clutch.eris.primitive_laws
This file proves the basic laws of the ProbLang weakest precondition by
applying the lifting lemmas.
From iris.proofmode Require Import proofmode.
From iris.algebra Require Import auth excl.
From iris.base_logic.lib Require Export ghost_map.
From clutch.base_logic Require Export error_credits.
From clutch.eris Require Export weakestpre ectx_lifting.
From clutch.prob_lang Require Export class_instances.
From clutch.prob_lang Require Import tactics lang notation.
From iris.prelude Require Import options.
Class erisGS Σ := HeapG {
erisGS_invG : invGS_gen HasNoLc Σ;
(* CMRA for the state *)
erisGS_heap : ghost_mapG Σ loc val;
erisGS_tapes : ghost_mapG Σ loc tape;
(* ghost names for the state *)
erisGS_heap_name : gname;
erisGS_tapes_name : gname;
(* CMRA and ghost name for the error *)
erisGS_error :: ecGS Σ;
}.
Definition progUR : ucmra := optionUR (exclR exprO).
Definition cfgO : ofe := prodO exprO stateO.
Definition cfgUR : ucmra := optionUR (exclR cfgO).
Definition heap_auth `{erisGS Σ} :=
@ghost_map_auth _ _ _ _ _ erisGS_heap erisGS_heap_name.
Definition tapes_auth `{erisGS Σ} :=
@ghost_map_auth _ _ _ _ _ erisGS_tapes erisGS_tapes_name.
Global Instance erisGS_erisWpGS `{!erisGS Σ} : erisWpGS prob_lang Σ := {
erisWpGS_invGS := erisGS_invG;
state_interp σ := (heap_auth 1 σ.(heap) ∗ tapes_auth 1 σ.(tapes))%I;
err_interp ε := (ec_supply ε);
}.
From iris.algebra Require Import auth excl.
From iris.base_logic.lib Require Export ghost_map.
From clutch.base_logic Require Export error_credits.
From clutch.eris Require Export weakestpre ectx_lifting.
From clutch.prob_lang Require Export class_instances.
From clutch.prob_lang Require Import tactics lang notation.
From iris.prelude Require Import options.
Class erisGS Σ := HeapG {
erisGS_invG : invGS_gen HasNoLc Σ;
(* CMRA for the state *)
erisGS_heap : ghost_mapG Σ loc val;
erisGS_tapes : ghost_mapG Σ loc tape;
(* ghost names for the state *)
erisGS_heap_name : gname;
erisGS_tapes_name : gname;
(* CMRA and ghost name for the error *)
erisGS_error :: ecGS Σ;
}.
Definition progUR : ucmra := optionUR (exclR exprO).
Definition cfgO : ofe := prodO exprO stateO.
Definition cfgUR : ucmra := optionUR (exclR cfgO).
Definition heap_auth `{erisGS Σ} :=
@ghost_map_auth _ _ _ _ _ erisGS_heap erisGS_heap_name.
Definition tapes_auth `{erisGS Σ} :=
@ghost_map_auth _ _ _ _ _ erisGS_tapes erisGS_tapes_name.
Global Instance erisGS_erisWpGS `{!erisGS Σ} : erisWpGS prob_lang Σ := {
erisWpGS_invGS := erisGS_invG;
state_interp σ := (heap_auth 1 σ.(heap) ∗ tapes_auth 1 σ.(tapes))%I;
err_interp ε := (ec_supply ε);
}.
Heap
Notation "l ↦{ dq } v" := (@ghost_map_elem _ _ _ _ _ erisGS_heap erisGS_heap_name l dq v)
(at level 20, format "l ↦{ dq } v") : bi_scope.
Notation "l ↦□ v" := (l ↦{ DfracDiscarded } v)%I
(at level 20, format "l ↦□ v") : bi_scope.
Notation "l ↦{# q } v" := (l ↦{ DfracOwn q } v)%I
(at level 20, format "l ↦{# q } v") : bi_scope.
Notation "l ↦ v" := (l ↦{ DfracOwn 1 } v)%I
(at level 20, format "l ↦ v") : bi_scope.
(at level 20, format "l ↦{ dq } v") : bi_scope.
Notation "l ↦□ v" := (l ↦{ DfracDiscarded } v)%I
(at level 20, format "l ↦□ v") : bi_scope.
Notation "l ↦{# q } v" := (l ↦{ DfracOwn q } v)%I
(at level 20, format "l ↦{# q } v") : bi_scope.
Notation "l ↦ v" := (l ↦{ DfracOwn 1 } v)%I
(at level 20, format "l ↦ v") : bi_scope.
