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Require Import Relations.
Require Import FSets.
Require Import Arith.
Require Import Omega.
Set Keyed Unification.
Lemma Bool_elim_bool : forall (b:bool),b=true \/ b=false.
destruct b;try tauto.
Qed.
Module OrderedPair (X:OrderedType) (Y:OrderedType) <: OrderedType with
Definition t := (X.t * Y.t)%type.
Definition t := (X.t * Y.t)%type.
Definition eq (xy1:t) (xy2:t) :=
let (x1,y1) := xy1 in
let (x2,y2) := xy2 in
(X.eq x1 x2) /\ (Y.eq y1 y2).
Definition lt (xy1:t) (xy2:t) :=
let (x1,y1) := xy1 in
let (x2,y2) := xy2 in
(X.lt x1 x2) \/ ((X.eq x1 x2) /\ (Y.lt y1 y2)).
Lemma eq_refl : forall (x:t),(eq x x).
destruct x.
unfold eq.
split;[apply X.eq_refl | apply Y.eq_refl].
Qed.
Lemma eq_sym : forall (x y:t),(eq x y)->(eq y x).
destruct x;destruct y;unfold eq;intro.
elim H;clear H;intros.
split;[apply X.eq_sym | apply Y.eq_sym];trivial.
Qed.
Lemma eq_trans : forall (x y z:t),(eq x y)->(eq y z)->(eq x z).
unfold eq;destruct x;destruct y;destruct z;intros.
elim H;clear H;intros.
elim H0;clear H0;intros.
split;[eapply X.eq_trans | eapply Y.eq_trans];eauto.
Qed.
Lemma lt_trans : forall (x y z:t),(lt x y)->(lt y z)->(lt x z).
unfold lt;destruct x;destruct y;destruct z;intros.
case H;clear H;intro.
case H0;clear H0;intro.
left.
eapply X.lt_trans;eauto.
elim H0;clear H0;intros.
left.
case (X.compare t0 t4);trivial;intros.
generalize (X.eq_sym H0);intro.
generalize (X.eq_trans e H2);intro.
elim (X.lt_not_eq H H3).
generalize (X.lt_trans l H);intro.
generalize (X.eq_sym H0);intro.
elim (X.lt_not_eq H2 H3).
elim H;clear H;intros.
case H0;clear H0;intro.
left.
case (X.compare t0 t4);trivial;intros.
generalize (X.eq_sym H);intro.
generalize (X.eq_trans H2 e);intro.
elim (X.lt_not_eq H0 H3).
generalize (X.lt_trans H0 l);intro.
generalize (X.eq_sym H);intro.
elim (X.lt_not_eq H2 H3).
elim H0;clear H0;intros.
right.
split.
eauto.
eauto.
Qed.
Lemma lt_not_eq : forall (x y:t),(lt x y)->~(eq x y).
unfold lt, eq;destruct x;destruct y;intro;intro.
elim H0;clear H0;intros.
case H.
intro.
apply (X.lt_not_eq H2 H0).
intro.
elim H2;clear H2;intros.
apply (Y.lt_not_eq H3 H1).
Qed.
Definition compare : forall (x y:t),(Compare lt eq x y).
destruct x;destruct y.
case (X.compare t0 t2);intro.
apply LT.
left;trivial.
case (Y.compare t1 t3);intro.
apply LT.
right.
tauto.
apply EQ.
split;trivial.
apply GT.
right;auto.
apply GT.
left;trivial.
Defined.
Definition eq_dec : forall (x y: t), { eq x y } + { ~ eq x y}.
Proof.
intros [xa xb] [ya yb]; simpl.
destruct (X.eq_dec xa ya).
destruct (Y.eq_dec xb yb).
+ left; now split.
+ right. now intros [eqa eqb].
+ right. now intros [eqa eqb].
Defined.
Hint Immediate eq_sym.
Hint Resolve eq_refl eq_trans lt_not_eq lt_trans.
End OrderedPair.
Module MessageSpi.
Inductive message : Set :=
| MNam : nat -> message.
Definition t := message.
Fixpoint message_lt (m n:message) {struct m} : Prop :=
match (m,n) with
| (MNam n1,MNam n2) => n1 < n2
end.
Module Ord <: OrderedType with Definition t := message with Definition eq :=
@eq message.
Definition t := message.
Definition eq := @eq message.
Definition lt := message_lt.
Lemma eq_refl : forall (x:t),eq x x.
unfold eq;auto.
Qed.
Lemma eq_sym : forall (x y:t),(eq x y )->(eq y x).
unfold eq;auto.
Qed.
