bedrock.lang.cpp.logic.zstring
(*
* Copyright (C) BedRock Systems Inc. 2019-2021
* This software is distributed under the terms of the BedRock Open-Source License.
* See the LICENSE-BedRock file in the repository root for details.
*)
Require Import Stdlib.ZArith.BinInt.
Require Import Stdlib.Lists.List.
Require Import bedrock.lang.proofmode.proofmode.
Require Import bedrock.prelude.base.
Require Import bedrock.lang.bi.prelude.
Require Import bedrock.lang.bi.observe.
Require Import bedrock.lang.cpp.semantics.values.
Require Import bedrock.lang.cpp.logic.arr.
Require Import bedrock.lang.cpp.logic.heap_pred.
Import ChargeNotation.
#[local] Open Scope Z_scope.
* Copyright (C) BedRock Systems Inc. 2019-2021
* This software is distributed under the terms of the BedRock Open-Source License.
* See the LICENSE-BedRock file in the repository root for details.
*)
Require Import Stdlib.ZArith.BinInt.
Require Import Stdlib.Lists.List.
Require Import bedrock.lang.proofmode.proofmode.
Require Import bedrock.prelude.base.
Require Import bedrock.lang.bi.prelude.
Require Import bedrock.lang.bi.observe.
Require Import bedrock.lang.cpp.semantics.values.
Require Import bedrock.lang.cpp.logic.arr.
Require Import bedrock.lang.cpp.logic.heap_pred.
Import ChargeNotation.
#[local] Open Scope Z_scope.
zstrings
Module zstring.
Definition t := list N.
Bind Scope list_scope with t.
Section with_ct.
Variable ct : char_type.
#[local] Notation Tchar := (Tchar_ ct).
(* size reflects the in-memory footprint of the null-terminated string (i.e. the
length is one greater than that of the string as given by `std::strlen`).
*)
Definition _size (zs : t) : Z :=
Z.of_nat (List.length zs).
#[global] Arguments _size zs : simpl never.
Notation size zs := (_size zs%Z).
(* strlen mirrors the behavior of the standard-library function of the same name
(i.e. the length DOES NOT include the null-terminator).
*)
Definition _strlen (zs : t) : Z :=
(size zs - 1)%Z.
#[global] Arguments _strlen zs : simpl never.
Notation strlen zs := (_strlen zs%Z).
Definition _WF_alt {σ : genv} (zs : t) : Prop :=
let zs' := take (length zs - 1) zs in
last zs = Some 0%N /\ 0%N ∉ zs' /\
List.Forall (λ c, 0 <= c < 2 ^ char_type.bitsN ct)%N zs'.
#[global] Instance _WF_alt_dec {σ : genv} zs : Decision (_WF_alt zs) := _.
Definition _WF {σ : genv} (zs : t) : Prop :=
exists zs', zs = zs' ++ [0]%N /\ not (In 0%N zs') /\
List.Forall (λ c : N, has_type_prop (Vchar c) Tchar) zs.
#[global] Arguments _WF {σ} zs : simpl never.
Lemma _WF_WF_alt {σ : genv} zs : _WF zs <-> _WF_alt zs.
Proof.
rewrite /_WF /_WF_alt elem_of_list_In; setoid_rewrite <-has_type_prop_char'; split. {
move=> [] zs' [] -> [].
rewrite last_app /= length_app Nat.add_sub take_app_length Forall_app Forall_singleton.
naive_solver.
}
move=> [] Hl [] Hin Hall; exists (take (length zs - 1) zs).
destruct zs as [|z zs] using rev_ind => //= {IHzs}.
move: Hl Hin Hall;
rewrite last_app length_app /=. rewrite Nat.add_sub take_app_length.
rewrite Forall_app Forall_singleton.
intros [= ->]; intros; split_and! => //.
lia.
Qed.
Notation WF zs := (_WF zs%N).
#[global, program] Instance _WF_dec {σ : genv} zs : Decision (_WF zs) :=
cast_if (decide (_WF_alt zs)).
Solve All Obligations with intros; exact /_WF_WF_alt.
(* this definition is less intensional, and seems to work more
smoothly wrt. automation: *)
Definition _WF' {σ : genv} (zs : t) : Prop :=
(exists v, nth_error zs 0 = Some v) /\
(forall i, Z.of_nat i <= strlen zs ->
exists z, nth_error zs i = Some z /\ has_type_prop (Vchar z) Tchar) /\
(forall i, Z.of_nat i < strlen zs ->
exists z, nth_error zs i = Some z /\ z <> 0%N) /\
forall i, Z.of_nat i = strlen zs -> nth_error zs i = Some 0%N.
#[global] Arguments _WF' {σ} zs : simpl never.
Notation WF' zs := (_WF' zs%Z).
Section WF_Theory.
Context {σ : genv}.
Remark not_WF_nil : not (WF []).
Proof.
rewrite /_WF; intro CONTRA;
destruct CONTRA as [? [CONTRA ?]];
eapply app_cons_not_nil; eauto.
Qed.
Remark WF_singleton : WF [0%N].
Proof.
rewrite /_WF; exists []; repeat split.
- intro CONTRA; by inversion CONTRA.
- repeat constructor. apply has_type_prop_char_0.
Qed.
Lemma WF_singleton_inj :
forall (z : N),
WF [z] ->
z = 0%N.
Proof.
intros * [zs [Hzs [Hin Hbounds]]].
destruct zs; simpl in *; inversion Hzs.
- reflexivity.
- exfalso; eapply app_cons_not_nil; by eauto.
Qed.
Lemma WF_nonempty_inj :
forall (zs : t),
WF zs ->
exists z zs', zs = z :: zs'.
Proof.
move=> zs Hzstring; induction zs; [ | by eauto].
unfold WF in Hzstring.
destruct Hzstring as [? [CONTRA ?]].
now apply app_cons_not_nil in CONTRA.
Qed.
Lemma WF_cons :
forall (z : N) (zs : t),
(z <> 0)%N ->
has_type_prop (Vchar z) Tchar ->
WF (z :: zs) <->
WF zs.
Proof.
intros * Hnonzero Hz; split; [intro WF_zs | intro WF_z]; unfold WF in *.
- destruct WF_zs as [zs' [Hzs' [Hin Hbounds]]].
exists (tail zs'); split; destruct zs'=> //=;
[rewrite app_nil_l in Hzs' | | |];
inversion Hzs'=> //.
split.
+ intro; apply Hin=> //=; auto.
+ inversion Hbounds; by subst.
- destruct WF_z as [zs' [Hzs' [Hin Hbounds]]];
exists (z :: zs'); split.
+ by subst.
+ split.
* intro CONTRA; apply Hin; by inversion CONTRA.
* by constructor.
Qed.
(* Unlike WF_cons, here the explicit tail ++ [0] lets us write a more elegant lemma. *)
Lemma WF_cons' (z : N) (zs : t) :
WF (z :: zs ++ [0%N]) <->
(z <> 0)%N /\ has_type_prop (Vchar z) Tchar /\ WF (zs ++ [0%N]).
Proof.
rewrite /_WF/=. setoid_rewrite Forall_cons.
split.
- case => [[|z' zs'] [/= [-> Hzs']]]; first by destruct zs.
move: Hzs' => /(app_inj_2 _ _ _ _ eq_refl) [-> _].
move=> [/Decidable.not_or [Hz Hin] [Hz'Bounds Hrest]].
split_and! => //.
by exists zs'.
- case => Hz [HzBounds] [] zs' [] /(app_inj_2 _ _ _ _ eq_refl) [-> _] [Hin].
exists (z :: zs'); split_and! => //.
intros ?%in_inv. naive_solver.
Qed.
(* WF is prefix-free predicate *)
Lemma WF_eq_prefix_eq a1 a2 :
WF a1
-> WF a2
-> a1 = take (length a1) a2
-> a1 = a2.
Proof.
move=>[xs [-> [Nin _]]] [xs' [-> [Nin' _]]] H.
f_equal; move: xs' H Nin'.
rewrite length_app //= Nat.add_1_r.
induction xs.
- by case=>[//|??[<-][]];left.
- rewrite -app_comm_cons.
case; first by case: (xs).
move=>a' l' [<-] ? Hin'; f_equal.
apply: IHxs=>//= ?.
+ by apply: Nin; right.
+ by apply: Hin'; right.
Qed.
End WF_Theory.
Section size_Theory.
Context {σ : genv}.
(* Use rewrite size_unfold/= to reduce away a size application to a
concrete string; if complete reduction is not possible, this is
equivalent to unfolding the size notation.
*)
Lemma size_unfold (zs : t) :
size zs = Unfold _size (size zs).
Proof. by unfold size. Qed.
Lemma size_strlen (zs : t) :
size zs = strlen zs + 1.
Proof. unfold strlen; by rewrite Z.sub_simpl_r. Qed.
Remark size_nil : size [] = 0.
Proof. by rewrite size_unfold. Qed.
Lemma size_cons :
forall (z : N) (zs : t),
size (z :: zs) = 1 + size zs.
Proof. intros *; rewrite !size_unfold/=; by lia. Qed.
Lemma size_cons':
forall (z : N) (zs : t),
(z <> 0)%N ->
WF (z :: zs) ->
Z.to_nat (size (z :: zs)) = S (Z.to_nat (size zs)).
Proof.
move=> z zs Hz Hzstring; inversion Hzstring as [zs' [Hzs' HIn]];
unfold strlen, size; simpl.
replace ((length zs + 1)%nat - 1)%Z
with (Z.of_nat (length zs))
by lia.
destruct zs=> //=.
lia.
Qed.
Lemma size_app :
forall (zs zs' : t),
size (zs ++ zs') = size zs + size zs'.
Proof.
intros *; induction zs as [| z zs'' IHzs'']; simpl in *.
- by rewrite size_nil Z.add_0_l.
- by rewrite !size_cons IHzs'' Z.add_assoc.
Qed.
Lemma size_nil_cons_contra :
forall (z : _) (zs : t),
not (size (z :: zs) = size []).
Proof. intros *; intro CONTRA; unfold size in CONTRA; by inversion CONTRA. Qed.
Lemma size_nonneg : forall (zs : t), (0 <= size zs).
Proof. intros *; destruct zs; unfold size; simpl; by lia. Qed.
Lemma size_neg_inj : forall (zs : t), (0 = size zs)%Z -> zs = [].
Proof. move=> [ | a s] Hlen=> //. Qed.
End size_Theory.
Section strlen_Theory.
Context {σ : genv}.
(* Use rewrite strlen_unfold/= to reduce away a strlen application to a
concrete string; if complete reduction is not possible, this is
equivalent to unfolding the strlen notation.
*)
Lemma strlen_unfold (zs : t) :
strlen zs = Unfold _size (Unfold _strlen (strlen zs)).
Proof. by unfold strlen. Qed.
Lemma strlen_size (zs : t) :
strlen zs = size zs - 1.
Proof. by unfold strlen. Qed.
Remark strlen_nil : strlen [] = -1.
