Start of DataAbstraction: queue examples

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Adam Chlipala 2017-02-12 15:19:48 -05:00
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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 3: Data Abstraction
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Require Import Frap.
Set Implicit Arguments.
Module Algebraic.
Module Type QUEUE.
Parameter t : Set -> Set.
Parameter empty : forall A, t A.
Parameter enqueue : forall A, t A -> A -> t A.
Parameter dequeue : forall A, t A -> option (t A * A).
Axiom dequeue_empty : forall A,
dequeue (empty A) = None.
Axiom empty_dequeue : forall A (q : t A),
dequeue q = None -> q = empty A.
Axiom dequeue_enqueue : forall A (q : t A) x,
dequeue (enqueue q x) = Some (match dequeue q with
| None => (empty A, x)
| Some (q', y) => (enqueue q' x, y)
end).
End QUEUE.
Module ListQueue : QUEUE.
Definition t : Set -> Set := list.
Definition empty A : t A := nil.
Definition enqueue A (q : t A) (x : A) : t A := x :: q.
Fixpoint dequeue A (q : t A) : option (t A * A) :=
match q with
| [] => None
| x :: q' =>
match dequeue q' with
| None => Some ([], x)
| Some (q'', y) => Some (x :: q'', y)
end
end.
Theorem dequeue_empty : forall A, dequeue (empty A) = None.
Proof.
simplify.
equality.
Qed.
Theorem empty_dequeue : forall A (q : t A),
dequeue q = None -> q = empty A.
Proof.
simplify.
cases q.
simplify.
equality.
simplify.
cases (dequeue q).
cases p.
equality.
equality.
Qed.
Theorem dequeue_enqueue : forall A (q : t A) x,
dequeue (enqueue q x) = Some (match dequeue q with
| None => (empty A, x)
| Some (q', y) => (enqueue q' x, y)
end).
Proof.
simplify.
cases (dequeue q).
cases p.
equality.
equality.
Qed.
End ListQueue.
Module ReversedListQueue : QUEUE.
Definition t : Set -> Set := list.
Definition empty A : t A := [].
Definition enqueue A (q : t A) (x : A) : t A := q ++ [x].
Definition dequeue A (q : t A) : option (t A * A) :=
match q with
| [] => None
| x :: q' => Some (q', x)
end.
Theorem dequeue_empty : forall A, dequeue (empty A) = None.
Proof.
simplify.
equality.
Qed.
Theorem empty_dequeue : forall A (q : t A),
dequeue q = None -> q = empty A.
Proof.
simplify.
cases q.
simplify.
equality.
simplify.
equality.
Qed.
Theorem dequeue_enqueue : forall A (q : t A) x,
dequeue (enqueue q x) = Some (match dequeue q with
| None => (empty A, x)
| Some (q', y) => (enqueue q' x, y)
end).
Proof.
simplify.
unfold dequeue, enqueue.
cases q; simplify.
equality.
equality.
Qed.
End ReversedListQueue.
Module DelayedSum (Q : QUEUE).
Fixpoint makeQueue (n : nat) (q : Q.t nat) : Q.t nat :=
match n with
| 0 => q
| S n' => makeQueue n' (Q.enqueue q n')
end.
Fixpoint computeSum (n : nat) (q : Q.t nat) : nat :=
match n with
| 0 => 0
| S n' => match Q.dequeue q with
| None => 0
| Some (q', v) => v + computeSum n' q'
end
end.
Fixpoint sumUpto (n : nat) : nat :=
match n with
| 0 => 0
| S n' => n' + sumUpto n'
end.
Lemma dequeue_makeQueue : forall n q,
Q.dequeue (makeQueue n q)
= match Q.dequeue q with
| Some (q', v) => Some (makeQueue n q', v)
| None =>
match n with
| 0 => None
| S n' => Some (makeQueue n' q, n')
end
end.
Proof.
induct n.
simplify.
cases (Q.dequeue q).
cases p.
equality.
equality.
simplify.
rewrite IHn.
rewrite Q.dequeue_enqueue.
cases (Q.dequeue q).
cases p.
equality.
rewrite (Q.empty_dequeue (q := q)).
equality.
assumption.
Qed.
Theorem computeSum_ok : forall n,
computeSum n (makeQueue n (Q.empty nat)) = sumUpto n.
Proof.
induct n.
simplify.
equality.
simplify.
rewrite dequeue_makeQueue.
rewrite Q.dequeue_enqueue.
rewrite Q.dequeue_empty.
rewrite IHn.
equality.
Qed.
End DelayedSum.
End Algebraic.
Module AlgebraicWithEquivalenceRelation.
Module Type QUEUE.
Parameter t : Set -> Set.
Parameter empty : forall A, t A.
