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(*                                                                    *)
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(* 02110-1301 USA                                                     *)
(** In this file, we prove properties about hedges. These properties are in section 6.1 of the paper "On Bisimulations for the Spi-Calculus" by J. Borgström and U. Nestmann. However, we consider here an extended message language. *)
(** Author: Sébastien Briais (sebastien.briais@epfl.ch) *)
Unset Standard Proposition Elimination Names.
(** We assume to have a (countably infinite) set of names. *)
Hypothesis Nam : Set.
(** We define three operators for building messages. *)
Inductive op : Set :=
| Pub : op
| Priv : op
| Hash : op.
simple destruct o1; simple destruct o2; try auto;
try (right; red in |- *; intro; inversion H).
Qed.
(** The inversion of an operator. *)
Definition inv_op (o : op) : op :=
match o with
| Pub => Priv
| Priv => Pub
| _ => o
end.
(** A message is either a name, or an encrypted message, or a pair of messages, or an operator applied to a message. *)
Inductive Msg : Set :=
| MNam : Nam -> Msg
| MEnc : Msg -> Msg -> Msg
| MPair : Msg -> Msg -> Msg
| Mop : op -> Msg -> Msg.
(** Inversion of a message. This is to manage public/private keys. *)
Definition inv (M : Msg) : Msg :=
match M with
| Mop op M => Mop (inv_op op) M
| _ => M
end.
simple induction M; simpl in |- *; trivial.
simple destruct o; simpl in |- *; trivial.
Qed.
(** A hedge is a set of pairs of message. *)
Definition hedge := Msg -> Msg -> Prop.
intros.
apply (classic (h M N)).
Qed.
(** g is included in h iff whenever (M,N) is in g then (M,N) is in h *)
Definition inclusion (g h : hedge) := forall M N : Msg, g M N -> h M N.
(** g = h iff g is included in h and h is included in g *)
Definition equal (g h : hedge) := inclusion g h /\ inclusion h g.
unfold inclusion in |- *.
trivial.
Qed.
unfold inclusion in |- *.
intros.
apply H0.
apply H; trivial.
Qed.
unfold equal in |- *.
tauto.
Qed.
unfold equal in |- *.
intro.
split; apply inclusion_refl.
Qed.
unfold equal in |- *.
intros.
elim H.
tauto.
Qed.
unfold equal in |- *.
intros.
elim H.
elim H0.
intros.
split; apply inclusion_trans with g; trivial.
Qed.
(** union and intersection of hedges *)
Definition union (g h : hedge) (M N : Msg) := g M N \/ h M N.
Definition inter (g h : hedge) (M N : Msg) := g M N /\ h M N.
unfold union in |- *.
unfold equal in |- *.
intros.
unfold inclusion in |- *.
tauto.
Qed.
unfold inter in |- *.
unfold equal in |- *.
intros.
unfold inclusion in |- *.
tauto.
Qed.
unfold union in |- *.
unfold inclusion in |- *.
tauto.
Qed.
unfold union in |- *.
unfold inclusion in |- *.
tauto.
Qed.
unfold inter in |- *.
unfold inclusion in |- *.
tauto.
Qed.
unfold inter in |- *.
unfold inclusion in |- *.
tauto.
Qed.
unfold inclusion in |- *.
unfold union in |- *.
intros.
case H1.
apply H.
apply H0.
Qed.
intros.
unfold union in |- *.
unfold inter in |- *.
unfold inclusion in |- *.
tauto.
Qed.
(** The synthesis S(h) of h : it contains all the messages that can be synthetised by the hedge. *)
Inductive synthesis (h : hedge) : hedge :=
| SynInc : forall M N : Msg, h M N -> synthesis h M N
| SynEnc :
forall M N K L : Msg,
synthesis h M N -> synthesis h K L -> synthesis h (MEnc M K) (MEnc N L)
| SynPair :
forall M1 N1 M2 N2 : Msg,
synthesis h M1 N1 ->
synthesis h M2 N2 -> synthesis h (MPair M1 M2) (MPair N1 N2)
| SynOp :
forall (o : op) (M N : Msg),
synthesis h M N -> synthesis h (Mop o M) (Mop o N).
intros.
inversion H.
trivial.
Qed.
intros.
inversion H.
trivial.
Qed.
unfold inclusion in |- *.
intros.
apply SynInc.
trivial.
Qed.
(** g <= h if S(g) is included in S(h) *)
Definition le_h (g h : hedge) := inclusion (synthesis g) (synthesis h).
(** g ~ h if g <= h and h <= g *)
Definition equiv_h (g h : hedge) := le_h g h /\ le_h h g.
unfold le_h in |- *.
intro.
apply inclusion_refl.
Qed.
unfold le_h in |- *.
intros.
apply inclusion_trans with (synthesis g); trivial.
Qed.
unfold equiv_h in |- *.
tauto.
Qed.
unfold equiv_h in |- *.
intro.
split; apply le_h_refl.
Qed.
unfold equiv_h in |- *.
intros.
elim H.
elim H0.
intros.
split; apply le_h_trans with g; trivial.
Qed.
unfold equiv_h in |- *.
tauto.
Qed.
unfold le_h in |- *.
intros.
apply inclusion_trans with (synthesis g); trivial.
apply h_incl_syn.
Qed.
unfold le_h in |- *.
unfold inclusion in |- *.
intros.
induction  H0
as
[M
N
H0|
M
N
K
L
H0_1
HrecH0_1
H0_0
HrecH0_0|
M1
N1
M2
N2
H0_1
HrecH0_1
H0_0
HrecH0_0|
o
M
N
H0
HrecH0].
apply H; trivial.
apply SynEnc; trivial.
apply SynPair; trivial.
apply SynOp; trivial.
Qed.
split.
apply le_h_impl_incl_syn.
apply incl_syn_impl_le_h.
Qed.
intros.
apply incl_syn_impl_le_h.
apply inclusion_trans with h; trivial.
apply h_incl_syn.
