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FSet library: subset is an operator rather than a predicate #3

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strub merged 1 commit into from
Feb 21, 2018
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FSet library: subset is an operator rather than a predicate #3

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33 changes: 17 additions & 16 deletions theories/datatypes/FSet.ec
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Original file line number Diff line number Diff line change
Expand Up @@ -25,6 +25,7 @@ axiom oflistK (s : 'a list): perm_eq (undup s) (elems (oflist s)).

axiom fset_eq (s1 s2 : 'a fset):
(perm_eq (elems s1) (elems s2)) => (s1 = s2).

(* -------------------------------------------------------------------- *)
lemma oflist_uniq (s : 'a list) : uniq s =>
perm_eq s (elems (oflist s)).
Expand Down Expand Up @@ -235,19 +236,19 @@ lemma filter_memC (A B : 'a fset): filter (predC (mem A)) B = B `\` A.
proof. by apply/fsetP=> x; rewrite in_fsetD in_filter andbC. qed.

(* -------------------------------------------------------------------- *)
pred (<=) (s1 s2 : 'a fset) = mem s1 <= mem s2.
pred (< ) (s1 s2 : 'a fset) = mem s1 < mem s2.
op (\subset) (s1 s2 : 'a fset) = forall x, x \in s1 => x \in s2.
op (\proper) (s1 s2 : 'a fset) = s1 \subset s2 /\ s1 <> s2.

lemma nosmt subsetP (s1 s2 : 'a fset):
(s1 <= s2) <=> (mem s1 <= mem s2). (* FIXME: This is the definition *)
(s1 \subset s2) <=> (mem s1 <= mem s2).
proof. by []. qed.

lemma nosmt subset_trans (s1 s2 s3 : 'a fset):
s1 <= s2 => s2 <= s3 => s1 <= s3.
s1 \subset s2 => s2 \subset s3 => s1 \subset s3.
proof. by move=> H1 H2 ? H3;apply /H2/H1. qed.

(* -------------------------------------------------------------------- *)
lemma nosmt eqEsubset (A B : 'a fset) : (A = B) <=> (A <= B) /\ (B <= A).
lemma nosmt eqEsubset (A B : 'a fset) : (A = B) <=> (A \subset B) /\ (B \subset A).
proof. by rewrite fsetP 2!subsetP subpred_eqP. qed.

(* -------------------------------------------------------------------- *)
Expand Down Expand Up @@ -416,18 +417,18 @@ lemma fsetDKv (A B : 'a fset) : (A `&` B) `\` B = fset0.
proof. by rewrite fsetDIl fsetDv fsetI0. qed.

(* -------------------------------------------------------------------- *)
lemma subsetIl (A B : 'a fset) : (A `&` B) <= A.
lemma subsetIl (A B : 'a fset) : (A `&` B) \subset A.
proof. by apply/subsetP=> x; rewrite inE; case. qed.

lemma subsetIr (A B : 'a fset) : (A `&` B) <= B.
lemma subsetIr (A B : 'a fset) : (A `&` B) \subset B.
proof. by apply/subsetP=> x; rewrite inE; case. qed.

(* -------------------------------------------------------------------- *)
lemma subsetDl (A B : 'a fset) : A `\` B <= A.
lemma subsetDl (A B : 'a fset) : A `\` B \subset A.
proof. by apply/subsetP=> x; rewrite !inE; case. qed.

(* -------------------------------------------------------------------- *)
lemma sub0set (A : 'a fset) : fset0 <= A.
lemma sub0set (A : 'a fset) : fset0 \subset A.
proof. by apply/subsetP=> x; rewrite !inE. qed.

(* -------------------------------------------------------------------- *)
Expand Down Expand Up @@ -495,21 +496,21 @@ qed.

(* -------------------------------------------------------------------- *)
lemma subset_fsetU_id (A B : 'a fset) :
A <= B => A `|` B = B.
A \subset B => A `|` B = B.
proof.
move=> A_le_B; apply/fsetP=> x; rewrite in_fsetU.
by split=> [[/A_le_B|->]|->].
qed.

lemma subset_fsetI_id (A B : 'a fset) :
B <= A => A `&` B = B.
B \subset A => A `&` B = B.
proof.
move=> B_le_A; apply/fsetP=> x; rewrite in_fsetI.
by split=> [[_ ->]|^x_in_B /B_le_A].
qed.

lemma subset_leq_fcard (A B : 'a fset) :
A <= B => card A <= card B.
A \subset B => card A <= card B.
proof.
move=> /subsetP le_AB; rewrite -(fcardID B A).
have ->: B `&` A = A; 1: apply/fsetP=> x.
Expand All @@ -519,7 +520,7 @@ qed.

(* -------------------------------------------------------------------- *)
lemma subset_cardP (A B : 'a fset) :
card A = card B => (A = B <=> A <= B).
card A = card B => (A = B <=> A \subset B).
proof. (* FIXME: views *)
move=> eqcAB; split=> [->// |]; rewrite eqEsubset.
move=> le_AB; split=> // x Bx; case: (mem A x) => // Ax.
Expand All @@ -531,7 +532,7 @@ qed.

(* -------------------------------------------------------------------- *)
lemma nosmt eqEcard (A B : 'a fset) :
(A = B) <=> (A <= B) /\ (card B <= card A).
(A = B) <=> (A \subset B) /\ (card B <= card A).
proof.
split=> [->// |]; case=> le_AB le_cAB; rewrite subset_cardP //.
by rewrite eqr_le le_cAB /=; apply/subset_leq_fcard.
Expand Down Expand Up @@ -583,15 +584,15 @@ proof.
qed.

lemma nosmt imageI (f : 'a -> 'b) (A B : 'a fset):
image f (A `&` B) <= image f A `&` image f B.
image f (A `&` B) \subset image f A `&` image f B.
proof.
move=> x; rewrite !inE !imageP=> -[a].
rewrite !inE=> -[[a_in_A a_in_B] <-].
by split; exists a.
qed.

lemma nosmt imageD (f : 'a -> 'b) (A B : 'a fset):
image f A `\` image f B <= image f (A `\` B).
image f A `\` image f B \subset image f (A `\` B).
proof.
move=> x; rewrite !inE !imageP=> -[[a] [a_in_A <-]].
move=> h; exists a; rewrite !inE a_in_A /= -negP=> a_in_B.
Expand Down