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DataAbstraction: FindDuplicates

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Adam Chlipala 2017-02-12 16:27:57 -05:00
parent d09f1abe92
commit f14ed26afa
1 changed files with 162 additions and 12 deletions
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174
DataAbstraction.v
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@ -989,16 +989,18 @@ Module Type FINITE_SET.
Parameter t : Set.
Parameter empty : t.
Parameter insert : t -> key -> t.
Parameter add : t -> key -> t.
Parameter member : t -> key -> bool.
Axiom member_empty : forall k, member empty k = false.
Axiom member_insert_eq : forall k s,
member (insert s k) k = true.
Axiom member_insert_noteq : forall k1 k2 s,
Axiom member_add_eq : forall k s,
member (add s k) k = true.
Axiom member_add_noteq : forall k1 k2 s,
k1 <> k2
-> member (insert s k1) k2 = member s k2.
-> member (add s k1) k2 = member s k2.
Axiom decidable_equality : forall a b : key, a = b \/ a <> b.
End FINITE_SET.
Module Type SET_WITH_EQUALITY.
@ -1008,12 +1010,12 @@ Module Type SET_WITH_EQUALITY.
Axiom equal_ok : forall a b, if equal a b then a = b else a <> b.
End SET_WITH_EQUALITY.
Module ListSet(SE : SET_WITH_EQUALITY) : FINITE_SET with Definition key := SE.t.
Module ListSet(SE : SET_WITH_EQUALITY) <: FINITE_SET with Definition key := SE.t.
Definition key := SE.t.
Definition t := list SE.t.
Definition empty : t := [].
Definition insert (s : t) (k : key) : t := k :: s.
Definition add (s : t) (k : key) : t := k :: s.
Fixpoint member (s : t) (k : key) : bool :=
match s with
| [] => false
@ -1026,8 +1028,8 @@ Module ListSet(SE : SET_WITH_EQUALITY) : FINITE_SET with Definition key := SE.t.
equality.
Qed.
Theorem member_insert_eq : forall k s,
member (insert s k) k = true.
Theorem member_add_eq : forall k s,
member (add s k) k = true.
Proof.
simplify.
pose proof (SE.equal_ok k k).
@ -1036,9 +1038,9 @@ Module ListSet(SE : SET_WITH_EQUALITY) : FINITE_SET with Definition key := SE.t.
equality.
Qed.
Theorem member_insert_noteq : forall k1 k2 s,
Theorem member_add_noteq : forall k1 k2 s,
k1 <> k2
-> member (insert s k1) k2 = member s k2.
-> member (add s k1) k2 = member s k2.
Proof.
simplify.
pose proof (SE.equal_ok k1 k2).
@ -1046,9 +1048,16 @@ Module ListSet(SE : SET_WITH_EQUALITY) : FINITE_SET with Definition key := SE.t.
equality.
equality.
Qed.
Theorem decidable_equality : forall a b : key, a = b \/ a <> b.
Proof.
simplify.
pose proof (SE.equal_ok a b).
cases (SE.equal a b); propositional.
Qed.
End ListSet.
Module NatWithEquality : SET_WITH_EQUALITY with Definition t := nat.
Module NatWithEquality <: SET_WITH_EQUALITY with Definition t := nat.
Definition t := nat.
Fixpoint equal (a b : nat) : bool :=
@ -1076,3 +1085,144 @@ Module NatWithEquality : SET_WITH_EQUALITY with Definition t := nat.
End NatWithEquality.
Module NatSet := ListSet(NatWithEquality).
Module FindDuplicates (FS : FINITE_SET).
Fixpoint noDuplicates' (ls : list FS.key) (s : FS.t) : bool :=
match ls with
| [] => true
| x :: ls' => negb (FS.member s x) && noDuplicates' ls' (FS.add s x)
end.
Definition noDuplicates (ls : list FS.key) := noDuplicates' ls FS.empty.
Definition hasDuplicate (ls : list FS.key) :=
exists ls1 a ls2 ls3, ls = ls1 ++ a :: ls2 ++ a :: ls3.
Definition contains (a : FS.key) (ls : list FS.key) :=
exists ls1 ls2, ls = ls1 ++ a :: ls2.
Lemma noDuplicates'_ok : forall ls s, if noDuplicates' ls s
then ~(hasDuplicate ls
\/ exists a, FS.member s a = true
/\ contains a ls)
else (hasDuplicate ls
\/ exists a, FS.member s a = true
/\ contains a ls).
Proof.
induct ls; simplify.
unfold hasDuplicate, contains.
propositional.
first_order.
cases x; simplify.
equality.
equality.
first_order.
cases x0; simplify.
equality.
equality.
cases (FS.member s a); simplify.
right.
exists a.
propositional.
unfold contains.
exists [].
exists ls.
simplify.
equality.
specialize (IHls (FS.add s a)).
cases (noDuplicates' ls (FS.add s a)).
propositional.
apply H1.
exists a.
propositional.
apply FS.member_add_eq.
unfold hasDuplicate, contains in *.
first_order.
cases x; simplify.
invert H0.
exists x1.
exists x2.
equality.
invert H0.
exfalso.
apply H with (x3 := x).
exists x0.
exists x1.
exists x2.
equality.
first_order.
apply H1 with x.
propositional.
pose proof (FS.decidable_equality a x).
propositional.
rewrite H4.
apply FS.member_add_eq.
rewrite FS.member_add_noteq.
assumption.
assumption.
cases x0; simplify.
equality.
invert H2.
exists x0.
exists x1.
equality.
first_order.
left.
exists (a :: x).
exists x0.
exists x1.
exists x2.
simplify.
equality.
cases (FS.member s x).
right.
exists x.
propositional.
exists (a :: x0).
exists x1.
simplify.
equality.
left.
pose proof (FS.decidable_equality a x).
propositional.
exists nil.
exists a.
exists x0.
exists x1.
simplify.
equality.
rewrite FS.member_add_noteq in H.
equality.
assumption.
Qed.
Theorem noDuplicates_ok : forall ls, if noDuplicates ls
then ~hasDuplicate ls
else hasDuplicate ls.
Proof.
simplify.
pose proof (noDuplicates'_ok ls FS.empty).
unfold noDuplicates.
cases (noDuplicates' ls FS.empty); first_order.
rewrite FS.member_empty in H.
equality.
Qed.
End FindDuplicates.
Module NatDuplicateFinder := FindDuplicates(NatSet).
Compute NatDuplicateFinder.noDuplicates [].
Compute NatDuplicateFinder.noDuplicates [1].
Compute NatDuplicateFinder.noDuplicates [1; 2].
Compute NatDuplicateFinder.noDuplicates [1; 2; 3].
Compute NatDuplicateFinder.noDuplicates [1; 2; 1; 3].