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Copy pathDomainSepInv.thy
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DomainSepInv.thy
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(*
* Copyright 2014, NICTA
*
* This software may be distributed and modified according to the terms of
* the GNU General Public License version 2. Note that NO WARRANTY is provided.
* See "LICENSE_GPLv2.txt" for details.
*
* @TAG(NICTA_GPL)
*)
theory DomainSepInv
imports "Ipc_AC" (* for transfer_caps_loop_pres_dest lec_valid_cap' set_endpoint_get_tcb thread_set_tcb_fault_update_valid_mdb *)
begin
text {*
We define and prove an invariant that is necessary to achieve domain
separation on seL4. In its strongest form, we require that all IRQs, other than
those for the timer, are inactive, and that no IRQControl or
IRQHandler caps are present (to prevent any inactive IRQs from becoming
active in the future).
It always requires that there are no domain caps.
*}
text {*
When @{term irqs} is @{term False} we require that non-timer IRQs are off permanently.
*}
definition domain_sep_inv where
"domain_sep_inv irqs st s \<equiv>
(\<forall> slot. \<not> cte_wp_at (op = DomainCap) slot s) \<and>
(irqs \<or> (\<forall> irq slot. \<not> cte_wp_at (op = IRQControlCap) slot s
\<and> \<not> cte_wp_at (op = (IRQHandlerCap irq)) slot s
\<and> interrupt_states s irq \<noteq> IRQNotifyAEP
\<and> interrupt_states s = interrupt_states st))"
definition domain_sep_inv_cap where
"domain_sep_inv_cap irqs cap \<equiv> case cap of
IRQControlCap \<Rightarrow> irqs
| IRQHandlerCap irq \<Rightarrow> irqs
| DomainCap \<Rightarrow> False
| _ \<Rightarrow> True"
lemma cte_wp_at_not_domain_sep_inv_cap:
"cte_wp_at (not domain_sep_inv_cap irqs) slot s \<longleftrightarrow>
((irqs \<longrightarrow> False) \<and>
(\<not> irqs \<longrightarrow> (cte_wp_at (op = IRQControlCap) slot s \<or>
(\<exists> irq. cte_wp_at (op = (IRQHandlerCap irq)) slot s)))
)
\<or> cte_wp_at (op = DomainCap) slot s"
apply(rule iffI)
apply(drule cte_wp_at_eqD)
apply clarsimp
apply(case_tac c, simp_all add: domain_sep_inv_cap_def pred_neg_def)
apply(auto elim: cte_wp_at_weakenE split: if_splits)
done
lemma domain_sep_inv_def2:
"domain_sep_inv irqs st s =
((\<forall> slot. \<not> cte_wp_at (op = DomainCap) slot s) \<and>
(irqs \<or> (\<forall> irq slot. \<not> cte_wp_at (op = IRQControlCap) slot s
\<and> \<not> cte_wp_at (op = (IRQHandlerCap irq)) slot s)) \<and>
(irqs \<or> (\<forall> irq.
interrupt_states s irq \<noteq> IRQNotifyAEP
\<and> interrupt_states s = interrupt_states st)))"
apply(fastforce simp: domain_sep_inv_def)
done
lemma domain_sep_inv_wp:
assumes nctrl: "\<And>slot. \<lbrace>(\<lambda>s. \<not> cte_wp_at (not domain_sep_inv_cap irqs) slot s) and P\<rbrace> f \<lbrace>\<lambda>_ s. \<not> cte_wp_at (not domain_sep_inv_cap irqs) slot s\<rbrace>"
assumes irq_pres: "\<And>P. \<not> irqs \<Longrightarrow> \<lbrace>(\<lambda>s. P (interrupt_states s)) and R\<rbrace> f \<lbrace>\<lambda>_ s. P (interrupt_states s)\<rbrace>"
shows "\<lbrace>domain_sep_inv irqs st and P and (\<lambda>s. irqs \<or> R s)\<rbrace> f \<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply (clarsimp simp: domain_sep_inv_def2 valid_def)
apply (subst conj_assoc[symmetric])
apply (rule conjI)
apply (rule conjI)
apply(intro allI)
apply(erule use_valid[OF _ hoare_strengthen_post[OF nctrl]])
apply(fastforce simp: cte_wp_at_not_domain_sep_inv_cap)
apply(fastforce simp: cte_wp_at_not_domain_sep_inv_cap)
apply(fastforce elim!: use_valid[OF _ hoare_strengthen_post[OF nctrl]] simp: cte_wp_at_not_domain_sep_inv_cap)
apply(case_tac "irqs")
apply blast
apply(rule disjI2)
apply simp
apply(intro allI conjI)
apply(erule_tac P1="\<lambda>x. x irq \<noteq> IRQNotifyAEP" in use_valid[OF _ irq_pres], assumption)
apply blast
apply(erule use_valid[OF _ irq_pres], assumption)
apply blast
done
lemma domain_sep_inv_triv:
assumes cte_pres: "\<And>P slot. \<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace> f \<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
assumes irq_pres: "\<And>P. \<lbrace>\<lambda>s. P (interrupt_states s)\<rbrace> f \<lbrace>\<lambda>_ s. P (interrupt_states s)\<rbrace>"
shows
"\<lbrace>domain_sep_inv irqs st\<rbrace> f \<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(rule domain_sep_inv_wp[where P="\<top>" and R="\<top>", simplified])
apply(rule cte_pres, rule irq_pres)
done
(* FIXME: clagged from FinalCaps *)
lemma set_object_wp:
"\<lbrace> \<lambda> s. P (s\<lparr>kheap := kheap s(ptr \<mapsto> obj)\<rparr>) \<rbrace>
set_object ptr obj
\<lbrace> \<lambda>_. P \<rbrace>"
unfolding set_object_def
apply (wp)
done
(* FIXME: following 3 lemmas clagged from FinalCaps *)
