cap_machine.rules_binary.rules_binary_Subseg
From cap_machine Require Export rules_Subseg rules_binary_base.
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.
Section cap_lang_spec_rules.
Context `{cfgSG Σ, MachineParameters, invG Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types σ : cap_lang.state.
Implicit Types a b : Addr.
Implicit Types r : RegName.
Implicit Types w : Word.
Implicit Types reg : gmap RegName Word.
Implicit Types ms : gmap Addr Word.
Lemma step_Subseg Ep K pc_p pc_b pc_e pc_a w dst src1 src2 regs :
decodeInstrW w = Subseg dst src1 src2 ->
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
regs !! PC = Some (WCap pc_p pc_b pc_e pc_a) →
regs_of (Subseg dst src1 src2) ⊆ 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', ⌜ Subseg_spec regs dst src1 src2 regs' retv ⌝ ∗ ⤇ fill K (of_val retv) ∗ pc_a ↣ₐ w ∗ ([∗ map] k↦y ∈ regs', k ↣ᵣ y).
Proof.
iIntros (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). rewrite /exec /= in Hstep.
specialize (indom_regs_incl _ _ _ Dregs Hregs) as Hri.
unfold regs_of in Hri, Dregs.
destruct (Hri dst) as [wdst [H'dst Hdst]]. by set_solver+.
destruct (addr_of_argument regs src1) as [a1|] eqn:Ha1;
pose proof Ha1 as H'a1; cycle 1.
{ destruct src1 as [| r1] eqn:?; cbn in Ha1, Hstep.
{ rewrite Hdst Ha1 /= in Hstep.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ repeat case_match; inv Hstep; auto. }
iMod (exprspec_mapsto_update _ _ (fill _ (Instr Failed)) with "Hown Hj") as "[Hown Hj]";
iMod ("Hclose" with "[Hown]") as "_";
[iNext;iExists _,_;iFrame;iPureIntro;eapply rtc_r;eauto;prim_step_from_exec|].
iExists (FailedV),_; iFrame;iModIntro;iFailCore Subseg_fail_src1_nonaddr.
}
subst src1. destruct (Hri r1) as [r1v [Hr'1 Hr1]].
by unfold regs_of_argument; set_solver+.
rewrite /addr_of_argument /= Hr'1 in Ha1.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct r1v.
all: unfold addr_of_argument, z_of_argument at 2 in Hstep.
all: rewrite /= Hdst Hr1 ?Ha1 /= in Hstep.
all: inv Hstep; auto.
}
repeat case_match; try congruence; try rename c into c0.
all: iFailStep Subseg_fail_src1_nonaddr.
}
apply (addr_of_arg_mono _ σr) in Ha1; auto. rewrite Ha1 /= in Hstep.
eapply addr_of_argument_Some_inv' in Ha1 as [z1 [Hz1 Hz1']]; eauto.
destruct (addr_of_argument regs src2) as [a2|] eqn:Ha2;
pose proof Ha2 as H'a2; cycle 1.
{ destruct src2 as [| r2] eqn:?; cbn in Ha2, Hstep.
{ rewrite Hdst Ha2 /= in Hstep.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ repeat case_match; inv Hstep; auto. }
iMod (exprspec_mapsto_update _ _ (fill _ (Instr Failed)) with "Hown Hj") as "[Hown Hj]";
iMod ("Hclose" with "[Hown]") as "_";
[iNext;iExists _,_;iFrame;iPureIntro;eapply rtc_r;eauto;prim_step_from_exec|].
iExists (FailedV),_; iFrame;iModIntro;iFailCore Subseg_fail_src2_nonaddr. }
subst src2. destruct (Hri r2) as [r2v [Hr'2 Hr2]].
by unfold regs_of_argument; set_solver+.
rewrite /addr_of_argument /= Hr'2 in Ha2.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct r2v.
all: unfold addr_of_argument, z_of_argument in Hstep.
all: rewrite /= Hdst Hr2 ?Ha2 /= in Hstep.
all: inv Hstep; auto. }
repeat case_match; try congruence; try rename c into c0.
all: iFailStep Subseg_fail_src2_nonaddr. }
apply (addr_of_arg_mono _ σr) in Ha2; auto. rewrite Ha2 /= in Hstep.
destruct wdst as [| cdst].
