MAYBE

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))
 isEmpty(empty()) -> true()
 isEmpty(edge(x,y,i)) -> false()
 from(edge(x,y,i)) -> x
 to(edge(x,y,i)) -> y
 rest(edge(x,y,i)) -> i
 rest(empty()) -> empty()
 reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
 if1(true(),b1,b2,b3,x,y,i,h) -> true()
 if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h)
 if2(true(),b2,b3,x,y,i,h) -> false()
 if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h)
 if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h))
 if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h)
 if4(true(),x,y,i,h) -> true()
 if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))

Proof:
 DP Processor:
  DPs:
   eq#(s(x),s(y)) -> eq#(x,y)
   union#(edge(x,y,i),h) -> union#(i,h)
   reach#(x,y,i,h) -> to#(i)
   reach#(x,y,i,h) -> eq#(y,to(i))
   reach#(x,y,i,h) -> from#(i)
   reach#(x,y,i,h) -> eq#(x,from(i))
   reach#(x,y,i,h) -> isEmpty#(i)
   reach#(x,y,i,h) -> eq#(x,y)
   reach#(x,y,i,h) -> if1#(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
   if1#(false(),b1,b2,b3,x,y,i,h) -> if2#(b1,b2,b3,x,y,i,h)
   if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h)
   if3#(false(),b3,x,y,i,h) -> to#(i)
   if3#(false(),b3,x,y,i,h) -> from#(i)
   if3#(false(),b3,x,y,i,h) -> rest#(i)
   if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h))
   if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h)
   if4#(false(),x,y,i,h) -> union#(rest(i),h)
   if4#(false(),x,y,i,h) -> to#(i)
   if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty())
   if4#(false(),x,y,i,h) -> rest#(i)
   if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h)
   if4#(false(),x,y,i,h) -> or#(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
  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))
   isEmpty(empty()) -> true()
   isEmpty(edge(x,y,i)) -> false()
   from(edge(x,y,i)) -> x
   to(edge(x,y,i)) -> y
   rest(edge(x,y,i)) -> i
   rest(empty()) -> empty()
   reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
   if1(true(),b1,b2,b3,x,y,i,h) -> true()
   if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h)
   if2(true(),b2,b3,x,y,i,h) -> false()
   if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h)
   if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h))
   if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h)
   if4(true(),x,y,i,h) -> true()
   if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
  TDG Processor:
   DPs:
    eq#(s(x),s(y)) -> eq#(x,y)
    union#(edge(x,y,i),h) -> union#(i,h)
    reach#(x,y,i,h) -> to#(i)
    reach#(x,y,i,h) -> eq#(y,to(i))
    reach#(x,y,i,h) -> from#(i)
    reach#(x,y,i,h) -> eq#(x,from(i))
    reach#(x,y,i,h) -> isEmpty#(i)
    reach#(x,y,i,h) -> eq#(x,y)
    reach#(x,y,i,h) -> if1#(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
    if1#(false(),b1,b2,b3,x,y,i,h) -> if2#(b1,b2,b3,x,y,i,h)
    if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h)
    if3#(false(),b3,x,y,i,h) -> to#(i)
    if3#(false(),b3,x,y,i,h) -> from#(i)
    if3#(false(),b3,x,y,i,h) -> rest#(i)
    if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h))
    if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h)
    if4#(false(),x,y,i,h) -> union#(rest(i),h)
    if4#(false(),x,y,i,h) -> to#(i)
    if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty())
    if4#(false(),x,y,i,h) -> rest#(i)
    if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h)
    if4#(false(),x,y,i,h) -> or#(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
   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))
    isEmpty(empty()) -> true()
    isEmpty(edge(x,y,i)) -> false()
    from(edge(x,y,i)) -> x
    to(edge(x,y,i)) -> y
    rest(edge(x,y,i)) -> i
    rest(empty()) -> empty()
    reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
    if1(true(),b1,b2,b3,x,y,i,h) -> true()
    if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h)
    if2(true(),b2,b3,x,y,i,h) -> false()
    if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h)
    if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h))
    if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h)
    if4(true(),x,y,i,h) -> true()
    if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
   graph:
    if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty()) ->
    reach#(x,y,i,h) -> if1#(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
    if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty()) ->
    reach#(x,y,i,h) -> eq#(x,y)
    if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty()) ->
    reach#(x,y,i,h) -> isEmpty#(i)
    if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty()) ->
    reach#(x,y,i,h) -> eq#(x,from(i))
    if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty()) ->
    reach#(x,y,i,h) -> from#(i)
    if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty()) ->
    reach#(x,y,i,h) -> eq#(y,to(i))
    if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty()) ->
    reach#(x,y,i,h) -> to#(i)
    if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h) ->
    reach#(x,y,i,h) -> if1#(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
    if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h) ->
    reach#(x,y,i,h) -> eq#(x,y)
    if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h) ->
    reach#(x,y,i,h) -> isEmpty#(i)
    if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h) ->
    reach#(x,y,i,h) -> eq#(x,from(i))
    if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h) ->
    reach#(x,y,i,h) -> from#(i)
    if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h) ->
    reach#(x,y,i,h) -> eq#(y,to(i))
    if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h) ->
    reach#(x,y,i,h) -> to#(i)
