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ObjectInitialised_SI.thy
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ObjectInitialised_SI.thy
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(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
*
* SPDX-License-Identifier: GPL-2.0-only
*)
(*
* Object predicates.
*
* This file contains the definitions of the state of the objects created
* by the initialiser - when they are empty, initialised and the states in between.
* It also contains the decompositions of these predicates.
*)
theory ObjectInitialised_SI
imports WellFormed_SI
begin
(************************************************************
* Definitions about the state of objects,
* when an object is newly created, completely set up, etc.
* Non-separation style definitions are labeled "classical".
************************************************************)
(* Translates the object_ids in a cap for a given transformation.
* If the object_id is not in the mapping, it is transformed to undefined.
*)
definition
cap_transform :: "(cdl_object_id \<rightharpoonup> cdl_object_id) \<Rightarrow> cdl_cap \<Rightarrow> cdl_cap"
where
"cap_transform t cap \<equiv>
let
t' = \<lambda> obj. case t obj of None \<Rightarrow> undefined | Some obj' \<Rightarrow> obj'
in
if is_untyped_cap cap
then update_cap_objects (t' ` (cap_objects cap)) cap
else update_cap_object (t' (cap_object cap)) cap"
(* Translates the object_ids in an object for a given transformation.
* This does *not* translate the cdl_tcb_fault_endpoint.
* The cdl_tcb_fault_endpoint specifies a cap pointer,
* which should be the same in the spec and the kernel
* (as both are looked up in the same way).
*)
definition
spec2s :: "(cdl_object_id \<rightharpoonup> cdl_object_id) \<Rightarrow> cdl_object \<Rightarrow> cdl_object"
where
"spec2s t object \<equiv> update_slots (cap_transform t \<circ>\<^sub>M object_slots object) object"
(* This is used to define object_empty, object_initialised (and others).
* Since we pass in the spec object cap transformation, we can specify
* objects with no caps, all their caps (or anything else).
*)
definition
object_initialised_general :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow>
(cdl_object \<Rightarrow> cdl_object) \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object \<Rightarrow> sep_pred) \<Rightarrow>
cdl_object_id \<Rightarrow> sep_pred"
where
"object_initialised_general spec t obj_trans arrow spec_object_id \<equiv> \<lambda>s.
let spec_objects = cdl_objects spec
in
\<exists>kernel_object_id. \<exists>spec_object.
t spec_object_id = Some kernel_object_id \<and>
(arrow kernel_object_id (obj_trans spec_object)) s \<and>
spec_objects spec_object_id = Some spec_object"
(* The object is set up (as per the spec). *)
definition
object_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"object_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t) sep_map_o spec_object_id"
(* The object is created and in it's default state. *)
definition
object_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"object_empty spec t spec_object_id \<equiv>
object_initialised_general spec t object_default_state sep_map_o spec_object_id"
definition
objects_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id set \<Rightarrow> sep_pred"
where
"objects_initialised spec t obj_ids \<equiv> \<And>* obj_id \<in> obj_ids. object_initialised spec t obj_id"
definition
objects_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id set \<Rightarrow> sep_pred"
where
"objects_empty spec t obj_ids \<equiv> \<And>* obj_id \<in> obj_ids. object_empty spec t obj_id"
(* The object's fields are set up (as per the spec). *)
definition
object_fields_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"object_fields_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t) sep_map_f spec_object_id"
(* The object's slots are set up (as per the spec). *)
definition
object_slots_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"object_slots_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t) sep_map_S spec_object_id"
definition
object_empty_slots_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"object_empty_slots_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t) sep_map_E spec_object_id"
(* A particular slot of an object is set up (as per the spec). *)
definition
object_slot_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> cdl_cnode_index \<Rightarrow> sep_pred"
where
"object_slot_initialised spec t spec_object_id slot \<equiv>
object_initialised_general spec t (spec2s t) (\<lambda>p. sep_map_s (p, slot)) spec_object_id"
(* The object's fields are in their default state. *)
definition
object_fields_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"object_fields_empty spec t spec_object_id \<equiv>
object_initialised_general spec t object_default_state sep_map_f spec_object_id"
(* The object's slots are in their default state. *)
definition
object_slots_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"object_slots_empty spec t spec_object_id \<equiv>
object_initialised_general spec t object_default_state sep_map_S spec_object_id"
(* A particular slot of an object is set up (as per the spec). *)
definition
object_slot_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> cdl_cnode_index \<Rightarrow> sep_pred"
where
"object_slot_empty spec t spec_object_id slot \<equiv>
object_initialised_general spec t object_default_state (\<lambda>p. sep_map_s (p, slot)) spec_object_id"
definition
object_empty_slots_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"object_empty_slots_empty spec t spec_object_id \<equiv>
object_initialised_general spec t object_default_state sep_map_E spec_object_id"
definition slots_in_object_empty ::
"(cdl_cap \<Rightarrow> bool) \<Rightarrow> cdl_object_id \<Rightarrow> cdl_state \<Rightarrow>
(cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> sep_pred" where
"slots_in_object_empty P obj_id spec t \<equiv>
sep_map_set_conj (object_empty spec t)
{obj. \<exists>slot. slot \<in> dom (slots_of obj_id spec)
\<and> cap_at P (obj_id, slot) spec
\<and> obj = cap_ref_object (obj_id, slot) spec}"
definition slots_in_object_init ::
"(cdl_cap \<Rightarrow> bool) \<Rightarrow> cdl_object_id \<Rightarrow> cdl_state \<Rightarrow>
(cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> sep_pred" where
"slots_in_object_init P obj_id spec t \<equiv>
sep_map_set_conj (object_initialised spec t)
{obj. \<exists>slot. slot \<in> dom (slots_of obj_id spec)
\<and> cap_at P (obj_id, slot) spec
\<and> obj = cap_ref_object (obj_id, slot) spec}"
(**********************************************
* Predicates about CNodes being initialised. *
**********************************************)
(* A cnode that has the original caps in it set to NullCap *)
definition
cnode_half :: "cdl_state \<Rightarrow> cdl_object_id \<Rightarrow> cdl_object \<Rightarrow> cdl_object"
where
"cnode_half spec obj_id obj = update_slots (\<lambda>slot.
