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Start of DataAbstraction: queue examples
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DataAbstraction.v
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DataAbstraction.v
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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
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* Chapter 3: Data Abstraction
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* Author: Adam Chlipala
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* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
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Require Import Frap.
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Set Implicit Arguments.
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Module Algebraic.
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Module Type QUEUE.
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Parameter t : Set -> Set.
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Parameter empty : forall A, t A.
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Parameter enqueue : forall A, t A -> A -> t A.
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Parameter dequeue : forall A, t A -> option (t A * A).
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Axiom dequeue_empty : forall A,
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dequeue (empty A) = None.
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Axiom empty_dequeue : forall A (q : t A),
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dequeue q = None -> q = empty A.
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Axiom dequeue_enqueue : forall A (q : t A) x,
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dequeue (enqueue q x) = Some (match dequeue q with
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| None => (empty A, x)
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| Some (q', y) => (enqueue q' x, y)
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end).
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End QUEUE.
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Module ListQueue : QUEUE.
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Definition t : Set -> Set := list.
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Definition empty A : t A := nil.
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Definition enqueue A (q : t A) (x : A) : t A := x :: q.
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Fixpoint dequeue A (q : t A) : option (t A * A) :=
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match q with
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| [] => None
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| x :: q' =>
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match dequeue q' with
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| None => Some ([], x)
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| Some (q'', y) => Some (x :: q'', y)
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end
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end.
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Theorem dequeue_empty : forall A, dequeue (empty A) = None.
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Proof.
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simplify.
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equality.
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Qed.
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Theorem empty_dequeue : forall A (q : t A),
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dequeue q = None -> q = empty A.
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Proof.
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simplify.
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cases q.
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simplify.
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equality.
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simplify.
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cases (dequeue q).
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cases p.
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equality.
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equality.
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Qed.
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Theorem dequeue_enqueue : forall A (q : t A) x,
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dequeue (enqueue q x) = Some (match dequeue q with
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| None => (empty A, x)
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| Some (q', y) => (enqueue q' x, y)
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end).
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Proof.
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simplify.
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cases (dequeue q).
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cases p.
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equality.
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equality.
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Qed.
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End ListQueue.
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Module ReversedListQueue : QUEUE.
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Definition t : Set -> Set := list.
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Definition empty A : t A := [].
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Definition enqueue A (q : t A) (x : A) : t A := q ++ [x].
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Definition dequeue A (q : t A) : option (t A * A) :=
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match q with
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| [] => None
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| x :: q' => Some (q', x)
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end.
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Theorem dequeue_empty : forall A, dequeue (empty A) = None.
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Proof.
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simplify.
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equality.
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Qed.
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Theorem empty_dequeue : forall A (q : t A),
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dequeue q = None -> q = empty A.
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Proof.
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simplify.
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cases q.
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simplify.
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equality.
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simplify.
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equality.
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Qed.
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Theorem dequeue_enqueue : forall A (q : t A) x,
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dequeue (enqueue q x) = Some (match dequeue q with
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| None => (empty A, x)
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| Some (q', y) => (enqueue q' x, y)
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end).
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Proof.
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simplify.
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unfold dequeue, enqueue.
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cases q; simplify.
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equality.
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equality.
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Qed.
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End ReversedListQueue.
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Module DelayedSum (Q : QUEUE).
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Fixpoint makeQueue (n : nat) (q : Q.t nat) : Q.t nat :=
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match n with
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| 0 => q
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| S n' => makeQueue n' (Q.enqueue q n')
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end.
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Fixpoint computeSum (n : nat) (q : Q.t nat) : nat :=
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match n with
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| 0 => 0
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| S n' => match Q.dequeue q with
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| None => 0
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| Some (q', v) => v + computeSum n' q'
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end
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end.
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Fixpoint sumUpto (n : nat) : nat :=
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match n with
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| 0 => 0
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| S n' => n' + sumUpto n'
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end.
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Lemma dequeue_makeQueue : forall n q,
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Q.dequeue (makeQueue n q)
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= match Q.dequeue q with
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| Some (q', v) => Some (makeQueue n q', v)
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| None =>
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match n with
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| 0 => None
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| S n' => Some (makeQueue n' q, n')
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end
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end.
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Proof.
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induct n.
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simplify.
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cases (Q.dequeue q).
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cases p.
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equality.
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equality.
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simplify.
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rewrite IHn.
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rewrite Q.dequeue_enqueue.
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cases (Q.dequeue q).
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cases p.
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equality.
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rewrite (Q.empty_dequeue (q := q)).
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equality.
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assumption.
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Qed.
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Theorem computeSum_ok : forall n,
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computeSum n (makeQueue n (Q.empty nat)) = sumUpto n.
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Proof.
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induct n.
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simplify.
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equality.
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simplify.
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rewrite dequeue_makeQueue.
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rewrite Q.dequeue_enqueue.
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rewrite Q.dequeue_empty.
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rewrite IHn.
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equality.
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Qed.
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End DelayedSum.