    if4#(false(),x,y,i,h) -> union#(rest(i),h) ->
    union#(edge(x,y,i),h) -> union#(i,h)
    if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h)) ->
    reach#(x,y,i,h) -> if1#(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
    if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h)) ->
    reach#(x,y,i,h) -> eq#(x,y)
    if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h)) ->
    reach#(x,y,i,h) -> isEmpty#(i)
    if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h)) ->
    reach#(x,y,i,h) -> eq#(x,from(i))
    if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h)) ->
    reach#(x,y,i,h) -> from#(i)
    if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h)) ->
    reach#(x,y,i,h) -> eq#(y,to(i))
    if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h)) ->
    reach#(x,y,i,h) -> to#(i)
    if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h) ->
    if4#(false(),x,y,i,h) -> or#(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
    if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h) ->
    if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h)
    if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h) ->
    if4#(false(),x,y,i,h) -> rest#(i)
    if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h) ->
    if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty())
    if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h) ->
    if4#(false(),x,y,i,h) -> to#(i)
    if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h) ->
    if4#(false(),x,y,i,h) -> union#(rest(i),h)
    if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h) ->
    if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h)
    if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h) ->
    if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h))
    if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h) ->
    if3#(false(),b3,x,y,i,h) -> rest#(i)
    if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h) ->
    if3#(false(),b3,x,y,i,h) -> from#(i)
    if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h) ->
    if3#(false(),b3,x,y,i,h) -> to#(i)
    if1#(false(),b1,b2,b3,x,y,i,h) -> if2#(b1,b2,b3,x,y,i,h) ->
    if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h)
    reach#(x,y,i,h) -> if1#(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h) ->
    if1#(false(),b1,b2,b3,x,y,i,h) -> if2#(b1,b2,b3,x,y,i,h)
    reach#(x,y,i,h) -> eq#(y,to(i)) -> eq#(s(x),s(y)) -> eq#(x,y)
    reach#(x,y,i,h) -> eq#(x,from(i)) -> eq#(s(x),s(y)) -> eq#(x,y)
    reach#(x,y,i,h) -> eq#(x,y) -> 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)
   Restore Modifier:
    DPs:
     eq#(s(x),s(y)) -> eq#(x,y)
     union#(edge(x,y,i),h) -> union#(i,h)
     reach#(x,y,i,h) -> to#(i)
     reach#(x,y,i,h) -> eq#(y,to(i))
     reach#(x,y,i,h) -> from#(i)
     reach#(x,y,i,h) -> eq#(x,from(i))
     reach#(x,y,i,h) -> isEmpty#(i)
     reach#(x,y,i,h) -> eq#(x,y)
     reach#(x,y,i,h) -> if1#(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
     if1#(false(),b1,b2,b3,x,y,i,h) -> if2#(b1,b2,b3,x,y,i,h)
     if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h)
     if3#(false(),b3,x,y,i,h) -> to#(i)
     if3#(false(),b3,x,y,i,h) -> from#(i)
     if3#(false(),b3,x,y,i,h) -> rest#(i)
     if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h))
     if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h)
     if4#(false(),x,y,i,h) -> union#(rest(i),h)
     if4#(false(),x,y,i,h) -> to#(i)
     if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty())
     if4#(false(),x,y,i,h) -> rest#(i)
     if4#(false(),x,y,i,h) -> reach#(x,y,rest(i),h)
     if4#(false(),x,y,i,h) -> or#(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
    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))
     isEmpty(empty()) -> true()
     isEmpty(edge(x,y,i)) -> false()
     from(edge(x,y,i)) -> x
     to(edge(x,y,i)) -> y
     rest(edge(x,y,i)) -> i
     rest(empty()) -> empty()
     reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
     if1(true(),b1,b2,b3,x,y,i,h) -> true()
     if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h)
     if2(true(),b2,b3,x,y,i,h) -> false()
     if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h)
     if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h))
     if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h)
     if4(true(),x,y,i,h) -> true()
     if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
    SCC Processor:
     #sccs: 3
     #rules: 9
     #arcs: 40/484
     DPs:
      if4#(false(),x,y,i,h) -> reach#(to(i),y,union(rest(i),h),empty())
      reach#(x,y,i,h) -> if1#(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
      if1#(false(),b1,b2,b3,x,y,i,h) -> if2#(b1,b2,b3,x,y,i,h)
      if2#(false(),b2,b3,x,y,i,h) -> if3#(b2,b3,x,y,i,h)
      if3#(false(),b3,x,y,i,h) -> reach#(x,y,rest(i),edge(from(i),to(i),h))
      if3#(true(),b3,x,y,i,h) -> if4#(b3,x,y,i,h)
      if4#(false(),x,y,i,h) -> reach#(x,y,rest(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))
      isEmpty(empty()) -> true()
      isEmpty(edge(x,y,i)) -> false()
      from(edge(x,y,i)) -> x
      to(edge(x,y,i)) -> y
      rest(edge(x,y,i)) -> i
      rest(empty()) -> empty()
      reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
      if1(true(),b1,b2,b3,x,y,i,h) -> true()
      if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h)
      if2(true(),b2,b3,x,y,i,h) -> false()
      if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h)
      if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h))
      if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h)
      if4(true(),x,y,i,h) -> true()
      if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
     Open
     
