cap_machine.rules_binary.rules_binary_Lea
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_Lea 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 r : RegName.
Implicit Types w : Word.
Implicit Types reg : gmap RegName Word.
Implicit Types ms : gmap Addr Word.
Lemma step_lea Ep K pc_p pc_b pc_e pc_a r1 w arg (regs: Reg) :
decodeInstrW w = Lea r1 arg →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
regs !! PC = Some (WCap pc_p pc_b pc_e pc_a) →
regs_of (Lea r1 arg) ⊆ 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) ∗ ⌜ Lea_spec regs r1 arg regs' 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.
pose proof (lookup_weaken _ _ _ _ HPC Hregs).
iDestruct (@spec_heap_valid with "[$Hown $Hpc_a]") as %Hpc_a; auto.
iDestruct (spec_expr_valid with "[$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. unfold regs_of in Hri.
odestruct (Hri r1) as [r1v [Hr'1 Hr1]]. by set_solver+.
cbn in Hstep.
rewrite Hr1 /= in Hstep.
destruct (z_of_argument regs arg) as [ argz |] eqn:Harg;
pose proof Harg as Harg'; cycle 1.
{ (* Failure: argument is not a constant (z_of_argument regs arg = None) *)
unfold z_of_argument in Harg, Hstep. destruct arg as [| r0]; [ congruence |].
odestruct (Hri r0) as [r0v [Hr'0 Hr0]].
{ unfold regs_of_argument. set_solver+. }
rewrite Hr0 Hr'0 in Harg Hstep.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct_word r0v; cbn in Hstep; try congruence; by simplify_pair_eq. }
iFailStep Lea_fail_rv_nonconst. }
apply (z_of_arg_mono _ σr) in Harg; auto. rewrite Harg in Hstep; cbn in Hstep.
destruct (is_mutable_range r1v) eqn:Hr1v.
2: { (* Failure: r1v is not of the right type *)
unfold is_mutable_range in Hr1v.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct r1v as [ | [p b e a | ] | ]; try by inversion Hr1v.
all: try by simplify_pair_eq.
destruct p; try congruence.
simplify_pair_eq; auto. }
iFailStep Lea_fail_allowed. }
(* Now the proof splits depending on the type of value in r1v *)
destruct r1v as [ | [p b e a | p b e a] | ].
1,4: inversion Hr1v.
(* First, the case where r1v is a capability *)
+ destruct (perm_eq_dec p E); [ subst p |].
{ rewrite /is_mutable_range in Hr1v; congruence. }
destruct (a + argz)%a as [ a' |] eqn:Hoffset; cycle 1.
{ (* Failure: offset is too large *)
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->)
by (destruct p; inversion Hstep; auto).
iFailStep Lea_fail_overflow_cap. }
rewrite /update_reg /= in Hstep.
destruct (incrementPC (<[ r1 := WCap p b e a' ]> regs)) as [ regs' |] eqn:Hregs';
pose proof Hregs' as Hregs'2; cycle 1.
{ (* Failure: incrementing PC overflows *)
assert (incrementPC (<[ r1 := WCap p b e a' ]> σr) = None) as HH.
{ eapply incrementPC_overflow_mono; first eapply Hregs'.
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 p; inversion Hstep; auto).
iFailStep Lea_fail_overflow_PC_cap. }
(* Success *)
eapply (incrementPC_success_updatePC _ σm) in Hregs'
as (p' & g' & b' & e' & a'' & a_pc' & HPC'' & HuPC & ->).
eapply updatePC_success_incl in HuPC. 2: by eapply insert_mono; eauto.
rewrite HuPC in Hstep; clear HuPC.
eassert ((c, σ2) = (NextI, _)) as HH.
{ destruct p; cbn in Hstep; eauto. congruence. }
simplify_pair_eq.
iMod ((regspec_heap_update_inSepM _ _ _ r1) with "Hown Hmap") as "[Hr Hmap]"; eauto.
iMod ((regspec_heap_update_inSepM _ _ _ PC) with "Hr Hmap") as "[Hr Hmap]"; eauto.
iMod (exprspec_pointsto_update _ _ (fill K (Instr NextI)) with "Hr 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.
eapply Lea_spec_success_cap; eauto.