{ rewrite /= Hdst in Hstep. repeat case_match; inv Hstep; simplify_pair_eq.
all: iFailStep Subseg_fail_dst_noncap. }
rewrite Hdst /= in Hstep.
destruct (decide (cdst = E)).
{ subst cdst. inv Hstep.
iFailStep Subseg_fail_pE. }
rewrite /update_reg /= in Hstep.
destruct (isWithin a1 a2 b e) eqn:Hiw; cycle 1.
{ destruct cdst; try congruence; inv Hstep; iFailStep Subseg_fail_not_iswithin. }
destruct (incrementPC (<[ dst := (WCap cdst a1 a2 a) ]> regs)) eqn:HX;
pose proof HX as H'X; cycle 1.
{
assert (incrementPC (<[ dst := (WCap cdst a1 a2 a) ]> σr) = None) as HH.
{ eapply incrementPC_overflow_mono; first eapply HX.
by rewrite lookup_insert_is_Some'; eauto.
by apply insert_mono; eauto. }
apply (incrementPC_fail_updatePC _ σm) in HH. rewrite HH in Hstep.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->)
by (destruct cdst; inversion Hstep; auto).
iFailStep Subseg_fail_incrPC. }
eapply (incrementPC_success_updatePC _ σm) in HX
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.
eassert ((c, σ2) = (NextI, _)) as HH.
{ destruct cdst; cbn in Hstep; eauto. congruence. }
simplify_pair_eq. iFrame.
iMod ((regspec_heap_update_inSepM _ _ _ dst) with "Hown Hmap") as "[Hown Hmap]"; eauto.
iMod ((regspec_heap_update_inSepM _ _ _ PC (WCap p' g' b' a'')) with "Hown Hmap") as "[Hown Hmap]"; eauto.
iMod (exprspec_mapsto_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.
Qed.
Lemma step_subseg_success E K pc_p pc_b pc_e pc_a w dst r1 r2 p b e a n1 n2 a1 a2 pc_a' :
decodeInstrW w = Subseg dst (inr r1) (inr r2) →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
z_to_addr n1 = Some a1 ∧ z_to_addr n2 = Some a2 →
p ≠ machine_base.E →
isWithin a1 a2 b e = true →
(pc_a + 1)%a = Some pc_a' →
nclose specN ⊆ E →
spec_ctx ∗ ⤇ fill K (Instr Executable)
∗ ▷ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↣ₐ w
∗ ▷ dst ↣ᵣ WCap p b e a
∗ ▷ r1 ↣ᵣ WInt n1
∗ ▷ r2 ↣ᵣ WInt n2
={E}=∗ ⤇ fill K (Instr NextI)
∗ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↣ₐ w
∗ r1 ↣ᵣ WInt n1
∗ r2 ↣ᵣ WInt n2
∗ dst ↣ᵣ WCap p a1 a2 a.
Proof.
iIntros (Hinstr Hvpc [Hn1 Hn2] Hpne Hwb Hpc_a' Hnclose) "(Hown & Hj & >HPC & >Hpc_a & >Hdst & >Hr1 & >Hr2)".
iDestruct (map_of_regs_4 with "HPC Hr1 Hr2 Hdst") as "[Hmap (%&%&%&%&%&%)]".
iMod (step_Subseg with "[$Hown $Hj $Hpc_a $Hmap]") as (retv regs' Hspec) "(Hj & Hpc_a & Hmap)";simplify_map_eq;eauto.
by unfold regs_of; rewrite !dom_insert; set_solver+.
destruct Hspec as [| * Hfail].
{ (* Success *)
iFrame. incrementPC_inv. unfold addr_of_argument, z_of_argument in *; simplify_map_eq_alt.
rewrite (insert_commute _ PC dst) // insert_insert (insert_commute _ r2 dst) //
(insert_commute _ r1 dst) // (insert_commute _ PC dst) // insert_insert.
iDestruct (regs_of_map_4 with "Hmap") as "(?&?&?&?)"; eauto; by iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; unfold addr_of_argument, z_of_argument in *; simplify_map_eq_alt. congruence.
incrementPC_inv;[|rewrite lookup_insert_ne// lookup_insert;eauto]. simplify_eq. }
Qed.