Qed.
unfold equal in |- *.
unfold equiv_h in |- *.
intros.
elim H.
intros.
split.
apply inclusion_impl_le_h; trivial.
apply inclusion_impl_le_h; trivial.
Qed.
intros.
apply incl_syn_impl_le_h.
apply union_intro; apply le_h_impl_incl_syn; trivial.
Qed.
intros.
apply le_h_le_h_impl_union_le_h.
apply le_h_trans with h1; trivial.
apply inclusion_impl_le_h; apply hedge_left_union.
apply le_h_trans with h2; trivial.
apply inclusion_impl_le_h; apply hedge_right_union.
Qed.
(** we define how to analyse 'one time' a hedge h: all the pairs are splitted and the encrypted messages are decrypted if the keys are in the synthesis of h *)
Inductive analysis (h : hedge) : hedge :=
| AnaInc : forall M N : Msg, h M N -> analysis h M N
| AnaSplitL :
forall M1 M2 N1 N2 : Msg,
analysis h (MPair M1 M2) (MPair N1 N2) -> analysis h M1 N1
| AnaSplitR :
forall M1 M2 N1 N2 : Msg,
analysis h (MPair M1 M2) (MPair N1 N2) -> analysis h M2 N2
| AnaDec :
forall M N K L : Msg,
analysis h (MEnc M K) (MEnc N L) ->
synthesis h (inv K) (inv L) -> analysis h M N.
intros g h.
unfold inclusion in |- *.
intros.
induction  H0
as
[M N H0| M1 M2 N1 N2 H0 HrecH0| M1 M2 N1 N2 H0 HrecH0| M N K L H0 HrecH0 H1].
apply AnaInc.
apply H; trivial.
apply (AnaSplitL h M1 M2 N1 N2).
trivial.
apply (AnaSplitR h M1 M2 N1 N2).
trivial.
apply (AnaDec h M N K L).
trivial.
generalize (inclusion_impl_le_h g h H).
unfold le_h in |- *.
unfold inclusion in |- *.
intros.
auto.
Qed.
(** We define an infinite sequence (analysis_seq h n) where the n-th term is the n-th analysis of h *)
Fixpoint analysis_seq (h : hedge) (n : nat) {struct n} : hedge :=
match n with
| O => h
| S n => analysis (analysis_seq h n)
end.
simple induction n.
simpl in |- *.
intros.
unfold inclusion in |- *.
intros.
apply AnaInc; trivial.
intros.
simpl in |- *.
unfold inclusion in |- *.
intros.
apply AnaInc.
trivial.
Qed.
simple induction p.
simpl in |- *.
intros.
apply inclusion_refl.
intros.
replace (S n + n0) with (S (n + n0)); try auto with arith.
apply inclusion_trans with (analysis_seq h (n + n0)); trivial.
apply analysis_seq_grow_S.
Qed.
intros.
cut (exists p : nat, m = p + n).
intro.
elim H0.
intro p.
intro.
rewrite H1.
apply analysis_seq_grows.
exists (m - n).
rewrite plus_comm.
apply le_plus_minus.
trivial.
Qed.
(** reduce remove from h all the redundancy: a pair (M,N) is kept in (reduce h) iff it cannot be 'strictly' synthetised by h *)
Definition reduce (h : hedge) (M N : Msg) :=
h M N /\
match M with
| MEnc M' K =>
match N with
| MEnc N' L => ~ synthesis h M' N' \/ ~ synthesis h K L
| _ => True
end
| MPair M1 M2 =>
match N with
| MPair N1 N2 => ~ synthesis h M1 N1 \/ ~ synthesis h M2 N2
| _ => True
end
| Mop opM M' =>
match N with
| Mop opN N' => opM <> opN \/ ~ synthesis h M' N'
| _ => True
end
| _ => True
end.
unfold inclusion in |- *.
unfold reduce in |- *.
tauto.
Qed.
unfold le_h in |- *.
unfold inclusion in |- *.
intro.
simple induction M.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H.
tauto.
intro M'.
intro.
intro K.
intro.
simple destruct N.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1.
tauto.
intros N' L.
intros.
case (hedge_mem_dec (synthesis h) M' N').
case (hedge_mem_dec (synthesis h) K L).
intros.
apply SynEnc.
apply H; trivial.
apply H0; trivial.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1.
split; tauto.
tauto.
intro.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intro M1.
intro.
intro M2.
intro.
simple destruct N.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros N1 N2.
intro.
case (hedge_mem_dec (synthesis h) M1 N1).
case (hedge_mem_dec (synthesis h) M2 N2).
intros.
apply SynPair.
apply H; trivial.
apply H0; trivial.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intro.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros oM M'.
intro.
simple destruct N.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H0; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H0; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H0; tauto.
intros oN N'.
intro.
case (op_eq_dec oM oN).
intro.
rewrite e in H0.
rewrite e.
case (hedge_mem_dec (synthesis h) M' N').
intro.
apply SynOp.
apply H; trivial.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H0; tauto.
intros.
apply SynInc.
unfold reduce in |- *.
inversion H0; tauto.
Qed.
unfold equiv_h in |- *.
intro.
split.
apply h_le_h_red.
apply inclusion_impl_le_h.
apply red_incl_h.
Qed.
unfold le_h in |- *.
unfold inclusion in |- *.
unfold reduce in |- *.
intros g h H H0.
intros M N.
case M.
intros.
elim H1; auto.
case N; intros; elim H1; auto.
intros.
split; auto.
case H3.
intro; left; red in |- *; intro; apply H4; auto.
intro; right; red in |- *; intro; apply H4; auto.
case N; intros; elim H1; auto.
split; auto.
case H3.
intro; left; red in |- *; intro; apply H4; auto.
intro; right; red in |- *; intro; apply H4; auto.
case N; intros; elim H1; auto.
split; auto.
case H3; try tauto.
intros.
right; red in |- *; intro; apply H4; auto.
