cap_machine.rules.rules_Get
From iris.base_logic Require Export invariants gen_heap.
From iris.program_logic Require Export weakestpre ectx_lifting.
From iris.proofmode Require Import tactics.
From iris.algebra Require Import frac.
From cap_machine Require Export rules_base.
Section cap_lang_rules.
Context `{memG Σ, regG Σ}.
Context `{MachineParameters}.
Implicit Types P Q : iProp Σ.
Implicit Types σ : ExecConf.
Implicit Types c : cap_lang.expr.
Implicit Types a b : Addr.
Implicit Types r : RegName.
Implicit Types v : cap_lang.val.
Implicit Types w : Word.
Implicit Types reg : gmap RegName Word.
Implicit Types ms : gmap Addr Word.
Definition denote (i: instr) p b e a: Z :=
match i with
| GetP _ _ => encodePerm p
| GetB _ _ => b
| GetE _ _ => e
| GetA _ _ => a
| _ => 0%Z
end.
Global Arguments denote : simpl nomatch.
Definition is_Get (i: instr) (dst src: RegName) :=
i = GetP dst src ∨
i = GetB dst src ∨
i = GetE dst src ∨
i = GetA dst src.
Lemma regs_of_is_Get i dst src :
is_Get i dst src →
regs_of i = {[ dst; src ]}.
Proof.
intros HH. destruct_or! HH; subst i; reflexivity.
Qed.
Inductive Get_failure (i: instr) (regs: Reg) (dst src: RegName) :=
| Get_fail_src_noncap : forall n,
regs !! src = Some (WInt n) →
Get_failure i regs dst src
| Get_fail_overflow_PC : forall p b e a,
regs !! src = Some (WCap p b e a) →
incrementPC (<[ dst := WInt (denote i p b e a) ]> regs) = None →
Get_failure i regs dst src.
Inductive Get_spec (i: instr) (regs: Reg) (dst src: RegName) (regs': Reg): cap_lang.val -> Prop :=
| Get_spec_success p b e a:
regs !! src = Some (WCap p b e a) →
incrementPC (<[ dst := WInt (denote i p b e a) ]> regs) = Some regs' →
Get_spec i regs dst src regs' NextIV
| Get_spec_failure:
Get_failure i regs dst src →
Get_spec i regs dst src regs' FailedV.
Lemma wp_Get Ep pc_p pc_b pc_e pc_a w get_i dst src regs :
decodeInstrW w = get_i →
is_Get get_i dst src →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
regs !! PC = Some (WCap pc_p pc_b pc_e pc_a) →
regs_of get_i ⊆ dom _ regs →
{{{ ▷ pc_a ↦ₐ w ∗
▷ [∗ map] k↦y ∈ regs, k ↦ᵣ y }}}
Instr Executable @ Ep
{{{ regs' retv, RET retv;
⌜ Get_spec (decodeInstrW w) regs dst src regs' retv ⌝ ∗
pc_a ↦ₐ w ∗
[∗ map] k↦y ∈ regs', k ↦ᵣ y }}}.
Proof.
iIntros (Hdecode Hinstr Hvpc HPC Dregs φ) "(>Hpc_a & >Hmap) Hφ".
iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 l1 l2 n) "Hσ1 /=". destruct σ1; simpl.
iDestruct "Hσ1" as "[Hr Hm]".
iPoseProof (gen_heap_valid_inclSepM with "Hr Hmap") as "#H".
iDestruct "H" as %Hregs.
have ? := lookup_weaken _ _ _ _ HPC Hregs.
iDestruct (@gen_heap_valid with "Hm Hpc_a") as %Hpc_a; auto.
iModIntro. iSplitR. by iPureIntro; apply normal_always_head_reducible.
iNext. iIntros (e2 σ2 efs Hpstep).
apply prim_step_exec_inv in Hpstep as (-> & -> & (c & -> & Hstep)).
iSplitR; auto. eapply step_exec_inv in Hstep; eauto.
unfold exec in Hstep.
specialize (indom_regs_incl _ _ _ Dregs Hregs) as Hri.
erewrite regs_of_is_Get in Hri; eauto.
destruct (Hri src) as [wsrc [H'src Hsrc]]. by set_solver+.
destruct (Hri dst) as [wdst [H'dst Hdst]]. by set_solver+.
destruct wsrc as [| p b e a ] eqn:Hwsrc.
