(*
 * 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: 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
                  split: option.splits)
   apply (fastforce simp: intent_reset_def spec2s_def object_slots_def cdl_tcb.splits
                   split: cdl_object.splits)
  apply clarsimp
  apply (clarsimp simp: sep_map_S_def sep_map_S'_def sep_map_E_def slots_of_def
                 split: option.splits)
  apply (fastforce simp: intent_reset_def spec2s_def object_slots_def cdl_tcb.splits
                  split: cdl_object.splits)
  done

lemma object_empty_slots_initialised_def2:
  "object_empty_slots_initialised spec t obj_id =
   object_initialised_general spec t (spec2s t) (\<lambda>obj_id'. sep_map_S' (obj_id', UNIV - dom (slots_of obj_id spec))) obj_id"
  apply (clarsimp simp: object_empty_slots_initialised_def object_initialised_general_def sep_map_E_def)
  apply (fastforce simp: slots_of_def
                  split: option.splits)
  done

lemma object_slots_initialised_decomp:
  "well_formed spec \<Longrightarrow>
  object_slots_initialised spec t obj_id =
  ((\<And>* slot \<in> dom (slots_of obj_id spec). (object_slot_initialised spec t obj_id) slot) \<and>*
    object_empty_slots_initialised spec t obj_id)"
  apply (drule well_formed_finite [where obj_id=obj_id])
  apply (case_tac "dom (slots_of obj_id spec) = {}")
   apply clarsimp
   apply (rule empty_slots_object_slots_initialised_object_empty_slots_initialised [THEN sym], simp)
  apply (subst object_slots_initialised_decomp_helper, assumption)
   apply clarsimp
  apply (clarsimp simp: object_empty_slots_initialised_def2)
  apply (drule_tac obj_id=obj_id and spec=spec and t=t in object_slots_initialised_decomp', simp)
  apply clarsimp
  done

lemma object_initialised_decomp_total:
  "\<lbrakk>well_formed spec\<rbrakk>
    \<Longrightarrow> object_initialised spec t obj_id =
        (object_fields_initialised spec t obj_id \<and>*
        (\<And>* slot \<in> dom (slots_of obj_id spec). object_slot_initialised spec t obj_id slot) \<and>*
         object_empty_slots_initialised spec t obj_id)"
  by (clarsimp simp: object_initialised_decomp object_slots_initialised_decomp sep_conj_assoc)

lemma object_slot_empty_initialised_NullCap:
  "\<lbrakk>well_formed spec; \<not>tcb_at obj_id spec; opt_cap (obj_id, slot) spec = Some NullCap\<rbrakk> \<Longrightarrow>
  object_slot_empty spec t obj_id slot = object_slot_initialised spec t obj_id slot"
  apply (clarsimp simp: object_slot_empty_def object_slot_initialised_def object_initialised_general_def)
  apply (rule ext)
  apply (rule iffI)
   apply (clarsimp simp: sep_conj_exists)
   apply (cut_tac obj="object_default_state spec_object" and obj_id=kernel_object_id and
                  obj'="spec2s t spec_object" and slot=slot
               in sep_map_s_object_slots_equal)
     apply (clarsimp simp: object_slots_opt_cap)
     apply (drule (3) object_slots_object_default_state_NullCap, simp)
    apply clarsimp
   apply clarsimp
  apply (clarsimp simp: sep_conj_exists)
  apply (cut_tac obj="spec2s t spec_object" and obj_id=kernel_object_id and
                 obj'="object_default_state spec_object" and slot=slot
              in sep_map_s_object_slots_equal)
    apply (drule (3) object_slots_object_default_state_NullCap)
    apply (clarsimp simp: object_slots_opt_cap)
   apply clarsimp
  apply clarsimp
  done