lemma set_cap_neg_cte_wp_at_other_helper':
"\<lbrakk>oslot \<noteq> slot;
ko_at (TCB x) (fst oslot) s;
tcb_cap_cases (snd oslot) = Some (ogetF, osetF, orestr);
kheap
(s\<lparr>kheap := kheap s(fst oslot \<mapsto>
TCB (osetF (\<lambda> x. cap) x))\<rparr>)
(fst slot) =
Some (TCB tcb);
tcb_cap_cases (snd slot) = Some (getF, setF, restr);
P (getF tcb)\<rbrakk> \<Longrightarrow>
cte_wp_at P slot s"
apply(case_tac "fst oslot = fst slot")
apply(rule cte_wp_at_tcbI)
apply(fastforce split: if_splits simp: obj_at_def)
apply assumption
apply(fastforce split: if_splits simp: tcb_cap_cases_def dest: prod_eqI)
apply(rule cte_wp_at_tcbI)
apply(fastforce split: if_splits simp: obj_at_def)
apply assumption
apply assumption
done
lemma set_cap_neg_cte_wp_at_other_helper:
"\<lbrakk>\<not> cte_wp_at P slot s; oslot \<noteq> slot; ko_at (TCB x) (fst oslot) s;
tcb_cap_cases (snd oslot) = Some (getF, setF, restr)\<rbrakk> \<Longrightarrow>
\<not> cte_wp_at P slot (s\<lparr>kheap := kheap s(fst oslot \<mapsto> TCB (setF (\<lambda> x. cap) x))\<rparr>)"
apply(rule notI)
apply(erule cte_wp_atE)
apply(fastforce elim: notE intro: cte_wp_at_cteI split: if_splits)
apply(fastforce elim: notE intro: set_cap_neg_cte_wp_at_other_helper')
done
lemma set_cap_neg_cte_wp_at_other:
"oslot \<noteq> slot \<Longrightarrow> \<lbrace> \<lambda> s. \<not> (cte_wp_at P slot s)\<rbrace> set_cap cap oslot \<lbrace> \<lambda>rv s. \<not> (cte_wp_at P slot s) \<rbrace>"
apply(rule hoare_pre)
unfolding set_cap_def
apply(wp set_object_wp get_object_wp | wpc | simp add: split_def)+
apply(intro allI impI conjI)
apply(rule notI)
apply(erule cte_wp_atE)
apply (fastforce split: if_splits dest: prod_eqI elim: notE intro: cte_wp_at_cteI simp: obj_at_def)
apply(fastforce split: if_splits elim: notE intro: cte_wp_at_tcbI)
apply(auto dest: set_cap_neg_cte_wp_at_other_helper)
done
lemma set_cap_neg_cte_wp_at:
"\<lbrace>(\<lambda>s. \<not> cte_wp_at P slot s) and K (\<not> P capa)\<rbrace>
set_cap capa slota
\<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
apply(case_tac "slot = slota")
apply simp
apply(simp add: set_cap_def set_object_def)
apply(rule hoare_pre)
apply(wp get_object_wp | wpc)+
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
apply(rule hoare_pre)
apply(rule set_cap_neg_cte_wp_at_other, simp+)
done
lemma domain_sep_inv_cap_IRQControlCap:
"\<lbrakk>domain_sep_inv_cap irqs cap; \<not> irqs\<rbrakk> \<Longrightarrow> cap \<noteq> IRQControlCap"
apply(auto simp: domain_sep_inv_cap_def)
done
lemma domain_sep_inv_cap_IRQHandlerCap:
"\<lbrakk>domain_sep_inv_cap irqs cap; \<not> irqs\<rbrakk> \<Longrightarrow> cap \<noteq> IRQHandlerCap irq"
apply(auto simp: domain_sep_inv_cap_def)
done
lemma set_cap_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and K (domain_sep_inv_cap irqs cap)\<rbrace>
set_cap cap slot
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(rule hoare_gen_asm)
apply(rule hoare_pre)
apply(rule domain_sep_inv_wp)
apply(wp set_cap_neg_cte_wp_at | simp add: pred_neg_def | blast)+
done
lemma cte_wp_at_domain_sep_inv_cap:
"\<lbrakk>domain_sep_inv irqs st s; cte_wp_at (op = cap) slot s\<rbrakk> \<Longrightarrow> domain_sep_inv_cap irqs cap"
apply(case_tac slot)
apply(auto simp: domain_sep_inv_def domain_sep_inv_cap_def split: cap.splits)
done
lemma weak_derived_irq_handler:
"weak_derived (IRQHandlerCap irq) cap \<Longrightarrow> cap = (IRQHandlerCap irq)"
apply(auto simp: weak_derived_def copy_of_def same_object_as_def split: cap.splits if_splits arch_cap.splits)
done
(* FIXME: move to CSpace_AI *)
lemma weak_derived_DomainCap:
"weak_derived c' c \<Longrightarrow> (c' = cap.DomainCap) = (c = cap.DomainCap)"
apply (clarsimp simp: weak_derived_def)
apply (erule disjE)
apply (clarsimp simp: copy_of_def split: split_if_asm)
apply (auto simp: is_cap_simps same_object_as_def
split: cap.splits arch_cap.splits)[1]
apply simp
done
lemma cte_wp_at_weak_derived_domain_sep_inv_cap:
"\<lbrakk>domain_sep_inv irqs st s; cte_wp_at (weak_derived cap) slot s\<rbrakk> \<Longrightarrow> domain_sep_inv_cap irqs cap"
apply (cases slot)
apply (force simp: domain_sep_inv_def domain_sep_inv_cap_def
split: cap.splits
dest: cte_wp_at_eqD weak_derived_irq_handler weak_derived_DomainCap)
done
lemma is_derived_IRQHandlerCap:
"is_derived m src (IRQHandlerCap irq) cap \<Longrightarrow> (cap = IRQHandlerCap irq)"
apply(clarsimp simp: is_derived_def)
apply(case_tac cap, simp_all add: is_cap_simps cap_master_cap_def)
done
(* FIXME: move to CSpace_AI *)
lemma DomainCap_is_derived:
"is_derived m src cap.DomainCap cap \<Longrightarrow> cap = DomainCap"
by (auto simp: is_derived_def is_reply_cap_def is_pg_cap_def is_master_reply_cap_def cap_master_cap_def split: cap.splits)