Qed.
unfold equal in |- *.
intros.
split.
apply inclusion_le_h_reduce_inclusion.
tauto.
apply inclusion_impl_le_h; tauto.
apply inclusion_le_h_reduce_inclusion; try tauto.
apply inclusion_impl_le_h; tauto.
Qed.
unfold le_h in |- *.
unfold inclusion in |- *.
intros.
generalize H1.
generalize (H M N H1).
unfold reduce in |- *.
case M; try tauto.
case N; try tauto.
intros.
elim H3.
split; trivial.
case H5.
intro.
left.
red in |- *; intro.
apply H6.
generalize (inclusion_impl_le_h (reduce g) g (red_incl_h g)).
unfold le_h in |- *.
unfold inclusion in |- *.
intros.
apply H8.
apply H0.
trivial.
intro.
right; red in |- *; intro; apply H6.
generalize (inclusion_impl_le_h (reduce g) g (red_incl_h g)).
unfold le_h in |- *.
unfold inclusion in |- *.
intros.
apply H8.
apply H0.
trivial.
case N; try tauto.
intros.
split; trivial.
elim H3; intros.
case H5.
left; red in |- *; intro; apply H6.
generalize (inclusion_impl_le_h (reduce g) g (red_incl_h g)).
unfold le_h in |- *.
unfold inclusion in |- *; auto.
right; red in |- *; intro; apply H6.
generalize (inclusion_impl_le_h (reduce g) g (red_incl_h g)).
unfold le_h in |- *.
unfold inclusion in |- *; auto.
case N; try tauto.
intros.
split; trivial.
elim H3.
intros.
case H5; try tauto.
right.
red in |- *; intro.
apply H6.
generalize (inclusion_impl_le_h (reduce g) g (red_incl_h g)).
unfold le_h in |- *.
unfold inclusion in |- *; auto.
Qed.
unfold equal in |- *.
intro.
split; try apply red_incl_h.
apply inclusion_le_h_impl_inclusion_red.
apply inclusion_refl.
apply le_h_refl.
Qed.
unfold equiv_h in |- *.
unfold inclusion in |- *.
intros.
elim H.
intros.
generalize (le_h_impl_incl_syn (reduce g) h H1).
unfold inclusion in |- *.
intros.
generalize (H3 M N H0).
generalize H0.
unfold reduce in |- *.
case M.
intros.
apply nam_left_syn; trivial.
case N.
intros.
apply nam_right_syn; trivial.
intros.
elim H4.
intros.
inversion H5.
trivial.
case H7.
intro.
cut False; try tauto.
apply H14.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
intro.
cut False; try tauto.
apply H14.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
case N.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
elim H4.
intros.
case H13.
intro.
cut False.
tauto.
apply H14.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
intro.
cut False; try tauto.
apply H14.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *; auto.
intros.
inversion H5; trivial.
case N.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
intros.
elim H4.
intros.
case (op_eq_dec o0 o).
intro.
rewrite e.
rewrite e in H5.
inversion H5; trivial.
case H7.
intro.
tauto.
intro.
cut False; try tauto.
apply H12.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
intro.
inversion H5; trivial.
tauto.
Qed.
intros.
apply le_h_trans with h.
apply le_h_trans with g; trivial.
apply inclusion_impl_le_h.
apply red_incl_h.
apply h_le_h_red.
Qed.
(** h is stable under analysis iff (analysis h) is included in h *)
Definition stable_analysis (h : hedge) := inclusion (analysis h) h.
intro.
unfold stable_analysis in |- *.
unfold equal in |- *.
intro.
split; trivial.
unfold inclusion in |- *.
intros.
apply AnaInc.
trivial.
Qed.
intros.
unfold stable_analysis in |- *.
apply inclusion_analysis_inclusion.
unfold stable_analysis in H; trivial.
Qed.
intros.
induction  p as [| p Hrecp].
simpl in |- *.
trivial.
simpl in |- *.
apply stable_impl_ana_stable.
trivial.
Qed.
intros.
induction  p as [| p Hrecp].
simpl in |- *.
apply equal_refl.
apply equal_trans with (analysis_seq h (p + n)).
trivial.
simpl in |- *.
apply stable_closed.
apply stable_ana_stable.
trivial.
Qed.
(** h could be an analysis of g iff g is included in h and h is stable under analysis *)
Definition analysis_cond (g h : hedge) := inclusion g h /\ stable_analysis h.
intros.
induction  n as [| n Hrecn].
simpl in |- *.
unfold analysis_cond in H.
tauto.
simpl in |- *.
generalize (inclusion_analysis_inclusion (analysis_seq g n) h Hrecn).
intro.
apply inclusion_trans with (analysis h); trivial.
unfold analysis_cond in H.
unfold stable_analysis in H.
tauto.
Qed.
(** h is an analysis of g if h could be an analysis of g and if for all other candidates h' which could be an analysis of g, h is included in h' *)
Definition is_analysis (g h : hedge) :=
analysis_cond g h /\
(forall h' : hedge, analysis_cond g h' -> inclusion h h').
unfold is_analysis in |- *.
intros.
apply analysis_seq_incl_analysis_cond.
tauto.
Qed.
unfold is_analysis in |- *.
unfold equal in |- *.
intros.
elim H.
elim H0.
intros.
split.
apply H4; trivial.
apply H2; trivial.
Qed.
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
intros.
elim H.
elim H0.
intros.
elim H1.
elim H3.
intros.
split.
split; trivial.
apply inclusion_trans with h; trivial.
intros.
apply inclusion_trans with h.
apply H2.
split; trivial.
apply inclusion_refl.
apply H4; tauto.
Qed.
intros.
apply analysis_unique with g; trivial.
apply analysis_trans with h; trivial.
Qed.