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End Algebraic.
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Module AlgebraicWithEquivalenceRelation.
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Module Type QUEUE.
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Parameter t : Set -> Set.
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Parameter empty : forall A, t A.
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Parameter enqueue : forall A, t A -> A -> t A.
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Parameter dequeue : forall A, t A -> option (t A * A).
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Parameter equiv : forall A, t A -> t A -> Prop.
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Infix "~=" := equiv (at level 70).
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Axiom equiv_refl : forall A (a : t A), a ~= a.
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Axiom equiv_sym : forall A (a b : t A), a ~= b -> b ~= a.
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Axiom equiv_trans : forall A (a b c : t A), a ~= b -> b ~= c -> a ~= c.
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Axiom equiv_enqueue : forall A (a b : t A) (x : A),
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a ~= b
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-> enqueue a x ~= enqueue b x.
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Definition dequeue_equiv A (a b : option (t A * A)) :=
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match a, b with
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| None, None => True
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| Some (qa, xa), Some (qb, xb) => qa ~= qb /\ xa = xb
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| _, _ => False
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end.
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Infix "~~=" := dequeue_equiv (at level 70).
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Axiom equiv_dequeue : forall A (a b : t A),
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a ~= b
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-> dequeue a ~~= dequeue b.
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Axiom dequeue_empty : forall A,
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dequeue (empty A) = None.
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Axiom empty_dequeue : forall A (q : t A),
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dequeue q = None -> q ~= empty A.
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Axiom dequeue_enqueue : forall A (q : t A) x,
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dequeue (enqueue q x)
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~~= match dequeue q with
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| None => Some (empty A, x)
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| Some (q', y) => Some (enqueue q' x, y)
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end.
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End QUEUE.
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Module ListQueue : QUEUE.
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Definition t : Set -> Set := list.
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Definition empty A : t A := nil.
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Definition enqueue A (q : t A) (x : A) : t A := x :: q.
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Fixpoint dequeue A (q : t A) : option (t A * A) :=
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match q with
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| [] => None
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| x :: q' =>
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match dequeue q' with
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| None => Some ([], x)
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| Some (q'', y) => Some (x :: q'', y)
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end
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end.
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Definition equiv A (a b : t A) := a = b.
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Infix "~=" := equiv (at level 70).
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Theorem equiv_refl : forall A (a : t A), a ~= a.
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Proof.
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equality.
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Qed.
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Theorem equiv_sym : forall A (a b : t A), a ~= b -> b ~= a.
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Proof.
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equality.
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Qed.
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Theorem equiv_trans : forall A (a b c : t A), a ~= b -> b ~= c -> a ~= c.
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Proof.
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equality.
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Qed.
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Theorem equiv_enqueue : forall A (a b : t A) (x : A),
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a ~= b
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-> enqueue a x ~= enqueue b x.
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Proof.
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unfold equiv; equality.
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Qed.
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Definition dequeue_equiv A (a b : option (t A * A)) :=
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match a, b with
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| None, None => True
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| Some (qa, xa), Some (qb, xb) => qa ~= qb /\ xa = xb
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| _, _ => False
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end.
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Infix "~~=" := dequeue_equiv (at level 70).
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Theorem equiv_dequeue : forall A (a b : t A),
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a ~= b
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-> dequeue a ~~= dequeue b.
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Proof.
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unfold equiv, dequeue_equiv; simplify.
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rewrite H.
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cases (dequeue b).
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cases p.
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equality.
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propositional.
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Qed.
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Theorem dequeue_empty : forall A, dequeue (empty A) = None.
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Proof.
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simplify.
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equality.
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Qed.
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Theorem empty_dequeue : forall A (q : t A),
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dequeue q = None -> q ~= empty A.
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Proof.
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simplify.
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cases q.
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simplify.
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unfold equiv.
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equality.
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simplify.
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cases (dequeue q).
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cases p.
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equality.
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equality.
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Qed.
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Theorem dequeue_enqueue : forall A (q : t A) x,
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dequeue (enqueue q x)
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~~= match dequeue q with
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| None => Some (empty A, x)
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| Some (q', y) => Some (enqueue q' x, y)
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end.
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Proof.
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unfold dequeue_equiv, equiv.
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induct q; simplify.
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equality.
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cases (dequeue q).
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cases p.
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equality.
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equality.
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Qed.
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End ListQueue.
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Module TwoStacksQueue : QUEUE.
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Record stackpair (A : Set) := {
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EnqueueHere : list A;
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DequeueHere : list A
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}.
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Definition t := stackpair.
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Definition empty A : t A := {|
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EnqueueHere := [];
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DequeueHere := []
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|}.
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Definition enqueue A (q : t A) (x : A) : t A := {|
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EnqueueHere := x :: q.(EnqueueHere);
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DequeueHere := q.(DequeueHere)
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|}.