lemma cte_wp_at_is_derived_domain_sep_inv_cap:
"\<lbrakk>domain_sep_inv irqs st s; cte_wp_at (is_derived (cdt s) slot cap) slot s\<rbrakk> \<Longrightarrow> domain_sep_inv_cap irqs cap"
apply (cases slot)
apply (fastforce simp: domain_sep_inv_def domain_sep_inv_cap_def
split: cap.splits
dest: cte_wp_at_eqD DomainCap_is_derived is_derived_IRQHandlerCap)
done
lemma domain_sep_inv_exst_update[simp]:
"domain_sep_inv irqs st (trans_state f s) = domain_sep_inv irqs st s"
apply(simp add: domain_sep_inv_def)
done
lemma domain_sep_inv_is_original_cap_update[simp]:
"domain_sep_inv irqs st (s\<lparr>is_original_cap := X\<rparr>) = domain_sep_inv irqs st s"
apply(simp add: domain_sep_inv_def)
done
lemma domain_sep_inv_cdt_update[simp]:
"domain_sep_inv irqs st (s\<lparr>cdt := X\<rparr>) = domain_sep_inv irqs st s"
apply(simp add: domain_sep_inv_def)
done
crunch domain_sep_inv[wp]: update_cdt "domain_sep_inv irqs st"
lemma set_untyped_cap_as_full_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace> set_untyped_cap_as_full a b c \<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(clarsimp simp: set_untyped_cap_as_full_def)
apply(rule hoare_pre)
apply(rule set_cap_domain_sep_inv)
apply(case_tac a, simp_all add: domain_sep_inv_cap_def)
done
lemma cap_insert_domain_sep_inv:
"\<lbrace> domain_sep_inv irqs st and (\<lambda>s. cte_wp_at (is_derived (cdt s) slot cap) slot s) \<rbrace>
cap_insert cap slot dest_slot
\<lbrace> \<lambda>_. domain_sep_inv irqs st \<rbrace>"
apply(simp add: cap_insert_def)
apply(wp set_cap_domain_sep_inv get_cap_wp set_original_wp dxo_wp_weak | simp split del: split_if)+
apply(blast dest: cte_wp_at_is_derived_domain_sep_inv_cap)
done
lemma domain_sep_inv_cap_NullCap[simp]:
"domain_sep_inv_cap irqs NullCap"
apply(simp add: domain_sep_inv_cap_def)
done
lemma cap_move_domain_sep_inv:
"\<lbrace> domain_sep_inv irqs st and (\<lambda>s. cte_wp_at (weak_derived cap) slot s) \<rbrace>
cap_move cap slot dest_slot
\<lbrace> \<lambda>_. domain_sep_inv irqs st \<rbrace>"
apply(simp add: cap_move_def)
apply(wp set_cap_domain_sep_inv get_cap_wp set_original_wp dxo_wp_weak | simp split del: split_if | blast dest: cte_wp_at_weak_derived_domain_sep_inv_cap)+
done
lemma domain_sep_inv_machine_state_update[simp]:
"domain_sep_inv irqs st (s\<lparr>machine_state := X\<rparr>) = domain_sep_inv irqs st s"
apply(simp add: domain_sep_inv_def)
done
lemma set_irq_state_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and (\<lambda>s. stt = interrupt_states s irq)\<rbrace>
set_irq_state stt irq
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(simp add: set_irq_state_def)
apply(wp | simp add: do_machine_op_def | wpc)+
done
lemma deleted_irq_handler_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and K irqs\<rbrace> deleted_irq_handler a \<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(rule hoare_gen_asm)
apply(simp add: deleted_irq_handler_def)
apply(simp add: set_irq_state_def)
apply wp
apply(rule domain_sep_inv_triv, wp)
apply(simp add: domain_sep_inv_def)
done
lemma empty_slot_domain_sep_inv:
"\<lbrace>\<lambda>s. domain_sep_inv irqs st s \<and> (\<not> irqs \<longrightarrow> b = None)\<rbrace>
empty_slot a b
\<lbrace>\<lambda>_ s. domain_sep_inv irqs st s\<rbrace>"
apply(simp add: empty_slot_def)
apply (wp | wpc)+
apply(wp get_cap_wp set_cap_domain_sep_inv set_original_wp dxo_wp_weak static_imp_wp deleted_irq_handler_domain_sep_inv | simp | blast)+
done
lemma set_endpoint_neg_cte_wp_at[wp]:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace> set_endpoint a b \<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
apply(simp add: set_endpoint_def)
apply(wp set_object_wp get_object_wp | simp)+
apply(case_tac "a = fst slot")
apply(clarsimp split: kernel_object.splits)
apply(fastforce elim: cte_wp_atE simp: obj_at_def)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
done
lemma set_async_ep_neg_cte_wp_at[wp]:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace> set_async_ep a b \<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
apply(simp add: set_async_ep_def)
apply(wp set_object_wp get_object_wp | simp)+
apply(case_tac "a = fst slot")
apply(clarsimp split: kernel_object.splits)
apply(fastforce elim: cte_wp_atE simp: obj_at_def)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
done
crunch domain_sep_inv[wp]: set_endpoint "domain_sep_inv irqs st"
(wp: domain_sep_inv_triv)
crunch domain_sep_inv[wp]: set_async_ep "domain_sep_inv irqs st"
(wp: domain_sep_inv_triv)
lemma set_thread_state_neg_cte_wp_at[wp]:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace> set_thread_state a b \<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