(* Now, the case where r1v is a sealrange *)
+ destruct (a + argz)%ot as [ a' |] eqn:Hoffset; cycle 1.
{ (* Failure: offset is too large *)
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->)
by (destruct p; inversion Hstep; auto).
iFailStep Lea_fail_overflow_sr. }
rewrite /update_reg /= in Hstep.
destruct (incrementPC (<[ r1 := WSealRange p b e a' ]> regs)) as [ regs' |] eqn:Hregs';
pose proof Hregs' as Hregs'2; cycle 1.
{ (* Failure: incrementing PC overflows *)
assert (incrementPC (<[ r1 := WSealRange p b e a' ]> σr) = None) as HH.
{ eapply incrementPC_overflow_mono; first eapply Hregs'.
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 p; inversion Hstep; auto).
iFailStep Lea_fail_overflow_PC_sr. }
(* Success *)
eapply (incrementPC_success_updatePC _ σm) in Hregs'
as (p' & g' & b' & e' & a'' & a_pc' & HPC'' & HuPC & ->).
eapply updatePC_success_incl in HuPC. 2: by eapply insert_mono; eauto.
rewrite HuPC in Hstep; clear HuPC.
eassert ((c, σ2) = (NextI, _)) as HH.
{ destruct p; cbn in Hstep; eauto. }
simplify_pair_eq.
iMod ((regspec_heap_update_inSepM _ _ _ r1) with "Hown Hmap") as "[Hr Hmap]"; eauto.
iMod ((regspec_heap_update_inSepM _ _ _ PC) with "Hr Hmap") as "[Hr Hmap]"; eauto.
iMod (exprspec_pointsto_update _ _ (fill K (Instr NextI)) with "Hr 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.
eapply Lea_spec_success_sr; eauto. Unshelve. all: auto.
Unshelve. Fail idtac. Admitted.
Lemma step_lea_success_z Ep K pc_p pc_b pc_e pc_a pc_a' w r1 p b e a z a' :
decodeInstrW w = Lea r1 (inl z) →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' →
(a + z)%a = Some a' →
p ≠ E →
nclose specN ⊆ Ep →
spec_ctx ∗ ⤇ fill K (Instr Executable)
∗ ▷ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↣ₐ w
∗ ▷ r1 ↣ᵣ WCap p b e a
={Ep}=∗ ⤇ fill K (of_val NextIV)
∗ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↣ₐ w
∗ r1 ↣ᵣ WCap p b e a'.
Proof.
iIntros (Hinstr Hvpc Hpca' Ha' Hnep Hnclose) "(Hown & Hj & >HPC & >Hpc_a & >Hr1)".
iDestruct (map_of_regs_2 with "HPC Hr1") as "[Hmap %]".
iMod (step_lea with "[$Hmap $Hown $Hj $Hpc_a]") as (retv regs') "(Hj & #Hspec & Hpc_a & Hmap)"; eauto;[rewrite lookup_insert;eauto|..].
by rewrite !dom_insert; set_solver+.
iDestruct "Hspec" as %Hspec.
destruct Hspec as [ | | * Hfail ].
{ (* Success *)
iFrame. incrementPC_inv; simplify_map_eq.
(* FIXME: tedious *)
rewrite insert_commute // insert_insert insert_commute // insert_insert.
iDestruct (regs_of_map_2 with "Hmap") as "[? ?]"; eauto. by iFrame. }
{ (* Success with WSealRange (contradiction) *)
simplify_map_eq. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto.
destruct p; congruence.
congruence.
Unshelve. all: auto.
}
Unshelve. Fail idtac. Admitted.