     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))
      isEmpty(empty()) -> true()
      isEmpty(edge(x,y,i)) -> false()
      from(edge(x,y,i)) -> x
      to(edge(x,y,i)) -> y
      rest(edge(x,y,i)) -> i
      rest(empty()) -> empty()
      reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
      if1(true(),b1,b2,b3,x,y,i,h) -> true()
      if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h)
      if2(true(),b2,b3,x,y,i,h) -> false()
      if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h)
      if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h))
      if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h)
      if4(true(),x,y,i,h) -> true()
      if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [union#](x0, x1) = x0 + 1,
       
       [if4](x0, x1, x2, x3, x4) = 0,
       
       [if3](x0, x1, x2, x3, x4, x5) = 0,
       
       [if2](x0, x1, x2, x3, x4, x5, x6) = 0,
       
       [if1](x0, x1, x2, x3, x4, x5, x6, x7) = 0,
       
       [reach](x0, x1, x2, x3) = 0,
       
       [rest](x0) = x0,
       
       [to](x0) = x0,
       
       [from](x0) = x0 + 1,
       
       [isEmpty](x0) = 0,
       
       [edge](x0, x1, x2) = x0 + x1 + x2 + 1,
       
       [union](x0, x1) = x0 + x1,
       
       [empty] = 0,
       
       [or](x0, x1) = x1,
       
       [false] = 0,
       
       [s](x0) = 0,
       
       [true] = 0,
       
       [eq](x0, x1) = 0,
       
       [0] = 0
      orientation:
       union#(edge(x,y,i),h) = i + x + y + 2 >= i + 1 = union#(i,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) = y >= 0 = true()
       
       or(false(),y) = y >= y = y
       
       union(empty(),h) = h >= h = h
       
       union(edge(x,y,i),h) = h + i + x + y + 1 >= h + i + x + y + 1 = edge(x,y,union(i,h))
       
       isEmpty(empty()) = 0 >= 0 = true()
       
       isEmpty(edge(x,y,i)) = 0 >= 0 = false()
       
       from(edge(x,y,i)) = i + x + y + 2 >= x = x
       
       to(edge(x,y,i)) = i + x + y + 1 >= y = y
       
       rest(edge(x,y,i)) = i + x + y + 1 >= i = i
       
       rest(empty()) = 0 >= 0 = empty()
       
       reach(x,y,i,h) = 0 >= 0 = if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
       
       if1(true(),b1,b2,b3,x,y,i,h) = 0 >= 0 = true()
       
       if1(false(),b1,b2,b3,x,y,i,h) = 0 >= 0 = if2(b1,b2,b3,x,y,i,h)
       
       if2(true(),b2,b3,x,y,i,h) = 0 >= 0 = false()
       
       if2(false(),b2,b3,x,y,i,h) = 0 >= 0 = if3(b2,b3,x,y,i,h)
       
       if3(false(),b3,x,y,i,h) = 0 >= 0 = reach(x,y,rest(i),edge(from(i),to(i),h))
       
