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Commit 5a7b2bf6 authored by Lennard Gäher's avatar Lennard Gäher Committed by Michael Sammler
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dynamic alloc/free

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theories/lang/ghost_state.v
View file @ 5a7b2bf6
......@@ -22,8 +22,8 @@ Definition heapUR : ucmra :=
Class heapG Σ := HeapG {
heap_heap_inG :> inG Σ (authR heapUR);
heap_heap_name : gname;
heap_alloc_range_map_inG :> ghost_mapG Σ alloc_id (Z * nat);
heap_alloc_range_map_name : gname;
heap_alloc_meta_map_inG :> ghost_mapG Σ alloc_id (Z * nat * alloc_kind);
heap_alloc_meta_map_name : gname;
heap_alloc_alive_map_inG :> ghost_mapG Σ alloc_id bool;
heap_alloc_alive_map_name : gname;
heap_fntbl_inG :> ghost_mapG Σ addr function;
......@@ -39,11 +39,11 @@ Definition to_heap_cellR (hc : heap_cell) : heap_cellR :=
Definition to_heapUR : heap → heapUR :=
fmap to_heap_cellR.
Definition to_alloc_rangeR (al : allocation) : (Z * nat) :=
(al.(al_start), al.(al_len)).
Definition to_alloc_metaR (al : allocation) : (Z * nat * alloc_kind) :=
(al.(al_start), al.(al_len), al.(al_kind)).
Definition to_alloc_range_map : allocs → gmap alloc_id (Z * nat) :=
fmap to_alloc_rangeR.
Definition to_alloc_meta_map : allocs → gmap alloc_id (Z * nat * alloc_kind) :=
fmap to_alloc_metaR.
Definition to_alloc_alive_map : allocs → gmap alloc_id bool :=
fmap al_alive.
......@@ -53,20 +53,20 @@ Section definitions.
(** * Allocation stuff. *)
(** [alloc_range id al] persistently records the information that allocation
(** [alloc_meta id al] persistently records the information that allocation
with identifier [id] has a range corresponding to that of [a]. *)
Definition alloc_range_def (id : alloc_id) (al : allocation) : iProp Σ :=
id ↪[ heap_alloc_range_map_name ]□ to_alloc_rangeR al.
Definition alloc_range_aux : seal (@alloc_range_def). by eexists. Qed.
Definition alloc_range := unseal alloc_range_aux.
Definition alloc_range_eq : @alloc_range = @alloc_range_def :=
seal_eq alloc_range_aux.
Definition alloc_meta_def (id : alloc_id) (al : allocation) : iProp Σ :=
id ↪[ heap_alloc_meta_map_name ]□ to_alloc_metaR al.
Definition alloc_meta_aux : seal (@alloc_meta_def). by eexists. Qed.
Definition alloc_meta := unseal alloc_meta_aux.
Definition alloc_meta_eq : @alloc_meta = @alloc_meta_def :=
seal_eq alloc_meta_aux.
Global Instance allocs_range_pers id al : Persistent (alloc_range id al).
Proof. rewrite alloc_range_eq. by apply _. Qed.
Global Instance allocs_range_pers id al : Persistent (alloc_meta id al).
Proof. rewrite alloc_meta_eq. by apply _. Qed.
Global Instance allocs_range_tl id al : Timeless (alloc_range id al).
Proof. rewrite alloc_range_eq. by apply _. Qed.
Global Instance allocs_range_tl id al : Timeless (alloc_meta id al).
Proof. rewrite alloc_meta_eq. by apply _. Qed.
(** [loc_in_bounds l n] persistently records the information that location
[l] and any of its positive offset up to [n] (included) are in range of the
......@@ -75,7 +75,7 @@ Section definitions.
Definition loc_in_bounds_def (l : loc) (n : nat) : iProp Σ :=
∃ (id : alloc_id) (al : allocation),
⌜l.1 = ProvAlloc (Some id)⌝ ∗ ⌜al.(al_start) ≤ l.2⌝ ∗ ⌜l.2 + n ≤ al_end al⌝ ∗
⌜allocation_in_range al⌝ ∗ alloc_range id al.
