theory Retype_C

lemma createObjects_ccorres_ep:
  defines "ko \<equiv> (KOEndpoint (makeObject :: endpoint))"
  shows "\<forall>\<sigma> x. (\<sigma>, x) \<in> rf_sr
  \<and> ptr \<noteq> 0
  \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma>
  \<and> pspace_no_overlap' ptr sz \<sigma>
  \<and> ret_zero ptr (n * (2 ^ objBitsKO ko)) \<sigma>
  \<and> region_is_zero_bytes ptr (n * (2 ^ objBitsKO ko)) x
  \<and> range_cover ptr sz (objBitsKO ko) n
  \<and> {ptr ..+ n * (2 ^ objBitsKO ko)} \<inter> kernel_data_refs = {}
  \<longrightarrow>
  (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>,
   x\<lparr>globals := globals x
                 \<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (Ptr ptr :: endpoint_C ptr) False)
                   (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr"
  (is "\<forall>\<sigma> x. ?P \<sigma> x \<longrightarrow>
    (\<sigma>\<lparr>ksPSpace := ?ks \<sigma>\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := ?ks' x\<rparr>\<rparr>) \<in> rf_sr") proof (intro impI allI) fix \<sigma> x let ?thesis = "(\<sigma>\<lparr>ksPSpace := ?ks \<sigma>\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := ?ks' x\<rparr>\<rparr>) \<in> rf_sr" let ?ks = "?ks \<sigma>" let ?ks' = "?ks' x" let ?ptr = "Ptr ptr :: endpoint_C ptr" assume "?P \<sigma> x" hence rf: "(\<sigma>, x) \<in> rf_sr"
    and cover: "range_cover ptr sz (objBitsKO ko) n"
    and al: "is_aligned ptr (objBitsKO ko)" and ptr0: "ptr \<noteq> 0"
    and sz: "objBitsKO ko \<le> sz"
    and szb: "sz < word_bits"
    and pal: "pspace_aligned' \<sigma>" and pdst: "pspace_distinct' \<sigma>"
    and pno: "pspace_no_overlap' ptr sz \<sigma>"
    and rzo: "ret_zero ptr (n * (2 ^ objBitsKO ko)) \<sigma>"
    and empty: "region_is_bytes ptr (n * (2 ^ objBitsKO ko)) x"
    and zero: "heap_list_is_zero (hrs_mem (t_hrs_' (globals x))) ptr (n * (2 ^ objBitsKO ko))"
    and rc: "range_cover ptr sz (objBitsKO ko) n"
    and kdr: "{ptr..+n * (2 ^ objBitsKO ko)} \<inter> kernel_data_refs = {}" by (clarsimp simp:range_cover_def[where 'a=32, folded word_bits_def])+ 

  (* obj specific *)
   have mko: "\<And>dev. makeObjectKO dev (Inr (APIObjectType ArchTypes_H.apiobject_type.EndpointObject)) = Some ko" by (simp add: ko_def makeObjectKO_def) have relrl:
    "cendpoint_relation (cslift x) makeObject (from_bytes (replicate (size_of TYPE(endpoint_C)) 0))" unfolding cendpoint_relation_def apply (simp add: Let_def makeObject_endpoint size_of_def endpoint_lift_def) apply (simp add: from_bytes_def) apply (simp add: typ_info_simps endpoint_C_tag_def endpoint_lift_def
      size_td_lt_final_pad size_td_lt_ti_typ_pad_combine Let_def size_of_def) apply (simp add: final_pad_def Let_def size_td_lt_ti_typ_pad_combine Let_def
      size_of_def padup_def align_td_array' size_td_array update_ti_adjust_ti
      ti_typ_pad_combine_def Let_def ti_typ_combine_def empty_typ_info_def) apply (simp add: typ_info_array array_tag_def eval_nat_numeral) apply (simp add: array_tag_n_eq) apply (simp add: final_pad_def Let_def size_td_lt_ti_typ_pad_combine
      size_of_def padup_def align_td_array' size_td_array update_ti_adjust_ti
      ti_typ_pad_combine_def ti_typ_combine_def empty_typ_info_def) apply (simp add: EPState_Idle_def update_ti_t_word32_0s) done 