{ (* Failure: src is not a capability *)
assert (c = Failed ∧ σ2 = (r, m)) as (-> & ->).
{ destruct_or! Hinstr; rewrite Hinstr in Hstep; cbn in Hstep.
all: rewrite Hsrc in Hstep; inversion Hstep; auto. }
iFailWP "Hφ" Get_fail_src_noncap. }
assert (exec_opt get_i (r, m) = updatePC (update_reg (r, m) dst (WInt (denote get_i p b e a)))) as HH.
{ destruct_or! Hinstr; rewrite Hinstr /= in Hstep |- *; auto; cbn in Hstep.
all: destruct b, e, a; rewrite /update_reg Hsrc /= in Hstep |-*; auto. }
rewrite HH in Hstep. rewrite /update_reg /= in Hstep.
destruct (incrementPC (<[ dst := WInt (denote get_i p b e a) ]> regs))
as [regs'|] eqn:Hregs'; pose proof Hregs' as H'regs'; cycle 1.
{ (* Failure: incrementing PC overflows *)
apply incrementPC_fail_updatePC with (m:=m) in Hregs'.
eapply updatePC_fail_incl with (m':=m) in Hregs'.
2: by apply lookup_insert_is_Some'; eauto.
2: by apply insert_mono; eauto.
simplify_pair_eq.
rewrite Hregs' in Hstep. inversion Hstep.
iFailWP "Hφ" Get_fail_overflow_PC. }
(* Success *)
eapply (incrementPC_success_updatePC _ m) in Hregs'
as (p' & g' & b' & e' & a'' & a_pc' & HPC'' & HuPC & ->).
eapply updatePC_success_incl with (m':=m) in HuPC. 2: by eapply insert_mono; eauto. rewrite HuPC in Hstep.
simplify_pair_eq. iFrame.
iMod ((gen_heap_update_inSepM _ _ dst) with "Hr Hmap") as "[Hr Hmap]"; eauto.
iMod ((gen_heap_update_inSepM _ _ PC) with "Hr Hmap") as "[Hr Hmap]"; eauto.
iFrame. iModIntro. iApply "Hφ". iFrame. iPureIntro. econstructor; eauto.
Qed.
Lemma wp_Get_PC_success E get_i dst pc_p pc_b pc_e pc_a w wdst pc_a' :
decodeInstrW w = get_i →
is_Get get_i dst PC →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' ->
{{{ ▷ PC ↦ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↦ₐ w
∗ ▷ dst ↦ᵣ wdst }}}
Instr Executable @ E
{{{ RET NextIV;
PC ↦ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↦ₐ w
∗ dst ↦ᵣ WInt (denote get_i pc_p pc_b pc_e pc_a) }}}.
Proof.
iIntros (Hdecode Hinstr Hvpc Hpca' φ) "(>HPC & >Hpc_a & >Hdst) Hφ".
iDestruct (map_of_regs_2 with "HPC Hdst") as "[Hmap %]".
iApply (wp_Get with "[$Hmap Hpc_a]"); eauto; simplify_map_eq; eauto.
by erewrite regs_of_is_Get; eauto; rewrite !dom_insert; set_solver+.
iNext. iIntros (regs' retv) "(#Hspec & Hpc_a & Hmap)". iDestruct "Hspec" as %Hspec.
destruct Hspec as [| * Hfail].
{ (* Success *)
iApply "Hφ". iFrame. incrementPC_inv; simplify_map_eq.
rewrite insert_commute // insert_insert insert_commute // insert_insert.
iDestruct (regs_of_map_2 with "Hmap") as "[? ?]"; eauto; iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto. congruence. }
Qed.