⌜allocation_in_range al⌝ ∗ alloc_meta id al.
Definition loc_in_bounds_aux : seal (@loc_in_bounds_def). by eexists. Qed.
Definition loc_in_bounds := unseal loc_in_bounds_aux.
Definition loc_in_bounds_eq : @loc_in_bounds = @loc_in_bounds_def :=
......@@ -156,8 +156,8 @@ Section definitions.
(** * Freeable *)
Definition freeable_def (l : loc) (n : nat) : iProp Σ :=
∃ id, ⌜l.1 = ProvAlloc (Some id)⌝ ∗ alloc_range id {| al_start := l.2; al_len := n; al_alive := true |} ∗
Definition freeable_def (l : loc) (n : nat) (k : alloc_kind) : iProp Σ :=
∃ id, ⌜l.1 = ProvAlloc (Some id)⌝ ∗ alloc_meta id {| al_start := l.2; al_len := n; al_alive := true; al_kind := k |} ∗
alloc_alive id (DfracOwn 1) true.
Definition freeable_aux : seal (@freeable_def). by eexists. Qed.
Definition freeable := unseal freeable_aux.
......@@ -169,8 +169,8 @@ Section definitions.
Definition heap_ctx (h : heap) : iProp Σ :=
own heap_heap_name (● to_heapUR h).
Definition alloc_range_ctx (ub : allocs) : iProp Σ :=
ghost_map_auth heap_alloc_range_map_name 1 (to_alloc_range_map ub).
Definition alloc_meta_ctx (ub : allocs) : iProp Σ :=
ghost_map_auth heap_alloc_meta_map_name 1 (to_alloc_meta_map ub).
Definition alloc_alive_ctx (ub : allocs) : iProp Σ :=
ghost_map_auth heap_alloc_alive_map_name 1 (to_alloc_alive_map ub).
......@@ -181,7 +181,7 @@ Section definitions.
Definition heap_state_ctx (st : heap_state) : iProp Σ :=
⌜heap_state_invariant st⌝ ∗
heap_ctx st.(hs_heap) ∗
alloc_range_ctx st.(hs_allocs) ∗
alloc_meta_ctx st.(hs_allocs) ∗
alloc_alive_ctx st.(hs_allocs).
Definition state_ctx (σ:state) : iProp Σ :=
......@@ -241,68 +241,70 @@ Section fntbl.
Qed.
End fntbl.
Section alloc_range.
Section alloc_meta.
Context `{!heapG Σ}.
Implicit Types am : allocs.
Lemma alloc_range_mono id a1 a2 :
Lemma alloc_meta_mono id a1 a2 :
alloc_same_range a1 a2 →
alloc_range id a1 -∗ alloc_range id a2.
Proof. destruct a1 as [???], a2 as [???] => -[/= <- <-]. by rewrite alloc_range_eq. Qed.
a1.(al_kind) = a2.(al_kind) →
alloc_meta id a1 -∗ alloc_meta id a2.
Proof. destruct a1 as [????], a2 as [????] => -[/= <- <-] <-. by rewrite alloc_meta_eq. Qed.
Lemma alloc_range_agree id a1 a2 :
alloc_range id a1 -∗ alloc_range id a2 -∗ ⌜alloc_same_range a1 a2⌝.
Lemma alloc_meta_agree id a1 a2 :
alloc_meta id a1 -∗ alloc_meta id a2 -∗ ⌜alloc_same_range a1 a2⌝.
Proof.
destruct a1 as [???], a2 as [???]. rewrite alloc_range_eq /alloc_same_range.
destruct a1 as [????], a2 as [????]. rewrite alloc_meta_eq /alloc_same_range.
iIntros "H1 H2". by iDestruct (ghost_map_elem_agree with "H1 H2") as %[=->->].
Qed.