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
intros.
split.
split; trivial.
apply inclusion_trans with (analysis_seq h 0).
simpl in |- *.
apply inclusion_refl.
replace n with (n + 0); try auto with arith.
apply analysis_seq_grows.
intros.
apply analysis_seq_incl_analysis_cond.
unfold analysis_cond in |- *.
trivial.
Qed.
(** we define the analysis A(h) of h as being the infinite union of the sequence (analysis_seq h) *)
Definition analysis_def (h : hedge) (M N : Msg) :=
exists n : nat, analysis_seq h n M N.
intros.
unfold analysis_def in |- *.
intros.
induction  H
as
[M
N
H|
M
N
K
L
H1
HrecH1
H0
HrecH0|
M1
N1
M2
N2
H1
HrecH1
H0
HrecH0|
o
M
N
H
HrecH].
elim H.
intro n.
intro.
exists n.
apply SynInc.
trivial.
elim HrecH1.
intro n1.
intro.
elim HrecH0.
intro n2.
intro.
exists (max n1 n2).
apply SynEnc.
generalize
(inclusion_impl_le_h (analysis_seq h n1) (analysis_seq h (max n1 n2))
(analysis_seq_increase n1 (max n1 n2) h (le_max_l n1 n2))).
unfold le_h in |- *; auto.
generalize
(inclusion_impl_le_h (analysis_seq h n2) (analysis_seq h (max n1 n2))
(analysis_seq_increase n2 (max n1 n2) h (le_max_r n1 n2))).
unfold le_h in |- *; auto.
elim HrecH0; elim HrecH1.
intro n1.
intro.
intro n2.
intro.
exists (max n1 n2).
apply SynPair.
generalize
(inclusion_impl_le_h (analysis_seq h n1) (analysis_seq h (max n1 n2))
(analysis_seq_increase n1 (max n1 n2) h (le_max_l n1 n2))).
unfold le_h in |- *; auto.
generalize
(inclusion_impl_le_h (analysis_seq h n2) (analysis_seq h (max n1 n2))
(analysis_seq_increase n2 (max n1 n2) h (le_max_r n1 n2))).
unfold le_h in |- *; auto.
elim HrecH.
intro n.
intro.
exists n.
apply SynOp.
trivial.
Qed.
unfold is_analysis in |- *.
split.
unfold analysis_cond in |- *.
split.
unfold inclusion in |- *.
intros.
unfold analysis_def in |- *.
exists 0.
simpl in |- *.
trivial.
unfold stable_analysis in |- *.
unfold inclusion in |- *.
unfold analysis_def in |- *.
intros.
induction  H
as [M N H| M1 M2 N1 N2 H HrecH| M1 M2 N1 N2 H HrecH| M N K L H HrecH H0].
trivial.
elim HrecH.
intro n.
intro.
exists (S n).
simpl in |- *.
apply (AnaSplitL (analysis_seq h n) M1 M2 N1 N2).
apply AnaInc.
trivial.
elim HrecH.
intro n.
intro.
exists (S n).
apply (AnaSplitR (analysis_seq h n) M1 M2 N1 N2).
apply AnaInc; trivial.
elim HrecH.
intro n1.
intro.
generalize (synthesis_analysis_seq h (inv K) (inv L) H0).
intro.
elim H2.
intro n2.
intro.
exists (S (max n1 n2)).
simpl in |- *.
apply (AnaDec (analysis_seq h (max n1 n2)) M N K L).
apply AnaInc.
generalize (analysis_seq_increase n1 (max n1 n2) h (le_max_l n1 n2)).
unfold inclusion in |- *; auto.
generalize
(inclusion_impl_le_h (analysis_seq h n2) (analysis_seq h (max n1 n2))
(analysis_seq_increase n2 (max n1 n2) h (le_max_r n1 n2))).
unfold le_h in |- *; auto.
intros.
unfold inclusion in |- *.
unfold analysis_def in |- *.
intros.
elim H0.
intro n.
intro.
generalize (analysis_seq_incl_analysis_cond h h' H n).
unfold inclusion in |- *; auto.
Qed.
(** the irreducibles I(h) of h is (reduce (A(h))) *)
Definition irreducible (h : hedge) := reduce (analysis_def h).
intros h M N K L.
unfold irreducible in |- *.
unfold reduce in |- *.
tauto.
Qed.
unfold irreducible in |- *.
unfold reduce in |- *.
tauto.
Qed.
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).
intro.
apply inclusion_impl_le_h.
generalize (analysis_is_analysis h).
intro.
unfold is_analysis in H.
unfold analysis_cond in H.
elim H.
intros.
tauto.
Qed.
unfold irreducible in |- *.
intro.
apply h_equiv_h_red.
Qed.
intro.
apply le_h_trans with (analysis_def h).
apply h_le_h_ana_h.
generalize (ana_h_equiv_irr_h h).
unfold equiv_h in |- *.
tauto.
Qed.
intro.
generalize (analysis_is_analysis (irreducible h)).
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
intros.
elim H.
intros.
apply H1.
split.
unfold irreducible in |- *.
apply red_incl_h.
generalize (analysis_is_analysis h).
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
tauto.
Qed.
unfold irreducible in |- *.
intro.
apply inclusion_le_h_reduce_inclusion.
replace (reduce (analysis_def h)) with (irreducible h).
apply ana_irr_incl_ana.
unfold irreducible in |- *.
trivial.
replace (reduce (analysis_def h)) with (irreducible h).
apply le_h_trans with (irreducible h).
generalize ana_h_equiv_irr_h.
unfold equiv_h in |- *.
intros.
elim (H h); tauto.
apply h_le_h_ana_h.
unfold irreducible in |- *.
trivial.
Qed.
unfold irreducible in |- *.
intros.
apply inclusion_le_h_impl_inclusion_red.
generalize (analysis_is_analysis (reduce (analysis_def h))).
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
tauto.
apply le_h_trans with (analysis_def h).
apply inclusion_impl_le_h.
fold (irreducible h) in |- *.
apply ana_irr_incl_ana.
fold (irreducible h) in |- *.
generalize ana_h_equiv_irr_h.
unfold equiv_h in |- *.
intros.
elim (H h); auto.
