cap_machine.rules_binary.rules_binary_Get
From iris.base_logic Require Export invariants gen_heap.
From iris.program_logic Require Export weakestpre ectx_lifting.
From iris.proofmode Require Import proofmode.
From iris.algebra Require Import frac.
From cap_machine Require Export rules_Get rules_binary_base.
Section cap_lang_spec_rules.
Context `{cfgSG Σ, MachineParameters, invGS Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types σ : cap_lang.state.
Implicit Types a b : Addr.
Implicit Types o : OType.
Implicit Types r : RegName.
Implicit Types w : Word.
Implicit Types reg : gmap RegName Word.
Implicit Types ms : gmap Addr Word.
Lemma step_Get Ep K 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 →
nclose specN ⊆ Ep →
spec_ctx ∗ ⤇ fill K (Instr Executable) ∗ ▷ pc_a ↣ₐ w ∗ ▷ ([∗ map] k↦y ∈ regs, k ↣ᵣ y)
={Ep}=∗ ∃ retv regs', ⤇ fill K (of_val 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 Hnclose) "(Hinv & Hj & >Hpc_a & >Hmap)".
iDestruct "Hinv" as (ρ) "Hinv". rewrite /spec_inv.
iInv specN as ">Hinv'" "Hclose". iDestruct "Hinv'" as (e [σr σm]) "[Hown %] /=".
iDestruct (regspec_heap_valid_inclSepM with "Hown Hmap") as %Hregs.
have ? := lookup_weaken _ _ _ _ HPC Hregs.
iDestruct (spec_heap_valid with "[$Hown $Hpc_a]") as %Hpc_a.
iDestruct (spec_expr_valid with "[$Hown $Hj]") as %Heq; subst e.
specialize (normal_always_step (σr,σm)) as [c [ σ2 Hstep]].
eapply step_exec_inv in Hstep; eauto.
pose proof (Hstep' := Hstep). 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 (denote get_i wsrc) as [z | ] eqn:Hwsrc.
2 : { (* Failure: src is not of the right word type *)
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct_or! Hinstr; rewrite Hinstr in Hstep; cbn in Hstep.
all: rewrite Hsrc /= in Hstep.
all : destruct wsrc as [ | [ | ] | ]; try (inversion Hstep; auto);
rewrite /denote /= in Hwsrc; rewrite Hinstr in Hwsrc; congruence. }
rewrite Hdecode. iFailStep Get_fail_src_denote. }
assert (exec_opt get_i (σr, σm) = updatePC (update_reg (σr, σm) dst (WInt z))) as HH.
{ destruct_or! Hinstr; clear Hdecode; subst get_i; cbn in Hstep |- *.
all: rewrite /update_reg Hsrc /= in Hstep |-*; auto.
all : destruct wsrc as [ | [ | ] | ]; inversion Hwsrc; auto.
}
rewrite HH in Hstep. rewrite /update_reg /= in Hstep.
destruct (incrementPC (<[ dst := WInt z ]> regs))
as [regs'|] eqn:Hregs'; pose proof Hregs' as H'regs'; cycle 1.
{ (* Failure: incrementing PC overflows *)
apply incrementPC_fail_updatePC with (m:=σm) in Hregs'.
eapply updatePC_fail_incl with (m':=σm) in Hregs'.
2: by apply lookup_insert_is_Some'; eauto.
2: by apply insert_mono; eauto.
simplify_pair_eq.
rewrite Hregs' in Hstep. inv Hstep.
iFailStep Get_fail_overflow_PC. }
(* Success *)
eapply (incrementPC_success_updatePC _ σm) in Hregs'
as (p' & g' & b' & e' & a'' & a_pc' & HPC'' & HuPC & ->).
eapply updatePC_success_incl with (m':=σm) in HuPC. 2: by eapply insert_mono; eauto. rewrite HuPC in Hstep.
simplify_pair_eq.
iMod ((regspec_heap_update_inSepM _ _ _ dst) with "Hown Hmap") as "[Hown Hmap]"; eauto.
iMod ((regspec_heap_update_inSepM _ _ _ PC) with "Hown Hmap") as "[Hown Hmap]"; eauto.
iMod (exprspec_pointsto_update _ _ (fill K (Instr NextI)) with "Hown Hj") as "[Hown Hj]".
iExists NextIV,_. iFrame.
iMod ("Hclose" with "[Hown]") as "_".