Lemma alloc_range_alloc am id al :
Lemma alloc_meta_alloc am id al :
am !! id = None →
alloc_range_ctx am ==∗
alloc_range_ctx (<[id := al]> am) ∗ alloc_range id al.
alloc_meta_ctx am ==∗
alloc_meta_ctx (<[id := al]> am) ∗ alloc_meta id al.
Proof.
move => Hid. rewrite alloc_range_eq. iIntros "Hctx".
move => Hid. rewrite alloc_meta_eq. iIntros "Hctx".
iMod (ghost_map_insert_persist with "Hctx") as "[? $]". { by rewrite lookup_fmap fmap_None. }
by rewrite -fmap_insert.
Qed.
Lemma alloc_range_to_loc_in_bounds l id (n : nat) al:
Lemma alloc_meta_to_loc_in_bounds l id (n : nat) al:
l.1 = ProvAlloc (Some id) →
al.(al_start) ≤ l.2 ∧ l.2 + n ≤ al_end al →
allocation_in_range al →
alloc_range id al -∗ loc_in_bounds l n.
alloc_meta id al -∗ loc_in_bounds l n.
Proof.
iIntros (?[??]?) "Hr". rewrite loc_in_bounds_eq.
iExists id, al. by iFrame "Hr".
Qed.
Lemma alloc_range_lookup am id al :
alloc_range_ctx am -∗
alloc_range id al -∗
⌜∃ al', am !! id = Some al' ∧ alloc_same_range al al'⌝.
Lemma alloc_meta_lookup am id al :
alloc_meta_ctx am -∗
alloc_meta id al -∗
⌜∃ al', am !! id = Some al' ∧ alloc_same_range al al' ∧ al.(al_kind) = al'.(al_kind)⌝.
Proof.
rewrite alloc_range_eq. iIntros "Htbl Hf".
rewrite alloc_meta_eq. iIntros "Htbl Hf".
iDestruct (ghost_map_lookup with "Htbl Hf") as %Hlookup.
iPureIntro. move: Hlookup. rewrite lookup_fmap fmap_Some => -[[???][?[??]]].
iPureIntro. move: Hlookup. rewrite lookup_fmap fmap_Some => -[[????][?[???]]].
by eexists _.
Qed.
Lemma alloc_range_ctx_same_range am id al1 al2 :
Lemma alloc_meta_ctx_same_range am id al1 al2 :
am !! id = Some al1 →
alloc_same_range al1 al2 →
alloc_range_ctx am = alloc_range_ctx (<[id := al2]> am).
al1.(al_kind) = al2.(al_kind) →
alloc_meta_ctx am = alloc_meta_ctx (<[id := al2]> am).
Proof.
move => Hid [Heq1 Heq2].
rewrite /alloc_range_ctx /to_alloc_range_map fmap_insert insert_id; first done.
move => Hid [Heq1 Heq2] Heq3.
rewrite /alloc_meta_ctx /to_alloc_meta_map fmap_insert insert_id; first done.
rewrite lookup_fmap Hid /=. destruct al1, al2; naive_solver.
Qed.
Lemma alloc_range_ctx_killed am id al :
Lemma alloc_meta_ctx_killed am id al :
am !! id = Some al →
alloc_range_ctx am = alloc_range_ctx (<[id := killed al]> am).
Proof. move => ?. by apply: alloc_range_ctx_same_range. Qed.
End alloc_range.
alloc_meta_ctx am = alloc_meta_ctx (<[id := killed al]> am).
Proof. move => ?. by apply: alloc_meta_ctx_same_range. Qed.
End alloc_meta.
Section alloc_alive.
Context `{!heapG Σ}.
......@@ -349,7 +351,7 @@ Section loc_in_bounds.
iDestruct "H2" as (?? Hl2 ? Hend ?) "#H2".
move: Hl1 Hl2 => /= Hl1 Hl2. iExists id, al.
destruct l. unfold al_end in *. simpl in *. simplify_eq.
iDestruct (alloc_range_agree with "H2 H1") as %[? <-].
iDestruct (alloc_meta_agree with "H2 H1") as %[? <-].
iFrame "H1". iPureIntro. rewrite /shift_loc /= in Hend. naive_solver lia.