Lemma eq_trans : forall (x y z:t),(eq x y)->(eq y z)->(eq x z).
unfold eq;auto;intros;congruence.
Qed.
Lemma lt_trans : forall (x y z:t),(lt x y)->(lt y z)->(lt x z).
unfold lt.
induction x;destruct y;simpl;try tauto;destruct z;try tauto;intros.
omega.
Qed.
Lemma lt_not_eq : forall (x y:t),(lt x y)->~(eq x y).
unfold eq;unfold lt.
induction x;destruct y;simpl;try tauto;intro;red;intro;try (discriminate
H0);injection H0;intros.
elim (lt_irrefl n);try omega.
Qed.
Definition compare : forall (x y:t),(Compare lt eq x y).
unfold lt, eq.
induction x;destruct y;intros;try (apply LT;simpl;trivial;fail);try
(apply
GT;simpl;trivial;fail).
case (lt_eq_lt_dec n n0);intros;try (case s;clear s;intros).
apply LT;trivial.
apply EQ;trivial.
rewrite e;trivial.
apply GT;trivial.
Defined.
Definition eq_dec : forall (x y: t), { eq x y } + { ~ eq x y}.
Proof.
intros [i] [j]. unfold eq.
destruct (eq_nat_dec i j).
+ left. now f_equal.
+ right. intros meq; now inversion meq.
Defined.
Hint Immediate eq_sym.
Hint Resolve eq_refl eq_trans lt_not_eq lt_trans.
End Ord.
Theorem eq_dec : forall (m n:message),{m=n}+{~(m=n)}.
intros.
case (Ord.compare m n);intro;[right | left | right];try (red;intro).
elim (Ord.lt_not_eq m n);auto.
rewrite e;auto.
elim (Ord.lt_not_eq n m);auto.
Defined.
End MessageSpi.
Module MessagePair := OrderedPair MessageSpi.Ord MessageSpi.Ord.
Module Type Hedge := FSetInterface.S with Module E := MessagePair.
Module A (H:Hedge).
Definition hedge := H.t.
Definition message_relation := relation MessageSpi.message.
Definition relation_of_hedge (h:hedge) (m n:MessageSpi.message) := H.In (m,n)
h.
Inductive hedge_synthesis_relation (h:message_relation) : message_relation :=
| SynInc : forall (m n:MessageSpi.message),(h m
n)->(hedge_synthesis_relation h m n).
Fixpoint hedge_in_synthesis (h:hedge) (m:MessageSpi.message)
(n:MessageSpi.message) {struct m} : bool :=
if H.mem (m,n) h
then true
else false.
Definition hedge_synthesis_spec (h:hedge) := hedge_synthesis_relation
(relation_of_hedge h).
Lemma hedge_in_synthesis_impl_hedge_synthesis_spec : forall (h:hedge),forall
(m n:MessageSpi.message),(hedge_in_synthesis h m n)=true->(hedge_synthesis_spec
h m n).
unfold hedge_synthesis_spec;unfold relation_of_hedge.
induction m;simpl;intro.
elim (Bool_elim_bool (H.mem (MessageSpi.MNam n,n0) h));intros.
apply SynInc;apply H.mem_2;trivial.
rewrite H in H0. (* !! possible here !! *)
discriminate H0.
Qed.
End A.
Module B (H:Hedge).
Definition hedge := H.t.
Definition message_relation := relation MessageSpi.t.
Definition relation_of_hedge (h:hedge) (m n:MessageSpi.t) := H.In (m,n) h.
Inductive hedge_synthesis_relation (h:message_relation) : message_relation :=
| SynInc : forall (m n:MessageSpi.t),(h m n)->(hedge_synthesis_relation h m
n).
Fixpoint hedge_in_synthesis (h:hedge) (m:MessageSpi.t) (n:MessageSpi.t)
{struct m} : bool :=
if H.mem (m,n) h
then true
else false.
Definition hedge_synthesis_spec (h:hedge) := hedge_synthesis_relation
(relation_of_hedge h).
Lemma hedge_in_synthesis_impl_hedge_synthesis_spec : forall (h:hedge),forall
(m n:MessageSpi.t),(hedge_in_synthesis h m n)=true->(hedge_synthesis_spec h m
n).
unfold hedge_synthesis_spec;unfold relation_of_hedge.
induction m;simpl;intro.
elim (Bool_elim_bool (H.mem (MessageSpi.MNam n,n0) h));intros.
apply SynInc;apply H.mem_2;trivial.
rewrite H in H0. discriminate. (* !! impossible here !! *)
Qed.
End B.
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