Proof. by rewrite strlen_unfold. Qed.
Lemma strlen_cons (zs : t) :
forall (z : _),
strlen (z :: zs) = 1 + strlen zs.
Proof. intros *; rewrite !strlen_unfold/=; by lia. Qed.
Lemma strlen_cons':
forall (z : N) (zs : t),
(z <> 0)%N ->
WF (z :: zs) ->
Z.to_nat (strlen (z :: zs)) = S (Z.to_nat (strlen zs)).
Proof.
move=> z zs Hz Hzstring; inversion Hzstring as [zs' [Hzs' HIn]];
unfold strlen, size; simpl.
replace ((length zs + 1)%nat - 1)%Z
with (Z.of_nat (length zs))
by lia.
destruct zs=> //=.
- apply WF_singleton_inj in Hzstring; subst; lia.
- lia.
Qed.
Lemma strlen_app :
forall (zs zs' : t),
strlen (zs ++ zs') = size zs + strlen zs'.
Proof.
intros *; induction zs as [| z zs'' IHzs'']; simpl in *.
- rewrite !size_unfold !strlen_unfold/=; by lia.
- by rewrite !strlen_cons !size_cons IHzs'' Z.add_assoc.
Qed.
Lemma strlen_nil_cons_contra :
forall (z : _) (zs : t),
not (strlen (z :: zs) = strlen []).
Proof.
intros *.
unfold strlen.
pose proof (size_nil_cons_contra z zs).
by lia.
Qed.
Lemma strlen_lowerbound : forall (zs : t), (-1 <= strlen zs)%Z.
Proof.
intros *; destruct zs;
unfold strlen, size;
simpl; by lia.
Qed.
Lemma strlen_nonneg : forall (zs : t),
WF zs ->
(0 <= strlen zs).
Proof.
intros * Hzstring.
inversion Hzstring as [zs' [Hzs' HIn]].
rewrite Hzs' strlen_app.
pose proof (strlen_lowerbound zs').
pose proof (size_nonneg zs').
unfold strlen; unfold size at 2.
simpl; by lia.
Qed.
Lemma strlen_neg_inj : forall (zs : t), (-1 = strlen zs)%Z -> zs = [].
Proof.
move=> [ | a s] Hlen=> //; exfalso.
pose proof (strlen_nil_cons_contra a s) as H.
apply H; by rewrite strlen_nil.
Qed.
End strlen_Theory.
Section WFs_equiv_Theory.
Context {σ : genv}.
Section helpers.
Lemma len0_take (zs : t)
(H0 : 0 < List.length zs)
(H : forall i : nat, Z.of_nat i = strlen zs -> nth_error zs i = Some 0%N) :
zs = take (List.length zs - 1) zs ++ [0%N].
Proof.
rewrite -[zs](firstn_skipn (List.length zs - 1)). f_equal.
{ rewrite length_app length_take length_drop.
set x := List.length zs.
have ->: ((x - 1) `min` x + (x - (x - 1)) - 1 = x - 1)%nat by lia.
by rewrite firstn_skipn. }
elim: zs H0 H; first by inversion 1.
move => z zs IH /= H0 H. have ->: (List.length zs - 0 = List.length zs)%nat by lia.
destruct zs; first by move: (H 0%nat eq_refl); inversion 1.
destruct zs; first by move: (H 1%nat eq_refl); inversion 1.
apply: IH; first by simpl; lia.
case; unfold strlen, size; rewrite /List.length -/List.length; first by lia.
move => n' ?. simpl. move: (H (S (S n'))). apply.
rewrite Nat2Z.inj_succ; unfold strlen, size; rewrite /List.length -/List.length. lia.
Qed.
Lemma forall_not0 (zs : t)
(H : ∀ i : nat, (i <= strlen zs)%Z →
∃ z : N, nth_error zs i = Some z ∧ has_type_prop (Vchar z) Tchar)
(H' : forall i : nat, (i < strlen zs)%Z ->
∃ z : N, nth_error zs i = Some z /\ z <> 0%N):
List.Forall (fun z : N => z <> 0%N /\ has_type_prop (Vchar z) Tchar) (take (List.length zs - 1) zs).
Proof.
induction zs; first by constructor.
simpl. have ->: (List.length zs - 0 = List.length zs)%nat by lia. destruct zs.
{ simpl. constructor. }
simpl. constructor.
{ move: (H 0%nat); case.
{ unfold strlen, size; rewrite /List.length -/List.length. lia. }
move => x /= []; inversion 1; subst; intuition.
specialize (H' 0%nat ltac:(rewrite !strlen_cons;
pose proof (strlen_lowerbound zs);
by lia)) as [? [? CONTRA]];
simpl in *; inversion H1; congruence. }
have <-: (List.length zs - 0 = List.length zs)%nat by lia. apply: IHzs.
case.
{ move => h1; exists n. split => //.
destruct zs. 1: {
specialize (H 1%nat ltac:(rewrite strlen_unfold/=; lia))
as [z' [Hnth Htype]].
simpl in Hnth; inversion Hnth; by subst.
}
move: (H 1%nat); unfold strlen, size; case => //.
{ rewrite /List.length -/List.length. lia. }
by move => x /=; case; inversion 1; subst. }
move => n'; unfold strlen, size; rewrite /List.length -/List.length => h.
move: (H (S (S n'))); case.
{ unfold strlen, size; rewrite /List.length -/List.length. lia. }
move => x /=; case => h1 h2; exists x; split => //.
intros i Hstrlen.
rewrite [strlen (_ :: _ :: _)]strlen_cons in H'.
specialize (H' (S i) ltac:(lia)) as [x [? ?]].
exists x; by intuition.
Qed.
Lemma nth_error_strlen_contra :
forall (n : nat) (zs : t),
strlen zs < Z.of_nat n ->
forall z, not (nth_error zs n = Some z).
Proof.
intros n; induction n; intros **.
- destruct zs=> //.
rewrite strlen_cons in H.
pose proof (strlen_lowerbound zs).
by lia.
- destruct zs=> //.
simpl; apply IHn.
rewrite strlen_cons in H.
pose proof (strlen_lowerbound zs).
by lia.
Qed.
End helpers.
Lemma WFs_equiv (zs : t) : WF' zs <-> WF zs.
Proof.
(* this is a horrible proof *)
unfold WF, WF'; split.
{ case; case => x h1 []h2 []h3 h4.
exists (take (List.length zs - 1) zs). repeat split.
{ apply: len0_take => //.
clear - h1; case: zs h1 => //= a l _; lia. }
2: {
apply List.Forall_forall; intros z Hin.
pose proof (In_nth_error zs z Hin) as [n Hnth].
assert (n < strlen zs \/ Z.of_nat n = strlen zs \/ strlen zs < n)
as [Hn | [Hn | Hn]] by lia.
- specialize (h2 n ltac:(lia)) as [? ?]; intuition; congruence.
- specialize (h4 n Hn).
rewrite h4 in Hnth; inversion Hnth.
apply has_type_prop_char_0.
- exfalso; eapply nth_error_strlen_contra; by eauto.
}
suff: List.Forall (fun x => x <> 0%N /\ has_type_prop (σ:=σ) (Vchar x) Tchar)
(take (Datatypes.length zs - 1) zs).
{ clear - zs. elim: zs => //= [_ [] //|a zs IH /=]. rewrite Nat.sub_0_r.
destruct zs as [|z zs]; first done. move => /= /Forall_cons [[??]?].
move => [//|]. simpl in IH. rewrite Nat.sub_0_r in IH. apply IH. done. }
apply: forall_not0. apply h2. apply h3. }
case => zs' []-> h1. split.
{ destruct zs'; simpl; first by exists 0%N.
by exists n. }
split.
{ move => i. unfold strlen, size; rewrite length_app Nat2Z.inj_add /= => h2.
assert (i < List.length zs' \/ i = List.length zs')%nat
as [h2' | h2'] by lia; clear h2. 2: {
subst; exists 0%N; clear h1; split.
- by induction zs'=> //=.
- apply has_type_prop_char_0.
}
rewrite nth_error_app1 => //.
elim: zs' i h1 h2' => /=; first by lia.
move => a lx IH i h1 h2. destruct i.
{ exists a; split => //.
inversion h1 as [? Hforall].
apply Forall_cons in Hforall.
by intuition. }
case: (IH i). 1: {
split.
- move => X; apply h1; by right.
- destruct h1 as [? Hforall]; by apply List.Forall_inv_tail in Hforall. }
1: by lia.
move => x []h3 h4; exists x => /=; split => //. }
split. 1: {
move => i; unfold strlen, size.
rewrite length_app/=; intros Hi.
assert (i < length zs')%nat as Hzs' by lia; clear Hi.
pose proof (nth_error_app1 zs' [0%N] Hzs') as Hnth.
pose proof (nth_In zs' 0%N Hzs') as Hin.
rewrite Hnth; erewrite nth_error_nth' with (d:=0%N); eauto; eexists;
split; eauto.
intro CONTRA; rewrite CONTRA in Hin.
inversion h1; by congruence.
}
move => i; unfold strlen, size.
rewrite length_app Nat2Z.inj_add /= => h2.
have ->: i = List.length zs' by lia. clear h2.
rewrite nth_error_app2 //.
have ->: (List.length zs' - List.length zs' = 0)%nat by lia. by [].
Qed.
Lemma WF'_singleton_inj :
forall (z : N),
WF' [z] ->
z = 0%N.
Proof. intros *; rewrite WFs_equiv; by apply WF_singleton_inj. Qed.
Remark not_WF'_nil : not (WF' []).
Proof. rewrite WFs_equiv; by apply not_WF_nil. Qed.
Lemma WF'_nonempty_inj :
forall (zs : t),
WF' zs ->
exists z zs', zs = z :: zs'.
Proof. intros *; rewrite WFs_equiv; by apply WF_nonempty_inj. Qed.
Lemma WF'_cons :
forall (z : _) (zs : t),
(z <> 0)%N ->
has_type_prop (Vchar z) Tchar ->
WF' (z :: zs) <->
WF' zs.
Proof. intros *; rewrite !WFs_equiv; by apply WF_cons. Qed.
End WFs_equiv_Theory.
Section with_Σ.
Context `{Σ : cpp_logic} `{σ : genv}.
(* The toplevel definition of zstring.bufR: *)
Definition bufR (q : cQp.t) (sz : Z) (zs : t) : Rep :=
[| size zs <= sz |] **
arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF zs |] **
.[Tchar ! size zs]
|-> arrayR Tchar (fun _ => primR Tchar q (Vchar 0)) (repeat () (Z.to_nat (sz - size zs))).
(* The toplevel definition of zstring.bufR': *)
Definition bufR' (q : cQp.t) (sz : Z) (zs : t) : Rep :=
[| size zs <= sz |] **
arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF' zs |] **
.[Tchar ! size zs]
|-> arrayR Tchar (fun _ => primR Tchar q (Vchar 0)) (repeat () (Z.to_nat (sz - size zs))).