{ iNext. iExists _,_;iFrame. iPureIntro. eapply rtc_r;eauto.
prim_step_from_exec. }
iModIntro. iPureIntro. econstructor; eauto.
Unshelve. Fail idtac. Admitted.
Lemma step_Get_success E K get_i dst src pc_p pc_b pc_e pc_a w wdst wsrc z 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' ->
denote get_i wsrc = Some z →
nclose specN ⊆ E →
spec_ctx ∗ ⤇ fill K (Instr Executable)
∗ ▷ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↣ₐ w
∗ ▷ src ↣ᵣ wsrc
∗ ▷ dst ↣ᵣ wdst
={E}=∗ ⤇ fill K (Instr NextI)
∗ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↣ₐ w
∗ src ↣ᵣ wsrc
∗ dst ↣ᵣ WInt z.
Proof.
iIntros (Hdecode Hinstr Hvpc Hpca' Hdenote Hnlose) "(#Hown & Hj & >HPC & >Hpc_a & >Hsrc & >Hdst)".
iDestruct (map_of_regs_3 with "HPC Hdst Hsrc") as "[Hmap (%&%&%)]".
iMod (step_Get with "[$Hmap $Hj $Hown $Hpc_a]") as (retv regs') "(Hj & #Hspec & Hpc_a & Hmap)"; simplify_map_eq; eauto.
by erewrite regs_of_is_Get; eauto; rewrite !dom_insert; set_solver+.
iDestruct "Hspec" as %Hspec.
destruct Hspec as [| * Hfail].
{ (* Success *)
iFrame. incrementPC_inv; simplify_map_eq.
rewrite insert_commute // insert_insert insert_commute // insert_insert.
iDestruct (regs_of_map_3 with "Hmap") as "[? [? ?]]"; eauto; by iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto. congruence. }
Unshelve. Fail idtac. Admitted.
End cap_lang_spec_rules.
From iris.program_logic Require Export weakestpre ectx_lifting.
From iris.proofmode Require Import proofmode.
From iris.algebra Require Import frac.
From cap_machine Require Export rules_Get rules_binary_base.
Section cap_lang_spec_rules.
Context `{cfgSG Σ, MachineParameters, invGS Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types σ : cap_lang.state.
Implicit Types a b : Addr.
Implicit Types o : OType.
Implicit Types r : RegName.
Implicit Types w : Word.
Implicit Types reg : gmap RegName Word.
Implicit Types ms : gmap Addr Word.
Lemma step_Get Ep K pc_p pc_b pc_e pc_a w get_i dst src regs :
decodeInstrW w = get_i →
is_Get get_i dst src →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
regs !! PC = Some (WCap pc_p pc_b pc_e pc_a) →
regs_of get_i ⊆ dom regs →
nclose specN ⊆ Ep →
spec_ctx ∗ ⤇ fill K (Instr Executable) ∗ ▷ pc_a ↣ₐ w ∗ ▷ ([∗ map] k↦y ∈ regs, k ↣ᵣ y)
={Ep}=∗ ∃ retv regs', ⤇ fill K (of_val retv) ∗ ⌜ Get_spec (decodeInstrW w) regs dst src regs' retv ⌝ ∗ pc_a ↣ₐ w ∗ ([∗ map] k↦y ∈ regs', k ↣ᵣ y).
Proof.
iIntros (Hdecode Hinstr Hvpc HPC Dregs Hnclose) "(Hinv & Hj & >Hpc_a & >Hmap)".
iDestruct "Hinv" as (ρ) "Hinv". rewrite /spec_inv.
iInv specN as ">Hinv'" "Hclose". iDestruct "Hinv'" as (e [σr σm]) "[Hown %] /=".
iDestruct (regspec_heap_valid_inclSepM with "Hown Hmap") as %Hregs.
have ? := lookup_weaken _ _ _ _ HPC Hregs.
iDestruct (spec_heap_valid with "[$Hown $Hpc_a]") as %Hpc_a.
iDestruct (spec_expr_valid with "[$Hown $Hj]") as %Heq; subst e.
specialize (normal_always_step (σr,σm)) as [c [ σ2 Hstep]].
eapply step_exec_inv in Hstep; eauto.
pose proof (Hstep' := Hstep). 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 (denote get_i wsrc) as [z | ] eqn:Hwsrc.