apply(simp add: set_thread_state_def)
apply(wp set_object_wp get_object_wp dxo_wp_weak| simp)+
apply(case_tac "a = fst slot")
apply(clarsimp split: kernel_object.splits)
apply(erule notE)
apply(erule cte_wp_atE)
apply(fastforce simp: obj_at_def)
apply(drule get_tcb_SomeD)
apply(rule cte_wp_at_tcbI)
apply(simp)
apply assumption
apply (fastforce simp: tcb_cap_cases_def split: if_splits)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
done
crunch domain_sep_inv[wp]: set_thread_state "domain_sep_inv irqs st"
(wp: domain_sep_inv_triv)
lemma thread_set_tcb_fault_update_neg_cte_wp_at[wp]:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace>
thread_set (tcb_fault_update blah) param_a
\<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
apply(simp add: thread_set_def)
apply(wp set_object_wp get_object_wp | simp)+
apply(case_tac "param_a = fst slot")
apply(clarsimp split: kernel_object.splits)
apply(erule notE)
apply(erule cte_wp_atE)
apply(fastforce simp: obj_at_def)
apply(drule get_tcb_SomeD)
apply(rule cte_wp_at_tcbI)
apply(simp)
apply assumption
apply (fastforce simp: tcb_cap_cases_def split: if_splits)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
done
lemma thread_set_tcb_fault_update_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace>
thread_set (tcb_fault_update blah) param_a
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(wp domain_sep_inv_triv)
done
crunch domain_sep_inv[wp]: cap_delete_one "domain_sep_inv irqs st"
(wp: mapM_x_wp' hoare_unless_wp dxo_wp_weak ignore: tcb_sched_action reschedule_required)
lemma reply_ipc_cancel_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace>
reply_ipc_cancel t
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(simp add: reply_ipc_cancel_def)
apply (wp select_wp)
apply(rule hoare_strengthen_post[OF thread_set_tcb_fault_update_domain_sep_inv])
apply auto
done
lemma domain_sep_inv_arch_state_update[simp]:
"domain_sep_inv irqs st (s\<lparr>arch_state := blah\<rparr>) = domain_sep_inv irqs st s"
apply(simp add: domain_sep_inv_def)
done
lemma set_pt_neg_cte_wp_at[wp]:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace>
set_pt ptr pt
\<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
unfolding set_pt_def
apply(wp set_object_wp get_object_wp | simp)+
apply(case_tac "ptr = fst slot")
apply(clarsimp split: kernel_object.splits)
apply(fastforce elim: cte_wp_atE simp: obj_at_def)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
done
crunch domain_sep_inv[wp]: set_pt "domain_sep_inv irqs st"
(wp: domain_sep_inv_triv)
lemma set_pd_neg_cte_wp_at[wp]:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace>
set_pd ptr pt
\<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
unfolding set_pd_def
apply(wp set_object_wp get_object_wp | simp)+
apply(case_tac "ptr = fst slot")
apply(clarsimp split: kernel_object.splits)
apply(fastforce elim: cte_wp_atE simp: obj_at_def)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
done
crunch domain_sep_inv[wp]: set_pd "domain_sep_inv irqs st"
(wp: domain_sep_inv_triv)
lemma set_asid_pool_neg_cte_wp_at[wp]:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace>
set_asid_pool ptr pt
\<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
unfolding set_asid_pool_def
apply(wp set_object_wp get_object_wp | simp)+
apply(case_tac "ptr = fst slot")
apply(clarsimp split: kernel_object.splits)
apply(fastforce elim: cte_wp_atE simp: obj_at_def)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
done
crunch domain_sep_inv[wp]: set_asid_pool "domain_sep_inv irqs st"
(wp: domain_sep_inv_triv)
crunch domain_sep_inv[wp]: finalise_cap "domain_sep_inv irqs st"
(wp: crunch_wps dxo_wp_weak simp: crunch_simps ignore: set_object tcb_sched_action)
lemma finalise_cap_domain_sep_inv_cap:
"\<lbrace>\<lambda>s. domain_sep_inv_cap irqs cap\<rbrace> finalise_cap cap b \<lbrace>\<lambda>rv s. domain_sep_inv_cap irqs (fst rv)\<rbrace>"
apply(case_tac cap)
apply(wp | simp add: o_def split del: split_if split: cap.splits arch_cap.splits | fastforce split: if_splits simp: domain_sep_inv_cap_def)+
apply(rule hoare_pre, wp, fastforce)
apply(rule hoare_pre, simp, wp, fastforce simp: domain_sep_inv_cap_def)
apply(simp add: arch_finalise_cap_def)
apply(rule hoare_pre)
apply(wpc | wp | simp)+
done
lemma get_cap_domain_sep_inv_cap:
"\<lbrace>domain_sep_inv irqs st\<rbrace> get_cap cap \<lbrace>\<lambda>rv s. domain_sep_inv_cap irqs rv\<rbrace>"
apply(wp get_cap_wp)
apply(blast dest: cte_wp_at_domain_sep_inv_cap)
done
crunch domain_sep_inv[wp]: cap_swap_for_delete "domain_sep_inv irqs st"
(wp: get_cap_domain_sep_inv_cap dxo_wp_weak simp: crunch_simps ignore: cap_swap_ext)
lemma finalise_cap_returns_None:
"\<lbrace>\<lambda>s. domain_sep_inv_cap irqs cap\<rbrace>