lemma object_empty_slots_empty_initialised:
  "well_formed spec
  \<Longrightarrow> object_empty_slots_empty spec t spec_object_id =
      object_empty_slots_initialised spec t spec_object_id"
  apply (clarsimp simp: object_empty_slots_initialised_def object_empty_slots_empty_def
                        object_initialised_general_def)
  apply (rule ext)
  apply (rule iffI)
   apply clarsimp
   apply (frule (1) well_formed_object_slots)
   apply (clarsimp simp: well_formed_def)
   apply (erule_tac x=spec_object_id in allE)
   apply (clarsimp simp: sep_map_E_object_default_state
                  split: option.splits)
  apply (clarsimp simp: well_formed_def)
  apply (erule_tac x=spec_object_id in allE)
  apply (clarsimp split: option.splits)
  apply (drule_tac obj=spec_object and p=kernel_object_id in sep_map_E_object_default_state, simp)
  done

lemma cnode_empty_slots_half_initialised_object_empty_slots_initialised:
  "cnode_empty_slots_half_initialised spec t spec_object_id =
   object_empty_slots_initialised spec t spec_object_id"
  apply (clarsimp simp: object_empty_slots_initialised_def cnode_empty_slots_half_initialised_def
                        object_initialised_general_def)
  apply (rule ext)
  apply (rule iffI)
   apply (clarsimp split: option.splits)
   apply (cut_tac p=kernel_object_id and obj=spec_object and
                  obj'="cnode_half spec spec_object_id spec_object" in
                  sep_map_E_eq [OF sym], simp+)
  apply (clarsimp split: option.splits)
  apply (cut_tac p=kernel_object_id and obj="cnode_half spec spec_object_id spec_object" and
                 obj'="spec_object" in
                 sep_map_E_eq [OF sym], simp+)
  done

lemma object_default_state_has_slots_not_empty:
  "has_slots obj \<Longrightarrow> dom (object_slots (object_default_state obj)) \<noteq> {}"
  apply (clarsimp simp: object_default_state_def2 has_slots_def object_slots_def
                        default_tcb_def tcb_pending_op_slot_def
                        empty_cnode_def empty_irq_node_def empty_cap_map_def
                 split: cdl_object.splits)
  apply (clarsimp simp: fun_eq_iff, erule_tac x=0 in allE, simp)+
  done

lemma well_formed_has_slots:
  "\<lbrakk>well_formed spec; cdl_objects spec obj_id = Some obj; object_slots obj = Map.empty; has_slots obj \<rbrakk> \<Longrightarrow> False"
  apply (clarsimp simp: well_formed_def)
  apply (erule_tac x=obj_id in allE)
  apply (clarsimp split: option.splits)
  apply (drule object_default_state_has_slots_not_empty, simp)
  done

lemma sep_map_S_object_default_state_no_slots:
  "\<not> has_slots obj \<Longrightarrow> (obj_id \<mapsto>S object_default_state obj) = (obj_id \<mapsto>S obj)"
  apply (clarsimp simp: sep_map_S_def sep_map_general_def)
  apply (intro ext conjI iffI |
         clarsimp simp: object_to_sep_state_def object_project_def
                        update_slots_def empty_cnode_def
                        object_slots_object_clean
                        object_default_state_def default_object_def
                        object_type_def has_slots_def
                 split: cdl_component_id.splits option.splits cdl_object.splits)+
  done

lemma sep_map_s_object_default_state_no_slots:
  "\<not> has_slots obj \<Longrightarrow> (obj_id, slot) \<mapsto>s object_default_state obj = (obj_id, slot) \<mapsto>s obj"
  apply (clarsimp simp: sep_map_s_def sep_map_general_def)
  apply (intro ext conjI iffI |
         clarsimp simp: object_to_sep_state_def object_project_def
                        update_slots_def empty_cnode_def
                        object_slots_object_clean
                        object_default_state_def default_object_def
                        object_type_def has_slots_def
                 split: cdl_component_id.splits option.splits cdl_object.splits)+
  done

