121 lines
3.1 KiB
Coq
121 lines
3.1 KiB
Coq
(* -------------------------------------------------------------------------- *)
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(* The [bind] combinator of the option monad. *)
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Definition bind {A B} (f : option A) (k : A -> option B) : option B :=
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match f with
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| None =>
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None
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| Some a =>
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k a
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end.
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(* The standard syntactic sugar for [bind]. [f >>= k] can be read as ``first
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do [f]; then, if successful, with the result of [f], do [k]''. *)
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Notation "f >>= k" := (bind f k) (at level 55).
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(* -------------------------------------------------------------------------- *)
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(* These lemmas help prove an equation of the form [f >>= k = b]. *)
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Lemma prove_bind_None:
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forall {A B} {f} {k : A -> option B},
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f = None ->
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f >>= k = None.
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Proof.
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intros. subst. eauto.
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Qed.
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Lemma prove_bind_Some:
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forall {A B} {f} {k : A -> option B} {a b},
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f = Some a ->
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k a = b ->
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f >>= k = b.
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Proof.
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intros. subst. eauto.
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Qed.
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(* This lemma helps exploit an equation of the form [f >>= k = Some b]. *)
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Lemma invert_bind_Some:
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forall {A B} {f} {k : A -> option B} {b},
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f >>= k = Some b ->
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exists a, f = Some a /\ k a = Some b.
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Proof.
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destruct f; simpl; intros.
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{ eauto. }
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{ congruence. }
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Qed.
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Ltac invert_bind_Some :=
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match goal with
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h: ?f >>= _ = Some _ |- _ =>
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let heq := fresh in
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generalize (invert_bind_Some h); clear h; intros (?&?&h)
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end.
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(* -------------------------------------------------------------------------- *)
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(* The standard ordering on options, where [None] is less defined then
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everything, and every element of the form [Some a] is less defined
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than itself only. *)
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Definition less_defined {A} (o1 o2 : option A) :=
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forall a, o1 = Some a -> o2 = Some a.
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(* -------------------------------------------------------------------------- *)
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(* This lemma exploits an assertion of the form [less_defined (Some _) _]. *)
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Lemma invert_less_defined_Some:
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forall {A} {a : A} {o : option A},
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less_defined (Some a) o ->
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Some a = o.
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Proof.
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unfold less_defined. intros. symmetry. eauto.
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Qed.
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Ltac invert_less_defined :=
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match goal with
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| h: less_defined (Some _) _ |- _ =>
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generalize (invert_less_defined_Some h); clear h; intro h
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end.
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(* -------------------------------------------------------------------------- *)
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(* These lemmas help prove assertions of the form [less_defined _ _]. *)
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Lemma prove_less_defined_None:
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forall {A} {o : option A},
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less_defined None o.
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Proof.
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unfold less_defined. intros. congruence.
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Qed.
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Lemma reflexive_less_defined:
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forall {A} {o : option A},
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less_defined o o.
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Proof.
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unfold less_defined. eauto.
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Qed.
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Local Hint Extern 1 (_ <> _) => congruence : congruence.
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Lemma prove_less_defined_bind:
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forall {A B} {f1 f2 : option A} {k1 k2 : A -> option B},
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less_defined f1 f2 ->
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(f1 <> None -> forall a, less_defined (k1 a) (k2 a)) ->
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less_defined (f1 >>= k1) (f2 >>= k2).
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Proof.
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intros. destruct f1; simpl in *.
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(* Case: [f1] is [Some _]. *)
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{ invert_less_defined. subst. simpl. eauto with congruence. }
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(* Case: [f1] is [None]. *)
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{ eauto using prove_less_defined_None. }
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Qed.
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Hint Resolve
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prove_less_defined_None reflexive_less_defined prove_less_defined_bind
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: less_defined.
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