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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Fset.v,v 1.1.2.1 2004/07/16 19:31:27 herbelin Exp $ i*)
(*s Sets operations on maps *)
Require Bool.
Require Sumbool.
Require ZArith.
Require Addr.
Require Adist.
Require Addec.
Require Map.
Section Dom.
Variable A, B : Set.
Fixpoint MapDomRestrTo [m:(Map A)] : (Map B) -> (Map A) :=
Cases m of
M0 => [_:(Map B)] (M0 A)
| (M1 a y) => [m':(Map B)] Cases (MapGet B m' a) of
NONE => (M0 A)
| _ => m
end
| (M2 m1 m2) => [m':(Map B)] Cases m' of
M0 => (M0 A)
| (M1 a' y') => Cases (MapGet A m a') of
NONE => (M0 A)
| (SOME y) => (M1 A a' y)
end
| (M2 m'1 m'2) => (makeM2 A (MapDomRestrTo m1 m'1)
(MapDomRestrTo m2 m'2))
end
end.
Lemma MapDomRestrTo_semantics : (m:(Map A)) (m':(Map B))
(eqm A (MapGet A (MapDomRestrTo m m'))
[a0:ad] Cases (MapGet B m' a0) of
NONE => (NONE A)
| _ => (MapGet A m a0)
end).
Proof.
Unfold eqm. Induction m. Simpl. Intros. Case (MapGet B m' a); Trivial.
Intros. Simpl. Elim (sumbool_of_bool (ad_eq a a1)). Intro H. Rewrite H.
Rewrite <- (ad_eq_complete ? ? H). Case (MapGet B m' a). Reflexivity.
Intro. Apply M1_semantics_1.
Intro H. Rewrite H. Case (MapGet B m' a).
Case (MapGet B m' a1); Reflexivity.
Case (MapGet B m' a1); Intros; Exact (M1_semantics_2 A a a1 a0 H).
Induction m'. Trivial.
Unfold MapDomRestrTo. Intros. Elim (sumbool_of_bool (ad_eq a a1)).
Intro H1.
Rewrite (ad_eq_complete ? ? H1). Rewrite (M1_semantics_1 B a1 a0).
Case (MapGet A (M2 A m0 m1) a1). Reflexivity.
Intro. Apply M1_semantics_1.
Intro H1. Rewrite (M1_semantics_2 B a a1 a0 H1). Case (MapGet A (M2 A m0 m1) a). Reflexivity.
Intro. Exact (M1_semantics_2 A a a1 a2 H1).
Intros. Change (MapGet A (makeM2 A (MapDomRestrTo m0 m2) (MapDomRestrTo m1 m3)) a)
=(Cases (MapGet B (M2 B m2 m3) a) of
NONE => (NONE A)
| (SOME _) => (MapGet A (M2 A m0 m1) a)
end).
Rewrite (makeM2_M2 A (MapDomRestrTo m0 m2) (MapDomRestrTo m1 m3) a).
Rewrite MapGet_M2_bit_0_if. Rewrite (H0 m3 (ad_div_2 a)). Rewrite (H m2 (ad_div_2 a)).
Rewrite (MapGet_M2_bit_0_if B m2 m3 a). Rewrite (MapGet_M2_bit_0_if A m0 m1 a).
Case (ad_bit_0 a); Reflexivity.
Qed.
Fixpoint MapDomRestrBy [m:(Map A)] : (Map B) -> (Map A) :=
Cases m of
M0 => [_:(Map B)] (M0 A)
| (M1 a y) => [m':(Map B)] Cases (MapGet B m' a) of
NONE => m
| _ => (M0 A)
end
| (M2 m1 m2) => [m':(Map B)] Cases m' of
M0 => m
| (M1 a' y') => (MapRemove A m a')
| (M2 m'1 m'2) => (makeM2 A (MapDomRestrBy m1 m'1)
(MapDomRestrBy m2 m'2))
end
end.
Lemma MapDomRestrBy_semantics : (m:(Map A)) (m':(Map B))
(eqm A (MapGet A (MapDomRestrBy m m'))
[a0:ad] Cases (MapGet B m' a0) of
NONE => (MapGet A m a0)
| _ => (NONE A)
end).
Proof.