lemma object_slots_empty_initialised_no_slots:
  "\<lbrakk>well_formed spec; slots_of obj_id spec = Map.empty\<rbrakk>
  \<Longrightarrow> object_slots_empty spec t obj_id = object_slots_initialised spec t obj_id"
  apply (clarsimp simp: slots_of_def split: option.splits)
   apply (clarsimp simp: object_slots_empty_def object_slots_initialised_def object_initialised_general_def)
  apply (rename_tac obj)
  apply (case_tac "has_slots obj")
   apply (drule (3) well_formed_has_slots, simp)
  apply (frule (1) well_formed_object_slots)
  apply (clarsimp simp: object_slots_empty_def object_slots_initialised_def object_initialised_general_def)
  apply (rule ext, rule iffI)
   apply (clarsimp simp: spec2s_def)
   apply (drule_tac obj_id=kernel_object_id in sep_map_S_object_default_state_no_slots, simp)
  apply clarsimp
  apply (clarsimp simp: spec2s_def)
  apply (drule_tac obj_id=kernel_object_id in sep_map_S_object_default_state_no_slots, simp)
  done

lemma object_empty_slots_empty_def2:
  "well_formed spec
  \<Longrightarrow> object_empty_slots_empty spec t obj_id =
      object_initialised_general spec t object_default_state (\<lambda>obj_id'. sep_map_S' (obj_id', UNIV - dom (slots_of obj_id spec))) obj_id"
  apply (clarsimp simp: object_empty_slots_empty_def object_initialised_general_def sep_map_E_def)
  apply (rule ext)
  apply (rule iffI)
   apply (clarsimp simp: well_formed_def)
   apply (erule_tac x=obj_id in allE)
   apply (clarsimp split: option.splits)
   apply (fastforce simp: slots_of_def split: option.splits)
  apply (clarsimp simp: well_formed_def)
  apply (erule_tac x=obj_id in allE)
  apply (clarsimp split: option.splits)
  apply (fastforce simp: slots_of_def split: option.splits)
  done

lemma cnode_empty_slots_half_initialised_def2:
  "cnode_empty_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', UNIV - dom (slots_of obj_id spec))) obj_id"
  apply (clarsimp simp: object_empty_slots_initialised_def cnode_empty_slots_half_initialised_def
                        object_initialised_general_def)
  apply (rule ext)
  apply (rule iffI)
   apply (clarsimp split: option.splits)
   apply (cut_tac p=kernel_object_id and
                  obj="cnode_half spec obj_id spec_object" and
                  obj'="spec2s t (cnode_half spec obj_id spec_object)" in
          sep_map_E_eq, simp, simp)
   apply (clarsimp simp: sep_map_E_def slots_of_def split: option.splits)
  apply clarsimp
  apply (cut_tac p=kernel_object_id and
                 obj="spec2s t (cnode_half spec obj_id spec_object)" and
                 obj'="cnode_half spec obj_id spec_object" in
         sep_map_E_eq, simp, simp)
  apply (clarsimp simp: sep_map_E_def slots_of_def)
  done

lemma object_slots_empty_decomp:
  "\<lbrakk>well_formed spec\<rbrakk>
  \<Longrightarrow> object_slots_empty spec t obj_id =
  ((\<And>* slot \<in> dom (slots_of obj_id spec).  object_slot_empty spec t obj_id slot) \<and>*
    object_empty_slots_empty spec t obj_id)"
  apply (frule well_formed_finite [where obj_id=obj_id])
  apply (case_tac "dom (slots_of obj_id spec) = {}")
   apply clarsimp
   apply (subst object_empty_slots_empty_initialised, simp)
   apply (subst empty_slots_object_slots_initialised_object_empty_slots_initialised, simp)
   apply (clarsimp simp: object_slots_empty_initialised_no_slots)
  apply (subst object_slots_empty_decomp_helper, assumption)
   apply clarsimp
  apply (clarsimp simp: object_empty_slots_empty_def2)
  apply (drule_tac obj_id=obj_id and spec=spec and t=t in object_slots_empty_decomp', simp)
  apply clarsimp
  done