finalise_cap cap b
\<lbrace>\<lambda>rv s. \<not> irqs \<longrightarrow> snd rv = None\<rbrace>"
apply(case_tac cap)
apply(simp add: o_def split del: split_if | wp | fastforce simp: domain_sep_inv_cap_def | rule hoare_pre)+
done
lemma rec_del_domain_sep_inv':
notes drop_spec_valid[wp_split del] drop_spec_validE[wp_split del]
rec_del.simps[simp del]
shows
"s \<turnstile> \<lbrace> domain_sep_inv irqs st\<rbrace>
(rec_del call)
\<lbrace>\<lambda> a s. (case call of (FinaliseSlotCall x y) \<Rightarrow> (y \<or> fst a) \<longrightarrow> \<not> irqs \<longrightarrow> snd a = None | _ \<Rightarrow> True) \<and> domain_sep_inv irqs st s\<rbrace>,\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
proof (induct s arbitrary: rule: rec_del.induct, simp_all only: rec_del_fails hoare_fail_any)
case (1 slot exposed s) show ?case
apply(simp add: split_def rec_del.simps)
apply(wp empty_slot_domain_sep_inv drop_spec_validE[OF returnOk_wp] drop_spec_validE[OF liftE_wp] | simp)+
apply(rule spec_strengthen_postE[OF "1.hyps"])
apply fastforce
done
next
case (2 slot exposed s) show ?case
apply(simp add: rec_del.simps split del: split_if)
apply(rule hoare_pre_spec_validE)
apply(wp drop_spec_validE[OF returnOk_wp] drop_spec_validE[OF liftE_wp] set_cap_domain_sep_inv
|simp add: split_def split del: split_if)+
apply(rule spec_strengthen_postE)
apply(rule "2.hyps", fastforce+)
apply(rule drop_spec_validE, (wp preemption_point_inv| simp)+)[1]
apply simp
apply(rule spec_strengthen_postE)
apply(rule "2.hyps", fastforce+)
apply(wp finalise_cap_domain_sep_inv_cap get_cap_wp
finalise_cap_returns_None
drop_spec_validE[OF liftE_wp] set_cap_domain_sep_inv
|simp add: without_preemption_def split del: split_if
|wp_once hoare_drop_imps)+
apply(blast dest: cte_wp_at_domain_sep_inv_cap)
done
next
case (3 ptr bits n slot s) show ?case
apply(simp add: rec_del.simps)
apply (wp drop_spec_validE[OF returnOk_wp] drop_spec_validE[OF liftE_wp])
apply(rule hoare_pre_spec_validE)
apply (wp drop_spec_validE[OF assertE_wp])
apply(fastforce)
done
next
case (4 ptr bits n slot s) show ?case
apply(simp add: rec_del.simps)
apply (wp drop_spec_validE[OF returnOk_wp] drop_spec_validE[OF liftE_wp] set_cap_domain_sep_inv
drop_spec_validE[OF assertE_wp] get_cap_wp | simp add: without_preemption_def)+
apply (rule spec_strengthen_postE[OF "4.hyps"])
apply(simp add: returnOk_def return_def)
apply(clarsimp simp: domain_sep_inv_cap_def)
done
qed
lemma rec_del_domain_sep_inv:
"\<lbrace> domain_sep_inv irqs st\<rbrace>
(rec_del call)
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>,\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(rule use_spec)
apply(rule spec_strengthen_postE[OF rec_del_domain_sep_inv'])
by fastforce
crunch domain_sep_inv[wp]: cap_delete "domain_sep_inv irqs st"
lemma preemption_point_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace> preemption_point \<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
by (wp preemption_point_inv | simp)+
lemma cap_revoke_domain_sep_inv':
notes drop_spec_valid[wp_split del] drop_spec_validE[wp_split del]
shows
"s \<turnstile> \<lbrace> domain_sep_inv irqs st\<rbrace>
cap_revoke slot
\<lbrace> \<lambda>_. domain_sep_inv irqs st\<rbrace>, \<lbrace> \<lambda>_. domain_sep_inv irqs st\<rbrace>"
proof(induct rule: cap_revoke.induct[where ?a1.0=s])
case (1 slot s)
show ?case
apply(subst cap_revoke.simps)
apply(rule hoare_pre_spec_validE)
apply (wp "1.hyps", assumption+)
apply(wp spec_valid_conj_liftE2 | simp)+
apply(wp drop_spec_validE[OF valid_validE[OF preemption_point_domain_sep_inv]]
drop_spec_validE[OF valid_validE[OF cap_delete_domain_sep_inv]]
drop_spec_validE[OF assertE_wp] drop_spec_validE[OF returnOk_wp]
drop_spec_validE[OF liftE_wp] select_wp
| simp | wp_once hoare_drop_imps)+
done
qed
lemmas cap_revoke_domain_sep_inv[wp] = use_spec(2)[OF cap_revoke_domain_sep_inv']
(* FIXME: clagged from FinalCaps *)
lemma cap_move_cte_wp_at_other:
"\<lbrace> cte_wp_at P slot and K (slot \<noteq> dest_slot \<and> slot \<noteq> src_slot) \<rbrace>
cap_move cap src_slot dest_slot
\<lbrace> \<lambda>_. cte_wp_at P slot \<rbrace>"
unfolding cap_move_def
apply (rule hoare_pre)
apply (wp set_cdt_cte_wp_at set_cap_cte_wp_at' dxo_wp_weak static_imp_wp | simp)+
done
(* FIXME: clagged from FinalCaps *)
lemma cte_wp_at_weak_derived_ReplyCap:
"cte_wp_at (op = (ReplyCap x False)) slot s
\<Longrightarrow> cte_wp_at (weak_derived (ReplyCap x False)) slot s"
apply(erule cte_wp_atE)
apply(rule cte_wp_at_cteI)
apply assumption
apply assumption
apply assumption
apply simp
apply(rule cte_wp_at_tcbI)
apply auto
done
lemma thread_set_tcb_registers_caps_merge_default_tcb_neg_cte_wp_at[wp]:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace>
