unfold irreducible in |- *.
unfold reduce in |- *.
intros.
red in |- *.
intro.
elim H.
intro.
intro.
cut (analysis_def h M1 N1 /\ analysis_def h M2 N2).
intro.
elim H2.
intros.
case H1; intro H5; apply H5; trivial.
apply SynInc; trivial.
apply SynInc; trivial.
cut (is_analysis h (analysis_def h)); try apply analysis_is_analysis.
unfold is_analysis in |- *.
intros.
elim H2.
unfold analysis_cond in |- *.
intros.
elim H3.
intros.
unfold stable_analysis in H6.
unfold inclusion in H6.
split; apply H6.
apply (AnaSplitL (analysis_def h) M1 M2 N1 N2); apply AnaInc; trivial.
apply (AnaSplitR (analysis_def h) M1 M2 N1 N2); apply AnaInc; trivial.
Qed.
(** h is irreducible iff h = I(h) *)
Definition is_irreducible (h : hedge) := equal h (irreducible h).

(** h <= A(h) *)
Theorem h_le_h_ana_h : forall h : hedge, le_h h (analysis_def h).
intro.