- iIntros "H". iDestruct "H" as (id al ????) "#H".
iSplit; iExists id, al; iFrame "H"; iPureIntro; split_and! => //=; lia.
......@@ -387,7 +389,7 @@ Section loc_in_bounds.
Proof.
rewrite loc_in_bounds_eq.
iIntros "Hb ((?&?&Hctx&?)&?)". iDestruct "Hb" as (id al ????) "Hb".
iDestruct (alloc_range_lookup with "Hctx Hb") as %[al' [?[??]]].
iDestruct (alloc_meta_lookup with "Hctx Hb") as %[al' [?[[??]?]]].
iExists id, al'. iPureIntro. unfold allocation_in_range, al_end in *.
naive_solver lia.
Qed.
......@@ -524,6 +526,10 @@ Section heap.
by iDestruct (heap_mapsto_mbyte_agree with "[$]") as %->.
Qed.
Lemma heap_mapsto_layout_has_layout l ly :
l ↦|ly| -∗ ⌜l `has_layout_loc` ly⌝.
Proof. iIntros "(% & % & % & ?)". done. Qed.
Lemma heap_alloc_st l h v aid :
l.1 = ProvAlloc (Some aid) →
heap_range_free h l.2 (length v) →
......@@ -555,12 +561,12 @@ Section heap.
al.(al_start) = l.2 →
al.(al_len) = length v →
allocation_in_range al →
alloc_range id al -∗
alloc_meta id al -∗
alloc_alive id (DfracOwn 1) true -∗
heap_ctx h ==∗
heap_ctx (heap_alloc l.2 v id h) ∗
l ↦ v ∗
freeable l (length v).
freeable l (length v) al.(al_kind).
Proof.
iIntros (Hid Hfree Hstart Hlen Hrange) "#Hr Hal Hctx".
iMod (heap_alloc_st with "Hctx") as "[$ Hl]" => //.
......@@ -568,7 +574,7 @@ Section heap.
rewrite heap_mapsto_mbyte_eq /heap_mapsto_mbyte_def.
iSplitR "Hal"; last first; last iSplit.
- rewrite freeable_eq. iExists id. iFrame. iSplit => //.
by iApply (alloc_range_mono with "Hr").
by iApply (alloc_meta_mono with "Hr").
- rewrite loc_in_bounds_eq. iExists id, al. iFrame "Hr".
rewrite /al_end. iPureIntro. naive_solver lia.
- iApply (big_sepL_impl with "Hl").
......@@ -839,25 +845,25 @@ End alloc_alive.
Section alloc_new_blocks.
Context `{!heapG Σ}.
Lemma heap_alloc_new_block_upd σ1 l v σ2:
alloc_new_block σ1 l v σ2 →
heap_state_ctx σ1 ==∗ heap_state_ctx σ2 ∗ l ↦ v ∗ freeable l (length v).
Lemma heap_alloc_new_block_upd σ1 l v kind σ2:
alloc_new_block σ1 kind l v σ2 →
heap_state_ctx σ1 ==∗ heap_state_ctx σ2 ∗ l ↦ v ∗ freeable l (length v) kind.
Proof.
move => []; clear. move => σ l aid v alloc Haid ???; subst alloc.
move => []; clear. move => σ l aid kind v alloc Haid ???; subst alloc.
iIntros "Hctx". iDestruct "Hctx" as (Hinv) "(Hhctx&Hrctx&Hsctx)".
iMod (alloc_range_alloc with "Hrctx") as "[$ #Hb]" => //.
iMod (alloc_meta_alloc with "Hrctx") as "[$ #Hb]" => //.
iMod (alloc_alive_alloc with "Hsctx") as "[$ Hs]" => //.
iDestruct (alloc_range_to_loc_in_bounds l aid (length v) with "[Hb]")
iDestruct (alloc_meta_to_loc_in_bounds l aid (length v) with "[Hb]")
as "#Hinb" => //; [done|..].
iMod (heap_alloc with "Hb Hs Hhctx") as "[Hhctx [Hv Hal]]" => //.
iModIntro. iFrame. iPureIntro. by eapply alloc_new_block_invariant.