#[global] Instance bufR_WF_observe :
forall q (sz : Z) (zs : t),
Observe [| WF zs |] (bufR q sz zs).
Proof. refine _. Qed.
#[global] Instance bufR'_WF_observe :
forall q (sz : Z) (zs : t),
Observe [| WF' zs |] (bufR' q sz zs).
Proof. refine _. Qed.
Lemma bufRs_equiv (q : cQp.t) (sz : Z) (zs : t) :
bufR q sz zs -|- bufR' q sz zs.
Proof. intros *; by rewrite /bufR/bufR' WFs_equiv. Qed.
(* The toplevel definition of zstring.R: *)
Definition R (q : cQp.t) (zs : t) : Rep :=
arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF zs |].
(* The toplevel definition of zstring.R': *)
Definition R' (q : cQp.t) (zs : t) : Rep :=
arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF' zs |].
#[global] Instance R_WF_observe :
forall q (zs : t),
Observe [| WF zs |] (R q zs).
Proof. refine _. Qed.
#[global] Instance R'_WF'_observe :
forall q (zs : t),
Observe [| WF' zs |] (R' q zs).
Proof. refine _. Qed.
Lemma Rs_equiv :
forall (q : cQp.t) (zs : t),
R q zs -|- R' q zs.
Proof. intros *; by rewrite /R/R' WFs_equiv. Qed.
Section Theory.
#[local] Open Scope string_scope.
Section bufR.
Remark bufR_nil : forall (q : cQp.t) (sz : Z), bufR q sz [] |-- False.
Proof.
intros *.
iIntros "rest"; rewrite /bufR.
iDestruct "rest" as "(?&?& %CONTRA &?)".
by apply not_WF_nil in CONTRA.
Qed.
(* TODO (AUTO): Investigate whether or not this hint actually fires. *)
#[global] Instance bufR_sz_contra :
forall (q : cQp.t) (sz : Z) (zs : t),
Observe2 False [| sz < size zs |] (bufR q sz zs).
Proof. intros *; rewrite /Observe2/bufR; iIntros "%H [%CONTRA ?]"; by lia. Qed.
#[global] Instance bufR_singleton :
forall (q : cQp.t) (sz : Z) (z : _),
Observe [| z = 0%N |] (bufR q sz [z]).
Proof.
intros *; rewrite /Observe/bufR; iIntros "(% & Hz & %HWF & rest)".
iModIntro; iPureIntro; by apply WF_singleton_inj.
Qed.
Lemma bufR_cons :
forall (q : cQp.t) (sz : Z) (z : _) (zs : t),
z <> 0%N ->
bufR q sz (z :: zs) -|-
primR Tchar q (Vchar z) ** .[Tchar ! 1] |-> bufR q (sz - 1) zs.
Proof.
intros * Hz; split'.
- rewrite /bufR arrayR_cons.
iIntros "[%Hsz [[? [Hz Hzs]] [%HWF Hrest]]]".
assert (size zs <= sz - 1) by (rewrite size_cons in Hsz; by lia).
iDestruct (observe [| has_type_prop (Vchar z) _ |] with "Hz") as "%".
assert (WF zs) by (apply WF_cons in HWF; assumption).
rewrite !_offsetR_sep !_offsetR_only_provable
size_cons -_offsetR_sub_sub
Z.sub_add_distr.
iFrame "∗%".
- rewrite /bufR arrayR_cons !_offsetR_sep !_offsetR_only_provable;
iIntros "[H [%Hsz [? [%HWF Hrest]]]]".
iDestruct (observe (type_ptrR Tchar) with "H") as "#?".
assert (size (z :: zs) <= sz) by (rewrite size_cons; by lia).
iDestruct (observe [| has_type_prop (Vchar z) _ |] with "H") as "%".
assert (WF (z :: zs)) by (apply WF_cons; assumption).
rewrite size_cons -!_offsetR_sub_sub Z.sub_add_distr.
iFrame "∗#%"; iPureIntro; by lia.
Qed.
#[global] Instance bufR_cons_cons_head_nonzero :
forall (q : cQp.t) (sz : Z) (z z' : _) (zs : t),
Observe [| z <> 0 |]%N (bufR q sz (z :: z' :: zs)).
Proof.
intros *.
apply: observe_intro_only_provable.
rewrite /bufR; unfold WF; iIntros "[%Hsz [Hzs [%WF Hrest]]]"; iPureIntro.
destruct WF as [zs' [Hzs' HIn]].
destruct zs'=> //=.
inversion Hzs'.
intro CONTRA; apply HIn; subst.
by constructor.
Qed.
Lemma bufR_has_type_prop :
forall (q : cQp.t) (sz : Z) (zs : t),
strlen zs < sz ->
bufR q sz zs
|-- bufR q sz zs ** [| List.Forall (fun c => has_type_prop (Vchar c) Tchar) zs |].
Proof.
intros **; generalize dependent sz; induction zs; intros **.
- rewrite {1}bufR_nil; by iIntros.
- destruct zs=> //=.
+ iIntros "H"; iDestruct (observe [| a = 0 |]%N with "H") as "%H'".
iFrame "∗"; iPureIntro.
try repeat constructor; subst.
apply has_type_prop_char_0.
+ iIntros "H"; iDestruct (observe [| WF (a :: n :: zs) |] with "H") as "%H'".
assert (1 <= size (a :: n :: zs)). {
rewrite -> size_cons in *.
pose proof (size_nonneg (n :: zs)).
by lia.
}
iDestruct (observe [| a <> 0 |]%N with "H") as "%H''"; iStopProof.
rewrite {1}bufR_cons; auto; rewrite primR_has_type_prop IHzs.
iIntros "[[? %has_type_prop] ?]"; iStopProof.
rewrite _offsetR_sep _offsetR_only_provable.
iIntros "[head [tail %has_type_props]]"; iCombine "head tail" as "H".
rewrite -bufR_cons; [| done]; iFrame "∗"; iPureIntro.
apply Forall_cons_2; auto.
rewrite -> !strlen_cons in *.
pose proof (size_nonneg (n :: zs)).
by lia.
Qed.
(* TODO (AUTO): Fix this once we add the correct observations for arrayR *)
#[global] Instance bufR_type_ptrR_observe :
forall q (sz : Z) (zs : t),
Observe (type_ptrR Tchar) (bufR q sz zs).
Proof.
move=> q sz zs; rewrite /bufR; destruct zs.
- rewrite /Observe. iIntros "[? [? [%CONTRA ?]]]";
exfalso; now apply not_WF_nil.
- rewrite arr.arrayR_cons; refine _.
Qed.
#[global] Instance bufR_validR_end_observe :
forall q (sz : Z) (zs : t),
Observe (.[Tchar ! sz] |-> validR) (bufR q sz zs).
Proof.
move=> q sz zs; rewrite /bufR/Observe.
iIntros "[%Hsz [array [%HWF rest]]]".
iAssert (arrayR Tchar (fun c => primR Tchar q (Vchar c))
(zs ++ repeat 0%N (Z.to_nat (sz - size zs))))
with "[array rest]" as "array'". 2: {
assert (Z.to_nat sz <= length (zs ++ repeat 0%N (Z.to_nat (sz - size zs))))%nat
as AUX. 1: {
rewrite length_app repeat_length.
unfold strlen in Hsz.
unfold size in *.
lia.
}
pose proof (arrayR_valid_obs (λ c : N, primR Tchar q (Vchar c)) Tchar (Z.to_nat sz)
(zs ++ repeat 0%N (Z.to_nat (sz - size zs)))
AUX).
iDestruct (observe (.[ Tchar ! sz ] |-> validR) with "array'")
as "#?"; auto.
rewrite Z2Nat.id in H; auto.
pose proof (size_nonneg zs).
lia.
}
rewrite arrayR_eq/arrayR_def fmap_app arrR_append length_fmap; unfold size.
iFrame.
rewrite arrR_proper; [by iFrame |].
generalize (Z.to_nat (sz - length zs)); intros n.
induction n=> //=.
setoid_rewrite IHn.
rewrite /fmap.
reflexivity.
Qed.
#[global] Instance bufR_validR_inbounds_observe :
forall q sz (z : Z) (zs : t),
(0 <= z <= sz)%Z ->
Observe (.[Tchar ! z] |-> validR) (bufR q sz zs).
Proof.
intros **; generalize dependent zs; induction z; intros **; simpl in *.
- rewrite ->o_sub_0 by eauto; rewrite _offsetR_id; refine _.
- assert (Z.pos p < sz \/ Z.pos p = sz)%Z
as [Hp | Hp] by lia; last by (rewrite Hp; refine _).
unfold Observe, bufR.
move: H Hp; generalize (Z.pos p).
iIntros (z H Hz) "[%SZ [zs [_ zeros]]]".
assert (z <= length zs \/ length zs < z)%Z
as [Hz' | Hz'] by lia.
+ iClear "zeros".
iDestruct (observe (.[ Tchar ! z] |-> validR) with "zs")
as "#valid"; last by iFrame "#".
pose proof (arrayR_valid_obs
(fun c => primR Tchar q (Vchar c))
Tchar (Z.to_nat z) zs ltac:(lia)).
by rewrite ->Z2Nat.id in H0 by lia.
+ iClear "zs".
iDestruct (observe (.[Tchar ! z] |-> validR) with "zeros")
as "#valid"; last by iFrame "#".
assert (exists z', size zs + z' = z /\ 0 <= z')
as [z' [Hz'' Hneg]]
by (exists (z - size zs)%Z; unfold size; lia);
subst.
rewrite -_offsetR_sub_sub; apply _offsetR_observe.
pose proof (arrayR_valid_obs
(fun c => primR Tchar q (Vchar 0))
Tchar (Z.to_nat z')
(repeat () (Z.to_nat (sz - size zs)))
ltac:(rewrite repeat_length; lia)).
by rewrite ->Z2Nat.id in H0 by lia.
- by lia.
Qed.
End bufR.
Section bufR'.
#[local] Ltac lift_WF2WF' H :=
intros **; rewrite -!bufRs_equiv; by apply H.
Remark bufR'_nil : forall (q : cQp.t) (sz : Z), bufR' q sz [] |-- False.
Proof. lift_WF2WF' bufR_nil. Qed.
(* TODO (AUTO): Investigate whether or not this hint actually fires. *)
#[global] Instance bufR'_sz_contra :
forall (q : cQp.t) (sz : Z) (zs : t),
Observe2 False [| sz < size zs |] (bufR' q sz zs).
Proof. lift_WF2WF' bufR_sz_contra. Qed.
#[global] Instance bufR'_singleton :
forall (q : cQp.t) (sz : Z) (z : _),
0 <= sz ->
Observe [| z = 0 |]%N (bufR' q sz [z]).
Proof. lift_WF2WF' bufR_singleton. Qed.
Lemma bufR'_cons :
forall (q : cQp.t) (sz : Z) (z : _) (zs : t),
z <> 0%N ->
bufR' q sz (z :: zs) -|-
primR Tchar q (Vchar z) ** .[Tchar ! 1] |-> bufR' q (sz - 1) zs.
