# Omega Calculator v1.2 (based on Omega Library 1.2, August, 2000):
# symbolic p(2), n, m1, m2, low;
# 
# 
# R  := { [ir,jr] : 1 <= ir <= n && 1 <= jr <= m1 };
# 
# 
# f1i := { [iw,jw] -> [ir,jr] : 1 <= ir <= n && 1 <= jr <= m1 &&
# 			      1 <= iw <= n && 1 <= jw <= m2 &&
# 			      jw = jr &&
# 			      iw = ir };
# 
# f1f := { [iw,jw] -> [ir,jr] : 1 <= ir <= n && 1 <= jr <= m1 &&
# 			      1 <= iw <= n && 1 <= jw <= m2 &&
# 			      jw = jr &&
# 			      iw < ir };
# 
# 
# f3i := { [iw,jw] -> [ir,jr] : 1 <= ir <= n && 1 <= jr <= m1 &&
# 			      1 <= iw <= n && 1 < low <= jw <= m1 &&
# 			      jw-1 = jr &&
# 			      iw = ir };
# 
# f3f := { [iw,jw] -> [ir,jr] : 1 <= ir <= n && 1 <= jr <= m1 &&
# 			      1 <= iw <= n && 1 < low <= jw <= m1 &&
# 			      jw-1 = jr &&
# 			      iw < ir };
# 
# 
# o31i := { [iw3,jw3] -> [iw1,jw1] : 1 <= iw1 <= n && 1 <= jw1 <= m2 &&
# 				   1 <= iw3 <= n && 1 < low <= jw3 <= m1 &&
# 				   jw1 = jw3-1 &&
# 				   iw1 = iw3 };
# 
# 
# o13f := { [iw1,jw1] -> [iw3,jw3] : 1 <= iw1 <= n && 1 <= jw1 <= m2 &&
# 				   1 <= iw3 <= n && 1 < low <= jw3 <= m1 &&
# 				   jw1 = jw3-1 &&
# 				   iw1 < iw3 };
# 
# 
# o31f := { [iw3,jw3] -> [iw1,jw1] : 1 <= iw1 <= n && 1 <= jw1 <= m2 &&
# 				   1 <= iw3 <= n && 1 < low <= jw3 <= m1 &&
# 				   jw1 = jw3-1 &&
# 				   iw3 < iw1 };
# 
# 
# o11f := { [iw1a,jw1a] -> [iw1b,jw1b] : 1 <=iw1a<= n && 1 <=jw1a<= m2 &&
# 				       1 <=iw1b<= n && 1 <=jw1b<= m2 &&
# 				       jw1a = jw1b &&
# 				       iw1a < iw1b };
# 
# o33f := { [iw3a,jw3a] -> [iw3b,jw3b] : 1 <=iw3a<= n && 1 < low <=jw3a<= m1 &&
# 				       1 <=iw3b<= n && 1 < low <=jw3b<= m1 &&
# 				       jw3a-1 = jw3b-1 &&
# 				       iw3a < iw3b };
# 
# 
# 
# # FIRST GROUP - 1i and 2i (NO POSSIBLE OUTPUT DEPS. BETWEEN)
# v1i := f1i;
# 
# v1i;

{[iw,jw] -> [iw,jw] : 1 <= jw <= m1, m2 && 1 <= iw <= n}

# 
# 
# Exposed12i := R intersection complement domain f1i;
# 
# Exposed12i;

{[In_1,In_2]: 1, m2+1 <= In_2 <= m1 && 1 <= In_1 <= n}

# 
# 
# # SECOND GROUP - 3i
# v3i := f3i / Exposed12i;
#   # / is restrictRange
# v3i;

{[iw,jw] -> [iw,jw-1] : 2 <= low <= jw <= m1 && 1 <= iw <= n && m2 <= jw-2}

# 
# 
# Exposed3i := R intersection complement domain f3i;
# 
# Exposed3i;

{[In_1,In_2]: 1 <= In_1 <= n && 1 <= In_2 <= m1 && low <= 1} union
 {[In_1,In_2]: 1 <= In_2 <= m1, low-1 && 1 <= In_1 <= n}

# 
# 
# # THIRD GROUP - 1,2,3f (THERE ARE POSSIBLE OUTPUT DEPS BETWEEN THEM)
# 
# v1f := ( f1f / Exposed3i )
# 	intersection complement ( f3i compose o13f )
# 	intersection complement ( f3f compose o13f )
# 	intersection complement ( f1i compose o11f )
# 	intersection complement ( f1f compose o11f );
# 
# v1f;

{[iw1a,jw1a] -> [ir,jr]  : FALSE }

# 
# 
# # WE SHOULD BE ABLE TO DO v1f WITH SOME VALUE-BASED FLOW DD'S
# 
# v1f_val := ( f1f / Exposed3i )
# 	intersection complement ( v3i compose o13f )
# 	intersection complement ( f3f compose o13f )
# 	intersection complement ( v1i compose o11f )
# 	intersection complement ( f1f compose o11f );
# 
# v1f_val;

{[iw1a,jw1a] -> [ir,jr]  : FALSE }

# 
# 
# #
# # The effects of loop-independent flow have been taken out already,
# # so this should work.  But it does not.  Probably I am stupyd.
# #
# 
# v1f_clever := ( f1f / Exposed3i )
# 	intersection complement ( f3f compose o13f )
# 	intersection complement ( f1f compose o11f );
# 
# v1f_clever;

{[iw1a,jw1a] -> [iw1a+1,jw1a] : 1 <= jw1a <= m2, m1 && 1 <= iw1a < n && low <= 1} union
 {[iw1a,jw1a] -> [iw1a+1,jw1a] : 1 <= jw1a <= m2, m1, low-1 && 1 <= iw1a < n}

# 
# 
# #
# # NOW CHECK FOR EQUIVALENCES - THESE SHOULD BE TRUE
# #
# 
# v1f subset v1f_val;

True
# 
# v1f_val subset v1f;

True
# 
# # v1f subset v1f_clever;
# # v1f_clever subset v1f;