       if3(true(),b3,x,y,i,h) = 0 >= 0 = if4(b3,x,y,i,h)
       
       if4(true(),x,y,i,h) = 0 >= 0 = true()
       
       if4(false(),x,y,i,h) = 0 >= 0 = or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
      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))
        isEmpty(empty()) -> true()
        isEmpty(edge(x,y,i)) -> false()
        from(edge(x,y,i)) -> x
        to(edge(x,y,i)) -> y
        rest(edge(x,y,i)) -> i
        rest(empty()) -> empty()
        reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
        if1(true(),b1,b2,b3,x,y,i,h) -> true()
        if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h)
        if2(true(),b2,b3,x,y,i,h) -> false()
        if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h)
        if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h))
        if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h)
        if4(true(),x,y,i,h) -> true()
        if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
      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))
      isEmpty(empty()) -> true()
      isEmpty(edge(x,y,i)) -> false()
      from(edge(x,y,i)) -> x
      to(edge(x,y,i)) -> y
      rest(edge(x,y,i)) -> i
      rest(empty()) -> empty()
      reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
      if1(true(),b1,b2,b3,x,y,i,h) -> true()
      if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h)
      if2(true(),b2,b3,x,y,i,h) -> false()
      if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h)
      if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h))
      if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h)
      if4(true(),x,y,i,h) -> true()
      if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [eq#](x0, x1) = x1 + 1,
       
       [if4](x0, x1, x2, x3, x4) = 0,
       
       [if3](x0, x1, x2, x3, x4, x5) = 0,
       
       [if2](x0, x1, x2, x3, x4, x5, x6) = 0,
       
       [if1](x0, x1, x2, x3, x4, x5, x6, x7) = 0,
       
       [reach](x0, x1, x2, x3) = 0,
       
       [rest](x0) = x0,
       
       [to](x0) = x0,
       
       [from](x0) = x0,
       
       [isEmpty](x0) = 0,
       
       [edge](x0, x1, x2) = x0 + x1 + x2,
       
       [union](x0, x1) = x0 + x1,
       
       [empty] = 0,
       
       [or](x0, x1) = x1,
       
       [false] = 0,
       
       [s](x0) = x0 + 1,
       
       [true] = 0,
       
       [eq](x0, x1) = 0,
       
       [0] = 0
      orientation:
       eq#(s(x),s(y)) = y + 2 >= y + 1 = eq#(x,y)
       
       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) = y >= 0 = true()
       
       or(false(),y) = y >= y = y
       
       union(empty(),h) = h >= h = h
       
       union(edge(x,y,i),h) = h + i + x + y >= h + i + x + y = edge(x,y,union(i,h))
       
       isEmpty(empty()) = 0 >= 0 = true()
       
       isEmpty(edge(x,y,i)) = 0 >= 0 = false()
       
       from(edge(x,y,i)) = i + x + y >= x = x
       
       to(edge(x,y,i)) = i + x + y >= y = y
       
       rest(edge(x,y,i)) = i + x + y >= i = i
       
       rest(empty()) = 0 >= 0 = empty()
       
       reach(x,y,i,h) = 0 >= 0 = if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
       
       if1(true(),b1,b2,b3,x,y,i,h) = 0 >= 0 = true()
       
       if1(false(),b1,b2,b3,x,y,i,h) = 0 >= 0 = if2(b1,b2,b3,x,y,i,h)
       
       if2(true(),b2,b3,x,y,i,h) = 0 >= 0 = false()
       
       if2(false(),b2,b3,x,y,i,h) = 0 >= 0 = if3(b2,b3,x,y,i,h)
       
       if3(false(),b3,x,y,i,h) = 0 >= 0 = reach(x,y,rest(i),edge(from(i),to(i),h))
       
       if3(true(),b3,x,y,i,h) = 0 >= 0 = if4(b3,x,y,i,h)
       
       if4(true(),x,y,i,h) = 0 >= 0 = true()
       
       if4(false(),x,y,i,h) = 0 >= 0 = or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
      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))
        isEmpty(empty()) -> true()
        isEmpty(edge(x,y,i)) -> false()
        from(edge(x,y,i)) -> x
        to(edge(x,y,i)) -> y
        rest(edge(x,y,i)) -> i
        rest(empty()) -> empty()
        reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h)
        if1(true(),b1,b2,b3,x,y,i,h) -> true()
        if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h)
        if2(true(),b2,b3,x,y,i,h) -> false()
        if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h)
        if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h))
        if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h)
        if4(true(),x,y,i,h) -> true()
        if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty()))
      Qed