  (* /obj specific *)

  (* s/obj/obj'/ *)
   have szo: "size_of TYPE(endpoint_C) = 2 ^ objBitsKO ko" by (simp add: size_of_def objBits_simps' ko_def) have szo': "n * (2 ^ objBitsKO ko) = n * size_of TYPE(endpoint_C)" by (metis szo) note rl' = cslift_ptr_retyp_other_inst[OF empty cover[simplified] szo' szo] note rl = projectKO_opt_retyp_other [OF rc pal pno ko_def] note cterl = retype_ctes_helper [OF pal pdst pno al sz szb mko rc, simplified] note ht_rl = clift_eq_h_t_valid_eq[OF rl', OF tag_disj_via_td_name, simplified] have guard:
    "\<forall>b < n. c_guard (CTypesDefs.ptr_add ?ptr (of_nat b))" apply (rule retype_guard_helper [where m = 2, OF cover ptr0 szo]) apply (simp add: ko_def objBits_simps') apply (simp add: align_of_def) done from rf have "cpspace_relation (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>)) (t_hrs_' (globals x))" unfolding rf_sr_def cstate_relation_def by (simp add: Let_def) hence "cpspace_relation ?ks  (underlying_memory (ksMachineState \<sigma>)) ?ks'" unfolding cpspace_relation_def apply - supply image_cong_simp [cong del] apply (clarsimp simp: rl' cterl tag_disj_via_td_name foldr_upd_app_if [folded data_map_insert_def]
      heap_to_user_data_def cte_C_size heap_to_device_data_def) apply (subst clift_ptr_retyps_gen_prev_memset_same[OF guard _ _ szo' _ zero],
      simp_all only: szo empty, simp_all) apply (rule range_cover.strong_times_32[OF cover refl]) apply (simp add: ptr_add_to_new_cap_addrs [OF szo] ht_rl) apply (simp add: rl projectKO_opt_retyp_same projectKOs) apply (simp add: ko_def projectKO_opt_retyp_same projectKOs cong: if_cong) apply (erule cmap_relation_retype) apply (rule relrl[simplified szo ko_def]) done thus ?thesis using rf empty kdr rzo apply (simp add: rf_sr_def cstate_relation_def Let_def rl'
                   tag_disj_via_td_name) apply (simp add: carch_state_relation_def cmachine_state_relation_def) apply (simp add: rl' cterl tag_disj_via_td_name h_t_valid_clift_Some_iff) apply (clarsimp simp: hrs_htd_update ptr_retyps_htd_safe_neg szo
                        kernel_data_refs_domain_eq_rotate
                        ht_rl foldr_upd_app_if [folded data_map_insert_def]
                        rl projectKOs cvariable_array_ptr_retyps[OF szo]
                        zero_ranges_ptr_retyps
              simp del: endpoint_C_size) done qed

(* Proof state trace *)
proof (prove)
goal (1 subgoal):
 1. \<forall>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
this:
(\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {}

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
using this:
(\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {}

goal (1 subgoal):
 1. (((\<sigma>, x) \<in> rf_sr &&& range_cover ptr sz (objBitsKO ko) n &&& is_aligned ptr (objBitsKO ko)) &&& (ptr \<noteq> 0 &&& objBitsKO ko \<le> sz) &&& sz < word_bits &&& pspace_aligned' \<sigma>) &&& (pspace_distinct' \<sigma> &&& pspace_no_overlap' ptr sz \<sigma> &&& (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma>) &&& (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x &&& heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) &&& range_cover ptr sz (objBitsKO ko) n &&& {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} proof (state)
this:
(\<sigma>, x) \<in> rf_sr
range_cover ptr sz (objBitsKO ko) n
is_aligned ptr (objBitsKO ko)
ptr \<noteq> 0
objBitsKO ko \<le> sz
sz < word_bits
pspace_aligned' \<sigma>
pspace_distinct' \<sigma>
pspace_no_overlap' ptr sz \<sigma>
(valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma>
region_is_bytes ptr (n * 2 ^ objBitsKO ko) x
heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0
range_cover ptr sz (objBitsKO ko) n
{ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {}