Qed.
Lemma heap_alloc_new_blocks_upd σ1 ls vs σ2:
alloc_new_blocks σ1 ls vs σ2 →
Lemma heap_alloc_new_blocks_upd σ1 ls vs kind σ2:
alloc_new_blocks σ1 kind ls vs σ2 →
heap_state_ctx σ1 ==∗
heap_state_ctx σ2 ∗
[∗ list] l; v ∈ ls; vs, l ↦ v ∗ freeable l (length v).
[∗ list] l; v ∈ ls; vs, l ↦ v ∗ freeable l (length v) kind.
Proof.
move => alloc.
iInduction alloc as [σ|] "IH"; first by iIntros "$ !>". iIntros "Hσ".
......@@ -869,27 +875,27 @@ End alloc_new_blocks.
Section free_blocks.
Context `{!heapG Σ}.
Lemma heap_free_block_upd σ1 l ly:
Lemma heap_free_block_upd σ1 l ly kind:
l ↦|ly| -∗
freeable l (ly_size ly) -∗
heap_state_ctx σ1 ==∗ ∃ σ2, ⌜free_block σ1 l ly σ2⌝ ∗ heap_state_ctx σ2.
freeable l (ly_size ly) kind -∗
heap_state_ctx σ1 ==∗ ∃ σ2, ⌜free_block σ1 kind l ly σ2⌝ ∗ heap_state_ctx σ2.
Proof.
iIntros "Hl Hfree (Hinv&Hhctx&Hrctx&Hsctx)". iDestruct "Hinv" as %Hinv.
rewrite freeable_eq. iDestruct "Hfree" as (aid Haid) "[#Hrange Hkill]".
iDestruct "Hl" as (v Hv ?) "Hl".
iDestruct (alloc_alive_lookup with "Hsctx Hkill") as %[[???] [??]].
iDestruct (alloc_range_lookup with "Hrctx Hrange") as %[al'' [?[??]]]. simplify_eq/=.
iDestruct (alloc_alive_lookup with "Hsctx Hkill") as %[[????k] [??]].
iDestruct (alloc_meta_lookup with "Hrctx Hrange") as %[al'' [?[[??]?]]]. simplify_eq/=.
iDestruct (heap_mapsto_lookup_1 (λ st : lock_state, st = RSt 0) with "Hhctx Hl") as %? => //.
iExists _. iSplitR. { iPureIntro. by econstructor. }
iMod (heap_free_free with "Hhctx Hl") as "Hhctx". rewrite Hv. iFrame => /=.
iMod (alloc_alive_kill _ _ ({| al_start := l.2; al_len := ly_size ly; al_alive := true |}) with "Hsctx Hkill") as "[$ Hd]".
erewrite alloc_range_ctx_same_range; [iFrame |done..].
iMod (alloc_alive_kill _ _ ({| al_start := l.2; al_len := ly_size ly; al_alive := true; al_kind := k |}) with "Hsctx Hkill") as "[$ Hd]".
erewrite alloc_meta_ctx_same_range; [iFrame |done..].
iPureIntro. eapply free_block_invariant => //. by eapply FreeBlock.
Qed.
Lemma heap_free_blocks_upd ls σ1:
([∗ list] l ∈ ls, l.1 ↦|l.2| ∗ freeable l.1 (ly_size l.2)) -∗
heap_state_ctx σ1 ==∗ ∃ σ2, ⌜free_blocks σ1 ls σ2⌝ ∗ heap_state_ctx σ2.