Lemma wp_Get_same_success E get_i r pc_p pc_b pc_e pc_a w p' b' e' a' pc_a' :
decodeInstrW w = get_i →
is_Get get_i r r →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' ->
{{{ ▷ PC ↦ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↦ₐ w
∗ ▷ r ↦ᵣ WCap p' b' e' a' }}}
Instr Executable @ E
{{{ RET NextIV;
PC ↦ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↦ₐ w
∗ r ↦ᵣ WInt (denote get_i p' b' e' a') }}}.
Proof.
iIntros (Hdecode Hinstr Hvpc Hpca' φ) "(>HPC & >Hpc_a & >Hr) Hφ".
iDestruct (map_of_regs_2 with "HPC Hr") as "[Hmap %]".
iApply (wp_Get with "[$Hmap Hpc_a]"); eauto; simplify_map_eq; eauto.
by erewrite regs_of_is_Get; eauto; rewrite !dom_insert; set_solver+.
iNext. iIntros (regs' retv) "(#Hspec & Hpc_a & Hmap)". iDestruct "Hspec" as %Hspec.
destruct Hspec as [| * Hfail].
{ (* Success *)
iApply "Hφ". iFrame. incrementPC_inv; simplify_map_eq.
rewrite insert_commute // insert_insert insert_commute // insert_insert.
iDestruct (regs_of_map_2 with "Hmap") as "[? ?]"; eauto; iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto. congruence. }
Qed.
Lemma wp_Get_success E get_i dst src pc_p pc_b pc_e pc_a w wdst p' b' e' a' pc_a' :
decodeInstrW w = get_i →
is_Get get_i dst src →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' ->
{{{ ▷ PC ↦ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↦ₐ w
∗ ▷ src ↦ᵣ WCap p' b' e' a'
∗ ▷ dst ↦ᵣ wdst }}}
Instr Executable @ E
{{{ RET NextIV;
PC ↦ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↦ₐ w
∗ src ↦ᵣ WCap p' b' e' a'
∗ dst ↦ᵣ WInt (denote get_i p' b' e' a') }}}.
Proof.
iIntros (Hdecode Hinstr Hvpc Hpca' φ) "(>HPC & >Hpc_a & >Hsrc & >Hdst) Hφ".
iDestruct (map_of_regs_3 with "HPC Hdst Hsrc") as "[Hmap (%&%&%)]".
iApply (wp_Get with "[$Hmap Hpc_a]"); eauto; simplify_map_eq; eauto.
by erewrite regs_of_is_Get; eauto; rewrite !dom_insert; set_solver+.
iNext. iIntros (regs' retv) "(#Hspec & Hpc_a & Hmap)". iDestruct "Hspec" as %Hspec.
destruct Hspec as [| * Hfail].
{ (* Success *)
iApply "Hφ". iFrame. incrementPC_inv; simplify_map_eq.
rewrite insert_commute // insert_insert (insert_commute _ PC dst) // insert_insert.
iDestruct (regs_of_map_3 with "Hmap") as "(?&?&?)"; eauto; iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto. congruence. }
Qed.
End cap_lang_rules.
(* Hints to automate proofs of is_Get *)
Lemma is_Get_GetP dst src : is_Get (GetP dst src) dst src.
Proof. intros; unfold is_Get; eauto. Qed.
Lemma is_Get_GetB dst src : is_Get (GetB dst src) dst src.
Proof. intros; unfold is_Get; eauto. Qed.
Lemma is_Get_GetE dst src : is_Get (GetE dst src) dst src.
Proof. intros; unfold is_Get; eauto. Qed.
Lemma is_Get_GetA dst src : is_Get (GetA dst src) dst src.
Proof. intros; unfold is_Get; eauto. Qed.
Global Hint Resolve is_Get_GetP : core.
Global Hint Resolve is_Get_GetB : core.
Global Hint Resolve is_Get_GetE : core.