Lemma step_lea_success_reg Ep K pc_p pc_b pc_e pc_a pc_a' w r1 rv p b e a z a' :
decodeInstrW w = Lea r1 (inr rv) →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' →
(a + z)%a = Some a' →
p ≠ E →
nclose specN ⊆ Ep →
spec_ctx ∗ ⤇ fill K (Instr Executable)
∗ ▷ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↣ₐ w
∗ ▷ r1 ↣ᵣ WCap p b e a
∗ ▷ rv ↣ᵣ WInt z
={Ep}=∗ ⤇ fill K (Instr NextI)
∗ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↣ₐ w
∗ rv ↣ᵣ WInt z
∗ r1 ↣ᵣ WCap p b e a'.
Proof.
iIntros (Hinstr Hvpc Hpca' Ha' Hnep Hnclose) "(Hown & Hj & >HPC & >Hpc_a & >Hr1 & >Hrv)".
iDestruct (map_of_regs_3 with "HPC Hrv Hr1") as "[Hmap (%&%&%)]".
iMod (step_lea with "[$Hmap $Hown $Hj $Hpc_a]") as (retv regs') "(Hj & #Hspec & Hpc_a & Hmap)"; eauto;[rewrite lookup_insert;eauto|..].
by rewrite !dom_insert; set_solver+.
iDestruct "Hspec" as %Hspec.
destruct Hspec as [ | | * Hfail ].
{ (* Success *)
iFrame. incrementPC_inv; simplify_map_eq.
(* FIXME: tedious *)
rewrite (insert_commute _ PC r1) // insert_insert.
rewrite (insert_commute _ r1 PC) // (insert_commute _ r1 rv) // insert_insert.
iApply (regs_of_map_3 with "Hmap"); eauto. }
{ (* Success with WSealRange (contradiction) *)
simplify_map_eq. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto.
destruct p; congruence.
congruence.
}
Unshelve. all: auto.
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_Lea 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 r : RegName.
Implicit Types w : Word.
Implicit Types reg : gmap RegName Word.
Implicit Types ms : gmap Addr Word.
Lemma step_lea Ep K pc_p pc_b pc_e pc_a r1 w arg (regs: Reg) :
decodeInstrW w = Lea r1 arg →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
regs !! PC = Some (WCap pc_p pc_b pc_e pc_a) →
regs_of (Lea r1 arg) ⊆ 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) ∗ ⌜ Lea_spec regs r1 arg regs' 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.
pose proof (lookup_weaken _ _ _ _ HPC Hregs).
iDestruct (@spec_heap_valid with "[$Hown $Hpc_a]") as %Hpc_a; auto.
iDestruct (spec_expr_valid with "[$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. unfold regs_of in Hri.
odestruct (Hri r1) as [r1v [Hr'1 Hr1]]. by set_solver+.
cbn in Hstep.
rewrite Hr1 /= in Hstep.
destruct (z_of_argument regs arg) as [ argz |] eqn:Harg;
pose proof Harg as Harg'; cycle 1.
{ (* Failure: argument is not a constant (z_of_argument regs arg = None) *)
unfold z_of_argument in Harg, Hstep. destruct arg as [| r0]; [ congruence |].
odestruct (Hri r0) as [r0v [Hr'0 Hr0]].
{ unfold regs_of_argument. set_solver+. }
rewrite Hr0 Hr'0 in Harg Hstep.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct_word r0v; cbn in Hstep; try congruence; by simplify_pair_eq. }
iFailStep Lea_fail_rv_nonconst. }
apply (z_of_arg_mono _ σr) in Harg; auto. rewrite Harg in Hstep; cbn in Hstep.
destruct (is_mutable_range r1v) eqn:Hr1v.
2: { (* Failure: r1v is not of the right type *)
unfold is_mutable_range in Hr1v.
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->).
{ destruct r1v as [ | [p b e a | ] | ]; try by inversion Hr1v.
all: try by simplify_pair_eq.
destruct p; try congruence.
simplify_pair_eq; auto. }
iFailStep Lea_fail_allowed. }
(* Now the proof splits depending on the type of value in r1v *)
destruct r1v as [ | [p b e a | p b e a] | ].