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Definition dequeue A (q : t A) : option (t A * A) :=
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match q.(DequeueHere) with
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| x :: dq => Some ({| EnqueueHere := q.(EnqueueHere);
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DequeueHere := dq |}, x)
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| [] =>
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match rev q.(EnqueueHere) with
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| [] => None
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| x :: eq => Some ({| EnqueueHere := [];
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DequeueHere := eq |}, x)
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end
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end.
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Definition elements A (q : t A) : list A :=
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q.(EnqueueHere) ++ rev q.(DequeueHere).
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Definition equiv A (a b : t A) :=
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elements a = elements b.
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Infix "~=" := equiv (at level 70).
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Theorem equiv_refl : forall A (a : t A), a ~= a.
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Proof.
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equality.
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Qed.
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Theorem equiv_sym : forall A (a b : t A), a ~= b -> b ~= a.
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Proof.
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equality.
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Qed.
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Theorem equiv_trans : forall A (a b c : t A), a ~= b -> b ~= c -> a ~= c.
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Proof.
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equality.
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Qed.
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Theorem equiv_enqueue : forall A (a b : t A) (x : A),
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a ~= b
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-> enqueue a x ~= enqueue b x.
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Proof.
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unfold equiv, elements; simplify.
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equality.
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Qed.
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Definition dequeue_equiv A (a b : option (t A * A)) :=
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match a, b with
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| None, None => True
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| Some (qa, xa), Some (qb, xb) => qa ~= qb /\ xa = xb
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| _, _ => False
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end.
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Infix "~~=" := dequeue_equiv (at level 70).
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Theorem equiv_dequeue : forall A (a b : t A),
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a ~= b
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-> dequeue a ~~= dequeue b.
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Proof.
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unfold equiv, dequeue_equiv, elements, dequeue; simplify.
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cases (DequeueHere a); simplify.
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cases (rev (EnqueueHere a)); simplify.
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cases (DequeueHere b); simplify.
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cases (rev (EnqueueHere b)); simplify.
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propositional.
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SearchRewrite (_ ++ []).
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rewrite app_nil_r in H.
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rewrite app_nil_r in H.
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equality.
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cases (EnqueueHere a); simplify.
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cases (EnqueueHere b); simplify.
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cases (rev l); simplify.
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equality.
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equality.
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equality.
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cases (rev l0); simplify.
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equality.
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equality.
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cases (DequeueHere b); simplify.
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cases (rev (EnqueueHere b)); simplify.
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rewrite app_nil_r in H.
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rewrite app_nil_r in H.
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equality.
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rewrite app_nil_r in H.
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rewrite app_nil_r in H.
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equality.
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rewrite app_nil_r in H.
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rewrite H in Heq0.
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SearchRewrite (rev (_ ++ _)).
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rewrite rev_app_distr in Heq0.
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rewrite rev_app_distr in Heq0.
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simplify.
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invert Heq0.
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unfold equiv, elements.
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simplify.
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rewrite rev_app_distr.
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SearchRewrite (rev (rev _)).
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rewrite rev_involutive.
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rewrite rev_involutive.
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equality.
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cases (DequeueHere b); simplify.
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cases (rev (EnqueueHere b)); simplify.
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rewrite app_nil_r in H.
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rewrite <- H in Heq1.
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cases (EnqueueHere a); simplify.
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cases (rev l); simplify.
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equality.
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rewrite rev_app_distr in Heq1.
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simplify.
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equality.
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rewrite rev_app_distr in Heq1.
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rewrite rev_app_distr in Heq1.
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simplify.
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equality.
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unfold equiv, elements.
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simplify.
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rewrite app_nil_r in H.
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rewrite <- H in Heq1.
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rewrite rev_app_distr in Heq1. rewrite rev_app_distr in Heq1.
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simplify.
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invert Heq1.
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rewrite rev_involutive.
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rewrite rev_app_distr.
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rewrite rev_involutive.
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equality.
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unfold equiv, elements.
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simplify.
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SearchAbout app cons nil.
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apply app_inj_tail.
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rewrite <- app_assoc.
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rewrite <- app_assoc.
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assumption.
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Qed.
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Theorem dequeue_empty : forall A, dequeue (empty A) = None.
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Proof.
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simplify.
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equality.
|
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Qed.
|
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|
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Theorem empty_dequeue : forall A (q : t A),
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dequeue q = None -> q ~= empty A.
|
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Proof.
|
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simplify.
|
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cases q.
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unfold dequeue in *.
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simplify.
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cases DequeueHere0.
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cases (rev EnqueueHere0).
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cases EnqueueHere0.
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equality.
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simplify.
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cases (rev EnqueueHere0); simplify.
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equality.
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equality.
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equality.
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equality.
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Qed.
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|
||||
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.
|
||||
|
|
@ -12,6 +12,7 @@ BasicSyntax_template.v
|
|||
BasicSyntax.v
|
||||
Polymorphism.v
|
||||
Polymorphism_template.v
|
||||
DataAbstraction.v
|
||||
Interpreters_template.v
|
||||
Interpreters.v
|
||||
TransitionSystems_template.v
|
||||
|
|
|
|||
Loading…
Reference in a new issue