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
this:
(\<sigma>, x) \<in> rf_sr
range_cover ptr sz (objBitsKO ko) n
is_aligned ptr (objBitsKO ko)
ptr \<noteq> 0
objBitsKO ko \<le> sz
sz < word_bits
pspace_aligned' \<sigma>
pspace_distinct' \<sigma>
pspace_no_overlap' ptr sz \<sigma>
(valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma>
region_is_bytes ptr (n * 2 ^ objBitsKO ko) x
heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0
range_cover ptr sz (objBitsKO ko) n
{ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {}

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
goal (1 subgoal):
 1. \<And>dev. makeObjectKO dev (Inr (APIObjectType ArchTypes_H.apiobject_type.EndpointObject)) = Some ko proof (state)
this:
makeObjectKO ?dev2 (Inr (APIObjectType ArchTypes_H.apiobject_type.EndpointObject)) = Some ko

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
goal (1 subgoal):
 1. cendpoint_relation (cslift x) makeObject (from_bytes (replicate (size_of TYPE(endpoint_C)) 0)) proof (prove)
goal (1 subgoal):
 1. let cstate = endpoint_CL.state_CL (endpoint_lift (from_bytes (replicate (size_of TYPE(endpoint_C)) 0))); chead = (tcb_Ptr \<circ> epQueue_head_CL \<circ> endpoint_lift) (from_bytes (replicate (size_of TYPE(endpoint_C)) 0)); cend = (tcb_Ptr \<circ> epQueue_tail_CL \<circ> endpoint_lift) (from_bytes (replicate (size_of TYPE(endpoint_C)) 0)) in case makeObject of Structures_H.endpoint.RecvEP q \<Rightarrow> cstate = SCAST(32 signed \<rightarrow> 32) EPState_Recv \<and> ep_queue_relation' (cslift x) q chead cend | Structures_H.endpoint.IdleEP \<Rightarrow> cstate = SCAST(32 signed \<rightarrow> 32) EPState_Idle \<and> ep_queue_relation' (cslift x) [] chead cend | Structures_H.endpoint.SendEP q \<Rightarrow> cstate = SCAST(32 signed \<rightarrow> 32) EPState_Send \<and> ep_queue_relation' (cslift x) q chead cend proof (prove)
goal (1 subgoal):
 1. endpoint_C.words_C (from_bytes [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]).[0] && mask 2 = SCAST(32 signed \<rightarrow> 32) EPState_Idle \<and> endpoint_C.words_C (from_bytes [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]).[0] && ~~ mask 4 = 0 \<and> endpoint_C.words_C (from_bytes [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]).[Suc 0] && ~~ mask 4 = 0 proof (prove)
goal (1 subgoal):
 1. endpoint_C.words_C (update_ti_t (typ_info_t TYPE(endpoint_C)) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] undefined).[0] && mask 2 = SCAST(32 signed \<rightarrow> 32) EPState_Idle \<and> endpoint_C.words_C (update_ti_t (typ_info_t TYPE(endpoint_C)) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] undefined).[0] && ~~ mask 4 = 0 \<and> endpoint_C.words_C (update_ti_t (typ_info_t TYPE(endpoint_C)) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] undefined).[Suc 0] && ~~ mask 4 = 0 proof (prove)
goal (1 subgoal):
 1. endpoint_C.words_C (update_ti_t (final_pad (ti_typ_pad_combine TYPE(32 word[4]) endpoint_C.words_C (endpoint_C.words_C_update \<circ> (\<lambda>x _. x)) ''words_C'' (empty_typ_info ''endpoint_C''))) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] undefined).[0] && mask 2 = SCAST(32 signed \<rightarrow> 32) EPState_Idle \<and> endpoint_C.words_C (update_ti_t (final_pad (ti_typ_pad_combine TYPE(32 word[4]) endpoint_C.words_C (endpoint_C.words_C_update \<circ> (\<lambda>x _. x)) ''words_C'' (empty_typ_info ''endpoint_C''))) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] undefined).[0] && ~~ mask 4 = 0 \<and> endpoint_C.words_C (update_ti_t (final_pad (ti_typ_pad_combine TYPE(32 word[4]) endpoint_C.words_C (endpoint_C.words_C_update \<circ> (\<lambda>x _. x)) ''words_C'' (empty_typ_info ''endpoint_C''))) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] undefined).[Suc 0] && ~~ mask 4 = 0 proof (prove)