Lemma heap_free_blocks_upd ls σ1 kind:
([∗ list] l ∈ ls, l.1 ↦|l.2| ∗ freeable l.1 (ly_size l.2) kind ) -∗
heap_state_ctx σ1 ==∗ ∃ σ2, ⌜free_blocks σ1 kind ls σ2⌝ ∗ heap_state_ctx σ2.
Proof.
iInduction ls as [|[l ly] ls] "IH" forall (σ1).
{ iIntros "_ ? !>". iExists σ1. iFrame. iPureIntro. by constructor. }
......
theories/lang/heap.v
View file @ 5a7b2bf6
......@@ -185,11 +185,18 @@ Qed.
(** ** Representation of allocations. *)
(** An allocation can either be a stack allocation or a heap allocation. *)
Inductive alloc_kind : Set :=
| HeapAlloc
| StackAlloc
| GlobalAlloc.
Record allocation :=
Allocation {
al_start : Z; (* First valid address. *)
al_len : nat; (* Number of allocated byte. *)
al_alive : bool; (* Is the allocation still alive. *)
al_start : Z; (* First valid address. *)
al_len : nat; (* Number of allocated byte. *)
al_alive : bool; (* Is the allocation still alive. *)
al_kind : alloc_kind; (* On the heap or on the stack? *)
}.
Definition al_end (al : allocation) : Z :=
......@@ -201,7 +208,7 @@ Definition alloc_same_range (al1 al2 : allocation) : Prop :=
al1.(al_start) = al2.(al_start) ∧ al1.(al_len) = al2.(al_len).
Definition killed (al : allocation) : allocation :=
{| al_start := al.(al_start); al_len := al.(al_len); al_alive := false; |}.
{| al_start := al.(al_start); al_len := al.(al_len); al_alive := false; al_kind := al.(al_kind) |}.
(** Smallest allocatable address (we reserve 0 for NULL). *)
Definition min_alloc_start : Z := 1.
......@@ -262,6 +269,8 @@ Proof. rewrite /valid_ptr => ?. apply: heap_state_loc_in_bounds_has_alloc_id. na
Definition addr_in_range_alloc (a : addr) (aid : alloc_id) (st : heap_state) : Prop :=
∃ alloc, st.(hs_allocs) !! aid = Some alloc ∧ a ∈ alloc.
Global Instance alloc_kind_eq_dec : EqDecision alloc_kind.
Proof. solve_decision. Qed.
Global Instance allocation_eq_dec : EqDecision (allocation).
Proof. solve_decision. Qed.
Global Instance alloc_id_alive_dec aid st : Decision (alloc_id_alive aid st).
......@@ -474,57 +483,57 @@ Arguments mem_cast : simpl never.
(** ** Allocation and deallocation. *)
Inductive alloc_new_block : heap_state → loc → val → heap_state → Prop :=
| AllocNewBlock σ l aid v:
let alloc := Allocation l.2 (length v) true in
Inductive alloc_new_block : heap_state → alloc_kind → loc → val → heap_state → Prop :=
| AllocNewBlock σ l aid kind v:
let alloc := Allocation l.2 (length v) true kind in
l.1 = ProvAlloc (Some aid) →
σ.(hs_allocs) !! aid = None →
allocation_in_range alloc →
heap_range_free σ.(hs_heap) l.2 (length v) →
alloc_new_block σ l v {|
alloc_new_block σ kind l v {|
hs_heap := heap_alloc l.2 v aid σ.(hs_heap);
hs_allocs := <[aid := alloc]> σ.(hs_allocs);
|}.
Inductive alloc_new_blocks : heap_state → list loc → list val → heap_state → Prop :=
| AllocNewBlock_nil σ :
alloc_new_blocks σ [] [] σ
| AllocNewBlock_cons σ σ' σ'' l v ls vs :
alloc_new_block σ l v σ' →
alloc_new_blocks σ' ls vs σ'' →
alloc_new_blocks σ (l :: ls) (v :: vs) σ''.