lemma well_formed_cnode_not_empty:
  "\<lbrakk>well_formed spec; slots_of obj_id spec = Map.empty; cnode_at obj_id spec\<rbrakk> \<Longrightarrow> P"
  apply (clarsimp simp: slots_of_def object_at_def
                 split: option.splits)
  apply (rename_tac obj)
  apply (case_tac "has_slots obj")
   apply (drule (3) well_formed_has_slots, simp)
  apply (clarsimp simp: is_cnode_def has_slots_def split: cdl_object.splits)
  done

lemma cnode_slots_half_initialised_decomp:
  "\<lbrakk>well_formed spec; cnode_at obj_id spec\<rbrakk>
  \<Longrightarrow> cnode_slots_half_initialised spec t obj_id =
  ((\<And>* slot \<in> dom (slots_of obj_id spec). cnode_slot_half_initialised spec t obj_id slot) \<and>*
    cnode_empty_slots_half_initialised spec t obj_id)"
  apply (frule well_formed_finite [where obj_id=obj_id])
  apply (case_tac "dom (slots_of obj_id spec) = {}")
   apply clarsimp
   apply (erule (2) well_formed_cnode_not_empty)
  apply (subst cnode_slots_half_initialised_decomp_helper, assumption)
   apply clarsimp
   apply (drule_tac obj_id=obj_id in well_formed_finite, clarsimp)
  apply (subst cnode_slots_half_initialised_decomp', simp+)
  apply (clarsimp simp: cnode_empty_slots_half_initialised_def2)
  done

lemma distinct_singleton_set:
  "\<lbrakk>distinct xs; set xs = {x}\<rbrakk> \<Longrightarrow> xs = [x]"
  by (metis set_simps(2) distinct.simps(2) distinct_singleton
            insert_iff insert_not_empty list.exhaust set_empty2)

lemma irq_slots_initialised_decomp_helper:
  "well_formed spec
  \<Longrightarrow> irq_slots_initialised spec t irq =
  ((\<And>* slot \<in> dom (slots_of (cdl_irq_node spec irq) spec). irq_slot_initialised spec t irq slot) \<and>*
    object_empty_slots_initialised spec t (cdl_irq_node spec irq))"
  apply (clarsimp simp: irq_slots_initialised_def irq_slot_initialised_def [abs_def]
                        irq_initialised_general_def [abs_def]
                        object_empty_slots_initialised_def object_initialised_general_def
                        sep_conj_exists slots_of_def
                 split: option.splits)
  apply (subst sep_map_S_decomp, simp+)
   apply (erule (1) well_formed_finite_object_slots)
  apply (subst well_formed_object_slots_irq_node, assumption+)+
  apply (fastforce simp: sep_conj_ac)
  done

lemma irq_slots_empty_decomp_helper:
  "well_formed spec
  \<Longrightarrow> irq_slots_empty spec t irq =
  ((\<And>* slot \<in> dom (slots_of (cdl_irq_node spec irq) spec). irq_slot_empty spec t irq slot) \<and>*
    object_empty_slots_empty spec t (cdl_irq_node spec irq))"
  apply (clarsimp simp: irq_slots_empty_def irq_slot_empty_def [abs_def]
                        irq_initialised_general_def [abs_def]
                        object_empty_slots_empty_def object_initialised_general_def
                        sep_conj_exists slots_of_def
                 split: option.splits)
  apply (frule (1) well_formed_object_slots_default_irq_node)
  apply (subst sep_map_S_decomp, simp+)
  apply (subst well_formed_object_slots_irq_node, assumption+)+
  apply (fastforce simp: sep_conj_ac)
  done

(* This rule uses object_empty_slots_initialised, to make it easier to cancel. *)
lemma irq_slots_initialised_decomp:
  "\<lbrakk>well_formed spec; irq \<in> used_irqs spec\<rbrakk>
  \<Longrightarrow> irq_slots_initialised spec t irq = (irq_slot_initialised spec t irq 0 \<and>* object_empty_slots_initialised spec t (cdl_irq_node spec irq))"
  apply (subst irq_slots_initialised_decomp_helper, assumption)
  apply (subst well_formed_slots_of_used_irq_node, assumption+)
  apply clarsimp
  done