goal (1 subgoal):
 1. update_ti_t (typ_info_t TYPE(32 word[4])) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] (endpoint_C.words_C undefined).[0] && mask 2 = SCAST(32 signed \<rightarrow> 32) EPState_Idle \<and> update_ti_t (typ_info_t TYPE(32 word[4])) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] (endpoint_C.words_C undefined).[0] && ~~ mask 4 = 0 \<and> update_ti_t (typ_info_t TYPE(32 word[4])) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] (endpoint_C.words_C undefined).[Suc 0] && ~~ mask 4 = 0 proof (prove)
goal (1 subgoal):
 1. update_ti_t (array_tag_n (Suc (Suc (Suc (Suc 0))))) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] (endpoint_C.words_C undefined).[0] && mask (Suc (Suc 0)) = SCAST(32 signed \<rightarrow> 32) EPState_Idle \<and> update_ti_t (array_tag_n (Suc (Suc (Suc (Suc 0))))) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] (endpoint_C.words_C undefined).[0] && ~~ mask (Suc (Suc (Suc (Suc 0)))) = 0 \<and> update_ti_t (array_tag_n (Suc (Suc (Suc (Suc 0))))) [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] (endpoint_C.words_C undefined).[Suc 0] && ~~ mask (Suc (Suc (Suc (Suc 0)))) = 0 proof (prove)
goal (1 subgoal):
 1. update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[0]) (\<lambda>x f. Arrays.update f 0 x)) [0, 0, 0, 0] (update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[Suc 0]) (\<lambda>x f. Arrays.update f (Suc 0) x)) [0, 0, 0, 0] (update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[Suc (Suc 0)]) (\<lambda>x f. Arrays.update f (Suc (Suc 0)) x)) [0, 0, 0, 0] (update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[Suc (Suc (Suc 0))]) (\<lambda>x f. Arrays.update f (Suc (Suc (Suc 0))) x)) [0, 0, 0, 0] (endpoint_C.words_C undefined)))).[0] && mask (Suc (Suc 0)) = SCAST(32 signed \<rightarrow> 32) EPState_Idle \<and> update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[0]) (\<lambda>x f. Arrays.update f 0 x)) [0, 0, 0, 0] (update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[Suc 0]) (\<lambda>x f. Arrays.update f (Suc 0) x)) [0, 0, 0, 0] (update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[Suc (Suc 0)]) (\<lambda>x f. Arrays.update f (Suc (Suc 0)) x)) [0, 0, 0, 0] (update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[Suc (Suc (Suc 0))]) (\<lambda>x f. Arrays.update f (Suc (Suc (Suc 0))) x)) [0, 0, 0, 0] (endpoint_C.words_C undefined)))).[0] && ~~ mask (Suc (Suc (Suc (Suc 0)))) = 0 \<and> update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[0]) (\<lambda>x f. Arrays.update f 0 x)) [0, 0, 0, 0] (update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[Suc 0]) (\<lambda>x f. Arrays.update f (Suc 0) x)) [0, 0, 0, 0] (update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[Suc (Suc 0)]) (\<lambda>x f. Arrays.update f (Suc (Suc 0)) x)) [0, 0, 0, 0] (update_ti_t (adjust_ti (typ_info_t TYPE(32 word)) (\<lambda>x. x.[Suc (Suc (Suc 0))]) (\<lambda>x f. Arrays.update f (Suc (Suc (Suc 0))) x)) [0, 0, 0, 0] (endpoint_C.words_C undefined)))).[Suc 0] && ~~ mask (Suc (Suc (Suc (Suc 0)))) = 0 proof (prove)
goal (1 subgoal):
 1. update_ti_t (typ_info_t TYPE(32 word)) [0, 0, 0, 0] (endpoint_C.words_C undefined.[0]) && mask (Suc (Suc 0)) = SCAST(32 signed \<rightarrow> 32) EPState_Idle \<and> update_ti_t (typ_info_t TYPE(32 word)) [0, 0, 0, 0] (endpoint_C.words_C undefined.[0]) && ~~ mask (Suc (Suc (Suc (Suc 0)))) = 0 \<and> update_ti_t (typ_info_t TYPE(32 word)) [0, 0, 0, 0] (endpoint_C.words_C undefined.[Suc 0]) && ~~ mask (Suc (Suc (Suc (Suc 0)))) = 0 proof (prove)
goal:
No subgoals! proof (state)
this:
cendpoint_relation (cslift x) makeObject (from_bytes (replicate (size_of TYPE(endpoint_C)) 0))