Proof. lift_WF2WF' bufR_cons. Qed.
#[global] Instance bufR'_cons_cons_head_nonzero :
forall (q : cQp.t) (sz : Z) (z z' : _) (zs : t),
Observe [| z <> 0 |]%N (bufR' q sz (z :: z' :: zs)).
Proof. lift_WF2WF' bufR_cons_cons_head_nonzero. Qed.
Lemma bufR'_has_type_prop :
forall (q : cQp.t) (sz : Z) (zs : t),
strlen zs < sz ->
bufR' q sz zs
|-- bufR' q sz zs ** [| List.Forall (fun c => has_type_prop (Vchar c) Tchar) zs |].
Proof. lift_WF2WF' bufR_has_type_prop. Qed.
#[global] Instance bufR'_type_ptrR_observe :
forall q (sz : Z) (zs : t),
Observe (type_ptrR Tchar) (bufR' q sz zs).
Proof. lift_WF2WF' bufR_type_ptrR_observe. Qed.
#[global] Instance bufR'_validR_end_observe :
forall q (sz : Z) (zs : t),
Observe (.[Tchar ! sz] |-> validR) (bufR' q sz zs).
Proof. lift_WF2WF' bufR_validR_end_observe. Qed.
#[global] Instance bufR'_validR_inbounds_observe :
forall q sz (z : Z) (zs : t),
(0 <= z <= sz)%Z ->
Observe (.[Tchar ! z] |-> validR) (bufR' q sz zs).
Proof. lift_WF2WF' bufR_validR_inbounds_observe. Qed.
End bufR'.
Lemma bufR_unfold :
forall (q : cQp.t) (sz : Z) (zs : t),
bufR q sz zs -|-
[| size zs <= sz |] ** R q zs **
.[ Tchar ! size zs] |-> arrayR Tchar (fun _ => primR Tchar q (Vchar 0))
(repeat () (Z.to_nat (sz - size zs))).
Proof.
intros **; split'.
- rewrite /bufR/R; iIntros "(%&?&%&?)"; by iFrame "∗%".
- rewrite /bufR/R; iIntros "[% [[? %] ?]]"; by iFrame "∗%".
Qed.
Lemma bufR'_unfold :
forall (q : cQp.t) (sz : Z) (cstr : t),
bufR' q sz cstr -|-
[| size cstr <= sz |] ** R' q cstr **
.[ Tchar ! size cstr] |-> arrayR Tchar (fun _ => primR Tchar q (Vchar 0))
(repeat () (Z.to_nat (sz - size cstr))).
Proof. intros **; rewrite -!bufRs_equiv -!Rs_equiv; by apply bufR_unfold. Qed.
Section R_Theory.
Lemma R_bufR_equiv :
forall q (zs : t),
R q zs ≡ bufR q (size zs) zs.
Proof.
intros **; rewrite /R/bufR; split'.
- iIntros "[array %HWF]".
assert (size zs <= size zs) by lia.
rewrite Z.sub_diag/= arrayR_nil.
iDestruct (observe (.[Tchar ! size zs] |-> validR) with "array") as "#?". 1: {
unfold size; apply arrayR_valid_obs; by lia.
}
rewrite _offsetR_sep _offsetR_only_provable;
iFrame "#∗%"; iPureIntro; by eauto.
- iIntros "[%Hzs [array [%HWF rest]]]"; by iFrame "∗%".
Qed.
#[local] Ltac try_lift_bufR H :=
intros **; rewrite !R_bufR_equiv; eapply H; eauto.
Remark R_nil : forall (q : cQp.t), R q [] |-- False.
Proof. try_lift_bufR bufR_nil. Qed.
#[global] Instance R_singleton :
forall (q : cQp.t) (z : _),
Observe [| z = 0 |]%N (R q [z]).
Proof. try_lift_bufR bufR_singleton. Qed.
Lemma R_cons :
forall (q : cQp.t) (z : _) (zs : t),
z <> 0%N ->
R q (z :: zs) -|-
primR Tchar q (Vchar z) ** .[Tchar ! 1] |-> R q zs.
Proof.
intros **; rewrite !R_bufR_equiv.
replace (size zs) with ((size (z :: zs)) - 1)
by (rewrite size_cons; lia).
eapply bufR_cons; eauto.
Qed.
#[global] Instance R_cons_cons_head_nonzero :
forall (q : cQp.t) (z z' : _) (zs : t),
Observe [| z <> 0 |]%N (R q (z :: z' :: zs)).
Proof. try_lift_bufR bufR_cons_cons_head_nonzero. Qed.
Lemma R_has_type_prop :
forall (q : cQp.t) (zs : t),
R q zs
|-- R q zs ** [| List.Forall (fun c => has_type_prop (Vchar c) Tchar) zs |].
Proof.
try_lift_bufR bufR_has_type_prop.
unfold strlen; by lia.
Qed.
(* TODO (AUTO): Fix this once we add the correct observations for arrayR *)
#[global] Instance R_type_ptrR_observe :
forall q (zs : t),
Observe (type_ptrR Tchar) (R q zs).
Proof. try_lift_bufR bufR_type_ptrR_observe. Qed.
#[global] Instance R_validR_end_observe :
forall q (zs : t),
Observe (.[Tchar ! size zs] |-> validR) (R q zs).
Proof. try_lift_bufR bufR_validR_end_observe. Qed.
#[global] Instance R_validR_end_observe' :
forall q (zs : t),
Observe (.[Tchar ! strlen zs] |-> .[Tchar ! 1] |-> validR) (R q zs).
Proof.
intros *; pose proof (R_validR_end_observe q zs).
rewrite _offsetR_sub_sub; unfold size, strlen in *.
unfold size in *.
by replace (length zs - 1 + 1)%Z
with (Z.of_nat (length zs))
by lia.
Qed.
#[global] Instance R_validR_inbounds_observe :
forall q (z : Z) (zs : t),
(0 <= z <= size zs)%Z ->
Observe (.[Tchar ! z] |-> validR) (R q zs).
Proof.
intros * Hsize; unfold R; unfold size in Hsize.
apply observe_sep_l.
pose proof (arrayR_valid_obs
(fun c => primR Tchar q (Vchar c))
Tchar (Z.to_nat z) zs ltac:(lia)).
by rewrite ->Z2Nat.id in H by lia.
Qed.
End R_Theory.
Section R'_Theory.
#[local] Ltac lift_WF2WF' H :=
intros **; rewrite -!Rs_equiv; by apply H.
Lemma R'_bufR'_equiv :
forall q (zs : t),
R' q zs ≡ bufR' q (size zs) zs.
Proof. intros **; rewrite -Rs_equiv -bufRs_equiv; by apply R_bufR_equiv. Qed.
Remark R'_nil : forall (q : cQp.t), R' q [] |-- False.
Proof. lift_WF2WF' R_nil. Qed.
#[global] Instance R'_singleton :
forall (q : cQp.t) (z : _),
Observe [| z = 0 |]%N (R' q [z]).
Proof. lift_WF2WF' R_singleton. Qed.
Lemma R'_cons :
forall (q : cQp.t) (z : _) (zs : t),
z <> 0%N ->
R' q (z :: zs) -|-
primR Tchar q (Vchar z) ** .[Tchar ! 1] |-> R' q zs.
Proof. lift_WF2WF' R_cons. Qed.
#[global] Instance R'_cons_cons_head_nonzero :
forall (q : cQp.t) (z z' : _) (zs : t),
Observe [| z <> 0 |]%N (R' q (z :: z' :: zs)).
Proof. lift_WF2WF' R_cons_cons_head_nonzero. Qed.
Lemma R'_has_type_prop :
forall (q : cQp.t) (zs : t),
R' q zs
|-- R' q zs ** [| List.Forall (fun c => has_type_prop (Vchar c) Tchar) zs |].
Proof. lift_WF2WF' R_has_type_prop. Qed.
#[global] Instance R'_type_ptrR_observe :
forall q (zs : t),
Observe (type_ptrR Tchar) (R' q zs).
Proof. lift_WF2WF' R_type_ptrR_observe. Qed.
#[global] Instance R'_validR_end_observe :
forall q (zs : t),
Observe (.[Tchar ! size zs] |-> validR) (R' q zs).
Proof. lift_WF2WF' R_validR_end_observe. Qed.
#[global] Instance R'_validR_end_observe' :
forall q (zs : t),
Observe (.[Tchar ! strlen zs] |-> .[Tchar ! 1] |-> validR) (R' q zs).
Proof. lift_WF2WF' R_validR_end_observe'. Qed.
#[global] Instance R'_validR_inbounds_observe :
forall q (z : Z) (zs : t),
(0 <= z <= size zs)%Z ->
Observe (.[Tchar ! z] |-> validR) (R' q zs).
Proof. lift_WF2WF' R_validR_inbounds_observe. Qed.
End R'_Theory.
Lemma R_unfold :
forall (q : cQp.t) (zs : t),
R q zs
-|- arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF zs |].
Proof. intros **; split'; by rewrite /R. Qed.
Lemma R'_unfold :
forall (q : cQp.t) (zs : t),
R' q zs
-|- arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF' zs |].
Proof. intros **; split'; by rewrite /R'. Qed.
End Theory.
End with_Σ.
End with_ct.
(* refresh notation outside of the section *)
Notation size zs := (_size zs%Z).
Notation strlen zs := (_strlen zs%Z).
Notation WF zs := (_WF zs%Z).
Notation WF' zs := (_WF' zs%Z).
End zstring.
Section zstring_pure_hint_test_pre.
(* TODO (AUTO; cc/ @paolo): globally setting this prevents the use
of the Fail vernacular.
*)
Goal not True.
Proof.
(* by auto. *)
(* > In nested Ltac calls to "by (tactic)" and "tactics.done", last call failed. *)
(* > No applicable tactic. *)
Fail (by auto).
(* > In nested Ltac calls to "by (tactic)" and "tactics.done", last call failed. *)
(* > No applicable tactic. *)
Abort.
#[local] Unset Ltac Backtrace.
Goal forall {σ : genv}, not (zstring.WF char_type.Cchar []).
Proof. Fail (by auto with pure). Abort.
Goal
forall z z' z'' zs n,
zstring.strlen zs = n ->
zstring.strlen (z :: z' :: z'' :: zs) = 3 + n.
Proof. intros * Hzs. Fail (by auto with pure). Abort.
End zstring_pure_hint_test_pre.
(* TODO (AUTO; JH): talk with @janno w.r.t the appropriate way to package these; hopefully
further porting work will elucidate the appropriate export strategy.
*)
#[global] Hint Resolve
(* WF lemmas *)
zstring.not_WF_nil zstring.WF_cons zstring.WFs_equiv
zstring.WF_singleton zstring.WF_singleton_inj
zstring.WF_nonempty_inj
(* Bound Lemmas *)
zstring.strlen_nonneg zstring.strlen_lowerbound
zstring.size_nonneg
(* Symbolic reasoning about size/strlen; check out the
zstring.{size, strlen}_unfold* lemmas if you wish to
manually reduce applications which involve ground terms.