if original_cap_at (obj_id,slot) spec \<and> object_slots obj slot \<noteq> None
then Some NullCap else object_slots obj slot) obj"
definition
cnode_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"cnode_half_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t \<circ> cnode_half spec spec_object_id) sep_map_o spec_object_id"
definition
cnodes_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id set \<Rightarrow> sep_pred"
where
"cnodes_half_initialised spec t obj_ids \<equiv> \<And>* obj_id \<in> obj_ids. cnode_half_initialised spec t obj_id"
(* The cnode's fields are half done (as per the spec). *)
definition
cnode_fields_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"cnode_fields_half_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t \<circ> cnode_half spec spec_object_id) sep_map_f spec_object_id"
(* The cnode's slots are half done (as per the spec). *)
definition
cnode_slots_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"cnode_slots_half_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t \<circ> cnode_half spec spec_object_id) sep_map_S spec_object_id"
(* A particular slot of an object is set up (as per the spec). *)
definition
cnode_slot_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> cdl_cnode_index \<Rightarrow> sep_pred"
where
"cnode_slot_half_initialised spec t spec_object_id slot \<equiv>
object_initialised_general spec t (spec2s t \<circ> cnode_half spec spec_object_id) (\<lambda>p. sep_map_s (p, slot)) spec_object_id"
definition
cnode_empty_slots_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"cnode_empty_slots_half_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t \<circ> cnode_half spec spec_object_id) sep_map_E spec_object_id"
(**********************************************
* Predicates about CNodes being initialised. *
**********************************************)
(* A TCB that isn't set to be schedulable. *)
definition
tcb_half :: "cdl_state \<Rightarrow> cdl_object \<Rightarrow> cdl_object"
where
"tcb_half spec obj = update_slots (\<lambda>slot.
if (slot = tcb_pending_op_slot \<or> slot = tcb_replycap_slot \<or> slot = tcb_boundntfn_slot) \<and>
object_slots obj slot \<noteq> None
then Some NullCap else object_slots obj slot) obj"
definition
tcb_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"tcb_half_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t \<circ> tcb_half spec) sep_map_o spec_object_id"
definition
tcbs_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id set \<Rightarrow> sep_pred"
where
"tcbs_half_initialised spec t obj_ids \<equiv> \<And>* obj_id \<in> obj_ids. tcb_half_initialised spec t obj_id"
(* The cnode's fields are half done (as per the spec). *)
definition
tcb_fields_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"tcb_fields_half_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t \<circ> tcb_half spec) sep_map_f spec_object_id"
(* The cnode's slots are half done (as per the spec). *)
definition
tcb_slots_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"tcb_slots_half_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t \<circ> tcb_half spec) sep_map_S spec_object_id"
(* A particular slot of an object is set up (as per the spec). *)
definition
tcb_slot_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> cdl_cnode_index \<Rightarrow> sep_pred"
where
"tcb_slot_half_initialised spec t spec_object_id slot \<equiv>
object_initialised_general spec t (spec2s t \<circ> tcb_half spec) (\<lambda>p. sep_map_s (p, slot)) spec_object_id"
definition
tcb_empty_slots_half_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_object_id \<Rightarrow> sep_pred"
where
"tcb_empty_slots_half_initialised spec t spec_object_id \<equiv>
object_initialised_general spec t (spec2s t \<circ> tcb_half spec) sep_map_E spec_object_id"
(********************************************
* Predicates about IRQs being initialised. *
********************************************)
definition
irq_initialised_general :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow>
(cdl_object \<Rightarrow> cdl_object) \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object \<Rightarrow> sep_pred) \<Rightarrow>
cdl_irq \<Rightarrow> sep_pred"
where
"irq_initialised_general spec t obj_trans arrow irq \<equiv> \<lambda>s.
\<exists>kernel_irq_id spec_irq_node spec_irq_id.
t spec_irq_id = Some kernel_irq_id \<and>
(irq \<mapsto>irq kernel_irq_id \<and>*
arrow kernel_irq_id (obj_trans spec_irq_node)) s \<and>
cdl_irq_node spec irq = spec_irq_id \<and>
cdl_objects spec spec_irq_id = Some spec_irq_node"
lemma irq_initialised_general_def2:
"irq_initialised_general spec t obj_trans arrow irq s =
(\<exists>kernel_irq_id spec_irq_id.
