Section effsemlemmas.
Context {G C:Type} (Sem: @EffectSem G C) (g:G).
Fixpoint effstepN (n:nat) : (block -> Z -> bool) -> C -> mem -> C -> mem -> Prop :=
match n with
| O => fun U c m c' m' => (c,m) = (c',m')
| S k => fun U c1 m1 c3 m3 => exists c2, exists m2,
effstep Sem g U c1 m1 c2 m2 /\
effstepN k (fun b ofs => U b ofs  || freshloc m1 m2 b)  c2 m2 c3 m3
end.
Definition effstep_plus U c m c' m' :=
exists n, effstepN (S n) U c m c' m'.
Definition effstep_star U c m c' m' :=
exists n, effstepN n U c m c' m'.
End effsemlemmas.
Definition EmptyEffect: Values.block -> Z -> bool := fun b z => false.
Definition FreeEffect m lo hi (sp b:Values.block) (ofs:Z): bool :=
if valid_block_dec m b
then eq_block b sp && zle lo ofs && zlt ofs hi
else false.
Definition FreelistEffect m (L: list (Values.block * Z * Z)) (b:Values.block) (ofs:Z): bool :=
List.fold_right (fun X E b z => match X with (bb,lo,hi) =>
E b z || FreeEffect m lo hi bb b z
end)
EmptyEffect L b ofs.

Lemma FreelistEffect_same: forall m bb lo hi mm L
(F:Mem.free m bb lo hi = Some mm)
b (VB: Mem.valid_block mm b) ofs,
FreelistEffect mm L b ofs = false <-> FreelistEffect m L b ofs = false.