Parameter enqueue : forall A, t A -> A -> t A.
Parameter dequeue : forall A, t A -> option (t A * A).
Parameter equiv : forall A, t A -> t A -> Prop.
Infix "~=" := equiv (at level 70).
Axiom equiv_refl : forall A (a : t A), a ~= a.
Axiom equiv_sym : forall A (a b : t A), a ~= b -> b ~= a.
Axiom equiv_trans : forall A (a b c : t A), a ~= b -> b ~= c -> a ~= c.
Axiom equiv_enqueue : forall A (a b : t A) (x : A),
a ~= b
-> enqueue a x ~= enqueue b x.
Definition dequeue_equiv A (a b : option (t A * A)) :=
match a, b with
| None, None => True
| Some (qa, xa), Some (qb, xb) => qa ~= qb /\ xa = xb
| _, _ => False
end.
Infix "~~=" := dequeue_equiv (at level 70).
Axiom equiv_dequeue : forall A (a b : t A),
a ~= b
-> dequeue a ~~= dequeue b.
Axiom dequeue_empty : forall A,
dequeue (empty A) = None.
Axiom empty_dequeue : forall A (q : t A),
dequeue q = None -> q ~= empty A.
Axiom dequeue_enqueue : forall A (q : t A) x,
dequeue (enqueue q x)
~~= match dequeue q with
| None => Some (empty A, x)
| Some (q', y) => Some (enqueue q' x, y)
end.
End QUEUE.
Module ListQueue : QUEUE.
Definition t : Set -> Set := list.
Definition empty A : t A := nil.
Definition enqueue A (q : t A) (x : A) : t A := x :: q.
Fixpoint dequeue A (q : t A) : option (t A * A) :=
match q with
| [] => None
| x :: q' =>
match dequeue q' with
| None => Some ([], x)
| Some (q'', y) => Some (x :: q'', y)
end
end.
Definition equiv A (a b : t A) := a = b.
Infix "~=" := equiv (at level 70).
Theorem equiv_refl : forall A (a : t A), a ~= a.
Proof.
equality.
Qed.
Theorem equiv_sym : forall A (a b : t A), a ~= b -> b ~= a.
Proof.
equality.
Qed.
Theorem equiv_trans : forall A (a b c : t A), a ~= b -> b ~= c -> a ~= c.
Proof.
equality.
Qed.
Theorem equiv_enqueue : forall A (a b : t A) (x : A),
a ~= b
-> enqueue a x ~= enqueue b x.
Proof.
unfold equiv; equality.
Qed.
Definition dequeue_equiv A (a b : option (t A * A)) :=
match a, b with
| None, None => True
| Some (qa, xa), Some (qb, xb) => qa ~= qb /\ xa = xb
| _, _ => False
end.
Infix "~~=" := dequeue_equiv (at level 70).
Theorem equiv_dequeue : forall A (a b : t A),
a ~= b
-> dequeue a ~~= dequeue b.
Proof.
unfold equiv, dequeue_equiv; simplify.
rewrite H.
cases (dequeue b).
cases p.
equality.
propositional.
Qed.
Theorem dequeue_empty : forall A, dequeue (empty A) = None.
Proof.
simplify.
equality.
Qed.
Theorem empty_dequeue : forall A (q : t A),
dequeue q = None -> q ~= empty A.
Proof.
simplify.
cases q.
simplify.
unfold equiv.
equality.
simplify.
cases (dequeue q).
cases p.
equality.
equality.
Qed.
Theorem dequeue_enqueue : forall A (q : t A) x,
dequeue (enqueue q x)
~~= match dequeue q with
| None => Some (empty A, x)
| Some (q', y) => Some (enqueue q' x, y)
end.
Proof.
unfold dequeue_equiv, equiv.
induct q; simplify.
equality.
cases (dequeue q).
cases p.
equality.
equality.
Qed.
End ListQueue.
Module TwoStacksQueue : QUEUE.
Record stackpair (A : Set) := {
EnqueueHere : list A;
DequeueHere : list A
}.
Definition t := stackpair.
Definition empty A : t A := {|
EnqueueHere := [];
DequeueHere := []
|}.
Definition enqueue A (q : t A) (x : A) : t A := {|
EnqueueHere := x :: q.(EnqueueHere);
DequeueHere := q.(DequeueHere)
|}.
Definition dequeue A (q : t A) : option (t A * A) :=
match q.(DequeueHere) with
| x :: dq => Some ({| EnqueueHere := q.(EnqueueHere);
DequeueHere := dq |}, x)
| [] =>
match rev q.(EnqueueHere) with
| [] => None
| x :: eq => Some ({| EnqueueHere := [];
DequeueHere := eq |}, x)
end
end.