Unfold eqm. Induction m. Simpl. Intros. Case (MapGet B m' a); Trivial.
Intros. Simpl. Elim (sumbool_of_bool (ad_eq a a1)). Intro H. Rewrite H.
Rewrite (ad_eq_complete ? ? H). Case (MapGet B m' a1). Apply M1_semantics_1.
Trivial.
Intro H. Rewrite H. Case (MapGet B m' a). Rewrite (M1_semantics_2 A a a1 a0 H).
Case (MapGet B m' a1); Trivial.
Case (MapGet B m' a1); Trivial.
Induction m'. Trivial.
Unfold MapDomRestrBy. Intros. Rewrite (MapRemove_semantics A (M2 A m0 m1) a a1).
Elim (sumbool_of_bool (ad_eq a a1)). Intro H1. Rewrite H1. Rewrite (ad_eq_complete ? ? H1).
Rewrite (M1_semantics_1 B a1 a0). Reflexivity.
Intro H1. Rewrite H1. Rewrite (M1_semantics_2 B a a1 a0 H1). Reflexivity.
Intros. Change (MapGet A (makeM2 A (MapDomRestrBy m0 m2) (MapDomRestrBy m1 m3)) a)
=(Cases (MapGet B (M2 B m2 m3) a) of
NONE => (MapGet A (M2 A m0 m1) a)
| (SOME _) => (NONE A)
end).
Rewrite (makeM2_M2 A (MapDomRestrBy m0 m2) (MapDomRestrBy m1 m3) a).
Rewrite MapGet_M2_bit_0_if. Rewrite (H0 m3 (ad_div_2 a)). Rewrite (H m2 (ad_div_2 a)).
Rewrite (MapGet_M2_bit_0_if B m2 m3 a). Rewrite (MapGet_M2_bit_0_if A m0 m1 a).
Case (ad_bit_0 a); Reflexivity.
Qed.
Definition in_dom := [a:ad; m:(Map A)]
Cases (MapGet A m a) of
NONE => false
| _ => true
end.
Lemma in_dom_M0 : (a:ad) (in_dom a (M0 A))=false.
Proof.
Trivial.
Qed.
Lemma in_dom_M1 : (a,a0:ad) (y:A) (in_dom a0 (M1 A a y))=(ad_eq a a0).
Proof.
Unfold in_dom. Intros. Simpl. Case (ad_eq a a0); Reflexivity.
Qed.
Lemma in_dom_M1_1 : (a:ad) (y:A) (in_dom a (M1 A a y))=true.
Proof.
Intros. Rewrite in_dom_M1. Apply ad_eq_correct.
Qed.
Lemma in_dom_M1_2 : (a,a0:ad) (y:A) (in_dom a0 (M1 A a y))=true -> a=a0.
Proof.
Intros. Apply (ad_eq_complete a a0). Rewrite (in_dom_M1 a a0 y) in H. Assumption.
Qed.
Lemma in_dom_some : (m:(Map A)) (a:ad) (in_dom a m)=true ->
{y:A | (MapGet A m a)=(SOME A y)}.
Proof.
Unfold in_dom. Intros. Elim (option_sum ? (MapGet A m a)). Trivial.
Intro H0. Rewrite H0 in H. Discriminate H.
Qed.
Lemma in_dom_none : (m:(Map A)) (a:ad) (in_dom a m)=false ->
(MapGet A m a)=(NONE A).
Proof.
Unfold in_dom. Intros. Elim (option_sum ? (MapGet A m a)). Intro H0. Elim H0.
Intros y H1. Rewrite H1 in H. Discriminate H.
Trivial.
Qed.
Lemma in_dom_put : (m:(Map A)) (a0:ad) (y0:A) (a:ad)
(in_dom a (MapPut A m a0 y0))=(orb (ad_eq a a0) (in_dom a m)).
Proof.
Unfold in_dom. Intros. Rewrite (MapPut_semantics A m a0 y0 a).
Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H.
Rewrite H. Rewrite orb_true_b. Reflexivity.
Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. Rewrite H. Rewrite orb_false_b.
Reflexivity.
Qed.
Lemma in_dom_put_behind : (m:(Map A)) (a0:ad) (y0:A) (a:ad)
(in_dom a (MapPut_behind A m a0 y0))=(orb (ad_eq a a0) (in_dom a m)).