Inductive free_block : heap_state → loc → layout → heap_state → Prop :=
| FreeBlock σ l aid ly v:
let al_alive := Allocation l.2 ly.(ly_size) true in
let al_dead := Allocation l.2 ly.(ly_size) false in
Inductive alloc_new_blocks : heap_state → alloc_kind → list loc → list val → heap_state → Prop :=
| AllocNewBlock_nil σ kind :
alloc_new_blocks σ kind [] [] σ
| AllocNewBlock_cons σ σ' σ'' l v ls kind vs :
alloc_new_block σ kind l v σ' →
alloc_new_blocks σ' kind ls vs σ'' →
alloc_new_blocks σ kind (l :: ls) (v :: vs) σ''.
Inductive free_block : heap_state → alloc_kind → loc → layout → heap_state → Prop :=
| FreeBlock σ l aid ly kind v:
let al_alive := Allocation l.2 ly.(ly_size) true kind in
let al_dead := Allocation l.2 ly.(ly_size) false kind in
l.1 = ProvAlloc (Some aid) →
σ.(hs_allocs) !! aid = Some al_alive →
length v = ly.(ly_size) →
heap_lookup_loc l v (λ st, st = RSt 0%nat) σ.(hs_heap) →
free_block σ l ly {|
free_block σ kind l ly {|
hs_heap := heap_free l.2 ly.(ly_size) σ.(hs_heap);
hs_allocs := <[aid := al_dead]> σ.(hs_allocs);
|}.
Inductive free_blocks : heap_state → list (loc * layout) → heap_state → Prop :=
| FreeBlocks_nil σ :
free_blocks σ [] σ
| FreeBlocks_cons σ σ' σ'' l ly ls :
free_block σ l ly σ' →
free_blocks σ' ls σ'' →
free_blocks σ ((l, ly) :: ls) σ''.
Inductive free_blocks : heap_state → alloc_kind → list (loc * layout) → heap_state → Prop :=
| FreeBlocks_nil σ kind :
free_blocks σ kind [] σ
| FreeBlocks_cons σ σ' σ'' l ly kind ls :
free_block σ kind l ly σ' →
free_blocks σ' kind ls σ'' →
free_blocks σ kind ((l, ly) :: ls) σ''.
Lemma free_block_inj hs l ly hs1 hs2:
free_block hs l ly hs1 → free_block hs l ly hs2 → hs1 = hs2.
Lemma free_block_inj hs l ly kind hs1 hs2:
free_block hs kind l ly hs1 → free_block hs kind l ly hs2 → hs1 = hs2.
Proof. destruct l. inversion 1; simplify_eq. by inversion 1; simplify_eq/=. Qed.
Lemma free_blocks_inj hs1 hs2 hs ls:
free_blocks hs ls hs1 → free_blocks hs ls hs2 → hs1 = hs2.
Lemma free_blocks_inj hs1 hs2 hs kind ls:
free_blocks hs kind ls hs1 → free_blocks hs kind ls hs2 → hs1 = hs2.
Proof.
move Heq: {1}(hs) => hs' Hb.
elim: Hb hs Heq. { move => ?? ->. by inversion 1. }
move => ?????? Hb1 ? IH ??.
elim: Hb hs Heq. { move => ??? ->. by inversion 1. }
move => ??????? Hb1 ? IH ??.
inversion 1; simplify_eq. apply: IH; [|done].
by apply: free_block_inj.
Qed.
......@@ -580,25 +589,25 @@ Definition heap_state_invariant (st : heap_state) : Prop :=
(** ** Lemmas about the heap state invariant. *)
Lemma heap_state_alloc_alive_free_disjoint σ id a n b alloc:
Lemma heap_state_alloc_alive_free_disjoint σ id a n b kind alloc:
heap_state_alloc_alive_in_heap σ →
alloc_id_alive id σ →
heap_range_free σ.(hs_heap) a n →
σ.(hs_allocs) !! id = Some alloc →
Allocation a n b ## alloc.
Allocation a n b kind ## alloc.