2 : { (* Failure: src is not of the right word type *)
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct_or! Hinstr; rewrite Hinstr in Hstep; cbn in Hstep.
all: rewrite Hsrc /= in Hstep.
all : destruct wsrc as [ | [ | ] | ]; try (inversion Hstep; auto);
rewrite /denote /= in Hwsrc; rewrite Hinstr in Hwsrc; congruence. }
rewrite Hdecode. iFailStep Get_fail_src_denote. }
assert (exec_opt get_i (σr, σm) = updatePC (update_reg (σr, σm) dst (WInt z))) as HH.
{ destruct_or! Hinstr; clear Hdecode; subst get_i; cbn in Hstep |- *.
all: rewrite /update_reg Hsrc /= in Hstep |-*; auto.
all : destruct wsrc as [ | [ | ] | ]; inversion Hwsrc; auto.
}
rewrite HH in Hstep. rewrite /update_reg /= in Hstep.
destruct (incrementPC (<[ dst := WInt z ]> regs))
as [regs'|] eqn:Hregs'; pose proof Hregs' as H'regs'; cycle 1.
{ (* Failure: incrementing PC overflows *)
apply incrementPC_fail_updatePC with (m:=σm) in Hregs'.
eapply updatePC_fail_incl with (m':=σm) in Hregs'.
2: by apply lookup_insert_is_Some'; eauto.
2: by apply insert_mono; eauto.
simplify_pair_eq.
rewrite Hregs' in Hstep. inv Hstep.
iFailStep Get_fail_overflow_PC. }
(* Success *)
eapply (incrementPC_success_updatePC _ σm) in Hregs'
as (p' & g' & b' & e' & a'' & a_pc' & HPC'' & HuPC & ->).
eapply updatePC_success_incl with (m':=σm) in HuPC. 2: by eapply insert_mono; eauto. rewrite HuPC in Hstep.
simplify_pair_eq.
iMod ((regspec_heap_update_inSepM _ _ _ dst) with "Hown Hmap") as "[Hown Hmap]"; eauto.
iMod ((regspec_heap_update_inSepM _ _ _ PC) with "Hown Hmap") as "[Hown Hmap]"; eauto.
iMod (exprspec_pointsto_update _ _ (fill K (Instr NextI)) with "Hown Hj") as "[Hown Hj]".
iExists NextIV,_. iFrame.
iMod ("Hclose" with "[Hown]") as "_".
{ iNext. iExists _,_;iFrame. iPureIntro. eapply rtc_r;eauto.
prim_step_from_exec. }
iModIntro. iPureIntro. econstructor; eauto.
Unshelve. Fail idtac. Admitted.
Lemma step_Get_success E K get_i dst src pc_p pc_b pc_e pc_a w wdst wsrc z 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' ->
denote get_i wsrc = Some z →
nclose specN ⊆ E →
spec_ctx ∗ ⤇ fill K (Instr Executable)
∗ ▷ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↣ₐ w
∗ ▷ src ↣ᵣ wsrc
∗ ▷ dst ↣ᵣ wdst
={E}=∗ ⤇ fill K (Instr NextI)
∗ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↣ₐ w
∗ src ↣ᵣ wsrc
∗ dst ↣ᵣ WInt z.
Proof.
iIntros (Hdecode Hinstr Hvpc Hpca' Hdenote Hnlose) "(#Hown & Hj & >HPC & >Hpc_a & >Hsrc & >Hdst)".
iDestruct (map_of_regs_3 with "HPC Hdst Hsrc") as "[Hmap (%&%&%)]".
iMod (step_Get with "[$Hmap $Hj $Hown $Hpc_a]") as (retv regs') "(Hj & #Hspec & Hpc_a & Hmap)"; simplify_map_eq; eauto.
by erewrite regs_of_is_Get; eauto; rewrite !dom_insert; set_solver+.
iDestruct "Hspec" as %Hspec.
destruct Hspec as [| * Hfail].
{ (* Success *)
iFrame. incrementPC_inv; simplify_map_eq.
rewrite insert_commute // insert_insert insert_commute // insert_insert.
iDestruct (regs_of_map_3 with "Hmap") as "[? [? ?]]"; eauto; by iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto. congruence. }
Unshelve. Fail idtac. Admitted.
End cap_lang_spec_rules.