Qed.
unfold is_irreducible in |- *.
unfold equal in |- *.
intros.
generalize (irr_incl_irr_irr h).
generalize (irr_irr_incl_irr h).
tauto.
Qed.
unfold is_irreducible in |- *.
unfold irreducible in |- *.
intros.
apply inclusion_trans with (reduce (analysis_def g)).
unfold equal in H0.
tauto.
apply red_equiv_impl_red_incl.
apply equiv_h_trans with g.
apply equal_impl_equiv_h.
apply equal_sym.
trivial.
trivial.
Qed.
intros.
unfold equal in |- *.
split.
apply equiv_irr_impl_incl; trivial.
apply equiv_irr_impl_incl; trivial.
apply equiv_h_sym; trivial.
Qed.
intro.
apply inclusion_impl_le_h.
unfold inclusion in |- *.
intros.
apply AnaInc.
trivial.
Qed.
unfold equal in |- *.
split.
unfold inclusion in |- *.
intros.
apply SynInc; trivial.
generalize (incl_syn_impl_le_h (synthesis h) h).
unfold le_h in |- *.
intros.
apply H.
apply inclusion_refl.
Qed.
unfold stable_analysis in |- *.
unfold inclusion in |- *.
intros.
induction  H0
as
[M N H0| M1 M2 N1 N2 H0 HrecH0| M1 M2 N1 N2 H0 HrecH0| M N K L H0 HrecH0 H1];
trivial.
inversion HrecH0; trivial.
apply SynInc.
apply H.
apply (AnaSplitL h M1 M2 N1 N2).
apply AnaInc; trivial.
inversion HrecH0; trivial.
apply SynInc.
apply H.
apply (AnaSplitR h M1 M2 N1 N2).
apply AnaInc; trivial.
inversion HrecH0; trivial.
apply SynInc.
apply H.
apply (AnaDec h M N K L); trivial.
apply AnaInc; trivial.
generalize (synthesis_idempotent h).
unfold equal in |- *.
intros.
elim H5.
unfold inclusion in |- *.
intros.
apply H7; trivial.
Qed.
intros.
generalize (le_h_impl_incl_syn g h H).
intro.
apply incl_syn_impl_le_h.
generalize (analysis_is_analysis g).
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
intros.
elim H1.
intros.
apply H3.
split.
apply inclusion_trans with (synthesis h).
trivial.
generalize h_le_h_ana_h.
unfold le_h in |- *.
trivial.
apply stable_ana_impl_stable_syn_ana.
generalize (analysis_is_analysis h).
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
tauto.
Qed.
intros.
apply le_h_trans with (analysis_def h).
apply le_h_trans with (analysis_def g).
generalize (ana_h_equiv_irr_h g).
unfold equiv_h in |- *; tauto.
apply le_h_ana_le_h; trivial.
generalize (ana_h_equiv_irr_h h).
unfold equiv_h in |- *; tauto.
Qed.
unfold equiv_h in |- *.
intros.
apply equiv_irr_irr_impl_equal; try apply irreducible_is_irreducible.
unfold equiv_h in |- *.
elim H; intros.
split; apply le_h_irr_le_h; trivial.
Qed.
intros.
apply equiv_irr_irr_impl_equal; try apply irreducible_is_irreducible.
unfold equiv_h in |- *.
split.
apply le_h_trans with (irreducible (irreducible (union h g))).
apply le_h_irr_le_h.
apply le_h_le_h_impl_union_le_h.
apply le_h_irr_le_h.
apply inclusion_impl_le_h.
apply hedge_left_union.
apply le_h_trans with (irreducible g).
apply h_le_h_irr_h.
apply le_h_irr_le_h.
apply inclusion_impl_le_h.
apply hedge_right_union.
generalize (irreducible_is_irreducible (union h g)).
unfold is_irreducible in |- *.
unfold equal in |- *.
intros.
apply inclusion_impl_le_h.
tauto.
apply le_h_irr_le_h.
apply le_h_le_h_impl_le_h_union.
apply h_le_h_irr_h.
apply le_h_refl.
Qed.
(** we define afterwards the notion of left consistency: we define first several predicates on hedges *)
(** if (M,N) is in h and M is a name then N is a name *)
Definition name_left_name (h : hedge) :=
forall M N : Msg,
h M N -> forall a : Nam, M = MNam a -> exists b : Nam, N = MNam b.
(** if (M,N) and (M',N') are in h and M = M' then N = N' *)
Definition injective_left (h : hedge) :=
forall M N : Msg, h M N -> forall M' N' : Msg, h M' N' -> M = M' -> N = N'.
(** if op(M) is in the left projection of h then M is not in the left projection of S(h) *)
Definition op_synthesis_left (h : hedge) :=
forall (o : op) (M N : Msg),
h M N ->
forall M' : Msg, M = Mop o M' -> forall N' : Msg, ~ synthesis h M' N'.
(** if {M}K is in the left projection of h then either M or K are not in the left projection of S(h) *)
Definition enc_synthesis_left (h : hedge) :=
forall M N : Msg,
h M N ->
forall M' K : Msg,
M = MEnc M' K ->
forall N' L : Msg, ~ synthesis h M' N' \/ ~ synthesis h K L.
(** the left projection of h does not contain any pair *)
Definition pair_free_left (h : hedge) :=
forall M1 M2 N : Msg, ~ h (MPair M1 M2) N.
(** if ({M}K,N') is in h and (inv K) is in the left projection of S(h) then N' = {N}L, (inv K,inv L) is in S(h) and (M,N) is in S(h) *)
Definition enc_analysis_left (h : hedge) :=
forall M N : Msg,
h M N ->
forall M' K : Msg,
M = MEnc M' K ->
forall L : Msg,
synthesis h (inv K) (inv L) ->
exists N' : Msg, N = MEnc N' L /\ synthesis h M' N'.