thread_set (tcb_registers_caps_merge default_tcb) word
\<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
unfolding thread_set_def
apply(wp set_object_wp | simp)+
apply(case_tac "word = fst slot")
apply(clarsimp split: kernel_object.splits)
apply(erule notE)
apply(erule cte_wp_atE)
apply(fastforce simp: obj_at_def)
apply(drule get_tcb_SomeD)
apply(rule cte_wp_at_tcbI)
apply(simp)
apply assumption
apply (clarsimp simp: tcb_cap_cases_def tcb_registers_caps_merge_def split: if_splits)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
done
lemma thread_set_tcb_registers_caps_merge_default_tcb_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace>
thread_set (tcb_registers_caps_merge default_tcb) word
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(wp domain_sep_inv_triv)
done
lemma arch_recycle_cap_domain_sep_inv_cap[wp]:
"\<lbrace>\<lambda>s. domain_sep_inv_cap irqs (ArchObjectCap arch_cap)\<rbrace>
arch_recycle_cap is_final arch_cap
\<lbrace>(\<lambda>rv s. domain_sep_inv_cap irqs rv) \<circ> ArchObjectCap\<rbrace>"
apply(simp add: arch_recycle_cap_def split del: split_if)
apply(rule hoare_pre)
apply(wp | wpc | simp split del: split_if)+
apply(auto simp: domain_sep_inv_cap_def)
done
lemma recycle_cap_domain_sep_inv_cap[wp]:
shows
"\<lbrace>\<lambda>s. domain_sep_inv_cap irqs cap\<rbrace>
recycle_cap is_final cap
\<lbrace>\<lambda>rv s. domain_sep_inv_cap irqs rv\<rbrace>"
apply(simp add: recycle_cap_def)
apply(rule hoare_pre)
apply(wp | wpc | simp | wp_once hoare_drop_imps)+
apply(auto simp: domain_sep_inv_cap_def)
done
lemma ep_cancel_badged_sends_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace>
ep_cancel_badged_sends epptr badge
\<lbrace>\<lambda>rv. domain_sep_inv irqs st\<rbrace>"
apply(simp add: ep_cancel_badged_sends_def)
apply(rule hoare_pre)
apply(wp dxo_wp_weak mapM_wp | wpc | simp add: filterM_mapM | rule subset_refl | wp_once hoare_drop_imps)+
done
crunch domain_sep_inv[wp]: arch_recycle_cap "domain_sep_inv irqs st"
(wp: crunch_wps)
lemma recycle_cap_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace>
recycle_cap is_final cap
\<lbrace>\<lambda>rv. domain_sep_inv irqs st\<rbrace>"
apply(simp add: recycle_cap_def)
apply(rule hoare_pre)
apply(wp dxo_wp_weak | wpc | simp | wp_once hoare_drop_imps)+
done
crunch domain_sep_inv[wp]: finalise_slot "domain_sep_inv irqs st"
lemma cap_recycle_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace> cap_recycle blah \<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(simp add: cap_recycle_def)
apply(wp hoare_unless_wp set_cap_domain_sep_inv
get_cap_domain_sep_inv_cap[where st=st]
| simp add: crunch_simps | wp_once hoare_drop_imps)+
done
lemma invoke_cnode_domain_sep_inv:
"\<lbrace> domain_sep_inv irqs st and valid_cnode_inv ci\<rbrace>
invoke_cnode ci
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
unfolding invoke_cnode_def
apply(case_tac ci)
apply(wp cap_insert_domain_sep_inv cap_move_domain_sep_inv | simp split del: split_if)+
apply(rule hoare_pre)
apply(wp cap_move_domain_sep_inv cap_move_cte_wp_at_other get_cap_wp | simp | blast dest: cte_wp_at_weak_derived_domain_sep_inv_cap | wpc)+
apply(fastforce dest: cte_wp_at_weak_derived_ReplyCap)
apply(wp | simp)+
done
lemma create_cap_domain_sep_inv[wp]:
"\<lbrace> domain_sep_inv irqs st\<rbrace>
create_cap tp sz p slot
\<lbrace> \<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(simp add: create_cap_def)
apply(rule hoare_pre)
apply(wp set_cap_domain_sep_inv dxo_wp_weak | wpc | simp)+
apply clarsimp
apply(case_tac tp, simp_all add: domain_sep_inv_cap_def)
done
lemma detype_interrupts_states[simp]:
"interrupt_states (detype S s) = interrupt_states s"
apply(simp add: detype_def)
done
lemma detype_domain_sep_inv[simp]:
"domain_sep_inv irqs st s \<longrightarrow> domain_sep_inv irqs st (detype S s)"
apply(fastforce simp: domain_sep_inv_def)
done
lemma domain_sep_inv_detype_lift:
"\<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv. domain_sep_inv irqs st\<rbrace> \<Longrightarrow>
\<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv s. domain_sep_inv irqs st (detype S s)\<rbrace>"
apply(strengthen detype_domain_sep_inv, assumption)
done
lemma retype_region_neg_cte_wp_at_not_domain_sep_inv_cap:
"\<lbrace>\<lambda>s. \<not> cte_wp_at (not domain_sep_inv_cap irqs) slot s \<rbrace>
retype_region base n sz ty
\<lbrace>\<lambda>rv s. \<not> cte_wp_at (not domain_sep_inv_cap irqs) slot s\<rbrace>"
apply(rule hoare_pre)
apply(simp only: retype_region_def retype_addrs_def
foldr_upd_app_if fun_app_def K_bind_def)
apply(wp dxo_wp_weak |simp)+
apply clarsimp
apply(case_tac "fst slot \<in> (\<lambda>p. ptr_add base (p * 2 ^ obj_bits_api ty sz)) `