lemma irq_slots_empty_decomp:
  "\<lbrakk>well_formed spec; irq \<in> used_irqs spec\<rbrakk>
  \<Longrightarrow> irq_slots_empty spec t irq = (irq_slot_empty spec t irq 0 \<and>* object_empty_slots_initialised spec t (cdl_irq_node spec irq))"
  apply (subst irq_slots_empty_decomp_helper, assumption)
  apply (subst well_formed_slots_of_used_irq_node, assumption+)
  apply (subst object_empty_slots_empty_initialised, assumption)
  apply clarsimp
  done

lemma irq_initialised_decomp_total:
  "\<lbrakk>well_formed spec; irq \<in> used_irqs spec\<rbrakk>
  \<Longrightarrow> irq_initialised spec t irq =
     (irq_slot_initialised spec t irq 0 \<and>*
      object_empty_slots_initialised spec t (cdl_irq_node spec irq) \<and>*
      object_fields_initialised spec t (cdl_irq_node spec irq))"
  apply (subst irq_initialised_decomp)
  apply (subst irq_slots_initialised_decomp, assumption+)
  apply (clarsimp simp: sep_conj_assoc)
  done

lemma irq_empty_decomp_total:
  "\<lbrakk>well_formed spec; irq \<in> used_irqs spec\<rbrakk>
  \<Longrightarrow> irq_empty spec t irq =
     (irq_slot_empty spec t irq 0 \<and>*
      object_empty_slots_initialised spec t (cdl_irq_node spec irq) \<and>*
      object_fields_empty spec t (cdl_irq_node spec irq))"
  apply (subst irq_empty_decomp)
  apply (subst irq_slots_empty_decomp, assumption+)
  apply (clarsimp simp: sep_conj_assoc)
  done

(****************************************************************************
 * Lemmas proving equality between object_fields predicates.                *
 * These show that the fields of a CNode are already initialised correctly. *
 ****************************************************************************)

lemma sep_map_f_object_default_state_cnode [simp]:
  "object_type obj = CNodeType \<Longrightarrow> obj_id \<mapsto>f object_default_state 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: object_type_def split: cdl_object.splits)
  apply (intro ext conjI iffI |
         clarsimp simp: object_to_sep_state_def object_project_def
                        intent_reset_def object_wipe_slots_def
                        object_default_state_def default_object_def
                        asid_reset_def object_type_def update_slots_def
                        empty_cnode_def object_size_bits_def object_clean_def)+
  done

lemma sep_map_f_object_default_state_irq_node [simp]:
  "object_type obj = IRQNodeType \<Longrightarrow> obj_id \<mapsto>f object_default_state 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: object_type_def split: cdl_object.splits)
  apply (intro ext conjI iffI |
         clarsimp simp: object_to_sep_state_def object_project_def
                        intent_reset_def object_wipe_slots_def
                        object_default_state_def default_object_def
                        asid_reset_def object_type_def update_slots_def
                        empty_cnode_def object_size_bits_def object_clean_def)+
  done

lemma object_to_sep_state_fields[simp]:
  "object_to_sep_state obj_id (update_slots slot obj) {Fields} = object_to_sep_state obj_id obj {Fields}"
  apply (rule ext)
  apply (case_tac obj,
    simp_all add:object_to_sep_state_def update_slots_def split_def
    object_project_def object_clean_def asid_reset_def
    object_wipe_slots_def intent_reset_def object_slots_def)
  done

lemma sep_map_f_cnode_half [simp]:
  "obj_id \<mapsto>f cnode_half spec obj_id' obj = obj_id \<mapsto>f obj "
  apply (rule ext)
  apply (clarsimp simp: cnode_half_def sep_map_f_def sep_map_general_def)
  done

lemma sep_map_f_tcb_half [simp]:
  "obj_id \<mapsto>f tcb_half spec tcb = obj_id \<mapsto>f tcb"
  by (clarsimp simp: tcb_half_def sep_map_f_def sep_map_general_def)