End cap_lang_spec_rules.
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.
Section cap_lang_spec_rules.
Context `{cfgSG Σ, MachineParameters, invG Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types σ : cap_lang.state.
Implicit Types a b : Addr.
Implicit Types r : RegName.
Implicit Types w : Word.
Implicit Types reg : gmap RegName Word.
Implicit Types ms : gmap Addr Word.
Lemma step_Subseg Ep K pc_p pc_b pc_e pc_a w dst src1 src2 regs :
decodeInstrW w = Subseg dst src1 src2 ->
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
regs !! PC = Some (WCap pc_p pc_b pc_e pc_a) →
regs_of (Subseg dst src1 src2) ⊆ 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', ⌜ Subseg_spec regs dst src1 src2 regs' retv ⌝ ∗ ⤇ fill K (of_val retv) ∗ pc_a ↣ₐ w ∗ ([∗ map] k↦y ∈ regs', k ↣ᵣ y).
Proof.
iIntros (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). rewrite /exec /= in Hstep.
specialize (indom_regs_incl _ _ _ Dregs Hregs) as Hri.
unfold regs_of in Hri, Dregs.
destruct (Hri dst) as [wdst [H'dst Hdst]]. by set_solver+.
destruct (addr_of_argument regs src1) as [a1|] eqn:Ha1;
pose proof Ha1 as H'a1; cycle 1.
{ destruct src1 as [| r1] eqn:?; cbn in Ha1, Hstep.
{ rewrite Hdst Ha1 /= in Hstep.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ repeat case_match; inv Hstep; auto. }
iMod (exprspec_mapsto_update _ _ (fill _ (Instr Failed)) with "Hown Hj") as "[Hown Hj]";
iMod ("Hclose" with "[Hown]") as "_";
[iNext;iExists _,_;iFrame;iPureIntro;eapply rtc_r;eauto;prim_step_from_exec|].
iExists (FailedV),_; iFrame;iModIntro;iFailCore Subseg_fail_src1_nonaddr.
}
subst src1. destruct (Hri r1) as [r1v [Hr'1 Hr1]].
by unfold regs_of_argument; set_solver+.
rewrite /addr_of_argument /= Hr'1 in Ha1.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct r1v.
all: unfold addr_of_argument, z_of_argument at 2 in Hstep.
all: rewrite /= Hdst Hr1 ?Ha1 /= in Hstep.
all: inv Hstep; auto.
}
repeat case_match; try congruence; try rename c into c0.
all: iFailStep Subseg_fail_src1_nonaddr.
}
apply (addr_of_arg_mono _ σr) in Ha1; auto. rewrite Ha1 /= in Hstep.
eapply addr_of_argument_Some_inv' in Ha1 as [z1 [Hz1 Hz1']]; eauto.
destruct (addr_of_argument regs src2) as [a2|] eqn:Ha2;
pose proof Ha2 as H'a2; cycle 1.
{ destruct src2 as [| r2] eqn:?; cbn in Ha2, Hstep.
{ rewrite Hdst Ha2 /= in Hstep.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ repeat case_match; inv Hstep; auto. }
iMod (exprspec_mapsto_update _ _ (fill _ (Instr Failed)) with "Hown Hj") as "[Hown Hj]";
iMod ("Hclose" with "[Hown]") as "_";
[iNext;iExists _,_;iFrame;iPureIntro;eapply rtc_r;eauto;prim_step_from_exec|].
iExists (FailedV),_; iFrame;iModIntro;iFailCore Subseg_fail_src2_nonaddr. }
subst src2. destruct (Hri r2) as [r2v [Hr'2 Hr2]].
by unfold regs_of_argument; set_solver+.
rewrite /addr_of_argument /= Hr'2 in Ha2.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct r2v.
all: unfold addr_of_argument, z_of_argument in Hstep.
all: rewrite /= Hdst Hr2 ?Ha2 /= in Hstep.
all: inv Hstep; auto. }
repeat case_match; try congruence; try rename c into c0.
all: iFailStep Subseg_fail_src2_nonaddr. }
apply (addr_of_arg_mono _ σr) in Ha2; auto. rewrite Ha2 /= in Hstep.
destruct wdst as [| cdst].