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
this:
cendpoint_relation (cslift x) makeObject (from_bytes (replicate (size_of TYPE(endpoint_C)) 0))

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
goal (1 subgoal):
 1. size_of TYPE(endpoint_C) = 2 ^ objBitsKO ko proof (state)
this:
size_of TYPE(endpoint_C) = 2 ^ objBitsKO ko

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
goal (1 subgoal):
 1. n * 2 ^ objBitsKO ko = n * size_of TYPE(endpoint_C) proof (state)
this:
n * 2 ^ objBitsKO ko = n * size_of TYPE(endpoint_C)

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
this:
\<lbrakk>typ_uinfo_t TYPE(?'b) \<bottom>\<^sub>t typ_uinfo_t TYPE(endpoint_C); typ_uinfo_t TYPE(?'b) \<noteq> typ_uinfo_t TYPE(8 word)\<rbrakk> \<Longrightarrow> clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) ?arr) (t_hrs_' (globals x))) = cslift x

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
this:
\<forall>v. projectKO_opt (KOEndpoint makeObject) \<noteq> Some v \<Longrightarrow> projectKO_opt \<circ>\<^sub>m (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) = projectKO_opt \<circ>\<^sub>m ksPSpace \<sigma>

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
this:
map_to_ctes (\<lambda>xa. if xa \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> xa) = ctes_of \<sigma>

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (state)
this:
\<lbrakk>typ_name (typ_info_t TYPE(?'b1)) \<noteq> ''!pad_typ''; typ_name (typ_info_t TYPE(?'b1)) \<noteq> ''endpoint_C'' \<and> typ_name (typ_info_t TYPE(?'b1)) \<noteq> ''word0000010'' \<and> typ_name (typ_info_t TYPE(?'b1)) \<noteq> ''word0000010_array_0010''; ''endpoint_C'' \<notin> td_names (typ_info_t TYPE(?'b1)); typ_uinfo_t TYPE(?'b1) \<noteq> typ_uinfo_t TYPE(8 word)\<rbrakk> \<Longrightarrow> h_t_valid (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) ?arr1) (t_hrs_' (globals x)))) c_guard = c_h_t_valid x

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
goal (1 subgoal):
 1. \<forall>b<n. c_guard (ep_Ptr ptr +\<^sub>p int b) proof (prove)
goal (2 subgoals):
 1. 2 \<le> objBitsKO ko
 2. align_of TYPE(endpoint_C) = 2\<^sup>2 proof (prove)
goal (1 subgoal):
 1. align_of TYPE(endpoint_C) = 2\<^sup>2 proof (prove)
goal:
No subgoals! proof (state)
this:
\<forall>b<n. c_guard (ep_Ptr ptr +\<^sub>p int b)