(object_initialised_general spec t obj_trans arrow spec_irq_id \<and>*
irq \<mapsto>irq kernel_irq_id) s \<and>
cdl_irq_node spec irq = spec_irq_id \<and>
t spec_irq_id = Some kernel_irq_id)"
by (fastforce simp: irq_initialised_general_def object_initialised_general_def
sep_conj_exists sep_conj_ac)
(* The irq is set up (as per the spec). *)
definition
irq_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> sep_pred"
where
"irq_initialised spec t irq \<equiv>
irq_initialised_general spec t (spec2s t) sep_map_o irq"
(* The irq is created and in it's default state. *)
definition
irq_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> sep_pred"
where
"irq_empty spec t irq \<equiv>
irq_initialised_general spec t object_default_state sep_map_o irq"
definition
irqs_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq set \<Rightarrow> sep_pred"
where
"irqs_initialised spec t irqs \<equiv> \<And>* irq \<in> irqs. irq_initialised spec t irq"
definition
irqs_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq set \<Rightarrow> sep_pred"
where
"irqs_empty spec t irqs \<equiv> \<And>* irq \<in> irqs. irq_empty spec t irq"
(* The object's fields are set up (as per the spec). *)
definition
irq_fields_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> sep_pred"
where
"irq_fields_initialised spec t irq \<equiv>
irq_initialised_general spec t (spec2s t) sep_map_f irq"
(* The object's slots are set up (as per the spec). *)
definition
irq_slots_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> sep_pred"
where
"irq_slots_initialised spec t irq \<equiv>
irq_initialised_general spec t (spec2s t) sep_map_S irq"
definition
irq_slot_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> cdl_cnode_index \<Rightarrow> sep_pred"
where
"irq_slot_initialised spec t irq slot \<equiv>
irq_initialised_general spec t (spec2s t) (\<lambda>p. sep_map_s (p, slot)) irq"
definition
irq_empty_slots_initialised :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> sep_pred"
where
"irq_empty_slots_initialised spec t irq \<equiv>
irq_initialised_general spec t (spec2s t) sep_map_E irq"
(* The object's fields are set up (as per the spec). *)
definition
irq_fields_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> sep_pred"
where
"irq_fields_empty spec t irq \<equiv>
irq_initialised_general spec t object_default_state sep_map_f irq"
(* The object's slots are set up (as per the spec). *)
definition
irq_slots_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> sep_pred"
where
"irq_slots_empty spec t irq \<equiv>
irq_initialised_general spec t object_default_state sep_map_S irq"
definition
irq_slot_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> cdl_cnode_index \<Rightarrow> sep_pred"
where
"irq_slot_empty spec t irq slot \<equiv>
irq_initialised_general spec t object_default_state (\<lambda>p. sep_map_s (p, slot)) irq"
definition
irq_empty_slots_empty :: "cdl_state \<Rightarrow> (cdl_object_id \<Rightarrow> cdl_object_id option) \<Rightarrow> cdl_irq \<Rightarrow> sep_pred"
where
"irq_empty_slots_empty spec t irq \<equiv>
irq_initialised_general spec t object_default_state sep_map_E irq"
(*********************************************************************
* Introduction, destruction, and elimination rules for object_initialised. *
*********************************************************************)
lemma object_slot_initialisedI:
"\<lbrakk>t obj_id = Some kernel_object_id; cdl_objects spec obj_id = Some spec_object;
((kernel_object_id, slot) \<mapsto>s (spec2s t spec_object)) s\<rbrakk>
\<Longrightarrow> object_slot_initialised spec t obj_id slot s"
by (fastforce simp: object_slot_initialised_def object_initialised_general_def)
lemma object_slot_emptyI:
"\<lbrakk>well_formed spec; t obj_id = Some kernel_object_id;
cdl_objects spec obj_id = Some spec_object;
((kernel_object_id, slot) \<mapsto>s (object_default_state spec_object)) s\<rbrakk>
\<Longrightarrow> object_slot_empty spec t obj_id slot s"
apply (drule (1) well_formed_object_slots)
apply (fastforce simp: object_slot_empty_def object_initialised_general_def)
done
lemma object_slot_initialisedD:
"object_slot_initialised spec t obj_id slot s \<Longrightarrow>
\<exists>kernel_object_id spec_object.
t obj_id = Some kernel_object_id \<and>
((kernel_object_id, slot) \<mapsto>s (spec2s t spec_object)) s \<and>
cdl_objects spec obj_id = Some spec_object"
by (clarsimp simp: object_slot_initialised_def object_initialised_general_def)
lemma object_slot_emptyD:
"object_slot_empty spec t obj_id slot s \<Longrightarrow>
\<exists>kernel_object_id kernel_object spec_object.
t obj_id = Some kernel_object_id \<and>
((kernel_object_id, slot) \<mapsto>s (object_default_state spec_object)) s \<and>
cdl_objects spec obj_id = Some spec_object"
by (clarsimp simp: object_slot_empty_def object_initialised_general_def)
lemma object_slot_initialisedE:
"\<lbrakk>object_slot_initialised spec t obj_id slot s;
\<And>kernel_object_id spec_object.
\<lbrakk>t obj_id = Some kernel_object_id \<and>
((kernel_object_id, slot) \<mapsto>s (spec2s t spec_object)) s \<and>
cdl_objects spec obj_id = Some spec_object\<rbrakk> \<Longrightarrow> X\<rbrakk> \<Longrightarrow> X"
by (fastforce simp: object_slot_initialised_def object_initialised_general_def)
lemma object_slot_emptyE:
"\<lbrakk>object_slot_empty spec t obj_id slot s;
\<And>kernel_object_id spec_object.