*)
zstring.strlen_cons zstring.strlen_cons'
zstring.size_cons zstring.size_cons'
: pure.
Section zstring_pure_hint_test_post.
#[local] Unset Ltac Backtrace.
Goal forall {σ : genv}, not (zstring.WF char_type.Cchar []).
Proof. by auto with pure. Abort.
Goal
forall z z' z'' zs n,
zstring.strlen zs = n ->
zstring.strlen (z :: z' :: z'' :: zs) = 3 + n.
Proof. intros * Hzs. Fail (by auto with pure). Abort.
End zstring_pure_hint_test_post.
Definition t := list N.
Bind Scope list_scope with t.
Section with_ct.
Variable ct : char_type.
#[local] Notation Tchar := (Tchar_ ct).
(* size reflects the in-memory footprint of the null-terminated string (i.e. the
length is one greater than that of the string as given by `std::strlen`).
*)
Definition _size (zs : t) : Z :=
Z.of_nat (List.length zs).
#[global] Arguments _size zs : simpl never.
Notation size zs := (_size zs%Z).
(* strlen mirrors the behavior of the standard-library function of the same name
(i.e. the length DOES NOT include the null-terminator).
*)
Definition _strlen (zs : t) : Z :=
(size zs - 1)%Z.
#[global] Arguments _strlen zs : simpl never.
Notation strlen zs := (_strlen zs%Z).
Definition _WF_alt {σ : genv} (zs : t) : Prop :=
let zs' := take (length zs - 1) zs in
last zs = Some 0%N /\ 0%N ∉ zs' /\
List.Forall (λ c, 0 <= c < 2 ^ char_type.bitsN ct)%N zs'.
#[global] Instance _WF_alt_dec {σ : genv} zs : Decision (_WF_alt zs) := _.
Definition _WF {σ : genv} (zs : t) : Prop :=
exists zs', zs = zs' ++ [0]%N /\ not (In 0%N zs') /\
List.Forall (λ c : N, has_type_prop (Vchar c) Tchar) zs.
#[global] Arguments _WF {σ} zs : simpl never.
Lemma _WF_WF_alt {σ : genv} zs : _WF zs <-> _WF_alt zs.
Proof.
rewrite /_WF /_WF_alt elem_of_list_In; setoid_rewrite <-has_type_prop_char'; split. {
move=> [] zs' [] -> [].
rewrite last_app /= length_app Nat.add_sub take_app_length Forall_app Forall_singleton.
naive_solver.
}
move=> [] Hl [] Hin Hall; exists (take (length zs - 1) zs).
destruct zs as [|z zs] using rev_ind => //= {IHzs}.
move: Hl Hin Hall;
rewrite last_app length_app /=. rewrite Nat.add_sub take_app_length.
rewrite Forall_app Forall_singleton.
intros [= ->]; intros; split_and! => //.
lia.
Qed.
Notation WF zs := (_WF zs%N).
#[global, program] Instance _WF_dec {σ : genv} zs : Decision (_WF zs) :=
cast_if (decide (_WF_alt zs)).
Solve All Obligations with intros; exact /_WF_WF_alt.
(* this definition is less intensional, and seems to work more
smoothly wrt. automation: *)
Definition _WF' {σ : genv} (zs : t) : Prop :=
(exists v, nth_error zs 0 = Some v) /\
(forall i, Z.of_nat i <= strlen zs ->
exists z, nth_error zs i = Some z /\ has_type_prop (Vchar z) Tchar) /\
(forall i, Z.of_nat i < strlen zs ->
exists z, nth_error zs i = Some z /\ z <> 0%N) /\
forall i, Z.of_nat i = strlen zs -> nth_error zs i = Some 0%N.
#[global] Arguments _WF' {σ} zs : simpl never.
Notation WF' zs := (_WF' zs%Z).
Section WF_Theory.
Context {σ : genv}.
Remark not_WF_nil : not (WF []).
Proof.
rewrite /_WF; intro CONTRA;
destruct CONTRA as [? [CONTRA ?]];
eapply app_cons_not_nil; eauto.
Qed.
Remark WF_singleton : WF [0%N].
Proof.
rewrite /_WF; exists []; repeat split.
- intro CONTRA; by inversion CONTRA.
- repeat constructor. apply has_type_prop_char_0.
Qed.
Lemma WF_singleton_inj :
forall (z : N),
WF [z] ->
z = 0%N.
Proof.
intros * [zs [Hzs [Hin Hbounds]]].
destruct zs; simpl in *; inversion Hzs.
- reflexivity.
- exfalso; eapply app_cons_not_nil; by eauto.
Qed.
Lemma WF_nonempty_inj :
forall (zs : t),
WF zs ->
exists z zs', zs = z :: zs'.
Proof.
move=> zs Hzstring; induction zs; [ | by eauto].
unfold WF in Hzstring.
destruct Hzstring as [? [CONTRA ?]].
now apply app_cons_not_nil in CONTRA.
Qed.
Lemma WF_cons :
forall (z : N) (zs : t),
(z <> 0)%N ->
has_type_prop (Vchar z) Tchar ->
WF (z :: zs) <->
WF zs.
Proof.
intros * Hnonzero Hz; split; [intro WF_zs | intro WF_z]; unfold WF in *.
- destruct WF_zs as [zs' [Hzs' [Hin Hbounds]]].
exists (tail zs'); split; destruct zs'=> //=;
[rewrite app_nil_l in Hzs' | | |];
inversion Hzs'=> //.
split.
+ intro; apply Hin=> //=; auto.
+ inversion Hbounds; by subst.
- destruct WF_z as [zs' [Hzs' [Hin Hbounds]]];
exists (z :: zs'); split.
+ by subst.
+ split.
* intro CONTRA; apply Hin; by inversion CONTRA.
* by constructor.
Qed.
(* Unlike WF_cons, here the explicit tail ++ [0] lets us write a more elegant lemma. *)
Lemma WF_cons' (z : N) (zs : t) :
WF (z :: zs ++ [0%N]) <->
(z <> 0)%N /\ has_type_prop (Vchar z) Tchar /\ WF (zs ++ [0%N]).
Proof.
rewrite /_WF/=. setoid_rewrite Forall_cons.
split.
- case => [[|z' zs'] [/= [-> Hzs']]]; first by destruct zs.
move: Hzs' => /(app_inj_2 _ _ _ _ eq_refl) [-> _].
move=> [/Decidable.not_or [Hz Hin] [Hz'Bounds Hrest]].
split_and! => //.
by exists zs'.
- case => Hz [HzBounds] [] zs' [] /(app_inj_2 _ _ _ _ eq_refl) [-> _] [Hin].
exists (z :: zs'); split_and! => //.
intros ?%in_inv. naive_solver.
Qed.
(* WF is prefix-free predicate *)
Lemma WF_eq_prefix_eq a1 a2 :
WF a1
-> WF a2
-> a1 = take (length a1) a2
-> a1 = a2.
Proof.
move=>[xs [-> [Nin _]]] [xs' [-> [Nin' _]]] H.
f_equal; move: xs' H Nin'.
rewrite length_app //= Nat.add_1_r.
induction xs.
- by case=>[//|??[<-][]];left.
- rewrite -app_comm_cons.
case; first by case: (xs).
move=>a' l' [<-] ? Hin'; f_equal.
apply: IHxs=>//= ?.
+ by apply: Nin; right.
+ by apply: Hin'; right.
Qed.
End WF_Theory.
Section size_Theory.
Context {σ : genv}.
(* Use rewrite size_unfold/= to reduce away a size application to a
concrete string; if complete reduction is not possible, this is
equivalent to unfolding the size notation.
*)
Lemma size_unfold (zs : t) :
size zs = Unfold _size (size zs).
Proof. by unfold size. Qed.
Lemma size_strlen (zs : t) :
size zs = strlen zs + 1.
Proof. unfold strlen; by rewrite Z.sub_simpl_r. Qed.
Remark size_nil : size [] = 0.
Proof. by rewrite size_unfold. Qed.
Lemma size_cons :
forall (z : N) (zs : t),
size (z :: zs) = 1 + size zs.
Proof. intros *; rewrite !size_unfold/=; by lia. Qed.
Lemma size_cons':
forall (z : N) (zs : t),
(z <> 0)%N ->
WF (z :: zs) ->
Z.to_nat (size (z :: zs)) = S (Z.to_nat (size zs)).
Proof.
move=> z zs Hz Hzstring; inversion Hzstring as [zs' [Hzs' HIn]];
unfold strlen, size; simpl.
replace ((length zs + 1)%nat - 1)%Z
with (Z.of_nat (length zs))
by lia.
destruct zs=> //=.
lia.
Qed.
Lemma size_app :
forall (zs zs' : t),
size (zs ++ zs') = size zs + size zs'.
Proof.
intros *; induction zs as [| z zs'' IHzs'']; simpl in *.
- by rewrite size_nil Z.add_0_l.
- by rewrite !size_cons IHzs'' Z.add_assoc.
Qed.
Lemma size_nil_cons_contra :
forall (z : _) (zs : t),
not (size (z :: zs) = size []).
Proof. intros *; intro CONTRA; unfold size in CONTRA; by inversion CONTRA. Qed.
Lemma size_nonneg : forall (zs : t), (0 <= size zs).
Proof. intros *; destruct zs; unfold size; simpl; by lia. Qed.
Lemma size_neg_inj : forall (zs : t), (0 = size zs)%Z -> zs = [].
Proof. move=> [ | a s] Hlen=> //. Qed.
End size_Theory.
Section strlen_Theory.
Context {σ : genv}.
(* Use rewrite strlen_unfold/= to reduce away a strlen application to a
concrete string; if complete reduction is not possible, this is
equivalent to unfolding the strlen notation.
*)
Lemma strlen_unfold (zs : t) :
strlen zs = Unfold _size (Unfold _strlen (strlen zs)).
Proof. by unfold strlen. Qed.
Lemma strlen_size (zs : t) :
strlen zs = size zs - 1.
Proof. by unfold strlen. Qed.
Remark strlen_nil : strlen [] = -1.
Proof. by rewrite strlen_unfold. Qed.
Lemma strlen_cons (zs : t) :
forall (z : _),
strlen (z :: zs) = 1 + strlen zs.
Proof. intros *; rewrite !strlen_unfold/=; by lia. Qed.
Lemma strlen_cons':
forall (z : N) (zs : t),
(z <> 0)%N ->
WF (z :: zs) ->
Z.to_nat (strlen (z :: zs)) = S (Z.to_nat (strlen zs)).
Proof.
move=> z zs Hz Hzstring; inversion Hzstring as [zs' [Hzs' HIn]];
unfold strlen, size; simpl.
replace ((length zs + 1)%nat - 1)%Z
with (Z.of_nat (length zs))
by lia.
destruct zs=> //=.
- apply WF_singleton_inj in Hzstring; subst; lia.
