DataAbstraction: range sets

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Adam Chlipala 2017-02-12 17:49:30 -05:00
parent f14ed26afa
commit cf4d06c222

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@ -1226,3 +1226,331 @@ Compute NatDuplicateFinder.noDuplicates [1].
Compute NatDuplicateFinder.noDuplicates [1; 2]. Compute NatDuplicateFinder.noDuplicates [1; 2].
Compute NatDuplicateFinder.noDuplicates [1; 2; 3]. Compute NatDuplicateFinder.noDuplicates [1; 2; 3].
Compute NatDuplicateFinder.noDuplicates [1; 2; 1; 3]. Compute NatDuplicateFinder.noDuplicates [1; 2; 1; 3].
Module NatRangeSet <: FINITE_SET with Definition key := nat.
Definition key := nat.
Inductive rangeSet : Set :=
| Empty
| Range (from to : nat)
| AdHoc (s : NatSet.t).
Definition t := rangeSet.
Fixpoint fromRange' (from to : nat) : NatSet.t :=
match to with
| 0 => NatSet.add NatSet.empty 0
| S to' => if NatWithEquality.equal to from
then NatSet.add NatSet.empty to
else NatSet.add (fromRange' from to') (S to')
end.
Definition fromRange (from to : nat) : NatSet.t :=
if Compare_dec.leb from to
then fromRange' from to
else NatSet.empty.
Definition empty : t := Empty.
Definition add (s : t) (k : key) : t :=
match s with
| Empty => Range k k
| Range from to =>
if Compare_dec.leb from k && Compare_dec.leb k to
then s
else if NatWithEquality.equal k (from - 1) && Compare_dec.leb from to
then Range k to
else if NatWithEquality.equal k (to + 1) && Compare_dec.leb from to
then Range from k
else AdHoc (NatSet.add (fromRange from to) k)
| AdHoc s' => AdHoc (NatSet.add s' k)
end.
Definition member (s : t) (k : key) : bool :=
match s with
| Empty => false
| Range from to => Compare_dec.leb from to && Compare_dec.leb from k && Compare_dec.leb k to
| AdHoc s' => NatSet.member s' k
end.
Theorem member_empty : forall k, member empty k = false.
Proof.
simplify.
equality.
Qed.
Lemma member_fromRange' : forall k from to,
from <= to
-> NatSet.member (fromRange' from to) k = Compare_dec.leb from k && Compare_dec.leb k to.
Proof.
induct to; simplify.
cases k; simplify.
rewrite Compare_dec.leb_correct by assumption.
equality.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
cases from; simplify.
cases k; simplify.
apply IHto.
linear_arithmetic.
pose proof (NatWithEquality.equal_ok to k).
cases (NatWithEquality.equal to k); simplify.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
rewrite IHto by linear_arithmetic.
cases to.
rewrite Compare_dec.leb_correct_conv by linear_arithmetic.
equality.
cases (Compare_dec.leb k to).
apply Compare_dec.leb_complete in Heq0.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
apply Compare_dec.leb_complete_conv in Heq0.
rewrite Compare_dec.leb_correct_conv by linear_arithmetic.
equality.
pose proof (NatWithEquality.equal_ok to from).
cases (NatWithEquality.equal to from); simplify.
cases k; simplify.
equality.
pose proof (NatWithEquality.equal_ok to k).
cases (NatWithEquality.equal to k); simplify.
rewrite Compare_dec.leb_correct by linear_arithmetic.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
cases (Compare_dec.leb from k); simplify.
apply Compare_dec.leb_complete in Heq1.
rewrite Compare_dec.leb_correct_conv by linear_arithmetic.
equality.
equality.
cases k; simplify.
apply IHto.
linear_arithmetic.
rewrite IHto by linear_arithmetic.
pose proof (NatWithEquality.equal_ok to k).
cases (NatWithEquality.equal to k); simplify.
rewrite Compare_dec.leb_correct by linear_arithmetic.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
cases to.
rewrite (Compare_dec.leb_correct_conv 0 k) by linear_arithmetic.
equality.
cases (Compare_dec.leb k to).
apply Compare_dec.leb_complete in Heq1.
rewrite (Compare_dec.leb_correct k (S to)) by linear_arithmetic.
equality.
apply Compare_dec.leb_complete_conv in Heq1.
rewrite (Compare_dec.leb_correct_conv (S to) k) by linear_arithmetic.
equality.