1,4: inversion Hr1v.
(* First, the case where r1v is a capability *)
+ destruct (perm_eq_dec p E); [ subst p |].
{ rewrite /is_mutable_range in Hr1v; congruence. }
destruct (a + argz)%a as [ a' |] eqn:Hoffset; cycle 1.
{ (* Failure: offset is too large *)
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->)
by (destruct p; inversion Hstep; auto).
iFailStep Lea_fail_overflow_cap. }
rewrite /update_reg /= in Hstep.
destruct (incrementPC (<[ r1 := WCap p b e a' ]> regs)) as [ regs' |] eqn:Hregs';
pose proof Hregs' as Hregs'2; cycle 1.
{ (* Failure: incrementing PC overflows *)
assert (incrementPC (<[ r1 := WCap p b e a' ]> σr) = None) as HH.
{ eapply incrementPC_overflow_mono; first eapply Hregs'.
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 p; inversion Hstep; auto).
iFailStep Lea_fail_overflow_PC_cap. }
(* Success *)
eapply (incrementPC_success_updatePC _ σm) in Hregs'
as (p' & g' & b' & e' & a'' & a_pc' & HPC'' & HuPC & ->).
eapply updatePC_success_incl in HuPC. 2: by eapply insert_mono; eauto.
rewrite HuPC in Hstep; clear HuPC.
eassert ((c, σ2) = (NextI, _)) as HH.
{ destruct p; cbn in Hstep; eauto. congruence. }
simplify_pair_eq.
iMod ((regspec_heap_update_inSepM _ _ _ r1) with "Hown Hmap") as "[Hr Hmap]"; eauto.
iMod ((regspec_heap_update_inSepM _ _ _ PC) with "Hr Hmap") as "[Hr Hmap]"; eauto.
iMod (exprspec_pointsto_update _ _ (fill K (Instr NextI)) with "Hr 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.
eapply Lea_spec_success_cap; eauto.
(* Now, the case where r1v is a sealrange *)
+ destruct (a + argz)%ot as [ a' |] eqn:Hoffset; cycle 1.
{ (* Failure: offset is too large *)
assert (c = Failed ∧ σ2 = (σr, σm)) as (-> & ->)
by (destruct p; inversion Hstep; auto).
iFailStep Lea_fail_overflow_sr. }
rewrite /update_reg /= in Hstep.
destruct (incrementPC (<[ r1 := WSealRange p b e a' ]> regs)) as [ regs' |] eqn:Hregs';
pose proof Hregs' as Hregs'2; cycle 1.
{ (* Failure: incrementing PC overflows *)
assert (incrementPC (<[ r1 := WSealRange p b e a' ]> σr) = None) as HH.
{ eapply incrementPC_overflow_mono; first eapply Hregs'.
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 p; inversion Hstep; auto).
iFailStep Lea_fail_overflow_PC_sr. }
(* Success *)
eapply (incrementPC_success_updatePC _ σm) in Hregs'
as (p' & g' & b' & e' & a'' & a_pc' & HPC'' & HuPC & ->).
eapply updatePC_success_incl in HuPC. 2: by eapply insert_mono; eauto.
rewrite HuPC in Hstep; clear HuPC.
eassert ((c, σ2) = (NextI, _)) as HH.
{ destruct p; cbn in Hstep; eauto. }
simplify_pair_eq.
iMod ((regspec_heap_update_inSepM _ _ _ r1) with "Hown Hmap") as "[Hr Hmap]"; eauto.
iMod ((regspec_heap_update_inSepM _ _ _ PC) with "Hr Hmap") as "[Hr Hmap]"; eauto.
iMod (exprspec_pointsto_update _ _ (fill K (Instr NextI)) with "Hr 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.
eapply Lea_spec_success_sr; eauto. Unshelve. all: auto.
Unshelve. Fail idtac. Admitted.