Tapes
Notation "l ↪{ dq } v" := (@ghost_map_elem _ _ _ _ _ erisGS_tapes erisGS_tapes_name l dq (v : tape))
(at level 20, format "l ↪{ dq } v") : bi_scope.
Notation "l ↪□ v" := (l ↪{ DfracDiscarded } v)%I
(at level 20, format "l ↪□ v") : bi_scope.
Notation "l ↪{# q } v" := (l ↪{ DfracOwn q } v)%I
(at level 20, format "l ↪{# q } v") : bi_scope.
Notation "l ↪ v" := (l ↪{ DfracOwn 1 } v)%I
(at level 20, format "l ↪ v") : bi_scope.
Section lifting.
Context `{!erisGS Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types Φ Ψ : val → iProp Σ.
Implicit Types σ : state.
Implicit Types v : val.
Implicit Types l : loc.
(at level 20, format "l ↪{ dq } v") : bi_scope.
Notation "l ↪□ v" := (l ↪{ DfracDiscarded } v)%I
(at level 20, format "l ↪□ v") : bi_scope.
Notation "l ↪{# q } v" := (l ↪{ DfracOwn q } v)%I
(at level 20, format "l ↪{# q } v") : bi_scope.
Notation "l ↪ v" := (l ↪{ DfracOwn 1 } v)%I
(at level 20, format "l ↪ v") : bi_scope.
Section lifting.
Context `{!erisGS Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types Φ Ψ : val → iProp Σ.
Implicit Types σ : state.
Implicit Types v : val.
Implicit Types l : loc.
Recursive functions: we do not use this lemmas as it is easier to use Löb
(* induction directly, but this demonstrates that we can state the expected *)
(* reasoning principle for recursive functions, without any visible ▷. *)
Lemma wp_rec_löb E f x e Φ Ψ :
□ ( □ (∀ v, Ψ v -∗ WP (rec: f x := e)%V v @ E {{ Φ }}) -∗
∀ v, Ψ v -∗ WP (subst' x v (subst' f (rec: f x := e) e)) @ E {{ Φ }}) -∗
∀ v, Ψ v -∗ WP (rec: f x := e)%V v @ E {{ Φ }}.
Proof.
iIntros "#Hrec". iLöb as "IH". iIntros (v) "HΨ".
iApply lifting.wp_pure_step_later; first done.
iNext. iApply ("Hrec" with "[] HΨ"). iIntros "!>" (w) "HΨ".
iApply ("IH" with "HΨ").
Qed.
(* reasoning principle for recursive functions, without any visible ▷. *)
Lemma wp_rec_löb E f x e Φ Ψ :
□ ( □ (∀ v, Ψ v -∗ WP (rec: f x := e)%V v @ E {{ Φ }}) -∗
∀ v, Ψ v -∗ WP (subst' x v (subst' f (rec: f x := e) e)) @ E {{ Φ }}) -∗
∀ v, Ψ v -∗ WP (rec: f x := e)%V v @ E {{ Φ }}.
Proof.
iIntros "#Hrec". iLöb as "IH". iIntros (v) "HΨ".
iApply lifting.wp_pure_step_later; first done.
iNext. iApply ("Hrec" with "[] HΨ"). iIntros "!>" (w) "HΨ".
iApply ("IH" with "HΨ").
Qed.
Heap
Lemma wp_alloc E v s :
{{{ True }}} Alloc (Val v) @ s; E {{{ l, RET LitV (LitLoc l); l ↦ v }}}.
Proof.
iIntros (Φ) "_ HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "[Hh Ht] !#".
solve_red.
iIntros "!> /=" (e2 σ2 Hs); inv_head_step.
iMod ((ghost_map_insert (fresh_loc σ1.(heap)) v) with "Hh") as "[? Hl]".
{ apply not_elem_of_dom, fresh_loc_is_fresh. }
iFrame.
rewrite map_union_empty -insert_union_singleton_l.
iFrame.
iIntros "!>". by iApply "HΦ".
Qed.
Lemma wp_allocN_seq (N : nat) (z : Z) E v s:
TCEq N (Z.to_nat z) →
(0 < N)%Z →
{{{ True }}}
AllocN (Val $ LitV $ LitInt $ z) (Val v) @ s; E
{{{ l, RET LitV (LitLoc l); [∗ list] i ∈ seq 0 N, (l +ₗ (i : nat)) ↦ v }}}.
Proof.
iIntros (-> Hn Φ) "_ HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "[Hh Ht] !#".
iSplit.