{ rewrite /= Hdst in Hstep. repeat case_match; inv Hstep; simplify_pair_eq.
all: iFailStep Subseg_fail_dst_noncap. }
rewrite Hdst /= in Hstep.
destruct (decide (cdst = E)).
{ subst cdst. inv Hstep.
iFailStep Subseg_fail_pE. }
rewrite /update_reg /= in Hstep.
destruct (isWithin a1 a2 b e) eqn:Hiw; cycle 1.
{ destruct cdst; try congruence; inv Hstep; iFailStep Subseg_fail_not_iswithin. }
destruct (incrementPC (<[ dst := (WCap cdst a1 a2 a) ]> regs)) eqn:HX;
pose proof HX as H'X; cycle 1.
{
assert (incrementPC (<[ dst := (WCap cdst a1 a2 a) ]> σr) = None) as HH.
{ eapply incrementPC_overflow_mono; first eapply HX.
by rewrite lookup_insert_is_Some'; eauto.
by apply insert_mono; eauto. }
apply (incrementPC_fail_updatePC _ σm) in HH. rewrite HH in Hstep.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->)
by (destruct cdst; inversion Hstep; auto).
iFailStep Subseg_fail_incrPC. }
eapply (incrementPC_success_updatePC _ σm) in HX
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.
eassert ((c, σ2) = (NextI, _)) as HH.
{ destruct cdst; cbn in Hstep; eauto. congruence. }
simplify_pair_eq. iFrame.
iMod ((regspec_heap_update_inSepM _ _ _ dst) with "Hown Hmap") as "[Hown Hmap]"; eauto.
iMod ((regspec_heap_update_inSepM _ _ _ PC (WCap p' g' b' a'')) with "Hown Hmap") as "[Hown Hmap]"; eauto.
iMod (exprspec_mapsto_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.
Qed.
Lemma step_subseg_success E K pc_p pc_b pc_e pc_a w dst r1 r2 p b e a n1 n2 a1 a2 pc_a' :
decodeInstrW w = Subseg dst (inr r1) (inr r2) →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
z_to_addr n1 = Some a1 ∧ z_to_addr n2 = Some a2 →
p ≠ machine_base.E →
isWithin a1 a2 b e = true →
(pc_a + 1)%a = Some pc_a' →
nclose specN ⊆ E →
spec_ctx ∗ ⤇ fill K (Instr Executable)
∗ ▷ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↣ₐ w
∗ ▷ dst ↣ᵣ WCap p b e a
∗ ▷ r1 ↣ᵣ WInt n1
∗ ▷ r2 ↣ᵣ WInt n2
={E}=∗ ⤇ fill K (Instr NextI)
∗ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↣ₐ w
∗ r1 ↣ᵣ WInt n1
∗ r2 ↣ᵣ WInt n2
∗ dst ↣ᵣ WCap p a1 a2 a.
Proof.
iIntros (Hinstr Hvpc [Hn1 Hn2] Hpne Hwb Hpc_a' Hnclose) "(Hown & Hj & >HPC & >Hpc_a & >Hdst & >Hr1 & >Hr2)".
iDestruct (map_of_regs_4 with "HPC Hr1 Hr2 Hdst") as "[Hmap (%&%&%&%&%&%)]".
iMod (step_Subseg with "[$Hown $Hj $Hpc_a $Hmap]") as (retv regs' Hspec) "(Hj & Hpc_a & Hmap)";simplify_map_eq;eauto.
by unfold regs_of; rewrite !dom_insert; set_solver+.
destruct Hspec as [| * Hfail].
{ (* Success *)
iFrame. incrementPC_inv. unfold addr_of_argument, z_of_argument in *; simplify_map_eq_alt.
rewrite (insert_commute _ PC dst) // insert_insert (insert_commute _ r2 dst) //
(insert_commute _ r1 dst) // (insert_commute _ PC dst) // insert_insert.
iDestruct (regs_of_map_4 with "Hmap") as "(?&?&?&?)"; eauto; by iFrame. }
{ (* Failure (contradiction) *)
destruct Hfail; unfold addr_of_argument, z_of_argument in *; simplify_map_eq_alt. congruence.
incrementPC_inv;[|rewrite lookup_insert_ne// lookup_insert;eauto]. simplify_eq. }
Qed.
End cap_lang_spec_rules.