- lia.
Qed.
Lemma strlen_app :
forall (zs zs' : t),
strlen (zs ++ zs') = size zs + strlen zs'.
Proof.
intros *; induction zs as [| z zs'' IHzs'']; simpl in *.
- rewrite !size_unfold !strlen_unfold/=; by lia.
- by rewrite !strlen_cons !size_cons IHzs'' Z.add_assoc.
Qed.
Lemma strlen_nil_cons_contra :
forall (z : _) (zs : t),
not (strlen (z :: zs) = strlen []).
Proof.
intros *.
unfold strlen.
pose proof (size_nil_cons_contra z zs).
by lia.
Qed.
Lemma strlen_lowerbound : forall (zs : t), (-1 <= strlen zs)%Z.
Proof.
intros *; destruct zs;
unfold strlen, size;
simpl; by lia.
Qed.
Lemma strlen_nonneg : forall (zs : t),
WF zs ->
(0 <= strlen zs).
Proof.
intros * Hzstring.
inversion Hzstring as [zs' [Hzs' HIn]].
rewrite Hzs' strlen_app.
pose proof (strlen_lowerbound zs').
pose proof (size_nonneg zs').
unfold strlen; unfold size at 2.
simpl; by lia.
Qed.
Lemma strlen_neg_inj : forall (zs : t), (-1 = strlen zs)%Z -> zs = [].
Proof.
move=> [ | a s] Hlen=> //; exfalso.
pose proof (strlen_nil_cons_contra a s) as H.
apply H; by rewrite strlen_nil.
Qed.
End strlen_Theory.
Section WFs_equiv_Theory.
Context {σ : genv}.
Section helpers.
Lemma len0_take (zs : t)
(H0 : 0 < List.length zs)
(H : forall i : nat, Z.of_nat i = strlen zs -> nth_error zs i = Some 0%N) :
zs = take (List.length zs - 1) zs ++ [0%N].
Proof.
rewrite -[zs](firstn_skipn (List.length zs - 1)). f_equal.
{ rewrite length_app length_take length_drop.
set x := List.length zs.
have ->: ((x - 1) `min` x + (x - (x - 1)) - 1 = x - 1)%nat by lia.
by rewrite firstn_skipn. }
elim: zs H0 H; first by inversion 1.
move => z zs IH /= H0 H. have ->: (List.length zs - 0 = List.length zs)%nat by lia.
destruct zs; first by move: (H 0%nat eq_refl); inversion 1.
destruct zs; first by move: (H 1%nat eq_refl); inversion 1.
apply: IH; first by simpl; lia.
case; unfold strlen, size; rewrite /List.length -/List.length; first by lia.
move => n' ?. simpl. move: (H (S (S n'))). apply.
rewrite Nat2Z.inj_succ; unfold strlen, size; rewrite /List.length -/List.length. lia.
Qed.
Lemma forall_not0 (zs : t)
(H : ∀ i : nat, (i <= strlen zs)%Z →
∃ z : N, nth_error zs i = Some z ∧ has_type_prop (Vchar z) Tchar)
(H' : forall i : nat, (i < strlen zs)%Z ->
∃ z : N, nth_error zs i = Some z /\ z <> 0%N):
List.Forall (fun z : N => z <> 0%N /\ has_type_prop (Vchar z) Tchar) (take (List.length zs - 1) zs).
Proof.
induction zs; first by constructor.
simpl. have ->: (List.length zs - 0 = List.length zs)%nat by lia. destruct zs.
{ simpl. constructor. }
simpl. constructor.
{ move: (H 0%nat); case.
{ unfold strlen, size; rewrite /List.length -/List.length. lia. }
move => x /= []; inversion 1; subst; intuition.
specialize (H' 0%nat ltac:(rewrite !strlen_cons;
pose proof (strlen_lowerbound zs);
by lia)) as [? [? CONTRA]];
simpl in *; inversion H1; congruence. }
have <-: (List.length zs - 0 = List.length zs)%nat by lia. apply: IHzs.
case.
{ move => h1; exists n. split => //.
destruct zs. 1: {
specialize (H 1%nat ltac:(rewrite strlen_unfold/=; lia))
as [z' [Hnth Htype]].
simpl in Hnth; inversion Hnth; by subst.
}
move: (H 1%nat); unfold strlen, size; case => //.
{ rewrite /List.length -/List.length. lia. }
by move => x /=; case; inversion 1; subst. }
move => n'; unfold strlen, size; rewrite /List.length -/List.length => h.
move: (H (S (S n'))); case.
{ unfold strlen, size; rewrite /List.length -/List.length. lia. }
move => x /=; case => h1 h2; exists x; split => //.
intros i Hstrlen.
rewrite [strlen (_ :: _ :: _)]strlen_cons in H'.
specialize (H' (S i) ltac:(lia)) as [x [? ?]].
exists x; by intuition.
Qed.
Lemma nth_error_strlen_contra :
forall (n : nat) (zs : t),
strlen zs < Z.of_nat n ->
forall z, not (nth_error zs n = Some z).
Proof.
intros n; induction n; intros **.
- destruct zs=> //.
rewrite strlen_cons in H.
pose proof (strlen_lowerbound zs).
by lia.
- destruct zs=> //.
simpl; apply IHn.
rewrite strlen_cons in H.
pose proof (strlen_lowerbound zs).
by lia.
Qed.
End helpers.
Lemma WFs_equiv (zs : t) : WF' zs <-> WF zs.
Proof.
(* this is a horrible proof *)
unfold WF, WF'; split.
{ case; case => x h1 []h2 []h3 h4.
exists (take (List.length zs - 1) zs). repeat split.
{ apply: len0_take => //.
clear - h1; case: zs h1 => //= a l _; lia. }
2: {
apply List.Forall_forall; intros z Hin.
pose proof (In_nth_error zs z Hin) as [n Hnth].
assert (n < strlen zs \/ Z.of_nat n = strlen zs \/ strlen zs < n)
as [Hn | [Hn | Hn]] by lia.
- specialize (h2 n ltac:(lia)) as [? ?]; intuition; congruence.
- specialize (h4 n Hn).
rewrite h4 in Hnth; inversion Hnth.
apply has_type_prop_char_0.
- exfalso; eapply nth_error_strlen_contra; by eauto.
}
suff: List.Forall (fun x => x <> 0%N /\ has_type_prop (σ:=σ) (Vchar x) Tchar)
(take (Datatypes.length zs - 1) zs).
{ clear - zs. elim: zs => //= [_ [] //|a zs IH /=]. rewrite Nat.sub_0_r.
destruct zs as [|z zs]; first done. move => /= /Forall_cons [[??]?].
move => [//|]. simpl in IH. rewrite Nat.sub_0_r in IH. apply IH. done. }
apply: forall_not0. apply h2. apply h3. }
case => zs' []-> h1. split.
{ destruct zs'; simpl; first by exists 0%N.
by exists n. }
split.
{ move => i. unfold strlen, size; rewrite length_app Nat2Z.inj_add /= => h2.
assert (i < List.length zs' \/ i = List.length zs')%nat
as [h2' | h2'] by lia; clear h2. 2: {
subst; exists 0%N; clear h1; split.
- by induction zs'=> //=.
- apply has_type_prop_char_0.
}
rewrite nth_error_app1 => //.
elim: zs' i h1 h2' => /=; first by lia.
move => a lx IH i h1 h2. destruct i.
{ exists a; split => //.
inversion h1 as [? Hforall].
apply Forall_cons in Hforall.
by intuition. }
case: (IH i). 1: {
split.
- move => X; apply h1; by right.
- destruct h1 as [? Hforall]; by apply List.Forall_inv_tail in Hforall. }
1: by lia.
move => x []h3 h4; exists x => /=; split => //. }
split. 1: {
move => i; unfold strlen, size.
rewrite length_app/=; intros Hi.
assert (i < length zs')%nat as Hzs' by lia; clear Hi.
pose proof (nth_error_app1 zs' [0%N] Hzs') as Hnth.
pose proof (nth_In zs' 0%N Hzs') as Hin.
rewrite Hnth; erewrite nth_error_nth' with (d:=0%N); eauto; eexists;
split; eauto.
intro CONTRA; rewrite CONTRA in Hin.
inversion h1; by congruence.
}
move => i; unfold strlen, size.
rewrite length_app Nat2Z.inj_add /= => h2.
have ->: i = List.length zs' by lia. clear h2.
rewrite nth_error_app2 //.
have ->: (List.length zs' - List.length zs' = 0)%nat by lia. by [].
Qed.
Lemma WF'_singleton_inj :
forall (z : N),
WF' [z] ->
z = 0%N.
Proof. intros *; rewrite WFs_equiv; by apply WF_singleton_inj. Qed.
Remark not_WF'_nil : not (WF' []).
Proof. rewrite WFs_equiv; by apply not_WF_nil. Qed.
Lemma WF'_nonempty_inj :
forall (zs : t),
WF' zs ->
exists z zs', zs = z :: zs'.
Proof. intros *; rewrite WFs_equiv; by apply WF_nonempty_inj. Qed.
Lemma WF'_cons :
forall (z : _) (zs : t),
(z <> 0)%N ->
has_type_prop (Vchar z) Tchar ->
WF' (z :: zs) <->
WF' zs.
Proof. intros *; rewrite !WFs_equiv; by apply WF_cons. Qed.
End WFs_equiv_Theory.
Section with_Σ.
Context `{Σ : cpp_logic} `{σ : genv}.
(* The toplevel definition of zstring.bufR: *)
Definition bufR (q : cQp.t) (sz : Z) (zs : t) : Rep :=
[| size zs <= sz |] **
arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF zs |] **
.[Tchar ! size zs]
|-> arrayR Tchar (fun _ => primR Tchar q (Vchar 0)) (repeat () (Z.to_nat (sz - size zs))).
(* The toplevel definition of zstring.bufR': *)
Definition bufR' (q : cQp.t) (sz : Z) (zs : t) : Rep :=
[| size zs <= sz |] **
arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF' zs |] **
.[Tchar ! size zs]
|-> arrayR Tchar (fun _ => primR Tchar q (Vchar 0)) (repeat () (Z.to_nat (sz - size zs))).
#[global] Instance bufR_WF_observe :
forall q (sz : Z) (zs : t),
Observe [| WF zs |] (bufR q sz zs).
Proof. refine _. Qed.
#[global] Instance bufR'_WF_observe :
forall q (sz : Z) (zs : t),
Observe [| WF' zs |] (bufR' q sz zs).
Proof. refine _. Qed.
Lemma bufRs_equiv (q : cQp.t) (sz : Z) (zs : t) :
bufR q sz zs -|- bufR' q sz zs.
Proof. intros *; by rewrite /bufR/bufR' WFs_equiv. Qed.
(* The toplevel definition of zstring.R: *)
Definition R (q : cQp.t) (zs : t) : Rep :=
arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF zs |].
(* The toplevel definition of zstring.R': *)
Definition R' (q : cQp.t) (zs : t) : Rep :=
arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF' zs |].