{ iPureIntro.
rewrite /head_reducible.
eexists.
apply head_step_support_equiv_rel.
econstructor; eauto.
lia.
}
iIntros "!> /=" (e2 σ2 Hs); inv_head_step.
iMod ((ghost_map_insert_big _ _ with "Hh")) as "[$ Hl]".
iIntros "!>". iFrame.
iApply "HΦ".
iInduction (H) as [ | ?] "IH" forall (σ1).
- simpl.
iSplit; auto.
rewrite map_union_empty.
rewrite loc_add_0.
by rewrite big_sepM_singleton.
- rewrite seq_S.
rewrite heap_array_replicate_S_end.
iPoseProof (big_sepM_union _ _ _ _ with "Hl") as "[H1 H2]".
iApply big_sepL_app.
iSplitL "H1".
+ iApply "IH".
{ iPureIntro. lia. }
iApply "H1".
+ simpl. iSplit; auto.
by rewrite big_sepM_singleton.
Unshelve.
{
apply heap_array_map_disjoint.
intros.
apply not_elem_of_dom_1.
by apply fresh_loc_offset_is_fresh.
}
apply heap_array_map_disjoint.
intros.
apply not_elem_of_dom_1.
rewrite dom_singleton.
apply not_elem_of_singleton_2.
intros H2.
apply loc_add_inj in H2.
rewrite length_replicate in H1.
lia.
Qed.
Lemma wp_load E l dq v s :
{{{ ▷ l ↦{dq} v }}} Load (Val $ LitV $ LitLoc l) @ s; E {{{ RET v; l ↦{dq} v }}}.
Proof.
iIntros (Φ) ">Hl HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "[Hh Ht] !#".
iDestruct (ghost_map_lookup with "Hh Hl") as %?.
solve_red.
iIntros "!> /=" (e2 σ2 Hs); inv_head_step.
iFrame. iModIntro. by iApply "HΦ".
Qed.
Lemma wp_store E l v' v s :
{{{ ▷ l ↦ v' }}} Store (Val $ LitV (LitLoc l)) (Val v) @ s; E
{{{ RET LitV LitUnit; l ↦ v }}}.
Proof.
iIntros (Φ) ">Hl HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "[Hh Ht] !#".
iDestruct (ghost_map_lookup with "Hh Hl") as %?.
solve_red.
iIntros "!> /=" (e2 σ2 Hs); inv_head_step.
iMod (ghost_map_update with "Hh Hl") as "[$ Hl]".
iFrame. iModIntro. by iApply "HΦ".
Qed.
Lemma wp_rand (N : nat) (z : Z) E s :
TCEq N (Z.to_nat z) →
{{{ True }}} rand #z @ s; E {{{ (n : fin (S N)), RET #n; True }}}.
Proof.
iIntros (-> Φ) "_ HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "Hσ !#".
solve_red.
iIntros "!>" (e2 σ2 Hs).
inv_head_step.
iFrame.
by iApply ("HΦ" $! x) .
Qed.
Lemma wp_rand_nat (N : nat) (z : Z) E s :
TCEq N (Z.to_nat z) →
{{{ True }}} rand #z @ s; E {{{ (n : nat), RET #n; ⌜ n < S N ⌝ }}}.
Proof.
iIntros (-> Φ) "_ HΦ".
iApply wp_rand; auto.
iModIntro.
iIntros (n) "_".
iApply "HΦ".
iPureIntro.
apply fin_to_nat_lt.
Qed.
Tapes
Lemma wp_alloc_tape N z E s :
TCEq N (Z.to_nat z) →
{{{ True }}} alloc #z @ s; E {{{ α, RET #lbl:α; α ↪ (N; []) }}}.
Proof.
iIntros (-> Φ) "_ HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "(Hh & Ht) !# /=".
solve_red.
iIntros "!>" (e2 σ2 Hs); inv_head_step.
iMod (ghost_map_insert (fresh_loc σ1.(tapes)) with "Ht") as "[$ Hl]".
{ apply not_elem_of_dom, fresh_loc_is_fresh. }
iFrame. iModIntro.
by iApply "HΦ".
Qed.
Lemma wp_rand_tape N α n ns z E s :
TCEq N (Z.to_nat z) →
{{{ ▷ α ↪ (N; n :: ns) }}} rand(#lbl:α) #z @ s; E {{{ RET #(LitInt n); α ↪ (N; ns) }}}.