Global Hint Resolve is_Get_GetA : core.
From iris.program_logic Require Export weakestpre ectx_lifting.
From iris.proofmode Require Import tactics.
From iris.algebra Require Import frac.
From cap_machine Require Export rules_base.
Section cap_lang_rules.
Context `{memG Σ, regG Σ}.
Context `{MachineParameters}.
Implicit Types P Q : iProp Σ.
Implicit Types σ : ExecConf.
Implicit Types c : cap_lang.expr.
Implicit Types a b : Addr.
Implicit Types r : RegName.
Implicit Types v : cap_lang.val.
Implicit Types w : Word.
Implicit Types reg : gmap RegName Word.
Implicit Types ms : gmap Addr Word.
Definition denote (i: instr) p b e a: Z :=
match i with
| GetP _ _ => encodePerm p
| GetB _ _ => b
| GetE _ _ => e
| GetA _ _ => a
| _ => 0%Z
end.
Global Arguments denote : simpl nomatch.
Definition is_Get (i: instr) (dst src: RegName) :=
i = GetP dst src ∨
i = GetB dst src ∨
i = GetE dst src ∨
i = GetA dst src.
Lemma regs_of_is_Get i dst src :
is_Get i dst src →
regs_of i = {[ dst; src ]}.
Proof.
intros HH. destruct_or! HH; subst i; reflexivity.
Qed.
Inductive Get_failure (i: instr) (regs: Reg) (dst src: RegName) :=
| Get_fail_src_noncap : forall n,
regs !! src = Some (WInt n) →
Get_failure i regs dst src
| Get_fail_overflow_PC : forall p b e a,
regs !! src = Some (WCap p b e a) →
incrementPC (<[ dst := WInt (denote i p b e a) ]> regs) = None →
Get_failure i regs dst src.
Inductive Get_spec (i: instr) (regs: Reg) (dst src: RegName) (regs': Reg): cap_lang.val -> Prop :=
| Get_spec_success p b e a:
regs !! src = Some (WCap p b e a) →
incrementPC (<[ dst := WInt (denote i p b e a) ]> regs) = Some regs' →
Get_spec i regs dst src regs' NextIV
| Get_spec_failure:
Get_failure i regs dst src →
Get_spec i regs dst src regs' FailedV.
Lemma wp_Get Ep pc_p pc_b pc_e pc_a w get_i dst src regs :
decodeInstrW w = get_i →
is_Get get_i dst src →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
regs !! PC = Some (WCap pc_p pc_b pc_e pc_a) →
regs_of get_i ⊆ dom _ regs →
{{{ ▷ pc_a ↦ₐ w ∗
▷ [∗ map] k↦y ∈ regs, k ↦ᵣ y }}}
Instr Executable @ Ep
{{{ regs' retv, RET retv;
⌜ Get_spec (decodeInstrW w) regs dst src regs' retv ⌝ ∗
pc_a ↦ₐ w ∗
[∗ map] k↦y ∈ regs', k ↦ᵣ y }}}.
Proof.
iIntros (Hdecode Hinstr Hvpc HPC Dregs φ) "(>Hpc_a & >Hmap) Hφ".
iApply wp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1 l1 l2 n) "Hσ1 /=". destruct σ1; simpl.
iDestruct "Hσ1" as "[Hr Hm]".
iPoseProof (gen_heap_valid_inclSepM with "Hr Hmap") as "#H".
iDestruct "H" as %Hregs.
have ? := lookup_weaken _ _ _ _ HPC Hregs.
iDestruct (@gen_heap_valid with "Hm Hpc_a") as %Hpc_a; auto.
iModIntro. iSplitR. by iPureIntro; apply normal_always_head_reducible.
iNext. iIntros (e2 σ2 efs Hpstep).
apply prim_step_exec_inv in Hpstep as (-> & -> & (c & -> & Hstep)).
iSplitR; auto. eapply step_exec_inv in Hstep; eauto.
unfold exec in Hstep.
specialize (indom_regs_incl _ _ _ Dregs Hregs) as Hri.
erewrite regs_of_is_Get in Hri; eauto.
destruct (Hri src) as [wsrc [H'src Hsrc]]. by set_solver+.
destruct (Hri dst) as [wdst [H'dst Hdst]]. by set_solver+.
destruct wsrc as [| p b e a ] eqn:Hwsrc.