(** if (M,N) is in h and (inv M,N') is in h then N'=(inv N) *)
Definition injective_inv_left (h : hedge) :=
forall M N : Msg,
h M N -> forall M' N' : Msg, h M' N' -> M' = inv M -> N' = inv N.
(** left consistency of h *)
Definition is_left_consistent (h : hedge) : Prop :=
name_left_name h /\
injective_left h /\
op_synthesis_left h /\
pair_free_left h /\
enc_synthesis_left h /\ enc_analysis_left h /\ injective_inv_left h.
Definition weak_left_consistent (h : hedge) :=
name_left_name h /\
injective_left h /\
op_synthesis_left h /\
pair_free_left h /\ enc_synthesis_left h /\ enc_analysis_left h.
unfold injective_left in |- *.
unfold enc_synthesis_left in |- *.
unfold op_synthesis_left in |- *.
unfold pair_free_left in |- *.
intro h.
intro.
intro.
intro.
intro.
intros M N H3.
induction  H3
as
[M
N
H3|
M
N
K
L
H3_1
HrecH3_1
H3_0
HrecH3_0|
M1
N1
M2
N2
H3_1
HrecH3_1
H3_0
HrecH3_0|
o
M
N
H3
HrecH3]; intros M' N' H4;
[ induction  H4
as
[M0
N0
H4|
M0
N0
K
L
H4_1
HrecH4_1
H4_0
HrecH4_0|
M1
N1
M2
N2
H4_1
HrecH4_1
H4_0
HrecH4_0|
o
M0
N0
H4
HrecH4]
| induction  H4
as
[M0
N0
H3|
M0
N0
K0
L0
H4_1
HrecH4_1
H4_0
HrecH4_0|
M1
N1
M2
N2
H4_1
HrecH4_1
H4_0
HrecH4_0|
o
M0
N0
H4
HrecH4]
| induction  H4
as
[M
N
H3|
M
N
K
L
H4_1
HrecH4_1
H4_0
HrecH4_0|
M0
N0
M3
N3
H4_1
HrecH4_1
H4_0
HrecH4_0|
o
M
N
H4
HrecH4]
| induction  H4
as
[M0
N0
H4|
M0
N0
K
L
H4_1
HrecH4_1
H4_0
HrecH4_0|
M1
N1
M2
N2
H4_1
HrecH4_1
H4_0
HrecH4_0|
o0
M0
N0
H4
HrecH4] ].
intro.
exact (H M N H3 M0 N0 H4 H5).
intro.
cut (~ synthesis h M0 N0 \/ ~ synthesis h K L).
intro.
case H5; intro H6; red in H6; tauto.
apply (H0 M N H3 M0 K H4).
intro.
rewrite H4 in H3.
cut False.
tauto.
apply (H2 M1 M2 N); trivial.
intro.
cut (~ synthesis h M0 N0).
intro.
tauto.
apply (H1 o M N H3 M0 H5); trivial.
intro.
cut (~ synthesis h M N \/ ~ synthesis h K L).
intro.
case H5; intro H6; red in H6; tauto.
apply (H0 M0 N0 H3 M K); auto.
intro.
inversion H3.
rewrite (HrecH3_1 M0 N0 H4_1 H5).
rewrite (HrecH3_0 K0 L0 H4_0 H6).
trivial.
intro.
inversion H3.
intro.
inversion H3.
intro.
rewrite <- H4 in H3.
cut False.
tauto.
apply (H2 M1 M2 N H3).
intro.
inversion H3.
intro.
inversion H3.
rewrite (HrecH3_1 M0 N0 H4_1 H5).
rewrite (HrecH3_0 M3 N3 H4_0 H6); trivial.
intro.
inversion H3.
intro.
cut (~ synthesis h M N).
intro.
tauto.
apply (H1 o M0 N0 H4 M); auto.
intro.
inversion H4.
intro.
inversion H4.
case (op_eq_dec o o0).
intro.
rewrite <- e.
intro.
inversion H5.
rewrite (HrecH3 M0 N0 H4 H7).
trivial.
intros.
inversion H5.
tauto.
Qed.
intro h.
intros H0 H1 H2 H3 H4.
generalize (inj_left_syn h H0 H1 H2 H3).
intro H4bis.
unfold injective_left in H0.
unfold enc_synthesis_left in H1.
unfold op_synthesis_left in H2.
unfold pair_free_left in H3.
unfold injective_inv_left in H4.
unfold injective_left in H4bis.
unfold injective_inv_left in |- *.
intros M N H5.
induction  H5
as
[M
N
H|
M
N
K
L
H5_1
HrecH5_1
H5_0
HrecH5_0|
M1
N1
M2
N2
H5_1
HrecH5_1
H5_0
HrecH5_0|
o
M
N
H5
HrecH5]; intros M' N' H6;
[ induction  H6
as
[M0
N0
H5|
M0
N0
K
L
H6_1
HrecH6_1
H6_0
HrecH6_0|
M1
N1
M2
N2
H6_1
HrecH6_1
H6_0
HrecH6_0|
o
M0
N0
H6
HrecH6]
| induction  H6
as
[M0
N0
H|
M0
N0
K0
L0
H6_1
HrecH6_1
H6_0
HrecH6_0|
M1
N1
M2
N2
H6_1
HrecH6_1
H6_0
HrecH6_0|
o
M0
N0
H6
HrecH6]
| induction  H6
as
[M
N
H|
M
N
K
L
H6_1
HrecH6_1
H6_0
HrecH6_0|
M0
N0
M3
N3
H6_1
HrecH6_1
H6_0
HrecH6_0|
o
M
N
H6
HrecH6]
| induction  H6
as
[M0
N0
H|
M0
N0
K
L
H6_1
HrecH6_1
H6_0
HrecH6_0|
M1
N1
M2
N2
H6_1
HrecH6_1
H6_0
HrecH6_0|
o0
M0
N0
H6
HrecH6] ].