\<lbrakk>t obj_id = Some kernel_object_id \<and>
((kernel_object_id, slot) \<mapsto>s (object_default_state spec_object)) s \<and>
cdl_objects spec obj_id = Some spec_object\<rbrakk> \<Longrightarrow> X\<rbrakk> \<Longrightarrow> X"
by (fastforce simp: object_slot_empty_def object_initialised_general_def)
(********************************************
* Decomposition of object_initialised into parts. *
********************************************)
lemma spec2s_objects [simp]:
"spec2s t Untyped = Untyped"
"spec2s t Endpoint = Endpoint"
"spec2s t Notification = Notification"
"spec2s t (Frame f) = Frame f"
by (clarsimp simp: spec2s_def update_slots_def)+
lemma object_initialised_general_decomp:
"\<forall>p v. ((arrowL p v) \<and>* (arrowR p v)) = (arrow p v)
\<Longrightarrow> object_initialised_general spec t obj_trans arrow spec_object_id
= (object_initialised_general spec t obj_trans arrowL spec_object_id \<and>*
object_initialised_general spec t obj_trans arrowR spec_object_id)"
by (fastforce simp: object_initialised_general_def sep_conj_exists)
(* This is slightly different, as irq_initialised consumes ownership of the IRQ mapping. *)
lemma irq_initialised_general_decomp:
"\<forall>p v. ((arrowL p v) \<and>* (arrowR p v)) = (arrow p v)
\<Longrightarrow> irq_initialised_general spec t obj_trans arrow irq
= (irq_initialised_general spec t obj_trans arrowL irq \<and>*
object_initialised_general spec t obj_trans arrowR (cdl_irq_node spec irq))"
by (fastforce simp: irq_initialised_general_def object_initialised_general_def
sep_conj_exists sep_conj_assoc)
lemma cap_transform_nullcap [simp]:
"cap_transform t NullCap = NullCap"
by (clarsimp simp: cap_transform_def cap_has_object_def
update_cap_object_def)
lemma cap_transform_pt_simp [simp]:
"cap_transform t (PageTableCap x y z) = PageTableCap (the (t x)) y z"
by (clarsimp simp: option.the_def cap_transform_def update_cap_object_def cap_object_def
split: option.splits)
lemma cap_transform_frame [simp]:
"cap_transform t (FrameCap x ptr rights n y z) = FrameCap x (the (t ptr)) rights n y z"
by (clarsimp simp: option.the_def cap_transform_def update_cap_object_def cap_object_def
split: option.splits)
lemma cap_type_cap_transform [simp]:
"cap_type (cap_transform t cap) = cap_type cap"
by (clarsimp simp: cap_transform_def cap_has_object_def)
lemma cap_has_object_cap_transform [simp]:
"cap_has_object (cap_transform t cap) = cap_has_object cap"
by (clarsimp simp: cap_transform_def)
lemma well_formed_cap_cap_transform [simp]:
"well_formed_cap (cap_transform t cap) = well_formed_cap cap"
by (clarsimp simp: cap_transform_def)
lemma is_default_cap_cap_transform [simp]:
"well_formed_cap cap \<Longrightarrow> is_default_cap (cap_transform t cap) = is_default_cap cap"
apply (clarsimp simp: is_default_cap_def well_formed_cap_def cap_type_def default_cap_def
cap_transform_def cap_has_object_def)
apply (cases cap, simp_all add: cap_object_simps update_cap_object_def cnode_cap_size_def)
done
lemma default_cap_cap_transform:
"\<lbrakk>is_default_cap cap; well_formed_cap cap; t (cap_object cap) = Some obj_id;
cap_type cap = Some type; type \<noteq> IRQNodeType\<rbrakk>
\<Longrightarrow> default_cap type {obj_id} (cnode_cap_size cap) (is_device_cap cap) = cap_transform t cap"
by (clarsimp simp: is_default_cap_def default_cap_def cap_transform_def cap_type_def
well_formed_cap_def cap_has_object_def
update_cap_object_def split: cdl_cap.splits)+
lemma cap_transform_update_cap_object:
"\<lbrakk>t obj_id = Some k_obj_id; cap_object cap = obj_id; cap_type cap \<noteq> Some UntypedType\<rbrakk>
\<Longrightarrow> cap_transform t cap = update_cap_object k_obj_id cap"
by (clarsimp simp: update_cap_object_def cap_transform_def
cap_object_def cap_has_object_def
split: cdl_cap.splits)
lemma is_default_cap_def2:
"is_default_cap cap =
((\<exists>type. cap_type cap = Some type \<and> cap = default_cap type (cap_objects cap) (cnode_cap_size cap) (is_device_cap cap)) \<or>
is_irqhandler_cap cap)"
apply (clarsimp simp:is_default_cap_def)
apply (case_tac cap)
apply (auto simp: default_cap_def cap_type_def)
done
lemma default_cap_update_cap_object:
"\<lbrakk>is_default_cap cap; cap_type cap = Some type; cnode_cap_size cap \<le> 32;
type \<noteq> UntypedType; type \<noteq> AsidPoolType; type \<noteq> IRQNodeType\<rbrakk>
\<Longrightarrow> default_cap type {obj_id} (cnode_cap_size cap) (is_device_cap cap) = update_cap_object obj_id cap"
apply (subst default_cap_cap_transform, simp_all)
apply (frule (1) default_cap_well_formed_cap2 [where obj_ids="cap_objects cap"
and sz = "(cnode_cap_size cap)" and dev = "is_device_cap cap"], simp+)
apply (fastforce simp: is_default_cap_def2)
apply (subst cap_transform_update_cap_object, simp_all)
done
lemma default_cap_update_cap_object_pd:
"\<lbrakk>is_pd_cap cap; \<not> vm_cap_has_asid cap; \<not> is_fake_vm_cap cap\<rbrakk>
\<Longrightarrow> default_cap PageDirectoryType {obj_id} (cnode_cap_size cap) dev = update_cap_object obj_id cap"
by (clarsimp simp: default_cap_def update_cap_object_def cap_type_def
vm_cap_has_asid_def is_fake_vm_cap_def not_Some_eq_tuple
split: cdl_cap.splits cdl_frame_cap_type.splits)