Proof.
iIntros (-> Φ) ">Hl HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "(Hh & Ht) !#".
iDestruct (ghost_map_lookup with "Ht Hl") as %?.
solve_red.
iIntros "!>" (e2 σ2 Hs).
inv_head_step.
iMod (ghost_map_update with "Ht Hl") as "[$ Hl]".
iFrame. iModIntro.
by iApply "HΦ".
Qed.
Lemma wp_rand_tape_empty N z α E s :
TCEq N (Z.to_nat z) →
{{{ ▷ α ↪ (N; []) }}} rand(#lbl:α) #z @ s; E {{{ (n : fin (S N)), RET #(LitInt n); α ↪ (N; []) }}}.
Proof.
iIntros (-> Φ) ">Hl HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "(Hh & Ht) !#".
iDestruct (ghost_map_lookup with "Ht Hl") as %?.
solve_red.
iIntros "!>" (e2 σ2 Hs).
inv_head_step.
iFrame.
iModIntro. iApply ("HΦ" with "[$Hl //]").
Qed.
Lemma wp_rand_tape_wrong_bound N M z α E ns s :
TCEq N (Z.to_nat z) →
N ≠ M →
{{{ ▷ α ↪ (M; ns) }}} rand(#lbl:α) #z @ s; E {{{ (n : fin (S N)), RET #(LitInt n); α ↪ (M; ns) }}}.
Proof.
iIntros (-> ? Φ) ">Hl HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "(Hh & Ht) !#".
iDestruct (ghost_map_lookup with "Ht Hl") as %?.
solve_red.
iIntros "!>" (e2 σ2 Hs).
inv_head_step.
iFrame.
iModIntro.
iApply ("HΦ" with "[$Hl //]").
Qed.
End lifting.
Global Hint Extern 0 (TCEq _ (Z.to_nat _ )) => rewrite Nat2Z.id : typeclass_instances.
TCEq N (Z.to_nat z) →
{{{ True }}} alloc #z @ s; E {{{ α, RET #lbl:α; α ↪ (N; []) }}}.
Proof.
iIntros (-> Φ) "_ HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "(Hh & Ht) !# /=".
solve_red.
iIntros "!>" (e2 σ2 Hs); inv_head_step.
iMod (ghost_map_insert (fresh_loc σ1.(tapes)) with "Ht") as "[$ Hl]".
{ apply not_elem_of_dom, fresh_loc_is_fresh. }
iFrame. iModIntro.
by iApply "HΦ".
Qed.
Lemma wp_rand_tape N α n ns z E s :
TCEq N (Z.to_nat z) →
{{{ ▷ α ↪ (N; n :: ns) }}} rand(#lbl:α) #z @ s; E {{{ RET #(LitInt n); α ↪ (N; ns) }}}.
Proof.
iIntros (-> Φ) ">Hl HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "(Hh & Ht) !#".
iDestruct (ghost_map_lookup with "Ht Hl") as %?.
solve_red.
iIntros "!>" (e2 σ2 Hs).
inv_head_step.
iMod (ghost_map_update with "Ht Hl") as "[$ Hl]".
iFrame. iModIntro.
by iApply "HΦ".
Qed.
Lemma wp_rand_tape_empty N z α E s :
TCEq N (Z.to_nat z) →
{{{ ▷ α ↪ (N; []) }}} rand(#lbl:α) #z @ s; E {{{ (n : fin (S N)), RET #(LitInt n); α ↪ (N; []) }}}.
Proof.
iIntros (-> Φ) ">Hl HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "(Hh & Ht) !#".
iDestruct (ghost_map_lookup with "Ht Hl") as %?.
solve_red.
iIntros "!>" (e2 σ2 Hs).
inv_head_step.
iFrame.
iModIntro. iApply ("HΦ" with "[$Hl //]").
Qed.
Lemma wp_rand_tape_wrong_bound N M z α E ns s :
TCEq N (Z.to_nat z) →
N ≠ M →
{{{ ▷ α ↪ (M; ns) }}} rand(#lbl:α) #z @ s; E {{{ (n : fin (S N)), RET #(LitInt n); α ↪ (M; ns) }}}.
Proof.
iIntros (-> ? Φ) ">Hl HΦ".
iApply wp_lift_atomic_head_step; [done|].
iIntros (σ1) "(Hh & Ht) !#".
iDestruct (ghost_map_lookup with "Ht Hl") as %?.
solve_red.
iIntros "!>" (e2 σ2 Hs).
inv_head_step.
iFrame.
iModIntro.
iApply ("HΦ" with "[$Hl //]").
Qed.
End lifting.
Global Hint Extern 0 (TCEq _ (Z.to_nat _ )) => rewrite Nat2Z.id : typeclass_instances.