Proof.
move => Hin_heap Halive Hfree Hal p Hp1 Hp2.
apply (Hin_heap _ _ Hal Halive) in Hp2 as [? Hp2].
rewrite Hfree in Hp2; first done. apply Hp1.
Qed.
Lemma alloc_new_block_invariant σ1 σ2 l v :
alloc_new_block σ1 l v σ2 →
Lemma alloc_new_block_invariant σ1 σ2 l v kind :
alloc_new_block σ1 kind l v σ2 →
heap_state_invariant σ1 →
heap_state_invariant σ2.
Proof.
move => []; clear.
move => σ1 l aid v alloc Haid Hfresh Halloc Hrange H.
move => σ1 l aid kind v alloc Haid Hfresh Halloc Hrange H.
destruct H as (Hi1&Hi2&Hi3&Hi4&Hi5). split_and!.
- move => a [id??] /= Ha. destruct (decide (aid = id)) as [->|Hne].
+ exists alloc. split => /=; first by rewrite lookup_insert.
......@@ -650,22 +659,22 @@ Proof.
eapply (Hi5 _ _ Hal); [by eexists | done |..].
Qed.
Lemma alloc_new_blocks_invariant σ1 σ2 ls vs :
alloc_new_blocks σ1 ls vs σ2 →
Lemma alloc_new_blocks_invariant σ1 σ2 ls vs kind :
alloc_new_blocks σ1 kind ls vs σ2 →
heap_state_invariant σ1 →
heap_state_invariant σ2.
Proof.
elim => [] // ??????? Hb Hbs IH H.
elim => [] // ???????? Hb Hbs IH H.
apply IH. by eapply alloc_new_block_invariant.
Qed.
Lemma free_block_invariant σ1 σ2 l ly:
free_block σ1 l ly σ2 →
Lemma free_block_invariant σ1 σ2 l ly kind :
free_block σ1 kind l ly σ2 →
heap_state_invariant σ1 →
heap_state_invariant σ2.
Proof.
move => []; clear.
move => σ l aid ly v al_a al_d Haid Hal_a Hlen Hlookup H.
move => σ l aid ly kind v al_a al_d Haid Hal_a Hlen Hlookup H.
destruct H as (Hi1&Hi2&Hi3&Hi4&Hi5). split_and!.
- move => a hc /= Hhc.
assert (¬ (l.2 ≤ a < l.2 + length v)) as Hnot_in.
......@@ -712,12 +721,12 @@ Proof.
erewrite elem_of_disjoint in Hdisj. by eapply Hdisj.
Qed.
Lemma free_blocks_invariant σ1 σ2 ls:
free_blocks σ1 ls σ2 →
Lemma free_blocks_invariant σ1 σ2 ls kind :
free_blocks σ1 kind ls σ2 →
heap_state_invariant σ1 →
heap_state_invariant σ2.
Proof.
elim => [] // ?????? Hb Hbs IH H.
elim => [] // ??????? Hb Hbs IH H.
apply IH. by eapply free_block_invariant.
Qed.
......
theories/lang/lang.v
View file @ 5a7b2bf6
......@@ -37,6 +37,7 @@ Inductive expr :=
| Call (f : expr) (args : list expr)
| Concat (es : list expr)
| IfE (ot : op_type) (e1 e2 e3 : expr)
| Alloc (ly : layout) (e : expr)
| SkipE (e : expr)
| StuckE (* stuck expression *)
.
......@@ -53,6 +54,7 @@ Lemma expr_ind (P : expr → Prop) :
(∀ (f : expr) (args : list expr), P f → Forall P args → P (Call f args)) →
(∀ (es : list expr), Forall P es → P (Concat es)) →
(∀ (ot : op_type) (e1 e2 e3 : expr), P e1 → P e2 → P e3 → P (IfE ot e1 e2 e3)) →
(∀ (ly : layout) (e : expr), P e → P (Alloc ly e)) →
(∀ (e : expr), P e → P (SkipE e)) →
(P StuckE) →