lemma irq_node_fields_empty_initialised:
  "irq_node_at obj_id spec
  \<Longrightarrow> object_fields_empty spec spec2s_ids obj_id = object_fields_initialised spec spec2s_ids obj_id"
  by (clarsimp simp: object_fields_empty_def object_fields_initialised_def
                     object_initialised_general_def object_at_def object_type_is_object)

lemma cnode_fields_empty_initialised:
  "cnode_at obj_id spec
  \<Longrightarrow> object_fields_empty spec t obj_id = object_fields_initialised spec t obj_id"
  by (clarsimp simp: object_fields_empty_def object_fields_initialised_def
                     object_initialised_general_def object_at_def object_type_is_object)

lemma cnode_fields_half_initialised_object_fields_initialised:
  "cnode_at obj_id spec
  \<Longrightarrow> cnode_fields_half_initialised spec t obj_id = object_fields_initialised spec t obj_id"
  by (clarsimp simp: cnode_fields_half_initialised_def object_fields_initialised_def object_initialised_general_def)

lemma object_fields_empty_half_initialised:
  "cnode_at obj_id spec
  \<Longrightarrow> cnode_fields_half_initialised spec t obj_id = object_fields_empty spec t obj_id"
  by (clarsimp simp: cnode_fields_half_initialised_object_fields_initialised cnode_fields_empty_initialised)

lemma object_default_state_frame [simp]:
  "is_frame object \<Longrightarrow> object_default_state object = object"
  by (clarsimp simp: object_default_state_def default_object_def
                     object_type_is_object object_type_def
              split: cdl_object.splits)

end


theory Leadsto_Impl
  imports Leadsto_Map Unified_PW_Impl Liveness_Subsumption_Impl
begin

(* XXX Move *)
definition
  "list_of_set S = SPEC (\<lambda>l. set l = S)"

(* XXX Move *)
lemma lso_id_hnr:
  "(return o id, list_of_set) \<in> (lso_assn A)\<^sup>d \<rightarrow>\<^sub>a list_assn A"
  unfolding list_of_set_def lso_assn_def hr_comp_def br_def by sepref_to_hoare sep_auto

sepref_register hm.op_hms_empty

context Worklist_Map2_Impl
begin

sepref_thm pw_algo_map2_impl is
  "uncurry0 (pw_algo_map2)" ::
  "unit_assn\<^sup>k \<rightarrow>\<^sub>a bool_assn \<times>\<^sub>a (hm.hms_assn' K (lso_assn A))"
  unfolding pw_algo_map2_def add_pw'_map2_alt_def PR_CONST_def TRACE'_def[symmetric]
  supply [[goals_limit = 1]]
  supply conv_to_is_Nil[simp]
  unfolding fold_lso_bex
  unfolding take_from_list_alt_def
  apply (rewrite in "{a\<^sub>0}" lso_fold_custom_empty)
  unfolding hm.hms_fold_custom_empty
  apply (rewrite in "[a\<^sub>0]" HOL_list.fold_custom_empty)
   apply (rewrite in "{}" lso_fold_custom_empty)
  unfolding F_split (* XXX Why? F only appears in the invariant *)
  by sepref

end (* Worklist Map 2 Impl *)

locale Leadsto_Search_Space_Key_Impl =
  Leadsto_Search_Space_Key a\<^sub>0 F _ empty _ E key F' P Q succs_Q succs1 +
  liveness: Liveness_Search_Space_Key_Impl a\<^sub>0 F _ V succs_Q "\<lambda> x y. E x y \<and> \<not> empty y \<and> Q y"
    _ key A succsi a\<^sub>0i Lei keyi copyi
  for key :: "'v \<Rightarrow> 'k"
  and a\<^sub>0 F F' copyi P Q V empty succs_Q succs1 E A succsi a\<^sub>0i Lei keyi +
  fixes succs1i and emptyi and Pi Qi and tracei
  assumes  succs1_impl: "(succs1i, (RETURN \<circ>\<circ> PR_CONST) succs1) \<in> A\<^sup>k \<rightarrow>\<^sub>a list_assn A"
    and empty_impl:
      "(emptyi,RETURN o PR_CONST empty) \<in> A\<^sup>k \<rightarrow>\<^sub>a bool_assn"
    assumes [sepref_fr_rules]:
      "(Pi,RETURN o PR_CONST P) \<in> A\<^sup>k \<rightarrow>\<^sub>a bool_assn" "(Qi,RETURN o PR_CONST Q) \<in> A\<^sup>k \<rightarrow>\<^sub>a bool_assn"
    assumes trace_impl:
      "(uncurry tracei,uncurry (\<lambda>(_ :: string) _. RETURN ())) \<in> id_assn\<^sup>k *\<^sub>a A\<^sup>k \<rightarrow>\<^sub>a id_assn"
begin