Lemma step_lea_success_z Ep K pc_p pc_b pc_e pc_a pc_a' w r1 p b e a z a' :
decodeInstrW w = Lea r1 (inl z) →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' →
(a + z)%a = Some a' →
p ≠ E →
nclose specN ⊆ Ep →
spec_ctx ∗ ⤇ fill K (Instr Executable)
∗ ▷ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↣ₐ w
∗ ▷ r1 ↣ᵣ WCap p b e a
={Ep}=∗ ⤇ fill K (of_val NextIV)
∗ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↣ₐ w
∗ r1 ↣ᵣ WCap p b e a'.
Proof.
iIntros (Hinstr Hvpc Hpca' Ha' Hnep Hnclose) "(Hown & Hj & >HPC & >Hpc_a & >Hr1)".
iDestruct (map_of_regs_2 with "HPC Hr1") as "[Hmap %]".
iMod (step_lea with "[$Hmap $Hown $Hj $Hpc_a]") as (retv regs') "(Hj & #Hspec & Hpc_a & Hmap)"; eauto;[rewrite lookup_insert;eauto|..].
by rewrite !dom_insert; set_solver+.
iDestruct "Hspec" as %Hspec.
destruct Hspec as [ | | * Hfail ].
{ (* Success *)
iFrame. incrementPC_inv; simplify_map_eq.
(* FIXME: tedious *)
rewrite insert_commute // insert_insert insert_commute // insert_insert.
iDestruct (regs_of_map_2 with "Hmap") as "[? ?]"; eauto. by iFrame. }
{ (* Success with WSealRange (contradiction) *)
simplify_map_eq. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto.
destruct p; congruence.
congruence.
Unshelve. all: auto.
}
Unshelve. Fail idtac. Admitted.
Lemma step_lea_success_reg Ep K pc_p pc_b pc_e pc_a pc_a' w r1 rv p b e a z a' :
decodeInstrW w = Lea r1 (inr rv) →
isCorrectPC (WCap pc_p pc_b pc_e pc_a) →
(pc_a + 1)%a = Some pc_a' →
(a + z)%a = Some a' →
p ≠ E →
nclose specN ⊆ Ep →
spec_ctx ∗ ⤇ fill K (Instr Executable)
∗ ▷ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a
∗ ▷ pc_a ↣ₐ w
∗ ▷ r1 ↣ᵣ WCap p b e a
∗ ▷ rv ↣ᵣ WInt z
={Ep}=∗ ⤇ fill K (Instr NextI)
∗ PC ↣ᵣ WCap pc_p pc_b pc_e pc_a'
∗ pc_a ↣ₐ w
∗ rv ↣ᵣ WInt z
∗ r1 ↣ᵣ WCap p b e a'.
Proof.
iIntros (Hinstr Hvpc Hpca' Ha' Hnep Hnclose) "(Hown & Hj & >HPC & >Hpc_a & >Hr1 & >Hrv)".
iDestruct (map_of_regs_3 with "HPC Hrv Hr1") as "[Hmap (%&%&%)]".
iMod (step_lea with "[$Hmap $Hown $Hj $Hpc_a]") as (retv regs') "(Hj & #Hspec & Hpc_a & Hmap)"; eauto;[rewrite lookup_insert;eauto|..].
by rewrite !dom_insert; set_solver+.
iDestruct "Hspec" as %Hspec.
destruct Hspec as [ | | * Hfail ].
{ (* Success *)
iFrame. incrementPC_inv; simplify_map_eq.
(* FIXME: tedious *)
rewrite (insert_commute _ PC r1) // insert_insert.
rewrite (insert_commute _ r1 PC) // (insert_commute _ r1 rv) // insert_insert.
iApply (regs_of_map_3 with "Hmap"); eauto. }
{ (* Success with WSealRange (contradiction) *)
simplify_map_eq. }
{ (* Failure (contradiction) *)
destruct Hfail; try incrementPC_inv; simplify_map_eq; eauto.
destruct p; congruence.
congruence.
}
Unshelve. all: auto.
Unshelve. Fail idtac. Admitted.
End cap_lang_spec_rules.