#[global] Instance R_WF_observe :
forall q (zs : t),
Observe [| WF zs |] (R q zs).
Proof. refine _. Qed.
#[global] Instance R'_WF'_observe :
forall q (zs : t),
Observe [| WF' zs |] (R' q zs).
Proof. refine _. Qed.
Lemma Rs_equiv :
forall (q : cQp.t) (zs : t),
R q zs -|- R' q zs.
Proof. intros *; by rewrite /R/R' WFs_equiv. Qed.
Section Theory.
#[local] Open Scope string_scope.
Section bufR.
Remark bufR_nil : forall (q : cQp.t) (sz : Z), bufR q sz [] |-- False.
Proof.
intros *.
iIntros "rest"; rewrite /bufR.
iDestruct "rest" as "(?&?& %CONTRA &?)".
by apply not_WF_nil in CONTRA.
Qed.
(* TODO (AUTO): Investigate whether or not this hint actually fires. *)
#[global] Instance bufR_sz_contra :
forall (q : cQp.t) (sz : Z) (zs : t),
Observe2 False [| sz < size zs |] (bufR q sz zs).
Proof. intros *; rewrite /Observe2/bufR; iIntros "%H [%CONTRA ?]"; by lia. Qed.
#[global] Instance bufR_singleton :
forall (q : cQp.t) (sz : Z) (z : _),
Observe [| z = 0%N |] (bufR q sz [z]).
Proof.
intros *; rewrite /Observe/bufR; iIntros "(% & Hz & %HWF & rest)".
iModIntro; iPureIntro; by apply WF_singleton_inj.
Qed.
Lemma bufR_cons :
forall (q : cQp.t) (sz : Z) (z : _) (zs : t),
z <> 0%N ->
bufR q sz (z :: zs) -|-
primR Tchar q (Vchar z) ** .[Tchar ! 1] |-> bufR q (sz - 1) zs.
Proof.
intros * Hz; split'.
- rewrite /bufR arrayR_cons.
iIntros "[%Hsz [[? [Hz Hzs]] [%HWF Hrest]]]".
assert (size zs <= sz - 1) by (rewrite size_cons in Hsz; by lia).
iDestruct (observe [| has_type_prop (Vchar z) _ |] with "Hz") as "%".
assert (WF zs) by (apply WF_cons in HWF; assumption).
rewrite !_offsetR_sep !_offsetR_only_provable
size_cons -_offsetR_sub_sub
Z.sub_add_distr.
iFrame "∗%".
- rewrite /bufR arrayR_cons !_offsetR_sep !_offsetR_only_provable;
iIntros "[H [%Hsz [? [%HWF Hrest]]]]".
iDestruct (observe (type_ptrR Tchar) with "H") as "#?".
assert (size (z :: zs) <= sz) by (rewrite size_cons; by lia).
iDestruct (observe [| has_type_prop (Vchar z) _ |] with "H") as "%".
assert (WF (z :: zs)) by (apply WF_cons; assumption).
rewrite size_cons -!_offsetR_sub_sub Z.sub_add_distr.
iFrame "∗#%"; iPureIntro; by lia.
Qed.
#[global] Instance bufR_cons_cons_head_nonzero :
forall (q : cQp.t) (sz : Z) (z z' : _) (zs : t),
Observe [| z <> 0 |]%N (bufR q sz (z :: z' :: zs)).
Proof.
intros *.
apply: observe_intro_only_provable.
rewrite /bufR; unfold WF; iIntros "[%Hsz [Hzs [%WF Hrest]]]"; iPureIntro.
destruct WF as [zs' [Hzs' HIn]].
destruct zs'=> //=.
inversion Hzs'.
intro CONTRA; apply HIn; subst.
by constructor.
Qed.
Lemma bufR_has_type_prop :
forall (q : cQp.t) (sz : Z) (zs : t),
strlen zs < sz ->
bufR q sz zs
|-- bufR q sz zs ** [| List.Forall (fun c => has_type_prop (Vchar c) Tchar) zs |].
Proof.
intros **; generalize dependent sz; induction zs; intros **.
- rewrite {1}bufR_nil; by iIntros.
- destruct zs=> //=.
+ iIntros "H"; iDestruct (observe [| a = 0 |]%N with "H") as "%H'".
iFrame "∗"; iPureIntro.
try repeat constructor; subst.
apply has_type_prop_char_0.
+ iIntros "H"; iDestruct (observe [| WF (a :: n :: zs) |] with "H") as "%H'".
assert (1 <= size (a :: n :: zs)). {
rewrite -> size_cons in *.
pose proof (size_nonneg (n :: zs)).
by lia.
}
iDestruct (observe [| a <> 0 |]%N with "H") as "%H''"; iStopProof.
rewrite {1}bufR_cons; auto; rewrite primR_has_type_prop IHzs.
iIntros "[[? %has_type_prop] ?]"; iStopProof.
rewrite _offsetR_sep _offsetR_only_provable.
iIntros "[head [tail %has_type_props]]"; iCombine "head tail" as "H".
rewrite -bufR_cons; [| done]; iFrame "∗"; iPureIntro.
apply Forall_cons_2; auto.
rewrite -> !strlen_cons in *.
pose proof (size_nonneg (n :: zs)).
by lia.
Qed.
(* TODO (AUTO): Fix this once we add the correct observations for arrayR *)
#[global] Instance bufR_type_ptrR_observe :
forall q (sz : Z) (zs : t),
Observe (type_ptrR Tchar) (bufR q sz zs).
Proof.
move=> q sz zs; rewrite /bufR; destruct zs.
- rewrite /Observe. iIntros "[? [? [%CONTRA ?]]]";
exfalso; now apply not_WF_nil.
- rewrite arr.arrayR_cons; refine _.
Qed.
#[global] Instance bufR_validR_end_observe :
forall q (sz : Z) (zs : t),
Observe (.[Tchar ! sz] |-> validR) (bufR q sz zs).
Proof.
move=> q sz zs; rewrite /bufR/Observe.
iIntros "[%Hsz [array [%HWF rest]]]".
iAssert (arrayR Tchar (fun c => primR Tchar q (Vchar c))
(zs ++ repeat 0%N (Z.to_nat (sz - size zs))))
with "[array rest]" as "array'". 2: {
assert (Z.to_nat sz <= length (zs ++ repeat 0%N (Z.to_nat (sz - size zs))))%nat
as AUX. 1: {
rewrite length_app repeat_length.
unfold strlen in Hsz.
unfold size in *.
lia.
}
pose proof (arrayR_valid_obs (λ c : N, primR Tchar q (Vchar c)) Tchar (Z.to_nat sz)
(zs ++ repeat 0%N (Z.to_nat (sz - size zs)))
AUX).
iDestruct (observe (.[ Tchar ! sz ] |-> validR) with "array'")
as "#?"; auto.
rewrite Z2Nat.id in H; auto.
pose proof (size_nonneg zs).
lia.
}
rewrite arrayR_eq/arrayR_def fmap_app arrR_append length_fmap; unfold size.
iFrame.
rewrite arrR_proper; [by iFrame |].
generalize (Z.to_nat (sz - length zs)); intros n.
induction n=> //=.
setoid_rewrite IHn.
rewrite /fmap.
reflexivity.
Qed.
#[global] Instance bufR_validR_inbounds_observe :
forall q sz (z : Z) (zs : t),
(0 <= z <= sz)%Z ->
Observe (.[Tchar ! z] |-> validR) (bufR q sz zs).
Proof.
intros **; generalize dependent zs; induction z; intros **; simpl in *.
- rewrite ->o_sub_0 by eauto; rewrite _offsetR_id; refine _.
- assert (Z.pos p < sz \/ Z.pos p = sz)%Z
as [Hp | Hp] by lia; last by (rewrite Hp; refine _).
unfold Observe, bufR.
move: H Hp; generalize (Z.pos p).
iIntros (z H Hz) "[%SZ [zs [_ zeros]]]".
assert (z <= length zs \/ length zs < z)%Z
as [Hz' | Hz'] by lia.
+ iClear "zeros".
iDestruct (observe (.[ Tchar ! z] |-> validR) with "zs")
as "#valid"; last by iFrame "#".
pose proof (arrayR_valid_obs
(fun c => primR Tchar q (Vchar c))
Tchar (Z.to_nat z) zs ltac:(lia)).
by rewrite ->Z2Nat.id in H0 by lia.
+ iClear "zs".
iDestruct (observe (.[Tchar ! z] |-> validR) with "zeros")
as "#valid"; last by iFrame "#".
assert (exists z', size zs + z' = z /\ 0 <= z')
as [z' [Hz'' Hneg]]
by (exists (z - size zs)%Z; unfold size; lia);
subst.
rewrite -_offsetR_sub_sub; apply _offsetR_observe.
pose proof (arrayR_valid_obs
(fun c => primR Tchar q (Vchar 0))
Tchar (Z.to_nat z')
(repeat () (Z.to_nat (sz - size zs)))
ltac:(rewrite repeat_length; lia)).
by rewrite ->Z2Nat.id in H0 by lia.
- by lia.
Qed.
End bufR.
Section bufR'.
#[local] Ltac lift_WF2WF' H :=
intros **; rewrite -!bufRs_equiv; by apply H.
Remark bufR'_nil : forall (q : cQp.t) (sz : Z), bufR' q sz [] |-- False.
Proof. lift_WF2WF' bufR_nil. Qed.
(* TODO (AUTO): Investigate whether or not this hint actually fires. *)
#[global] Instance bufR'_sz_contra :
forall (q : cQp.t) (sz : Z) (zs : t),
Observe2 False [| sz < size zs |] (bufR' q sz zs).
Proof. lift_WF2WF' bufR_sz_contra. Qed.
#[global] Instance bufR'_singleton :
forall (q : cQp.t) (sz : Z) (z : _),
0 <= sz ->
Observe [| z = 0 |]%N (bufR' q sz [z]).
Proof. lift_WF2WF' bufR_singleton. Qed.
Lemma bufR'_cons :
forall (q : cQp.t) (sz : Z) (z : _) (zs : t),
z <> 0%N ->
bufR' q sz (z :: zs) -|-
primR Tchar q (Vchar z) ** .[Tchar ! 1] |-> bufR' q (sz - 1) zs.
Proof. lift_WF2WF' bufR_cons. Qed.
#[global] Instance bufR'_cons_cons_head_nonzero :
forall (q : cQp.t) (sz : Z) (z z' : _) (zs : t),
Observe [| z <> 0 |]%N (bufR' q sz (z :: z' :: zs)).
Proof. lift_WF2WF' bufR_cons_cons_head_nonzero. Qed.
Lemma bufR'_has_type_prop :
forall (q : cQp.t) (sz : Z) (zs : t),
strlen zs < sz ->
bufR' q sz zs
|-- bufR' q sz zs ** [| List.Forall (fun c => has_type_prop (Vchar c) Tchar) zs |].
Proof. lift_WF2WF' bufR_has_type_prop. Qed.