sublocale Worklist_Map2_Impl _ _ "\<lambda> _. False" _ succs1 _ _ "\<lambda>_. False" _ succs1i _
  "\<lambda>_. return False" Lei
  apply standard
  unfolding A'.trace_def
           apply (rule liveness.refinements succs1_impl)
  subgoal
    by sepref_to_hoare sep_auto
  by (rule liveness.refinements succs1_impl empty_impl
      liveness.pure_K liveness.left_unique_K liveness.right_unique_K trace_impl)+

sepref_register pw_algo_map2_copy
sepref_register "PR_CONST P" "PR_CONST Q"

lemmas [sepref_fr_rules] =
  lso_id_hnr
  ran_of_map_impl.refine[OF pure_K left_unique_K right_unique_K]

lemma pw_algo_map2_copy_fold:
  "PR_CONST pw_algo_map2_copy = A'.pw_algo_map2"
  unfolding pw_algo_map2_copy_def by simp

lemmas [sepref_fr_rules] = pw_algo_map2_impl.refine_raw[folded pw_algo_map2_copy_fold]

definition "has_cycle_map_copy \<equiv> has_cycle_map"

lemma has_cycle_map_copy_fold:
  "PR_CONST has_cycle_map_copy = has_cycle_map"
  unfolding has_cycle_map_copy_def by simp

sepref_register has_cycle_map_copy

lemma has_cycle_map_fold:
  "has_cycle_map = liveness.dfs_map'"
  unfolding has_cycle_map_def liveness.dfs_map'_def
  by (subst Liveness_Search_Space_Key_Defs.dfs_map_alt_def) standard

lemmas [sepref_fr_rules] =
  liveness.dfs_map'_impl.refine_raw[folded has_cycle_map_fold, folded has_cycle_map_copy_fold]

sepref_thm leadsto_impl is
  "uncurry0 leadsto_map4'" :: "unit_assn\<^sup>k \<rightarrow>\<^sub>a bool_assn"
  unfolding leadsto_map4'_def
  apply (rewrite in P PR_CONST_def[symmetric])
  apply (rewrite in Q PR_CONST_def[symmetric])
  unfolding pw_algo_map2_copy_def[symmetric]
  unfolding has_cycle_map_copy_def[symmetric]
  unfolding hm.hms_fold_custom_empty
  unfolding list_of_set_def[symmetric]
  by sepref

concrete_definition (in -) leadsto_impl
  uses Leadsto_Search_Space_Key_Impl.leadsto_impl.refine_raw is "(uncurry0 ?f,_)\<in>_"

lemma leadsto_impl_hnr:
  "(uncurry0 (
    leadsto_impl copyi succsi a\<^sub>0i Lei keyi succs1i emptyi Pi Qi tracei
    ),
    uncurry0 leadsto_spec_alt
   ) \<in> unit_assn\<^sup>k \<rightarrow>\<^sub>a bool_assn" if "V a\<^sub>0"
  unfolding leadsto_spec_alt_def
  using leadsto_impl.refine[
    OF Leadsto_Search_Space_Key_Impl_axioms,
    FCOMP leadsto_map4'_correct[unfolded leadsto_spec_alt_def, THEN Id_SPEC_refine, THEN nres_relI]
    ] .

end (* Leadsto Search Space Key Impl *)

end (* End of theory *)