intro.
apply (H4 M N H M0 N0 H5 H6).
intro.
cut (M = inv (MEnc M0 K)).
simpl in |- *.
intro.
case (H1 M N H M0 K H6 N0 L); intro; tauto.
rewrite H5.
symmetry  in |- *.
apply inv_invol.
intro.
cut (M = inv (MPair M1 M2)).
simpl in |- *.
intro.
rewrite H6 in H.
generalize (H3 M1 M2 N).
tauto.
rewrite H5.
symmetry  in |- *.
apply inv_invol.
intro.
cut (M = inv (Mop o M0)).
simpl in |- *.
intro.
generalize (H2 (inv_op o) M N H M0 H7 N0).
tauto.
rewrite H5.
symmetry  in |- *.
apply inv_invol.
simpl in |- *.
cut (synthesis h M0 N0).
intro.
intro.
cut (synthesis h (MEnc M K) (MEnc N L)).
intro.
apply (H4bis M0 N0 H5 (MEnc M K) (MEnc N L) H7 H6).
apply SynEnc; trivial.
apply SynInc; trivial.
simpl in |- *.
intro.
inversion H.
cut (synthesis h (MEnc M0 K0) (MEnc N0 L0)); try (apply SynEnc; trivial).
intro.
cut (synthesis h (MEnc M K) (MEnc N L)); try (apply SynEnc; trivial).
intro.
apply (H4bis (MEnc M0 K0) (MEnc N0 L0) H5 (MEnc M K) (MEnc N L) H8 H).
simpl in |- *.
intro.
inversion H.
simpl in |- *.
intro.
inversion H.
simpl in |- *.
cut (synthesis h M N); try (apply SynInc; trivial).
intro.
cut (synthesis h (MPair M1 M2) (MPair N1 N2)); try (apply SynPair; trivial).
intro.
intro.
apply (H4bis M N H5 (MPair M1 M2) (MPair N1 N2) H6 H7).
simpl in |- *.
intro.
inversion H.
simpl in |- *.
intro.
cut (synthesis h (MPair M1 M2) (MPair N1 N2)); try (apply SynPair; trivial).
intro.
cut (synthesis h (MPair M0 M3) (MPair N0 N3)); try (apply SynPair; trivial).
intro.
apply (H4bis (MPair M0 M3) (MPair N0 N3) H6 (MPair M1 M2) (MPair N1 N2) H5 H).
simpl in |- *.
intro.
inversion H.
simpl in |- *.
cut (synthesis h M0 N0); try (apply SynInc; trivial).
intro.
cut (synthesis h (Mop (inv_op o) M) (Mop (inv_op o) N));
try (apply SynOp; trivial).
intro.
intro.
apply (H4bis M0 N0 H6 (Mop (inv_op o) M) (Mop (inv_op o) N) H7 H8).
simpl in |- *.
intro.
inversion H.
simpl in |- *.
intro.
inversion H.
simpl in |- *.
intro.
inversion H.
cut (N0 = N).
intro.
rewrite H7; trivial.
apply (H4bis M0 N0 H6 M N H5 H9).
Qed.
(** the transposition of h is the hedge where all the pairs (M,N) are reversed in (N,M) *)
Definition transpose (h : hedge) (M N : Msg) := h N M.
(** h is consistent if h is left consistent and if (transpose h) is left consistent (h is right consistent) *)
Definition is_consistent (h : hedge) :=
is_left_consistent h /\ is_left_consistent (transpose h).
Definition no_pairs (h : hedge) :=
forall M1 M2 N1 N2 : Msg, ~ h (MPair M1 M2) (MPair N1 N2).
Definition no_syn_ops (h : hedge) :=
forall (o : op) (M N : Msg), h (Mop o M) (Mop o N) -> ~ synthesis h M N.
Definition no_syn_cyph (h : hedge) :=
forall M N K L : Msg,
h (MEnc M K) (MEnc N L) -> ~ synthesis h M N \/ ~ synthesis h K L.
Definition cyph_dec (h : hedge) :=
forall M N K L : Msg,
h (MEnc M K) (MEnc N L) -> synthesis h (inv K) (inv L) -> synthesis h M N.
(** we show in the following that a hedge is irreducible iff it satisfies these conditions *)
Definition cond_irr (h : hedge) :=
no_pairs h /\ no_syn_ops h /\ no_syn_cyph h /\ cyph_dec h.
intros.
elim H.
intros.
elim H1.
intros.
elim H3.
intros.
elim H5.
intros.
elim H7.
intros.
unfold cond_irr in |- *.
split.
unfold no_pairs in |- *.
intros.
unfold pair_free_left in H6.
red in |- *.
intro.
apply (H6 M1 M2 (MPair N1 N2)).
trivial.
split.
unfold op_synthesis_left in H4.
unfold no_syn_ops in |- *.
intros.
apply (H4 o (Mop o M) (Mop o N)); trivial.
split.
unfold no_syn_cyph in |- *.
intros.
unfold enc_synthesis_left in H8.
apply (H8 (MEnc M K) (MEnc N L)); trivial.
unfold cyph_dec in |- *.
intros.
unfold enc_analysis_left in H9.
generalize (H9 (MEnc M K) (MEnc N L) H10 M K).
intros.
cut (exists N' : Msg, MEnc N L = MEnc N' L /\ synthesis h M N').
intro.
elim H13.
intro N'.
intro.
elim H14.
intros.
inversion H15; trivial.
apply H12; auto.
Qed.
intros.
apply left_consistent_impl_cond_irr_strong.
unfold is_left_consistent in H.
tauto.
Qed.
unfold is_consistent in |- *.
intros.
elim H.
intros.
apply left_consistent_impl_cond_irr.
trivial.
