- lfp_least_closed
-
|- !f. monotone f ==> closed f (lfp f) /\ !X. closed f X ==> lfp f SUBSET X
- gfp_greatest_dense
-
|- !f. monotone f ==> dense f (gfp f) /\ !X. dense f X ==> X SUBSET gfp f
- lfp_fixedpoint
-
|- !f. monotone f ==> (lfp f = f (lfp f)) /\ !X. (X = f X) ==> lfp f SUBSET X
- gfp_greatest_fixedpoint
-
|- !f. monotone f ==> (gfp f = f (gfp f)) /\ !X. (X = f X) ==> X SUBSET gfp f
- lfp_induction
-
|- !f. monotone f ==> !X. f X SUBSET X ==> lfp f SUBSET X
- gfp_coinduction
-
|- !f. monotone f ==> !X. X SUBSET f X ==> X SUBSET gfp f
- lfp_strong_induction
-
|- !f. monotone f ==> !X. f (X INTER lfp f) SUBSET X ==> lfp f SUBSET X
- gfp_strong_coinduction
-
|- !f. monotone f ==> !X. X SUBSET f (X UNION gfp f) ==> X SUBSET gfp f
- fnsum_monotone
-
|- !f1 f2. monotone f1 /\ monotone f2 ==> monotone (f1 ++ f2)
- empty_monotone
-
|- monotone empty
- fnsum_empty
-
|- !f. (f ++ empty = f) /\ (empty ++ f = f)
- fnsum_ASSOC
-
|- !f g h. f ++ (g ++ h) = f ++ g ++ h
- fnsum_COMM
-
|- !f g. f ++ g = g ++ f
- fnsum_SUBSET
-
|- !f g X. f X SUBSET (f ++ g) X /\ g X SUBSET (f ++ g) X
- lfp_fnsum
-
|- !f1 f2.
monotone f1 /\ monotone f2 ==>
lfp f1 SUBSET lfp (f1 ++ f2) /\ lfp f2 SUBSET lfp (f1 ++ f2)
- lfp_rule_applied
-
|- !f X y. monotone f /\ X SUBSET lfp f /\ y IN f X ==> y IN lfp f
- lfp_empty
-
|- !f x. monotone f /\ x IN f {} ==> x IN lfp f