(*  Title:      HOL/Auth/n_german_lemma_inv__51_on_rules.thy
    Author:     Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
    Copyright    2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)

header{*The n_german Protocol Case Study*}

theory n_german_lemma_inv__51_on_rules imports n_german_lemma_on_inv__51
begin
section{*All lemmas on causal relation between inv__51*}
lemma lemma_inv__51_on_rules:
  assumes b1: "r \<in> rules N" and b2: "(\<exists> p__Inv3 p__Inv4. p__Inv3\<le>N\<and>p__Inv4\<le>N\<and>p__Inv3~=p__Inv4\<and>f=inv__51  p__Inv3 p__Inv4)"
  shows "invHoldForRule s f r (invariants N)"
  proof -
  have c1: "(\<exists> j. j\<le>N\<and>r=n_SendReqS  j)\<or>
    (\<exists> i. i\<le>N\<and>r=n_SendReqEI  i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_SendReqES  i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_RecvReq N i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_SendInvE  i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_SendInvS  i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_SendInvAck  i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_RecvInvAck  i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_SendGntS  i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_SendGntE N i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_RecvGntS  i)\<or>
    (\<exists> i. i\<le>N\<and>r=n_RecvGntE  i)\<or>
    (\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store  i d)"
  apply (cut_tac b1, auto) done
    moreover {
      assume d1: "(\<exists> j. j\<le>N\<and>r=n_SendReqS  j)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_SendReqSVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendReqEI  i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_SendReqEIVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendReqES  i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_SendReqESVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvReq N i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_RecvReqVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendInvE  i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_SendInvEVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendInvS  i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_SendInvSVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendInvAck  i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_SendInvAckVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvInvAck  i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_RecvInvAckVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendGntS  i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_SendGntSVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_SendGntEVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntS  i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_RecvGntSVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntE  i)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_RecvGntEVsinv__51) done
    }

    moreover {
      assume d1: "(\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store  i d)"
      have "invHoldForRule s f r (invariants N)"
      apply (cut_tac b2 d1, metis n_StoreVsinv__51) done
    }

  ultimately show "invHoldForRule s f r (invariants N)"
  by satx
qed

end


(*  Title:      HOL/Auth/n_german_lemma_on_inv__28.thy
    Author:     Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
    Copyright    2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)

header{*The n_german Protocol Case Study*}

theory n_german_lemma_on_inv__28 imports n_german_base
begin
section{*All lemmas on causal relation between inv__28 and some rule r*}
lemma n_RecvReqSVsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvReqS N i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvReqS N i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_RecvReqEVsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvReqE N i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvReqE N i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendInv__part__0Vsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__0  i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInv__part__0  i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendInv__part__1Vsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__1  i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInv__part__1  i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendInvAckVsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInvAck  i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInvAck  i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P3 s"
  apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Ident ''CurCmd'')) (Const Empty)) (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv2) ''Cmd'')) (Const Inv))))" in exI, auto) done
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_RecvInvAckVsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvInvAck  i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvInvAck  i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
  moreover {
    assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  moreover {
    assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
  moreover {
    assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  moreover {
    assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
    have "?P2 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendGntSVsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntS  i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntS  i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendGntEVsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntE N i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_RecvGntSVsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntS  i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvGntS  i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P3 s"
  apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Ident ''ExGntd'')) (Const true)) (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv2) ''Cmd'')) (Const GntS))))" in exI, auto) done
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_RecvGntEVsinv__28:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntE  i)" and
a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvGntE  i" apply fastforce done
from a2 obtain p__Inv2 where a2:"p__Inv2\<le>N\<and>f=inv__28  p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendReqE__part__1Vsinv__28:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqE__part__1  i" and
  a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_StoreVsinv__28:
  assumes a1: "\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store  i d" and
  a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_SendReqE__part__0Vsinv__28:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqE__part__0  i" and
  a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

lemma n_SendReqSVsinv__28:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqS  i" and
  a2: "(\<exists> p__Inv2. p__Inv2\<le>N\<and>f=inv__28  p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done

end