Qed.
unfold cond_irr in |- *.
intros.
apply incl_syn_impl_le_h.
generalize (analysis_is_analysis h).
unfold is_analysis in |- *.
intro.
elim H0.
unfold analysis_cond in |- *.
intros.
apply H2.
split.
unfold inclusion in |- *.
intros.
apply SynInc.
trivial.
unfold stable_analysis in |- *.
unfold inclusion in |- *.
intros.
induction  H3
as
[M N H3| M1 M2 N1 N2 H3 HrecH3| M1 M2 N1 N2 H3 HrecH3| M N K L H3 HrecH3 H4];
trivial.
inversion HrecH3; trivial.
unfold no_pairs in H.
elim H.
intros.
generalize (H7 M1 M2 N1 N2).
tauto.
inversion HrecH3; trivial.
unfold no_pairs in H.
elim H.
intros.
generalize (H7 M1 M2 N1 N2).
tauto.
generalize (synthesis_idempotent h).
unfold equal in |- *.
intros.
elim H5.
intros.
unfold inclusion in H7.
generalize (H7 (inv K) (inv L) H4).
intro.
unfold cyph_dec in H.
elim H.
intros.
elim H10.
intros.
elim H12.
intros.
inversion HrecH3; trivial.
apply (H14 M N K L); trivial.
Qed.
intros.
apply equiv_h_trans with (analysis_def h).
unfold equiv_h in |- *.
split.
apply h_le_h_ana_h.
apply cond_irr_impl_le_h_ana; trivial.
apply ana_h_equiv_irr_h.
Qed.
intros.
apply equiv_irr_impl_incl.
apply equiv_h_sym.
apply cond_irr_impl_equiv_h_irr.
trivial.
apply irreducible_is_irreducible.
Qed.
intros.
unfold equal in |- *.
split; try apply red_incl_h.
unfold inclusion in |- *.
intros M N.
unfold reduce in |- *.
case M; try tauto.
case N; try tauto.
intros.
unfold cond_irr in H.
unfold no_syn_cyph in H.
elim H.
intros.
elim H2.
intros.
elim H4.
intros.
split; trivial.
apply H5; trivial.
case N; try tauto.
intros.
unfold cond_irr in H.
unfold no_pairs in H.
elim H.
intros.
generalize (H1 m1 m2 m m0).
tauto.
case N; try tauto.
intros.
split; trivial.
case (op_eq_dec o0 o).
intro.
right.
rewrite e in H0.
unfold cond_irr in H.
unfold no_syn_ops in H.
elim H.
intros.
elim H2; intros.
apply (H3 o).
trivial.
tauto.
Qed.
intros.
apply inclusion_trans with (reduce h).
generalize (cond_irr_impl_stable_reduce h H).
unfold equal in |- *; tauto.
apply red_equiv_impl_red_incl.
apply equiv_h_trans with h.
apply equiv_h_sym.
apply h_equiv_h_red.
apply cond_irr_impl_equiv_h_irr.
trivial.
Qed.
unfold is_irreducible in |- *.
unfold equal in |- *.
intros.
split.
apply cond_irr_impl_h_incl_irr; trivial.
apply cond_irr_impl_irr_incl_h; trivial.
Qed.
unfold cond_irr in |- *.
unfold is_irreducible in |- *.
unfold equal in |- *.
intros.
elim H.
unfold inclusion in |- *.
intros.
split.
unfold no_pairs in |- *.
intros.
red in |- *.
intro.
apply (irreducible_pair h M1 N1 M2 N2).
apply H0; trivial.
split.
unfold no_syn_ops in |- *.
intros.
red in |- *.
intro.
generalize (irreducible_op h o o M N).
intro.
elim H4.
intros.
generalize (H0 (Mop o M) (Mop o N) H2).
intro.
generalize (H5 H7).
intro.
elim H8.
intros.
case H10.
tauto.
intro.
apply H11.
generalize (h_le_h_ana_h h).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
split.
unfold no_syn_cyph in |- *.
intros.
generalize (irreducible_enc h M N K L).
intro.
elim H3.
intros.
generalize (H4 (H0 (MEnc M K) (MEnc N L) H2)).
intro.
elim H6.
intros.
case H8.
intro.
left.
red in |- *.
intro.
apply H9.
generalize (h_le_h_ana_h h).
unfold le_h in |- *; unfold inclusion in |- *; auto.
intro.
right.
red in |- *; intro.
apply H9.
generalize (h_le_h_ana_h h).
unfold le_h in |- *; unfold inclusion in |- *; auto.
unfold cyph_dec in |- *.
intros.
generalize (H0 (MEnc M K) (MEnc N L) H2).
unfold irreducible in |- *.
unfold reduce in |- *.
intros.
elim H4.
intros.
generalize (ana_h_equiv_irr_h h).
unfold equiv_h in |- *.
intros.
elim H7.
unfold le_h in |- *.
intros.
cut (inclusion (synthesis (irreducible h)) (synthesis h)).
unfold inclusion in |- *.
intros.
apply H10.
unfold inclusion in H8.
apply H8.
apply SynInc.
generalize (analysis_is_analysis h).
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
intros.
elim H11.
intros.
elim H12.
unfold stable_analysis in |- *.
intros.
unfold inclusion in H15.
apply H15.
apply (AnaDec (analysis_def h) M N K L).
apply AnaInc; trivial.
generalize (h_le_h_ana_h h).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
fold (le_h (irreducible h) h) in |- *.
apply inclusion_impl_le_h.
tauto.
Qed.
intros.
split.
apply cond_irr_impl_irreducible.
apply irreducible_impl_cond_irr.
Qed.
intros.
apply cond_irr_impl_irreducible.
apply consistent_impl_cond_irr.
trivial.
Qed.
intros.
apply equiv_irr_irr_impl_equal; try apply consistent_impl_irreducible;
trivial.
Qed.
unfold transpose in |- *.
unfold inclusion in |- *.
intros.
apply H; trivial.
Qed.

(** if g = h then (transpose g)=(transpose h) *)
Lemma equal_transpose_equal :
forall g h : hedge, equal g h -> equal (transpose g) (transpose h).