lemma object_type_spec2s [simp]:
"object_type (spec2s t obj) = object_type obj"
by (clarsimp simp: spec2s_def)
lemma dom_object_slots_spec2s [simp]:
"dom (object_slots (spec2s t spec_object)) = dom (object_slots spec_object)"
by (fastforce simp: spec2s_def update_slots_def object_slots_def
split: cdl_object.splits option.splits)
lemma object_slots_spec2s:
"\<lbrakk>has_slots obj; object_slots obj slot = Some cap;
t (cap_object cap) = Some cap_object_id;
cap_has_object cap; \<not>is_untyped_cap cap\<rbrakk>
\<Longrightarrow> object_slots (spec2s t obj) slot = Some (update_cap_object cap_object_id cap)"
apply (clarsimp simp: spec2s_def)
apply (clarsimp simp: cap_transform_def)
done
lemma object_slots_spec2s':
"object_slots obj slot = Some spec_cap
\<Longrightarrow> object_slots (spec2s t obj) slot = Some (cap_transform t spec_cap)"
by (auto simp: spec2s_def object_slots_def update_slots_def
split: cdl_object.splits)
lemma object_slots_spec2s_NullCap [simp]:
"object_slots obj slot = Some NullCap
\<Longrightarrow> object_slots (spec2s t obj) slot = Some NullCap"
apply (case_tac "has_slots obj")
apply (clarsimp simp: spec2s_def)+
done
lemma update_cap_object_irqhandler_cap [simp]:
"is_irqhandler_cap cap \<Longrightarrow> update_cap_object obj_id cap = cap"
by (clarsimp simp: update_cap_object_def cap_type_def split: cdl_cap.splits)
lemma cap_transform_irqhandler_cap [simp]:
"is_irqhandler_cap cap \<Longrightarrow> cap_transform t cap = cap"
by (clarsimp simp: cap_transform_def)
lemma object_slots_spec2s_irqhandler_cap [simp]:
"\<lbrakk>object_slots obj slot = Some cap; is_irqhandler_cap cap\<rbrakk>
\<Longrightarrow> object_slots (spec2s t obj) slot = Some cap"
apply (case_tac "has_slots obj")
apply (clarsimp simp: spec2s_def)+
done
lemma update_slots_empty_spec2s [simp]:
"update_slots Map.empty (spec2s t obj)
= update_slots Map.empty obj"
by (clarsimp simp: spec2s_def)
lemma object_to_sep_state_fields_spec2s [simp]:
"object_to_sep_state obj_id (spec2s t obj) {Fields}
= object_to_sep_state obj_id obj {Fields}"
apply (rule ext)
apply (clarsimp simp: object_to_sep_state_def object_project_def object_clean_def
asid_reset_def spec2s_def object_wipe_slots_def)
done
lemma sep_map_f_spec2s [simp]:
"obj_id \<mapsto>f spec2s t obj = obj_id \<mapsto>f obj"
by (auto simp: sep_map_f_def sep_map_general_def)
lemma object_type_cnode_half [simp]:
"object_type (cnode_half spec obj_id obj) = object_type obj"
by (clarsimp simp: cnode_half_def)
lemma object_type_tcb_half [simp]:
"object_type (tcb_half spec tcb) = object_type tcb"
by (simp add: tcb_half_def)
lemma dom_object_slots_cnode_half [simp]:
"dom (object_slots (cnode_half spec obj_id obj)) = dom (object_slots obj)"
apply (clarsimp simp: cnode_half_def)
apply (case_tac "has_slots obj")
apply (auto simp: dom_def)
done
lemma dom_object_slots_tcb_half [simp]:
"dom (object_slots (tcb_half spec tcb)) =
dom (object_slots tcb)"
apply (clarsimp simp: tcb_half_def)
apply (case_tac "has_slots tcb")
apply (auto simp: dom_def)
done
lemma object_slots_tcb_half:
"object_slots (tcb_half spec obj) =
(\<lambda>slot. if (slot = tcb_pending_op_slot \<or> slot = tcb_replycap_slot \<or> slot = tcb_boundntfn_slot) \<and> object_slots obj slot \<noteq> None
then Some NullCap else object_slots obj slot)"
by (case_tac "has_slots obj", auto simp: tcb_half_def split: if_split_asm)
lemma intent_reset_object_type:
"intent_reset obj = intent_reset obj' \<Longrightarrow> object_type obj = object_type obj'"
by (clarsimp simp: intent_reset_def object_type_def split: cdl_object.splits)
lemma intent_reset_object_slots:
"intent_reset obj = intent_reset obj' \<Longrightarrow> object_slots obj = object_slots obj'"
by (clarsimp simp: intent_reset_def object_slots_def cdl_tcb.splits split: cdl_object.splits)
lemma intent_reset_object_size_bits:
"intent_reset obj = intent_reset obj' \<Longrightarrow> object_size_bits obj = object_size_bits obj'"
by (clarsimp simp: intent_reset_def object_size_bits_def split: cdl_object.splits)
lemma intent_reset_cnode:
"\<lbrakk>intent_reset obj = intent_reset obj'; object_type obj = CNodeType\<rbrakk>
\<Longrightarrow> obj = obj'"
by (clarsimp simp: intent_reset_def object_type_def split: cdl_object.splits)
lemma intent_reset_object_slots_NullCap:
"\<lbrakk>intent_reset (object_default_state obj) = intent_reset obj';
slot < 2 ^ object_size_bits obj; has_slots obj\<rbrakk>
\<Longrightarrow> object_slots obj' slot = Some NullCap"
apply (frule intent_reset_object_slots [THEN sym])
apply (clarsimp simp: object_default_state_def2 object_type_def has_slots_def
object_size_bits_def object_slots_def default_tcb_def
empty_cnode_def empty_irq_node_def empty_cap_map_def pt_size_def pd_size_def
split: cdl_object.splits)
done
lemma object_slots_object_default_state_NullCap':
"\<lbrakk>slot < 2 ^ object_size_bits obj; has_slots obj\<rbrakk>
\<Longrightarrow> object_slots (object_default_state obj) slot = Some NullCap"
by (clarsimp simp: object_default_state_def2 object_type_def has_slots_def
object_size_bits_def object_slots_def default_tcb_def
empty_cnode_def empty_irq_node_def empty_cap_map_def pt_size_def pd_size_def
split: cdl_object.splits)