Proof.
Unfold in_dom. Intros. Rewrite (MapPut_behind_semantics A m a0 y0 a).
Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H.
Rewrite H. Case (MapGet A m a); Reflexivity.
Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. Rewrite H. Case (MapGet A m a); Trivial.
Qed.
Lemma in_dom_remove : (m:(Map A)) (a0:ad) (a:ad)
(in_dom a (MapRemove A m a0))=(andb (negb (ad_eq a a0)) (in_dom a m)).
Proof.
Unfold in_dom. Intros. Rewrite (MapRemove_semantics A m a0 a).
Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H.
Rewrite H. Reflexivity.
Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. Rewrite H.
Case (MapGet A m a); Reflexivity.
Qed.
Lemma in_dom_merge : (m,m':(Map A)) (a:ad)
(in_dom a (MapMerge A m m'))=(orb (in_dom a m) (in_dom a m')).
Proof.
Unfold in_dom. Intros. Rewrite (MapMerge_semantics A m m' a).
Elim (option_sum A (MapGet A m' a)). Intro H. Elim H. Intros y H0. Rewrite H0.
Case (MapGet A m a); Reflexivity.
Intro H. Rewrite H. Rewrite orb_b_false. Reflexivity.
Qed.
Lemma in_dom_delta : (m,m':(Map A)) (a:ad)
(in_dom a (MapDelta A m m'))=(xorb (in_dom a m) (in_dom a m')).
Proof.
Unfold in_dom. Intros. Rewrite (MapDelta_semantics A m m' a).
Elim (option_sum A (MapGet A m' a)). Intro H. Elim H. Intros y H0. Rewrite H0.
Case (MapGet A m a); Reflexivity.
Intro H. Rewrite H. Case (MapGet A m a); Reflexivity.
Qed.
End Dom.
Section InDom.
Variable A, B : Set.
Lemma in_dom_restrto : (m:(Map A)) (m':(Map B)) (a:ad)
(in_dom A a (MapDomRestrTo A B m m'))=(andb (in_dom A a m) (in_dom B a m')).
Proof.
Unfold in_dom. Intros. Rewrite (MapDomRestrTo_semantics A B m m' a).
Elim (option_sum B (MapGet B m' a)). Intro H. Elim H. Intros y H0. Rewrite H0.
Rewrite andb_b_true. Reflexivity.
Intro H. Rewrite H. Rewrite andb_b_false. Reflexivity.
Qed.
Lemma in_dom_restrby : (m:(Map A)) (m':(Map B)) (a:ad)
(in_dom A a (MapDomRestrBy A B m m'))=(andb (in_dom A a m) (negb (in_dom B a m'))).
Proof.
Unfold in_dom. Intros. Rewrite (MapDomRestrBy_semantics A B m m' a).
Elim (option_sum B (MapGet B m' a)). Intro H. Elim H. Intros y H0. Rewrite H0.
Unfold negb. Rewrite andb_b_false. Reflexivity.
Intro H. Rewrite H. Unfold negb. Rewrite andb_b_true. Reflexivity.
Qed.
End InDom.
Definition FSet := (Map unit).
Section FSetDefs.
Variable A : Set.
Definition in_FSet : ad -> FSet -> bool := (in_dom unit).
Fixpoint MapDom [m:(Map A)] : FSet :=
Cases m of
M0 => (M0 unit)
| (M1 a _) => (M1 unit a tt)
| (M2 m m') => (M2 unit (MapDom m) (MapDom m'))
end.
Lemma MapDom_semantics_1 : (m:(Map A)) (a:ad)
(y:A) (MapGet A m a)=(SOME A y) -> (in_FSet a (MapDom m))=true.
Proof.
Induction m. Intros. Discriminate H.
Unfold MapDom. Unfold in_FSet. Unfold in_dom. Unfold MapGet. Intros a y a0 y0.
Case (ad_eq a a0). Trivial.
Intro. Discriminate H.
Intros m0 H m1 H0 a y. Rewrite (MapGet_M2_bit_0_if A m0 m1 a). Simpl. Unfold in_FSet.
Unfold in_dom. Rewrite (MapGet_M2_bit_0_if unit (MapDom m0) (MapDom m1) a).