{ (* Failure: src is not a capability *)
assert (c = Failed ∧ σ2 = (r, m)) as (-> & ->).
{ destruct_or! Hinstr; rewrite Hinstr in Hstep; cbn in Hstep.
all: rewrite Hsrc in Hstep; inversion Hstep; auto. }
iFailWP "Hφ" Get_fail_src_noncap. }
assert (exec_opt get_i (r, m) = updatePC (update_reg (r, m) dst (WInt (denote get_i p b e a)))) as HH.
{ destruct_or! Hinstr; rewrite Hinstr /= in Hstep |- *; auto; cbn in Hstep.
all: destruct b, e, a; rewrite /update_reg Hsrc /= in Hstep |-*; auto. }
rewrite HH in Hstep. rewrite /update_reg /= in Hstep.
destruct (incrementPC (<[ dst := WInt (denote get_i p b e a) ]> regs))
as [regs'|] eqn:Hregs'; pose proof Hregs' as H'regs'; cycle 1.
{ (* Failure: incrementing PC overflows *)
apply incrementPC_fail_updatePC with (m:=m) in Hregs'.
eapply updatePC_fail_incl with (m':=m) in Hregs'.
2: by apply lookup_insert_is_Some'; eauto.
2: by apply insert_mono; eauto.
simplify_pair_eq.
rewrite Hregs' in Hstep. inversion Hstep.
iFailWP "Hφ" Get_fail_overflow_PC. }
(* Success *)
eapply (incrementPC_success_updatePC _ m) in Hregs'
as (p' & g' & b' & e' & a'' & a_pc' & HPC'' & HuPC & ->).
eapply updatePC_success_incl with (m':=m) in HuPC. 2: by eapply insert_mono; eauto. rewrite HuPC in Hstep.
simplify_pair_eq. iFrame.
iMod ((gen_heap_update_inSepM _ _ dst) with "Hr Hmap") as "[Hr Hmap]"; eauto.
iMod ((gen_heap_update_inSepM _ _ PC) with "Hr Hmap") as "[Hr Hmap]"; eauto.
iFrame. iModIntro. iApply "Hφ". iFrame. iPureIntro. econstructor; eauto.
Qed.
Lemma wp_Get_PC_success E get_i dst pc_p pc_b pc_e pc_a w wdst pc_a' :
decodeInstrW w = get_i →
is_Get get_i dst PC →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' ->
{{{ ▷ PC ↦ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↦ₐ w
∗ ▷ dst ↦ᵣ wdst }}}
Instr Executable @ E
{{{ RET NextIV;
PC ↦ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↦ₐ w
∗ dst ↦ᵣ WInt (denote get_i pc_p pc_b pc_e pc_a) }}}.
Proof.
iIntros (Hdecode Hinstr Hvpc Hpca' φ) "(>HPC & >Hpc_a & >Hdst) Hφ".
iDestruct (map_of_regs_2 with "HPC Hdst") as "[Hmap %]".
iApply (wp_Get with "[$Hmap Hpc_a]"); eauto; simplify_map_eq; eauto.
by erewrite regs_of_is_Get; eauto; rewrite !dom_insert; set_solver+.
iNext. iIntros (regs' retv) "(#Hspec & Hpc_a & Hmap)". iDestruct "Hspec" as %Hspec.
destruct Hspec as [| * Hfail].
{ (* Success *)
iApply "Hφ". iFrame. incrementPC_inv; simplify_map_eq.
rewrite insert_commute // insert_insert insert_commute // insert_insert.
iDestruct (regs_of_map_2 with "Hmap") as "[? ?]"; eauto; iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto. congruence. }
Qed.