lemma dom_range_upper:
"\<lbrakk>dom f = {0..<n}; f x = Some y\<rbrakk> \<Longrightarrow> x < n"
by fastforce
lemma object_slots_object_default_state_NullCap:
"\<lbrakk>well_formed spec; \<not>tcb_at obj_id spec; opt_cap (obj_id, slot) spec = Some cap;
cdl_objects spec obj_id = Some spec_object\<rbrakk>
\<Longrightarrow> object_slots (object_default_state spec_object) slot = Some NullCap"
apply (drule (1) well_formed_object_slots)
apply (clarsimp simp: object_default_state_def2
split: cdl_object.splits,
(fastforce simp: object_at_def is_cnode_def object_size_bits_def object_slots_def
empty_cnode_def empty_irq_node_def empty_cap_map_def
opt_cap_def slots_of_def
dest!: dom_range_upper)+)
done
lemma intent_reset_remove:
"obj = obj' \<Longrightarrow> intent_reset obj = intent_reset obj'"
by (rule arg_cong)
lemma sep_map_E_eq:
"\<lbrakk>object_type obj = object_type obj'; dom (object_slots obj) = dom (object_slots obj')\<rbrakk>
\<Longrightarrow> (p \<mapsto>E obj) = (p \<mapsto>E obj')"
apply (clarsimp simp: sep_map_E_def sep_map_S'_def sep_map_general_def)
apply (rule ext)
apply (subgoal_tac "object_to_sep_state p obj (Slot ` (UNIV - dom (object_slots obj')))
= object_to_sep_state p obj' (Slot ` (UNIV - dom (object_slots obj')))")
apply simp
apply (fastforce simp: object_to_sep_state_def split_def
object_project_def object_slots_object_clean
split: option.splits)
done
lemma sep_map_E_object_default_state:
"dom (object_slots (object_default_state obj)) = dom (object_slots obj)
\<Longrightarrow> (p \<mapsto>E object_default_state obj) = (p \<mapsto>E obj)"
using sep_map_E_eq [where obj="object_default_state obj" and obj'=obj]
by simp
lemma sep_map_E_intent_reset:
"\<lbrakk>intent_reset obj = intent_reset obj'\<rbrakk>
\<Longrightarrow> (p \<mapsto>E obj) = (p \<mapsto>E obj')"
apply (cut_tac obj=obj and obj'=obj' in sep_map_E_eq)
apply (erule intent_reset_object_type)
apply (drule intent_reset_object_slots, simp)
apply simp
done
lemma sep_map_E_spec2s [simp]:
"(p \<mapsto>E spec2s t obj) = (p \<mapsto>E obj)"
apply (cut_tac obj="spec2s t obj" and obj'=obj in sep_map_E_eq, simp)
apply (clarsimp simp: spec2s_def)
apply (case_tac "has_slots obj")
apply simp+
done
lemma sep_map_E_tcb_half [simp]:
"obj_id \<mapsto>E tcb_half spec tcb = obj_id \<mapsto>E tcb"
by (rule sep_map_E_eq, simp+)
lemma object_to_sep_state_fields_tcb_eq:
"\<lbrakk>cdl_tcb_fault_endpoint tcb = cdl_tcb_fault_endpoint tcb';
cdl_tcb_has_fault tcb = cdl_tcb_has_fault tcb';
cdl_tcb_domain tcb = cdl_tcb_domain tcb'\<rbrakk>
\<Longrightarrow> object_to_sep_state obj_id (Tcb tcb) {Fields}
= object_to_sep_state obj_id (Tcb tcb') {Fields}"
apply (rule ext)
apply (clarsimp simp: object_to_sep_state_def object_project_def object_clean_def
asid_reset_def spec2s_def object_wipe_slots_def
update_slots_def intent_reset_def cdl_tcb.splits)
done
lemma sep_map_f_eq_tcb:
"\<lbrakk>cdl_tcb_fault_endpoint tcb = cdl_tcb_fault_endpoint tcb';
cdl_tcb_has_fault tcb = cdl_tcb_has_fault tcb';
cdl_tcb_domain tcb = cdl_tcb_domain tcb'\<rbrakk>
\<Longrightarrow> obj_id \<mapsto>f Tcb tcb = obj_id \<mapsto>f Tcb tcb'"
apply (clarsimp simp: sep_map_f_def sep_map_general_def object_slots_def
object_clean_def intent_reset_def asid_reset_def update_slots_def)
apply (subst object_to_sep_state_fields_tcb_eq [where tcb'=tcb'], simp_all)
done
lemma sep_map_f_intent_reset_cnode:
"\<lbrakk>object_type obj = CNodeType; intent_reset obj = intent_reset obj'\<rbrakk>
\<Longrightarrow> obj_id \<mapsto>f obj = obj_id \<mapsto>f obj'"
apply (clarsimp simp: sep_map_f_def sep_map_general_def split: sep_state.splits)
apply (rule ext)
apply (clarsimp simp: intent_reset_def object_type_def
split: cdl_object.splits)
done
lemma sep_map_f_empty_cnode:
"obj_id \<mapsto>f CNode (empty_cnode sz) =
obj_id \<mapsto>f CNode \<lparr>cdl_cnode_caps = Map.empty, cdl_cnode_size_bits = sz\<rparr>"
apply (rule ext, rename_tac s)
apply (clarsimp simp: sep_map_f_def sep_map_general_def split: sep_state.splits)
apply (intro iffI ext |
clarsimp simp: object_to_sep_state_def object_clean_def
object_project_def object_slots_object_clean asid_reset_def
intent_reset_def object_wipe_slots_def
update_slots_def empty_cnode_def)+
done
lemma empty_cnode_object_size_bits:
"object_type obj = CNodeType \<Longrightarrow> obj_id \<mapsto>f CNode (empty_cnode (object_size_bits obj)) = obj_id \<mapsto>f obj"
apply (subst sep_map_f_empty_cnode)
apply (rule ext)
apply (clarsimp simp: sep_map_f_def sep_map_general_def split: sep_state.splits)
apply (intro iffI ext |
clarsimp simp: object_type_def object_size_bits_def
object_clean_def reset_cap_asid_def asid_reset_def
object_to_sep_state_def object_project_def intent_reset_def
object_wipe_slots_def update_slots_def cdl_cnode.splits
split: cdl_object.splits)+
done
lemma sep_map_f_object_size_bits_cnode:
"\<lbrakk>object_type obj = CNodeType; object_type obj' = CNodeType;
object_size_bits obj = object_size_bits obj'\<rbrakk>
\<Longrightarrow> obj_id \<mapsto>f obj = obj_id \<mapsto>f obj'"
apply (clarsimp simp: sep_map_f_def sep_map_general_def split: sep_state.splits)
apply (rule ext)
apply (intro iffI ext |
clarsimp simp: object_type_def object_size_bits_def