Case (ad_bit_0 a). Unfold in_FSet in_dom in H0. Intro. Apply H0 with y:=y. Assumption.
Unfold in_FSet in_dom in H. Intro. Apply H with y:=y. Assumption.
Qed.
Lemma MapDom_semantics_2 : (m:(Map A)) (a:ad)
(in_FSet a (MapDom m))=true -> {y:A | (MapGet A m a)=(SOME A y)}.
Proof.
Induction m. Intros. Discriminate H.
Unfold MapDom. Unfold in_FSet. Unfold in_dom. Unfold MapGet. Intros a y a0. Case (ad_eq a a0).
Intro. Split with y. Reflexivity.
Intro. Discriminate H.
Intros m0 H m1 H0 a. Rewrite (MapGet_M2_bit_0_if A m0 m1 a). Simpl. Unfold in_FSet.
Unfold in_dom. Rewrite (MapGet_M2_bit_0_if unit (MapDom m0) (MapDom m1) a).
Case (ad_bit_0 a). Unfold in_FSet in_dom in H0. Intro. Apply H0. Assumption.
Unfold in_FSet in_dom in H. Intro. Apply H. Assumption.
Qed.
Lemma MapDom_semantics_3 : (m:(Map A)) (a:ad)
(MapGet A m a)=(NONE A) -> (in_FSet a (MapDom m))=false.
Proof.
Intros. Elim (sumbool_of_bool (in_FSet a (MapDom m))). Intro H0.
Elim (MapDom_semantics_2 m a H0). Intros y H1. Rewrite H in H1. Discriminate H1.
Trivial.
Qed.
Lemma MapDom_semantics_4 : (m:(Map A)) (a:ad)
(in_FSet a (MapDom m))=false -> (MapGet A m a)=(NONE A).
Proof.
Intros. Elim (option_sum A (MapGet A m a)). Intro H0. Elim H0. Intros y H1.
Rewrite (MapDom_semantics_1 m a y H1) in H. Discriminate H.
Trivial.
Qed.
Lemma MapDom_Dom : (m:(Map A)) (a:ad) (in_dom A a m)=(in_FSet a (MapDom m)).
Proof.
Intros. Elim (sumbool_of_bool (in_FSet a (MapDom m))). Intro H.
Elim (MapDom_semantics_2 m a H). Intros y H0. Rewrite H. Unfold in_dom. Rewrite H0.
Reflexivity.
Intro H. Rewrite H. Unfold in_dom. Rewrite (MapDom_semantics_4 m a H). Reflexivity.
Qed.
Definition FSetUnion : FSet -> FSet -> FSet := [s,s':FSet] (MapMerge unit s s').
Lemma in_FSet_union : (s,s':FSet) (a:ad)
(in_FSet a (FSetUnion s s'))=(orb (in_FSet a s) (in_FSet a s')).
Proof.
Exact (in_dom_merge unit).
Qed.
Definition FSetInter : FSet -> FSet -> FSet := [s,s':FSet] (MapDomRestrTo unit unit s s').
Lemma in_FSet_inter : (s,s':FSet) (a:ad)
(in_FSet a (FSetInter s s'))=(andb (in_FSet a s) (in_FSet a s')).
Proof.
Exact (in_dom_restrto unit unit).
Qed.
Definition FSetDiff : FSet -> FSet -> FSet := [s,s':FSet] (MapDomRestrBy unit unit s s').
Lemma in_FSet_diff : (s,s':FSet) (a:ad)
(in_FSet a (FSetDiff s s'))=(andb (in_FSet a s) (negb (in_FSet a s'))).
Proof.
Exact (in_dom_restrby unit unit).
Qed.
Definition FSetDelta : FSet -> FSet -> FSet := [s,s':FSet] (MapDelta unit s s').
Lemma in_FSet_delta : (s,s':FSet) (a:ad)
(in_FSet a (FSetDelta s s'))=(xorb (in_FSet a s) (in_FSet a s')).
Proof.
Exact (in_dom_delta unit).
Qed.
End FSetDefs.
Lemma FSet_Dom : (s:FSet) (MapDom unit s)=s.
Proof.
Induction s. Trivial.
Simpl. Intros a t. Elim t. Reflexivity.
Intros. Simpl. Rewrite H. Rewrite H0. Reflexivity.
Qed.
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