x3 + 3x4,
       
       [reach#](x0, x1, x2, x3) = 2x0 + x1 + 3x2 + 3x3,
       
       [if_reach_2](x0, x1, x2, x3, x4) = 0,
       
       [if_reach_1](x0, x1, x2, x3, x4) = 0,
       
       [reach](x0, x1, x2, x3) = 0,
       
       [edge](x0, x1, x2) = 2x1 + x2 + 3,
       
       [union](x0, x1) = x0 + x1,
       
       [empty] = 0,
       
       [or](x0, x1) = 2x1,
       
       [false] = 0,
       
       [s](x0) = 0,
       
       [true] = 0,
       
       [eq](x0, x1) = 0,
       
       [0] = 0
      orientation:
       reach#(x,y,edge(u,v,i),h) = 3h + 3i + 6v + 2x + y + 9 >= 3h + 3i + 6v + 2x + y + 9 = if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
       
       if_reach_1#(true(),x,y,edge(u,v,i),h) = 3h + 3i + 6v + 2x + y + 9 >= 3h + 3i + 6v + y + 9 = if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
       
       if_reach_2#(false(),x,y,edge(u,v,i),h) = 3h + 3i + 6v + y + 9 >= 3h + 3i + 2v + y = reach#(v,y,union(i,h),empty())
       
       if_reach_1#(false(),x,y,edge(u,v,i),h) = 3h + 3i + 6v + 2x + y + 9 >= 3h + 3i + 6v + 2x + y + 9 = reach#(x,y,i,edge(u,v,h))
       
       eq(0(),0()) = 0 >= 0 = true()
       
       eq(0(),s(x)) = 0 >= 0 = false()
       
       eq(s(x),0()) = 0 >= 0 = false()
       
       eq(s(x),s(y)) = 0 >= 0 = eq(x,y)
       
       or(true(),y) = 2y >= 0 = true()
       
       or(false(),y) = 2y >= y = y
       
       union(empty(),h) = h >= h = h
       
       union(edge(x,y,i),h) = h + i + 2y + 3 >= h + i + 2y + 3 = edge(x,y,union(i,h))
       
       reach(x,y,empty(),h) = 0 >= 0 = false()
       
       reach(x,y,edge(u,v,i),h) = 0 >= 0 = if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
       
       if_reach_1(true(),x,y,edge(u,v,i),h) = 0 >= 0 = if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
       
       if_reach_2(true(),x,y,edge(u,v,i),h) = 0 >= 0 = true()
       
       if_reach_2(false(),x,y,edge(u,v,i),h) = 0 >= 0 = or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
       
       if_reach_1(false(),x,y,edge(u,v,i),h) = 0 >= 0 = reach(x,y,i,edge(u,v,h))
      problem:
       DPs:
        reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
        if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
        if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))
       TRS:
        eq(0(),0()) -> true()
        eq(0(),s(x)) -> false()
        eq(s(x),0()) -> false()
        eq(s(x),s(y)) -> eq(x,y)
        or(true(),y) -> true()
        or(false(),y) -> y
        union(empty(),h) -> h
        union(edge(x,y,i),h) -> edge(x,y,union(i,h))
        reach(x,y,empty(),h) -> false()
        reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
        if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
        if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
        if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
        if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
      SCC Processor:
       #sccs: 1
       #rules: 2
       #arcs: 7/9
       DPs:
        if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))
        reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
       TRS:
        eq(0(),0()) -> true()
        eq(0(),s(x)) -> false()
        eq(s(x),0()) -> false()
        eq(s(x),s(y)) -> eq(x,y)
        or(true(),y) -> true()
        or(false(),y) -> y
        union(empty(),h) -> h
        union(edge(x,y,i),h) -> edge(x,y,union(i,h))
        reach(x,y,empty(),h) -> false()
        reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
        if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
        if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
        if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
        if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
       Subterm Criterion Processor:
        simple projection:
         pi(reach#) = 2
         pi(if_reach_1#) = 3
        problem:
         DPs:
          reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
         TRS:
          eq(0(),0()) -> true()
          eq(0(),s(x)) -> false()
          eq(s(x),0()) -> false()
          eq(s(x),s(y)) -> eq(x,y)
          or(true(),y) -> true()
          or(false(),y) -> y
          union(empty(),h) -> h
          union(edge(x,y,i),h) -> edge(x,y,union(i,h))
          reach(x,y,empty(),h) -> false()
          reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
          if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
          if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
          if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
          if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 2/1
         
    
    DPs:
     union#(edge(x,y,i),h) -> union#(i,h)
    TRS:
     eq(0(),0()) -> true()
     eq(0(),s(x)) -> false()
     eq(s(x),0()) -> false()
     eq(s(x),s(y)) -> eq(x,y)
     or(true(),y) -> true()
     or(false(),y) -> y
     union(empty(),h) -> h
     union(edge(x,y,i),h) -> edge(x,y,union(i,h))
     reach(x,y,empty(),h) -> false()
     reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
     if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
     if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
     if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
     if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
    Subterm Criterion Processor:
     simple projection:
      pi(union#) = 0
     problem:
      DPs:
       
      TRS:
       eq(0(),0()) -> true()
       eq(0(),s(x)) -> false()
       eq(s(x),0()) -> false()
       eq(s(x),s(y)) -> eq(x,y)
       or(true(),y) -> true()
       or(false(),y) -> y
       union(empty(),h) -> h
       union(edge(x,y,i),h) -> edge(x,y,union(i,h))
       reach(x,y,empty(),h) -> false()
       reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
       if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
       if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
       if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
       if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
     Qed
    
    DPs:
     eq#(s(x),s(y)) -> eq#(x,y)
    TRS:
     eq(0(),0()) -> true()
     eq(0(),s(x)) -> false()
     eq(s(x),0()) -> false()
     eq(s(x),s(y)) -> eq(x,y)
     or(true(),y) -> true()
     or(false(),y) -> y
     union(empty(),h) -> h
     union(edge(x,y,i),h) -> edge(x,y,union(i,h))
     reach(x,y,empty(),h) -> false()
     reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
     if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
     if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
     if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
     if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
    Subterm Criterion Processor:
     simple projection:
      pi(eq#) = 1
     problem:
      DPs:
       
      TRS:
       eq(0(),0()) -> true()
       eq(0(),s(x)) -> false()
       eq(s(x),0()) -> false()
       eq(s(x),s(y)) -> eq(x,y)
       or(true(),y) -> true()
       or(false(),y) -> y
       union(empty(),h) -> h
       union(edge(x,y,i),h) -> edge(x,y,union(i,h))
       reach(x,y,empty(),h) -> false()
       reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
       if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
       if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
       if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
       if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
     Qed