{0..<n}")
apply clarsimp
apply(erule cte_wp_atE)
apply(clarsimp simp: default_object_def empty_cnode_def split: apiobject_type.splits if_splits simp: pred_neg_def)
apply(clarsimp simp: default_object_def default_tcb_def tcb_cap_cases_def split: apiobject_type.splits if_splits simp: pred_neg_def)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_tcbI cte_wp_at_cteI)
done
lemma retype_region_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace>
retype_region base n sz tp
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(rule domain_sep_inv_wp[where P="\<top>" and R="\<top>", simplified])
apply(rule retype_region_neg_cte_wp_at_not_domain_sep_inv_cap)
apply wp
done
lemma domain_sep_inv_cap_UntypedCap[simp]:
"domain_sep_inv_cap irqs (UntypedCap base sz n)"
apply(simp add: domain_sep_inv_cap_def)
done
crunch domain_sep_inv[wp]: invoke_untyped "domain_sep_inv irqs st"
(ignore: freeMemory retype_region wp: crunch_wps domain_sep_inv_detype_lift
get_cap_wp
simp: crunch_simps mapM_x_def_bak)
lemma perform_page_invocation_domain_sep_inv_get_cap_helper:
"\<lbrace>\<top>\<rbrace>
get_cap blah
\<lbrace>\<lambda>rv s.
domain_sep_inv_cap irqs
(ArchObjectCap (F rv))\<rbrace>"
apply(simp add: domain_sep_inv_cap_def)
apply(rule wp_post_taut)
done
lemma set_object_tcb_context_update_neg_cte_wp_at:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s \<and> obj_at (op = (TCB tcb)) ptr s\<rbrace>
set_object ptr (TCB (tcb\<lparr>tcb_context := X\<rparr>))
\<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
apply(wp set_object_wp)
apply clarsimp
apply(case_tac "ptr = fst slot")
apply(erule cte_wp_atE)
apply(fastforce simp: obj_at_def)
apply(erule notE)
apply(clarsimp simp: obj_at_def)
apply(rule cte_wp_at_tcbI)
apply(simp)
apply(fastforce)
apply(fastforce simp: tcb_cap_cases_def split: if_splits)
apply(fastforce elim: cte_wp_atE intro: cte_wp_at_cteI cte_wp_at_tcbI)
done
lemma as_user_neg_cte_wp_at[wp]:
"\<lbrace>\<lambda>s. \<not> cte_wp_at P slot s\<rbrace>
as_user t f
\<lbrace>\<lambda>_ s. \<not> cte_wp_at P slot s\<rbrace>"
unfolding as_user_def
apply(wp set_object_tcb_context_update_neg_cte_wp_at | simp add: split_def)+
apply(fastforce simp: obj_at_def)
done
crunch domain_sep_inv[wp]: as_user "domain_sep_inv irqs st"
(wp: domain_sep_inv_triv)
lemma set_object_tcb_context_update_domain_sep_inv:
"\<lbrace>\<lambda>s. domain_sep_inv irqs st s \<and> obj_at (op = (TCB tcb)) ptr s\<rbrace>
set_object ptr (TCB (tcb\<lparr>tcb_context := X\<rparr>))
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(rule hoare_pre)
apply(rule domain_sep_inv_wp)
apply(rule hoare_pre)
apply(rule set_object_tcb_context_update_neg_cte_wp_at)
apply(fastforce)
apply(wp | simp | elim conjE | assumption)+
apply blast
done
crunch domain_sep_inv[wp]: set_mrs "domain_sep_inv irqs st"
(ignore: set_object
wp: crunch_wps set_object_tcb_context_update_domain_sep_inv
simp: crunch_simps)
crunch domain_sep_inv[wp]: send_async_ipc "domain_sep_inv irqs st" (wp: dxo_wp_weak ignore: switch_if_required_to)
lemma perform_page_invocation_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and valid_page_inv pi\<rbrace>
perform_page_invocation pi
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(rule hoare_pre)
apply(wp mapM_wp[OF _ subset_refl] set_cap_domain_sep_inv
mapM_x_wp[OF _ subset_refl]
perform_page_invocation_domain_sep_inv_get_cap_helper static_imp_wp
| simp add: perform_page_invocation_def o_def | wpc)+
apply(clarsimp simp: valid_page_inv_def)
apply(case_tac xa, simp_all add: domain_sep_inv_cap_def is_pg_cap_def)
done
lemma perform_page_table_invocation_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and valid_pti pi\<rbrace>
perform_page_table_invocation pi
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(rule hoare_pre)
apply(simp add: perform_page_table_invocation_def)
apply(wp crunch_wps perform_page_invocation_domain_sep_inv_get_cap_helper
set_cap_domain_sep_inv
| wpc
| simp add: o_def)+
apply(clarsimp simp: valid_pti_def)
apply(case_tac x, simp_all add: domain_sep_inv_cap_def is_pt_cap_def)
done
lemma perform_page_directory_invocation_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st\<rbrace>
perform_page_directory_invocation pti
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply (simp add: perform_page_directory_invocation_def)
apply (cases pti)
apply (simp | wp)+
done
lemma cap_insert_domain_sep_inv':
"\<lbrace> domain_sep_inv irqs st and K (domain_sep_inv_cap irqs cap) \<rbrace>
cap_insert cap slot dest_slot
\<lbrace> \<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(simp add: cap_insert_def)
apply(wp set_cap_domain_sep_inv get_cap_wp dxo_wp_weak | simp split del: split_if)+