Definition elements A (q : t A) : list A :=
q.(EnqueueHere) ++ rev q.(DequeueHere).
Definition equiv A (a b : t A) :=
elements a = elements b.
Infix "~=" := equiv (at level 70).
Theorem equiv_refl : forall A (a : t A), a ~= a.
Proof.
equality.
Qed.
Theorem equiv_sym : forall A (a b : t A), a ~= b -> b ~= a.
Proof.
equality.
Qed.
Theorem equiv_trans : forall A (a b c : t A), a ~= b -> b ~= c -> a ~= c.
Proof.
equality.
Qed.
Theorem equiv_enqueue : forall A (a b : t A) (x : A),
a ~= b
-> enqueue a x ~= enqueue b x.
Proof.
unfold equiv, elements; simplify.
equality.
Qed.
Definition dequeue_equiv A (a b : option (t A * A)) :=
match a, b with
| None, None => True
| Some (qa, xa), Some (qb, xb) => qa ~= qb /\ xa = xb
| _, _ => False
end.
Infix "~~=" := dequeue_equiv (at level 70).
Theorem equiv_dequeue : forall A (a b : t A),
a ~= b
-> dequeue a ~~= dequeue b.
Proof.
unfold equiv, dequeue_equiv, elements, dequeue; simplify.
cases (DequeueHere a); simplify.
cases (rev (EnqueueHere a)); simplify.
cases (DequeueHere b); simplify.
cases (rev (EnqueueHere b)); simplify.
propositional.
SearchRewrite (_ ++ []).
rewrite app_nil_r in H.
rewrite app_nil_r in H.
equality.
cases (EnqueueHere a); simplify.
cases (EnqueueHere b); simplify.
cases (rev l); simplify.
equality.
equality.
equality.
cases (rev l0); simplify.
equality.
equality.
cases (DequeueHere b); simplify.
cases (rev (EnqueueHere b)); simplify.
rewrite app_nil_r in H.
rewrite app_nil_r in H.
equality.
rewrite app_nil_r in H.
rewrite app_nil_r in H.
equality.
rewrite app_nil_r in H.
rewrite H in Heq0.
SearchRewrite (rev (_ ++ _)).
rewrite rev_app_distr in Heq0.
rewrite rev_app_distr in Heq0.
simplify.
invert Heq0.
unfold equiv, elements.
simplify.
rewrite rev_app_distr.
SearchRewrite (rev (rev _)).
rewrite rev_involutive.
rewrite rev_involutive.
equality.
cases (DequeueHere b); simplify.
cases (rev (EnqueueHere b)); simplify.
rewrite app_nil_r in H.
rewrite <- H in Heq1.
cases (EnqueueHere a); simplify.
cases (rev l); simplify.
equality.
rewrite rev_app_distr in Heq1.
simplify.
equality.
rewrite rev_app_distr in Heq1.
rewrite rev_app_distr in Heq1.
simplify.
equality.
unfold equiv, elements.
simplify.
rewrite app_nil_r in H.
rewrite <- H in Heq1.
rewrite rev_app_distr in Heq1. rewrite rev_app_distr in Heq1.
simplify.
invert Heq1.
rewrite rev_involutive.
rewrite rev_app_distr.
rewrite rev_involutive.
equality.
unfold equiv, elements.
simplify.
SearchAbout app cons nil.
apply app_inj_tail.
rewrite <- app_assoc.
rewrite <- app_assoc.
assumption.
Qed.
Theorem dequeue_empty : forall A, dequeue (empty A) = None.
Proof.
simplify.
equality.
Qed.
Theorem empty_dequeue : forall A (q : t A),
dequeue q = None -> q ~= empty A.
Proof.
simplify.
cases q.
unfold dequeue in *.
simplify.
cases DequeueHere0.
cases (rev EnqueueHere0).
cases EnqueueHere0.
equality.
simplify.
cases (rev EnqueueHere0); simplify.
equality.
equality.
equality.
equality.
Qed.
Theorem dequeue_enqueue : forall A (q : t A) x,
dequeue (enqueue q x)
~~= match dequeue q with
| None => Some (empty A, x)
| Some (q', y) => Some (enqueue q' x, y)
end.
Proof.
unfold dequeue_equiv, equiv; simplify.
cases q; simplify.
unfold dequeue, enqueue; simplify.
cases DequeueHere0; simplify.
cases (rev EnqueueHere0); simplify.
equality.
unfold elements; simplify.
SearchRewrite (rev (_ ++ _)).
rewrite rev_app_distr.
simplify.
equality.
equality.
Qed.
End TwoStacksQueue.
Module DelayedSum (Q : QUEUE).