Qed.
Theorem member_add_eq : forall k s,
member (add s k) k = true.
Proof.
unfold member, add; simplify.
cases s.
SearchAbout Compare_dec.leb.
rewrite Compare_dec.leb_correct.
equality.
linear_arithmetic.
cases (Compare_dec.leb from k); simplify.
cases (Compare_dec.leb k to); simplify.
rewrite Heq.
rewrite Heq0.
apply Compare_dec.leb_complete in Heq.
apply Compare_dec.leb_complete in Heq0.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
pose proof (NatWithEquality.equal_ok k (from - 1)).
cases (NatWithEquality.equal k (from - 1)).
apply leb_complete in Heq.
apply leb_complete_conv in Heq0.
linear_arithmetic.
simplify.
pose proof (NatWithEquality.equal_ok k (to + 1)).
cases (NatWithEquality.equal k (to + 1)); simplify.
cases (Compare_dec.leb from to).
rewrite Heq.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
apply NatSet.member_add_eq.
pose proof (NatWithEquality.equal_ok k k).
cases (NatWithEquality.equal k k); simplify.
equality.
equality.
pose proof (NatWithEquality.equal_ok k (from - 1)).
cases (NatWithEquality.equal k (from - 1)); simplify.
cases (Compare_dec.leb from to).
apply Compare_dec.leb_complete in Heq1.
rewrite Compare_dec.leb_correct by linear_arithmetic.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
pose proof (NatWithEquality.equal_ok k (to + 1)).
cases (NatWithEquality.equal k (to + 1)); simplify.
pose proof (NatWithEquality.equal_ok k k).
cases (NatWithEquality.equal k k); simplify.
equality.
equality.
pose proof (NatWithEquality.equal_ok k k).
cases (NatWithEquality.equal k k); simplify.
equality.
equality.
pose proof (NatWithEquality.equal_ok k (to + 1)).
cases (NatWithEquality.equal k (to + 1)); simplify.
cases (Compare_dec.leb from to).
apply Compare_dec.leb_complete in Heq2.
apply Compare_dec.leb_complete_conv in Heq.
linear_arithmetic.
apply NatSet.member_add_eq.
pose proof (NatWithEquality.equal_ok k k).
cases (NatWithEquality.equal k k); simplify.
equality.
equality.
apply NatSet.member_add_eq.
Qed.
Theorem member_add_noteq : forall k1 k2 s,
k1 <> k2
-> member (add s k1) k2 = member s k2.
Proof.
simplify.
unfold member, add.
cases s.
cases (Compare_dec.leb k1 k2); simplify.
rewrite Compare_dec.leb_correct by linear_arithmetic.
apply Compare_dec.leb_complete in Heq.
rewrite Compare_dec.leb_correct_conv.
equality.
unfold key in *. (* Tricky step! Coq needs to see that we are really working with numbers. *)
linear_arithmetic.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
cases (Compare_dec.leb from k1); simplify.
cases (Compare_dec.leb k1 to); simplify.
equality.
pose proof (NatWithEquality.equal_ok k1 (from - 1)).
cases (NatWithEquality.equal k1 (from - 1)); simplify.
apply leb_complete in Heq.
apply leb_complete_conv in Heq0.
linear_arithmetic.
pose proof (NatWithEquality.equal_ok k1 (to + 1)).
cases (NatWithEquality.equal k1 (to + 1)); simplify.
cases (Compare_dec.leb from to).
rewrite H1.
cases (Compare_dec.leb from k2); simplify.