object_to_sep_state_def object_project_def intent_reset_def
object_wipe_slots_def update_slots_def
cdl_cnode.splits object_clean_def asid_reset_def
split: cdl_object.splits)+
done
lemma sep_map_f_object_size_bits_pt:
"\<lbrakk>object_type obj = PageTableType; object_type obj' = PageTableType\<rbrakk>
\<Longrightarrow> obj_id \<mapsto>f obj = obj_id \<mapsto>f obj'"
apply (clarsimp simp: sep_map_f_def sep_map_general_def split: sep_state.splits)
apply (rule ext)
apply (intro iffI ext |
clarsimp simp: object_type_def object_size_bits_def
object_to_sep_state_def object_project_def intent_reset_def
object_wipe_slots_def update_slots_def object_clean_def asid_reset_def
split: cdl_object.splits)+
done
lemma sep_map_f_object_size_bits_pd:
"\<lbrakk>object_type obj = PageDirectoryType; object_type obj' = PageDirectoryType\<rbrakk>
\<Longrightarrow> obj_id \<mapsto>f obj = obj_id \<mapsto>f obj'"
apply (clarsimp simp: sep_map_f_def sep_map_general_def split: sep_state.splits)
apply (rule ext)
apply (intro iffI ext |
clarsimp simp: object_type_def object_size_bits_def
object_to_sep_state_def object_project_def intent_reset_def
object_wipe_slots_def update_slots_def object_clean_def asid_reset_def
split: cdl_object.splits)+
done
(********************************************************
* Object done, and other such predicate decompositions *
********************************************************)
lemma object_initialised_decomp:
"object_initialised spec t spec_object_id =
(object_fields_initialised spec t spec_object_id \<and>*
object_slots_initialised spec t spec_object_id)"
apply (clarsimp simp: object_initialised_def object_fields_initialised_def object_slots_initialised_def)
apply (rule object_initialised_general_decomp)
apply (clarsimp simp: sep_map_o_decomp)
done
lemma object_empty_decomp:
"object_empty spec t spec_object_id =
(object_fields_empty spec t spec_object_id \<and>*
object_slots_empty spec t spec_object_id)"
apply (clarsimp simp: object_empty_def object_fields_empty_def object_slots_empty_def)
apply (rule object_initialised_general_decomp)
apply (clarsimp simp: sep_map_o_decomp)
done
lemma cnode_half_initialised_decomp:
"cnode_half_initialised spec t spec_object_id =
(cnode_fields_half_initialised spec t spec_object_id \<and>*
cnode_slots_half_initialised spec t spec_object_id)"
apply (clarsimp simp: cnode_half_initialised_def cnode_fields_half_initialised_def cnode_slots_half_initialised_def)
apply (rule object_initialised_general_decomp)
apply (clarsimp simp: sep_map_o_decomp)
done
lemma irq_initialised_decomp:
"irq_initialised spec t irq =
(irq_slots_initialised spec t irq \<and>*
object_fields_initialised spec t (cdl_irq_node spec irq))"
apply (clarsimp simp: irq_initialised_def object_fields_initialised_def irq_slots_initialised_def)
apply (rule irq_initialised_general_decomp)
apply (clarsimp simp: sep_map_o_decomp sep_conj_ac)
done
lemma irq_empty_decomp:
"irq_empty spec t irq =
(irq_slots_empty spec t irq \<and>*
object_fields_empty spec t (cdl_irq_node spec irq))"
apply (clarsimp simp: irq_empty_def object_fields_empty_def irq_slots_empty_def)
apply (rule irq_initialised_general_decomp)
apply (clarsimp simp: sep_map_o_decomp sep_conj_ac)
done
(************************************
* object_slots_initialised rewrite rules. *
************************************)
lemma object_slot_initialised_eq:
"\<lbrakk>t obj_id = Some kernel_object_id; cdl_objects spec obj_id = Some spec_object\<rbrakk>
\<Longrightarrow> object_slot_initialised spec t obj_id slot
= (kernel_object_id, slot) \<mapsto>s (spec2s t spec_object)"
apply (rule ext, rename_tac s)
apply (fastforce simp: object_slot_initialised_def object_initialised_general_def)
done
lemma object_slot_empty_eq:
"\<lbrakk>well_formed spec; t obj_id = Some kernel_object_id;
cdl_objects spec obj_id = Some spec_object\<rbrakk>
\<Longrightarrow> object_slot_empty spec t obj_id slot
= (kernel_object_id, slot) \<mapsto>s (object_default_state spec_object)"
apply (rule ext, rename_tac s)
apply (drule (1) well_formed_object_slots)
apply (fastforce simp: object_slot_empty_def object_initialised_general_def)
done
(****************************************************************************
* Lemmas decomposing object_slots_initialised. *
* These show that initialising an objects slots can be done slot by slot. *
****************************************************************************)
lemma object_slots_initialised_decomp_helper:
"\<lbrakk>slots \<noteq> {}; slots \<noteq> UNIV\<rbrakk>
\<Longrightarrow> object_slots_initialised spec t obj_id =
(object_initialised_general spec t (spec2s t) (\<lambda>obj_id. sep_map_S' (obj_id, slots)) obj_id \<and>*
object_initialised_general spec t (spec2s t) (\<lambda>obj_id. sep_map_S' (obj_id, UNIV-slots)) obj_id)"
apply (clarsimp simp: object_slots_initialised_def)
apply (rule object_initialised_general_decomp)
apply (clarsimp simp: sep_map_S_decomp')
done
lemma object_slots_empty_decomp_helper:
"\<lbrakk>slots \<noteq> {}; slots \<noteq> UNIV\<rbrakk>
\<Longrightarrow> object_slots_empty spec t obj_id =