Fixpoint makeQueue (n : nat) (q : Q.t nat) : Q.t nat :=
match n with
| 0 => q
| S n' => makeQueue n' (Q.enqueue q n')
end.
Fixpoint computeSum (n : nat) (q : Q.t nat) : nat :=
match n with
| 0 => 0
| S n' => match Q.dequeue q with
| None => 0
| Some (q', v) => v + computeSum n' q'
end
end.
Fixpoint sumUpto (n : nat) : nat :=
match n with
| 0 => 0
| S n' => n' + sumUpto n'
end.
Infix "~=" := Q.equiv (at level 70).
Infix "~~=" := Q.dequeue_equiv (at level 70).
Lemma makeQueue_congruence : forall n a b,
a ~= b
-> makeQueue n a ~= makeQueue n b.
Proof.
induct n; simplify.
assumption.
apply IHn.
apply Q.equiv_enqueue.
assumption.
Qed.
Lemma dequeue_makeQueue : forall n q,
Q.dequeue (makeQueue n q)
~~= match Q.dequeue q with
| Some (q', v) => Some (makeQueue n q', v)
| None =>
match n with
| 0 => None
| S n' => Some (makeQueue n' q, n')
end
end.
Proof.
induct n.
simplify.
cases (Q.dequeue q).
cases p.
unfold Q.dequeue_equiv.
propositional.
apply Q.equiv_refl.
unfold Q.dequeue_equiv.
propositional.
simplify.
unfold Q.dequeue_equiv in *.
specialize (IHn (Q.enqueue q n)).
cases (Q.dequeue (makeQueue n (Q.enqueue q n))).
cases p.
pose proof (Q.dequeue_enqueue q n).
unfold Q.dequeue_equiv in *.
cases (Q.dequeue (Q.enqueue q n)).
cases p.
cases (Q.dequeue q).
cases p.
propositional.
apply Q.equiv_trans with (b := makeQueue n t0).
assumption.
apply makeQueue_congruence.
assumption.
equality.
propositional.
apply Q.equiv_trans with (b := makeQueue n t0).
assumption.
apply makeQueue_congruence.
apply Q.equiv_trans with (b := Q.empty nat).
assumption.
apply Q.equiv_sym.
apply Q.empty_dequeue.
assumption.
equality.
cases (Q.dequeue q).
cases p.
propositional.
propositional.
pose proof (Q.dequeue_enqueue q n).
unfold Q.dequeue_equiv in H.
cases (Q.dequeue (Q.enqueue q n)).
cases p.
propositional.
cases (Q.dequeue q).
cases p.
propositional.
propositional.
Qed.
Theorem computeSum_congruence : forall n a b,
a ~= b
-> computeSum n a = computeSum n b.
Proof.
induct n.
simplify.
equality.
simplify.
pose proof (Q.equiv_dequeue H).
unfold Q.dequeue_equiv in H0.
cases (Q.dequeue a).
cases p.
cases (Q.dequeue b).
cases p.
rewrite IHn with (b := t0).
equality.
equality.
propositional.
cases (Q.dequeue b).
propositional.
equality.
Qed.
Theorem computeSum_ok : forall n,
computeSum n (makeQueue n (Q.empty nat)) = sumUpto n.
Proof.
induct n.
simplify.
equality.
simplify.
pose proof (dequeue_makeQueue n (Q.enqueue (Q.empty nat) n)).
unfold Q.dequeue_equiv in H.
cases (Q.dequeue (makeQueue n (Q.enqueue (Q.empty nat) n))).
cases p.
pose proof (Q.dequeue_enqueue (Q.empty nat) n).
unfold Q.dequeue_equiv in H0.
cases (Q.dequeue (Q.enqueue (Q.empty nat) n)).
cases p.
rewrite Q.dequeue_empty in H0.
propositional.
f_equal.
equality.
rewrite <- IHn.
apply computeSum_congruence.
apply Q.equiv_trans with (b := makeQueue n t0).
assumption.
apply makeQueue_congruence.
assumption.
rewrite Q.dequeue_empty in H0.
propositional.
pose proof (Q.dequeue_enqueue (Q.empty nat) n).
unfold Q.dequeue_equiv in H0.
cases (Q.dequeue (Q.enqueue (Q.empty nat) n)).
cases p.
propositional.
rewrite Q.dequeue_empty in H0.
propositional.
Qed.
End DelayedSum.
End AlgebraicWithEquivalenceRelation.

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@ -12,6 +12,7 @@ BasicSyntax_template.v
BasicSyntax.v BasicSyntax.v
Polymorphism.v Polymorphism.v
Polymorphism_template.v Polymorphism_template.v
DataAbstraction.v
Interpreters_template.v Interpreters_template.v
Interpreters.v Interpreters.v
TransitionSystems_template.v TransitionSystems_template.v