#[global] Instance bufR'_type_ptrR_observe :
forall q (sz : Z) (zs : t),
Observe (type_ptrR Tchar) (bufR' q sz zs).
Proof. lift_WF2WF' bufR_type_ptrR_observe. Qed.
#[global] Instance bufR'_validR_end_observe :
forall q (sz : Z) (zs : t),
Observe (.[Tchar ! sz] |-> validR) (bufR' q sz zs).
Proof. lift_WF2WF' bufR_validR_end_observe. Qed.
#[global] Instance bufR'_validR_inbounds_observe :
forall q sz (z : Z) (zs : t),
(0 <= z <= sz)%Z ->
Observe (.[Tchar ! z] |-> validR) (bufR' q sz zs).
Proof. lift_WF2WF' bufR_validR_inbounds_observe. Qed.
End bufR'.
Lemma bufR_unfold :
forall (q : cQp.t) (sz : Z) (zs : t),
bufR q sz zs -|-
[| size zs <= sz |] ** R q zs **
.[ Tchar ! size zs] |-> arrayR Tchar (fun _ => primR Tchar q (Vchar 0))
(repeat () (Z.to_nat (sz - size zs))).
Proof.
intros **; split'.
- rewrite /bufR/R; iIntros "(%&?&%&?)"; by iFrame "∗%".
- rewrite /bufR/R; iIntros "[% [[? %] ?]]"; by iFrame "∗%".
Qed.
Lemma bufR'_unfold :
forall (q : cQp.t) (sz : Z) (cstr : t),
bufR' q sz cstr -|-
[| size cstr <= sz |] ** R' q cstr **
.[ Tchar ! size cstr] |-> arrayR Tchar (fun _ => primR Tchar q (Vchar 0))
(repeat () (Z.to_nat (sz - size cstr))).
Proof. intros **; rewrite -!bufRs_equiv -!Rs_equiv; by apply bufR_unfold. Qed.
Section R_Theory.
Lemma R_bufR_equiv :
forall q (zs : t),
R q zs ≡ bufR q (size zs) zs.
Proof.
intros **; rewrite /R/bufR; split'.
- iIntros "[array %HWF]".
assert (size zs <= size zs) by lia.
rewrite Z.sub_diag/= arrayR_nil.
iDestruct (observe (.[Tchar ! size zs] |-> validR) with "array") as "#?". 1: {
unfold size; apply arrayR_valid_obs; by lia.
}
rewrite _offsetR_sep _offsetR_only_provable;
iFrame "#∗%"; iPureIntro; by eauto.
- iIntros "[%Hzs [array [%HWF rest]]]"; by iFrame "∗%".
Qed.
#[local] Ltac try_lift_bufR H :=
intros **; rewrite !R_bufR_equiv; eapply H; eauto.
Remark R_nil : forall (q : cQp.t), R q [] |-- False.
Proof. try_lift_bufR bufR_nil. Qed.
#[global] Instance R_singleton :
forall (q : cQp.t) (z : _),
Observe [| z = 0 |]%N (R q [z]).
Proof. try_lift_bufR bufR_singleton. Qed.
Lemma R_cons :
forall (q : cQp.t) (z : _) (zs : t),
z <> 0%N ->
R q (z :: zs) -|-
primR Tchar q (Vchar z) ** .[Tchar ! 1] |-> R q zs.
Proof.
intros **; rewrite !R_bufR_equiv.
replace (size zs) with ((size (z :: zs)) - 1)
by (rewrite size_cons; lia).
eapply bufR_cons; eauto.
Qed.
#[global] Instance R_cons_cons_head_nonzero :
forall (q : cQp.t) (z z' : _) (zs : t),
Observe [| z <> 0 |]%N (R q (z :: z' :: zs)).
Proof. try_lift_bufR bufR_cons_cons_head_nonzero. Qed.
Lemma R_has_type_prop :
forall (q : cQp.t) (zs : t),
R q zs
|-- R q zs ** [| List.Forall (fun c => has_type_prop (Vchar c) Tchar) zs |].
Proof.
try_lift_bufR bufR_has_type_prop.
unfold strlen; by lia.
Qed.
(* TODO (AUTO): Fix this once we add the correct observations for arrayR *)
#[global] Instance R_type_ptrR_observe :
forall q (zs : t),
Observe (type_ptrR Tchar) (R q zs).
Proof. try_lift_bufR bufR_type_ptrR_observe. Qed.
#[global] Instance R_validR_end_observe :
forall q (zs : t),
Observe (.[Tchar ! size zs] |-> validR) (R q zs).
Proof. try_lift_bufR bufR_validR_end_observe. Qed.
#[global] Instance R_validR_end_observe' :
forall q (zs : t),
Observe (.[Tchar ! strlen zs] |-> .[Tchar ! 1] |-> validR) (R q zs).
Proof.
intros *; pose proof (R_validR_end_observe q zs).
rewrite _offsetR_sub_sub; unfold size, strlen in *.
unfold size in *.
by replace (length zs - 1 + 1)%Z
with (Z.of_nat (length zs))
by lia.
Qed.
#[global] Instance R_validR_inbounds_observe :
forall q (z : Z) (zs : t),
(0 <= z <= size zs)%Z ->
Observe (.[Tchar ! z] |-> validR) (R q zs).
Proof.
intros * Hsize; unfold R; unfold size in Hsize.
apply observe_sep_l.
pose proof (arrayR_valid_obs
(fun c => primR Tchar q (Vchar c))
Tchar (Z.to_nat z) zs ltac:(lia)).
by rewrite ->Z2Nat.id in H by lia.
Qed.
End R_Theory.
Section R'_Theory.
#[local] Ltac lift_WF2WF' H :=
intros **; rewrite -!Rs_equiv; by apply H.
Lemma R'_bufR'_equiv :
forall q (zs : t),
R' q zs ≡ bufR' q (size zs) zs.
Proof. intros **; rewrite -Rs_equiv -bufRs_equiv; by apply R_bufR_equiv. Qed.
Remark R'_nil : forall (q : cQp.t), R' q [] |-- False.
Proof. lift_WF2WF' R_nil. Qed.
#[global] Instance R'_singleton :
forall (q : cQp.t) (z : _),
Observe [| z = 0 |]%N (R' q [z]).
Proof. lift_WF2WF' R_singleton. Qed.
Lemma R'_cons :
forall (q : cQp.t) (z : _) (zs : t),
z <> 0%N ->
R' q (z :: zs) -|-
primR Tchar q (Vchar z) ** .[Tchar ! 1] |-> R' q zs.
Proof. lift_WF2WF' R_cons. Qed.
#[global] Instance R'_cons_cons_head_nonzero :
forall (q : cQp.t) (z z' : _) (zs : t),
Observe [| z <> 0 |]%N (R' q (z :: z' :: zs)).
Proof. lift_WF2WF' R_cons_cons_head_nonzero. Qed.
Lemma R'_has_type_prop :
forall (q : cQp.t) (zs : t),
R' q zs
|-- R' q zs ** [| List.Forall (fun c => has_type_prop (Vchar c) Tchar) zs |].
Proof. lift_WF2WF' R_has_type_prop. Qed.
#[global] Instance R'_type_ptrR_observe :
forall q (zs : t),
Observe (type_ptrR Tchar) (R' q zs).
Proof. lift_WF2WF' R_type_ptrR_observe. Qed.
#[global] Instance R'_validR_end_observe :
forall q (zs : t),
Observe (.[Tchar ! size zs] |-> validR) (R' q zs).
Proof. lift_WF2WF' R_validR_end_observe. Qed.
#[global] Instance R'_validR_end_observe' :
forall q (zs : t),
Observe (.[Tchar ! strlen zs] |-> .[Tchar ! 1] |-> validR) (R' q zs).
Proof. lift_WF2WF' R_validR_end_observe'. Qed.
#[global] Instance R'_validR_inbounds_observe :
forall q (z : Z) (zs : t),
(0 <= z <= size zs)%Z ->
Observe (.[Tchar ! z] |-> validR) (R' q zs).
Proof. lift_WF2WF' R_validR_inbounds_observe. Qed.
End R'_Theory.
Lemma R_unfold :
forall (q : cQp.t) (zs : t),
R q zs
-|- arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF zs |].
Proof. intros **; split'; by rewrite /R. Qed.
Lemma R'_unfold :
forall (q : cQp.t) (zs : t),
R' q zs
-|- arrayR Tchar (fun c => primR Tchar q (Vchar c)) zs ** [| WF' zs |].
Proof. intros **; split'; by rewrite /R'. Qed.
End Theory.
End with_Σ.
End with_ct.
(* refresh notation outside of the section *)
Notation size zs := (_size zs%Z).
Notation strlen zs := (_strlen zs%Z).
Notation WF zs := (_WF zs%Z).
Notation WF' zs := (_WF' zs%Z).
End zstring.
Section zstring_pure_hint_test_pre.
(* TODO (AUTO; cc/ @paolo): globally setting this prevents the use
of the Fail vernacular.
*)
Goal not True.
Proof.
(* by auto. *)
(* > In nested Ltac calls to "by (tactic)" and "tactics.done", last call failed. *)
(* > No applicable tactic. *)
Fail (by auto).
(* > In nested Ltac calls to "by (tactic)" and "tactics.done", last call failed. *)
(* > No applicable tactic. *)
Abort.
#[local] Unset Ltac Backtrace.
Goal forall {σ : genv}, not (zstring.WF char_type.Cchar []).
Proof. Fail (by auto with pure). Abort.
Goal
forall z z' z'' zs n,
zstring.strlen zs = n ->
zstring.strlen (z :: z' :: z'' :: zs) = 3 + n.
Proof. intros * Hzs. Fail (by auto with pure). Abort.
End zstring_pure_hint_test_pre.
(* TODO (AUTO; JH): talk with @janno w.r.t the appropriate way to package these; hopefully
further porting work will elucidate the appropriate export strategy.
*)
#[global] Hint Resolve
(* WF lemmas *)
zstring.not_WF_nil zstring.WF_cons zstring.WFs_equiv
zstring.WF_singleton zstring.WF_singleton_inj
zstring.WF_nonempty_inj
(* Bound Lemmas *)
zstring.strlen_nonneg zstring.strlen_lowerbound
zstring.size_nonneg
(* Symbolic reasoning about size/strlen; check out the
zstring.{size, strlen}_unfold* lemmas if you wish to
manually reduce applications which involve ground terms.
*)
zstring.strlen_cons zstring.strlen_cons'
zstring.size_cons zstring.size_cons'
: pure.
Section zstring_pure_hint_test_post.
#[local] Unset Ltac Backtrace.
Goal forall {σ : genv}, not (zstring.WF char_type.Cchar []).
Proof. by auto with pure. Abort.
Goal
forall z z' z'' zs n,
zstring.strlen zs = n ->
zstring.strlen (z :: z' :: z'' :: zs) = 3 + n.
Proof. intros * Hzs. Fail (by auto with pure). Abort.
End zstring_pure_hint_test_post.