done
lemma domain_sep_inv_cap_max_free_index_update[simp]:
"domain_sep_inv_cap irqs (max_free_index_update cap) =
domain_sep_inv_cap irqs cap"
apply(simp add: max_free_index_def free_index_update_def split: cap.splits)
done
lemma domain_sep_inv_cap_ArchObjectCap[simp]:
"domain_sep_inv_cap irqs (ArchObjectCap arch_cap)"
by(simp add: domain_sep_inv_cap_def)
lemma perform_asid_control_invocation_domain_sep_inv:
notes blah[simp del] = atLeastAtMost_iff atLeastatMost_subset_iff atLeastLessThan_iff
shows
"\<lbrace>domain_sep_inv irqs st\<rbrace>
perform_asid_control_invocation blah
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
unfolding perform_asid_control_invocation_def
apply(rule hoare_pre)
apply (wp modify_wp cap_insert_domain_sep_inv' set_cap_domain_sep_inv
get_cap_domain_sep_inv_cap[where st=st] static_imp_wp
| wpc | simp )+
done
lemma perform_asid_pool_invocation_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st\<rbrace>
perform_asid_pool_invocation blah
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(simp add: perform_asid_pool_invocation_def)
apply(rule hoare_pre)
apply(wp set_cap_domain_sep_inv get_cap_wp | wpc | simp)+
done
lemma arch_perform_invocation_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and valid_arch_inv ai\<rbrace>
arch_perform_invocation ai
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
unfolding arch_perform_invocation_def
apply(rule hoare_pre)
apply(wp perform_page_table_invocation_domain_sep_inv
perform_page_directory_invocation_domain_sep_inv
perform_page_invocation_domain_sep_inv
perform_asid_control_invocation_domain_sep_inv
perform_asid_pool_invocation_domain_sep_inv
| wpc)+
apply(clarsimp simp: valid_arch_inv_def split: arch_invocation.splits)
done
(* when blah is AckIRQ the preconditions here contradict each other, which
is why this lemma is true *)
lemma invoke_irq_handler_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and irq_handler_inv_valid blah\<rbrace>
invoke_irq_handler blah
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(case_tac blah)
apply(wp cap_insert_domain_sep_inv' | simp)+
apply(rename_tac irq cap cslot_ptr s)
apply(case_tac cap, simp_all add: domain_sep_inv_cap_def)[1]
apply(fastforce simp: domain_sep_inv_def)
apply(wp | simp )+
done
(* similarly, the preconditions here tend to contradict one another *)
lemma invoke_control_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and irq_control_inv_valid blah\<rbrace>
invoke_irq_control blah
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(case_tac blah)
apply(wp cap_insert_domain_sep_inv' | simp )+
apply(case_tac irqs)
apply(simp add: set_irq_state_def, wp, simp)
apply(fastforce simp: domain_sep_inv_def domain_sep_inv_cap_def)
apply(fastforce simp: valid_def domain_sep_inv_def)
apply(wp | simp )+
done
crunch domain_sep_inv[wp]: receive_async_ipc "domain_sep_inv irqs st"
lemma domain_sep_inv_cap_ReplyCap[simp]:
"domain_sep_inv_cap irqs (ReplyCap param_a param_b)"
by(simp add: domain_sep_inv_cap_def)
lemma setup_caller_cap_domain_sep_inv[wp]:
"\<lbrace>domain_sep_inv irqs st\<rbrace>
setup_caller_cap a b
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply(simp add: setup_caller_cap_def)
apply(wp cap_insert_domain_sep_inv' | simp)+
done
crunch domain_sep_inv[wp]: set_extra_badge "domain_sep_inv irqs st"
lemma derive_cap_domain_sep_inv_cap:
"\<lbrace>\<lambda>s. domain_sep_inv_cap irqs cap\<rbrace>
derive_cap slot cap
\<lbrace>\<lambda>rv s. domain_sep_inv_cap irqs rv\<rbrace>,-"
apply(simp add: derive_cap_def)
apply(rule hoare_pre)
apply(wp | wpc | simp add: arch_derive_cap_def)+
apply auto
done
lemma transfer_caps_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and valid_objs and valid_mdb
and (\<lambda> s. (\<forall>x\<in>set caps.
s \<turnstile> fst x) \<and>
(\<forall>x\<in>set caps.
cte_wp_at
(\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow>
cp = fst x)
(snd x) s \<and>
real_cte_at (snd x) s))\<rbrace>
transfer_caps mi caps endpoint receiver receive_buffer diminish
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
apply (simp add: transfer_caps_def)
apply (wpc | wp)+
apply(rule transfer_caps_loop_pres_dest)
apply(wp cap_insert_domain_sep_inv | simp )+
apply(wp hoare_vcg_all_lift hoare_vcg_imp_lift | simp)+
apply(fastforce elim: cte_wp_at_weakenE)
apply simp
done
crunch domain_sep_inv[wp]: copy_mrs "domain_sep_inv irqs st"
(wp: crunch_wps)
lemma do_normal_transfer_domain_sep_inv:
"\<lbrace>domain_sep_inv irqs st and valid_objs and valid_mdb\<rbrace>
do_normal_transfer sender send_buffer ep badge grant receiver recv_buffer diminish
\<lbrace>\<lambda>_. domain_sep_inv irqs st\<rbrace>"
unfolding do_normal_transfer_def