(object_initialised_general spec t object_default_state (\<lambda>obj_id. sep_map_S' (obj_id, slots)) obj_id \<and>*
object_initialised_general spec t object_default_state (\<lambda>obj_id. sep_map_S' (obj_id, UNIV-slots)) obj_id)"
apply (clarsimp simp: object_slots_empty_def)
apply (rule object_initialised_general_decomp)
apply (clarsimp simp: sep_map_S_decomp')
done
lemma cnode_slots_half_initialised_decomp_helper:
"\<lbrakk>slots \<noteq> {}; slots \<noteq> UNIV\<rbrakk>
\<Longrightarrow> cnode_slots_half_initialised spec t obj_id =
(object_initialised_general spec t (spec2s t \<circ> cnode_half spec obj_id) (\<lambda>obj_id. sep_map_S' (obj_id, slots)) obj_id \<and>*
object_initialised_general spec t (spec2s t \<circ> cnode_half spec obj_id) (\<lambda>obj_id. sep_map_S' (obj_id, UNIV-slots)) obj_id)"
apply (clarsimp simp: cnode_slots_half_initialised_def)
apply (rule object_initialised_general_decomp)
apply (clarsimp simp: sep_map_S_decomp')
done
lemma sep_map_exists_rewrite':
"\<lbrakk>((obj_id, slots) \<mapsto>S' obj') s; intent_reset obj' = intent_reset obj\<rbrakk>
\<Longrightarrow> ((obj_id, slots) \<mapsto>S' obj) s"
apply (clarsimp simp: intent_reset_def sep_map_S'_def sep_map_general_def
split: cdl_object.splits)
apply (rename_tac cdl_tcb cdl_tcb')
apply (rule ext)
apply (clarsimp simp: sep_map_S'_def sep_map_general_def intent_reset_def
object_slots_object_clean object_to_sep_state_def object_project_def
split: if_split_asm)
apply (case_tac cdl_tcb,clarsimp)
apply (case_tac cdl_tcb',clarsimp simp:object_slots_def)
apply (intro conjI |
clarsimp simp: object_slots_object_clean |
clarsimp simp: object_slots_def)+
done
lemma sep_map_exists_rewrite:
"(\<lambda>s. \<exists>obj'. ((obj_id, slots) \<mapsto>S' obj') s \<and> intent_reset obj = intent_reset obj') =
(obj_id, slots) \<mapsto>S' obj"
apply (rule ext)
apply (rule iffI)
apply clarsimp
apply (erule sep_map_exists_rewrite', simp)
apply fastforce
done
lemma object_slots_general_decomp_list:
"\<lbrakk>distinct slots; slots \<noteq> []\<rbrakk>
\<Longrightarrow> (object_initialised_general spec t obj_trans (\<lambda>obj_id. sep_map_S' (obj_id, set slots)) obj_id) =
(\<And>* map (\<lambda>slot. object_initialised_general spec t obj_trans (\<lambda>p. sep_map_s (p, slot)) obj_id) slots)"
apply (induct slots)
apply clarsimp
apply (atomize)
apply (case_tac "slots = []")
apply (clarsimp simp: object_initialised_general_def sep_map_S'_def sep_map_s_def)
apply (clarsimp simp: object_initialised_general_def)
apply (rule ext)
apply (rule iffI)
apply clarsimp
apply (drule_tac obj_id=kernel_object_id and obj="obj_trans spec_object" in sep_map_S'_decomp', simp)
apply (fastforce simp: sep_conj_exists sep_conj_ac)
apply (clarsimp simp: sep_conj_exists)
apply (drule_tac obj_id=kernel_object_id and obj="obj_trans spec_object" in sep_map_S'_decomp', simp)
apply (fastforce simp: sep_conj_exists sep_conj_ac)
done
lemma object_slots_general_decomp_set:
"\<lbrakk>finite slots; slots \<noteq> {}\<rbrakk>
\<Longrightarrow> (object_initialised_general spec t obj_trans (\<lambda>obj_id. sep_map_S' (obj_id, slots)) obj_id) =
(\<And>* slot \<in> slots. object_initialised_general spec t obj_trans (\<lambda>p. sep_map_s (p, slot)) obj_id)"
apply (drule sep_map_set_conj_sep_list_conj [where
P="\<lambda>slot. object_initialised_general spec t obj_trans (\<lambda>p. sep_map_s (p, slot)) obj_id"])
apply (elim exE conjE)
apply simp
apply (subst object_slots_general_decomp_list [symmetric], clarsimp+)
done
lemma object_slots_initialised_decomp':
"\<lbrakk>finite slots; slots \<noteq> {}\<rbrakk>
\<Longrightarrow> (object_initialised_general spec t (spec2s t) (\<lambda>obj_id. sep_map_S' (obj_id, slots)) obj_id) =
(\<And>* slot \<in> slots. object_slot_initialised spec t obj_id slot)"
apply (clarsimp simp: object_slot_initialised_def [abs_def])
apply (erule (1) object_slots_general_decomp_set)
done
lemma object_slots_empty_decomp':
"\<lbrakk>finite slots; slots \<noteq> {}\<rbrakk>
\<Longrightarrow> (object_initialised_general spec t object_default_state (\<lambda>obj_id. sep_map_S' (obj_id, slots)) obj_id) =
(\<And>* slot \<in> slots. object_slot_empty spec t obj_id slot)"
apply (clarsimp simp: object_slot_empty_def [abs_def])
apply (erule (1) object_slots_general_decomp_set)
done
lemma cnode_slots_half_initialised_decomp':
"\<lbrakk>finite slots; slots \<noteq> {}\<rbrakk>
\<Longrightarrow> (object_initialised_general spec t (spec2s t \<circ> cnode_half spec obj_id) (\<lambda>obj_id. sep_map_S' (obj_id, slots)) obj_id) =
(\<And>* slot \<in> slots. cnode_slot_half_initialised spec t obj_id slot)"
apply (clarsimp simp: cnode_slot_half_initialised_def [abs_def])
apply (erule (1) object_slots_general_decomp_set)
done
lemma empty_slots_object_slots_initialised_object_empty_slots_initialised:
"dom (slots_of obj_id spec) = {} \<Longrightarrow> object_empty_slots_initialised spec t obj_id = object_slots_initialised spec t obj_id"
apply (rule ext, rename_tac s)
apply (clarsimp simp: object_slots_initialised_def object_empty_slots_initialised_def object_initialised_general_def)
apply (rule iffI)
apply clarsimp
apply (clarsimp simp: sep_map_S_def sep_map_S'_def sep_map_E_def slots_of_def