Lemma wp_Get_same_success E get_i r pc_p pc_b pc_e pc_a w p' b' e' a' pc_a' :
decodeInstrW w = get_i →
is_Get get_i r r →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' ->
{{{ ▷ PC ↦ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↦ₐ w
∗ ▷ r ↦ᵣ WCap p' b' e' a' }}}
Instr Executable @ E
{{{ RET NextIV;
PC ↦ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↦ₐ w
∗ r ↦ᵣ WInt (denote get_i p' b' e' a') }}}.
Proof.
iIntros (Hdecode Hinstr Hvpc Hpca' φ) "(>HPC & >Hpc_a & >Hr) Hφ".
iDestruct (map_of_regs_2 with "HPC Hr") as "[Hmap %]".
iApply (wp_Get with "[$Hmap Hpc_a]"); eauto; simplify_map_eq; eauto.
by erewrite regs_of_is_Get; eauto; rewrite !dom_insert; set_solver+.
iNext. iIntros (regs' retv) "(#Hspec & Hpc_a & Hmap)". iDestruct "Hspec" as %Hspec.
destruct Hspec as [| * Hfail].
{ (* Success *)
iApply "Hφ". iFrame. incrementPC_inv; simplify_map_eq.
rewrite insert_commute // insert_insert insert_commute // insert_insert.
iDestruct (regs_of_map_2 with "Hmap") as "[? ?]"; eauto; iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto. congruence. }
Qed.
Lemma wp_Get_success E get_i dst src pc_p pc_b pc_e pc_a w wdst p' b' e' a' pc_a' :
decodeInstrW w = get_i →
is_Get get_i dst src →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' ->
{{{ ▷ PC ↦ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↦ₐ w
∗ ▷ src ↦ᵣ WCap p' b' e' a'
∗ ▷ dst ↦ᵣ wdst }}}
Instr Executable @ E
{{{ RET NextIV;
PC ↦ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↦ₐ w
∗ src ↦ᵣ WCap p' b' e' a'
∗ dst ↦ᵣ WInt (denote get_i p' b' e' a') }}}.
Proof.
iIntros (Hdecode Hinstr Hvpc Hpca' φ) "(>HPC & >Hpc_a & >Hsrc & >Hdst) Hφ".
iDestruct (map_of_regs_3 with "HPC Hdst Hsrc") as "[Hmap (%&%&%)]".
iApply (wp_Get with "[$Hmap Hpc_a]"); eauto; simplify_map_eq; eauto.
by erewrite regs_of_is_Get; eauto; rewrite !dom_insert; set_solver+.
iNext. iIntros (regs' retv) "(#Hspec & Hpc_a & Hmap)". iDestruct "Hspec" as %Hspec.
destruct Hspec as [| * Hfail].
{ (* Success *)
iApply "Hφ". iFrame. incrementPC_inv; simplify_map_eq.
rewrite insert_commute // insert_insert (insert_commute _ PC dst) // insert_insert.
iDestruct (regs_of_map_3 with "Hmap") as "(?&?&?)"; eauto; iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto. congruence. }
Qed.
End cap_lang_rules.
(* Hints to automate proofs of is_Get *)
Lemma is_Get_GetP dst src : is_Get (GetP dst src) dst src.
Proof. intros; unfold is_Get; eauto. Qed.
Lemma is_Get_GetB dst src : is_Get (GetB dst src) dst src.
Proof. intros; unfold is_Get; eauto. Qed.
Lemma is_Get_GetE dst src : is_Get (GetE dst src) dst src.
Proof. intros; unfold is_Get; eauto. Qed.
Lemma is_Get_GetA dst src : is_Get (GetA dst src) dst src.
Proof. intros; unfold is_Get; eauto. Qed.
Global Hint Resolve is_Get_GetP : core.
Global Hint Resolve is_Get_GetB : core.
Global Hint Resolve is_Get_GetE : core.
Global Hint Resolve is_Get_GetA : core.