cases (Compare_dec.leb k2 to).
apply Compare_dec.leb_complete in Heq5.
apply Compare_dec.leb_complete in Heq3.
rewrite Compare_dec.leb_correct by linear_arithmetic.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
apply Compare_dec.leb_complete in Heq3.
rewrite Compare_dec.leb_correct by linear_arithmetic.
apply Compare_dec.leb_complete_conv in Heq5.
unfold key in *.
rewrite Compare_dec.leb_correct_conv by linear_arithmetic.
equality.
rewrite andb_false_r.
equality.
simplify.
pose proof (NatWithEquality.equal_ok k1 k2).
cases (NatWithEquality.equal k1 k2); simplify.
equality.
unfold fromRange.
rewrite Heq3.
apply NatSet.member_empty.
pose proof (NatWithEquality.equal_ok k1 k2).
cases (NatWithEquality.equal k1 k2); simplify.
equality.
unfold fromRange.
cases (Compare_dec.leb from to); simplify.
apply member_fromRange'.
apply Compare_dec.leb_complete.
assumption.
equality.
pose proof (NatWithEquality.equal_ok k1 (from - 1)).
cases (NatWithEquality.equal k1 (from - 1)); simplify.
cases (Compare_dec.leb from to).
apply Compare_dec.leb_complete in Heq1.
rewrite Compare_dec.leb_correct by linear_arithmetic.
f_equal.
f_equal.
cases (Compare_dec.leb k1 k2).
apply Compare_dec.leb_complete in Heq2.
apply Compare_dec.leb_complete_conv in Heq.
unfold key in *.
rewrite Compare_dec.leb_correct by linear_arithmetic.
equality.
apply Compare_dec.leb_complete_conv in Heq2.
apply Compare_dec.leb_complete_conv in Heq.
unfold key in *.
rewrite Compare_dec.leb_correct_conv by linear_arithmetic.
equality.
pose proof (NatWithEquality.equal_ok k1 (to + 1)).
cases (NatWithEquality.equal k1 (to + 1)); simplify.
pose proof (NatWithEquality.equal_ok k1 k2).
cases (NatWithEquality.equal k1 k2); simplify.
unfold key in *; linear_arithmetic.
unfold fromRange.
rewrite Heq1.
apply NatSet.member_empty.
pose proof (NatWithEquality.equal_ok k1 k2).
cases (NatWithEquality.equal k1 k2); simplify.
equality.
unfold fromRange.
rewrite Heq1.
apply NatSet.member_empty.
pose proof (NatWithEquality.equal_ok k1 (to + 1)).
cases (NatWithEquality.equal k1 (to + 1)); simplify.
cases (Compare_dec.leb from to).
rewrite Heq; simplify.
apply Compare_dec.leb_complete in Heq2.
apply Compare_dec.leb_complete_conv in Heq.
linear_arithmetic.
rewrite NatSet.member_add_noteq by assumption; simplify.
unfold fromRange.
rewrite Heq2.
apply NatSet.member_empty.
pose proof (NatWithEquality.equal_ok k1 k2).
cases (NatWithEquality.equal k1 k2); simplify.
equality.
unfold fromRange.
cases (Compare_dec.leb from to); simplify.
apply member_fromRange'.
apply Compare_dec.leb_complete; assumption.
equality.
apply NatSet.member_add_noteq.
assumption.
Qed.
Theorem decidable_equality : forall a b : key, a = b \/ a <> b.
Proof.
simplify.
pose proof (NatWithEquality.equal_ok a b).
cases (NatWithEquality.equal a b); propositional.
Qed.
End NatRangeSet.
Module FasterNatDuplicateFinder := FindDuplicates(NatRangeSet).
Fixpoint upto (n : nat) : list nat :=
match n with
| 0 => []
| S n' => n' :: upto n'
end.
Compute NatDuplicateFinder.noDuplicates (upto 1000).
Compute FasterNatDuplicateFinder.noDuplicates (upto 1000).