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (chain)
picking this:
(\<sigma>, x) \<in> rf_sr proof (prove)
using this:
(\<sigma>, x) \<in> rf_sr

goal (1 subgoal):
 1. cpspace_relation (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>)) (t_hrs_' (globals x)) proof (prove)
using this:
(\<sigma>, x) \<in> {(s, s'). let cheap = t_hrs_' (globals s') in cpspace_relation (ksPSpace s) (underlying_memory (ksMachineState s)) cheap \<and> cready_queues_relation (clift cheap) (ksReadyQueues_' (globals s')) (ksReadyQueues s) \<and> zero_ranges_are_zero (gsUntypedZeroRanges s) cheap \<and> cbitmap_L1_relation (ksReadyQueuesL1Bitmap_' (globals s')) (ksReadyQueuesL1Bitmap s) \<and> cbitmap_L2_relation (ksReadyQueuesL2Bitmap_' (globals s')) (ksReadyQueuesL2Bitmap s) \<and> ksCurThread_' (globals s') = tcb_ptr_to_ctcb_ptr (ksCurThread s) \<and> ksIdleThread_' (globals s') = tcb_ptr_to_ctcb_ptr (ksIdleThread s) \<and> cinterrupt_relation (ksInterruptState s) intStateIRQNode_array_Ptr (intStateIRQTable_' (globals s')) \<and> cscheduler_action_relation (ksSchedulerAction s) (ksSchedulerAction_' (globals s')) \<and> carch_state_relation (ksArchState s) (globals s') \<and> cmachine_state_relation (ksMachineState s) (globals s') \<and> cte_array_relation s (globals s') \<and> tcb_cte_array_relation s (globals s') \<and> apsnd fst (ghost'state_' (globals s')) = (gsUserPages s, gsCNodes s) \<and> ghost_size_rel (ghost'state_' (globals s')) (gsMaxObjectSize s) \<and> ksWorkUnitsCompleted_' (globals s') = ksWorkUnitsCompleted s \<and> s' \<Turnstile>\<^sub>c intStateIRQNode_array_Ptr \<and> ptr_span intStateIRQNode_array_Ptr \<subseteq> kernel_data_refs \<and> s' \<Turnstile>\<^sub>c armKSGlobalPD_Ptr \<and> ptr_span armKSGlobalPD_Ptr \<subseteq> kernel_data_refs \<and> htd_safe domain (hrs_htd (t_hrs_' (globals s'))) \<and> - domain \<subseteq> kernel_data_refs \<and> globals_list_distinct (- kernel_data_refs) symbol_table kernel_all_global_addresses.global_data_list \<and> cdom_schedule_relation (kernel_state.ksDomSchedule s) kernel_all_global_addresses.ksDomSchedule \<and> ksDomScheduleIdx_' (globals s') = word_of_nat (ksDomScheduleIdx s) \<and> ksCurDomain_' (globals s') = UCAST(8 \<rightarrow> 32) (ksCurDomain s) \<and> ksDomainTime_' (globals s') = ksDomainTime s}

goal (1 subgoal):
 1. cpspace_relation (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>)) (t_hrs_' (globals x)) proof (state)
this:
cpspace_relation (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>)) (t_hrs_' (globals x))

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
using this:
cpspace_relation (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>)) (t_hrs_' (globals x))

goal (1 subgoal):
 1. cpspace_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) proof (prove)
using this:
cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation \<and> cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation \<and> cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)) \<and> cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)) \<and> cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation \<and> cmap_relation (heap_to_user_data (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>))) (cslift x) PTR(user_data_C) cuser_user_data_relation \<and> cmap_relation (heap_to_device_data (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>))) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation \<and> carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr \<and> carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr

goal (1 subgoal):
 1. cmap_relation (map_to_ctes (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) cte_Ptr ccte_relation \<and> cpspace_tcb_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cmap_relation (map_to_eps (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ep_Ptr (cendpoint_relation (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))))) \<and> cmap_relation (map_to_ntfns (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ntfn_Ptr (cnotification_relation (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))))) \<and> cmap_relation (map_to_pdes (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ap_Ptr casid_pool_relation \<and> cpspace_user_data_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_device_data_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_pde_array_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_pte_array_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) proof (prove)
goal (1 subgoal):
 1. cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation \<and> cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation \<and> cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)) \<and> cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)) \<and> cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation \<and> cmap_relation (heap_to_user_data (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>))) (cslift x) PTR(user_data_C) cuser_user_data_relation \<and> cmap_relation (heap_to_device_data (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>))) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation \<and> carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr \<and> carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr \<Longrightarrow> cmap_relation (map_to_ctes (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) cte_Ptr ccte_relation \<and> cpspace_tcb_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cmap_relation (map_to_eps (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ep_Ptr (cendpoint_relation (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))))) \<and> cmap_relation (map_to_ntfns (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ntfn_Ptr (cnotification_relation (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))))) \<and> cmap_relation (map_to_pdes (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ap_Ptr casid_pool_relation \<and> cpspace_user_data_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_device_data_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_pde_array_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_pte_array_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) proof (prove)
goal (1 subgoal):
 1. cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation \<and> cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation \<and> cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)) \<and> cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)) \<and> cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation \<and> cmap_relation (heap_to_user_data (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>))) (cslift x) PTR(user_data_C) cuser_user_data_relation \<and> cmap_relation (heap_to_device_data (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>))) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation \<and> carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr \<and> carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr \<Longrightarrow> cmap_relation (map_to_ctes (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) cte_Ptr ccte_relation \<and> cpspace_tcb_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cmap_relation (map_to_eps (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ep_Ptr (cendpoint_relation (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))))) \<and> cmap_relation (map_to_ntfns (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ntfn_Ptr (cnotification_relation (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))))) \<and> cmap_relation (map_to_pdes (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ap_Ptr casid_pool_relation \<and> cpspace_user_data_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_device_data_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_pde_array_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_pte_array_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) proof (prove)
goal (1 subgoal):
 1. \<lbrakk>cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation; cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation; cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)); cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)); cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation; cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation; cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation; carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr; carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr\<rbrakk> \<Longrightarrow> cmap_relation (map_to_tcbs (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation \<and> cmap_relation (map_to_eps (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) ep_Ptr (cendpoint_relation (cslift x)) \<and> cmap_relation (map_to_ntfns (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)) \<and> cmap_relation (map_to_pdes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) ap_Ptr casid_pool_relation \<and> cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation \<and> cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation \<and> cpspace_pde_array_relation (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_pte_array_relation (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) proof (prove)
goal (2 subgoals):
 1. \<lbrakk>cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation; cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation; cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)); cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)); cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation; cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation; cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation; carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr; carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr\<rbrakk> \<Longrightarrow> n * 2 ^ objBitsKO ko < 2 ^ word_bits
 2. \<lbrakk>cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation; cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation; cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)); cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)); cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation; cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation; cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation; carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr; carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr\<rbrakk> \<Longrightarrow> cmap_relation (map_to_tcbs (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation \<and> cmap_relation (map_to_eps (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (\<lambda>y. if y \<in> ((+\<^sub>p) (ep_Ptr ptr) \<circ> int) ` {k. k < n} then Some (from_bytes (replicate (2 ^ objBitsKO ko) 0)) else cslift x y) ep_Ptr (cendpoint_relation (cslift x)) \<and> cmap_relation (map_to_ntfns (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)) \<and> cmap_relation (map_to_pdes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) ap_Ptr casid_pool_relation \<and> cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation \<and> cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation \<and> cpspace_pde_array_relation (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_pte_array_relation (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) proof (prove)
goal (1 subgoal):
 1. \<lbrakk>cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation; cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation; cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)); cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)); cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation; cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation; cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation; carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr; carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr\<rbrakk> \<Longrightarrow> cmap_relation (map_to_tcbs (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation \<and> cmap_relation (map_to_eps (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (\<lambda>y. if y \<in> ((+\<^sub>p) (ep_Ptr ptr) \<circ> int) ` {k. k < n} then Some (from_bytes (replicate (2 ^ objBitsKO ko) 0)) else cslift x y) ep_Ptr (cendpoint_relation (cslift x)) \<and> cmap_relation (map_to_ntfns (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)) \<and> cmap_relation (map_to_pdes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) ap_Ptr casid_pool_relation \<and> cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation \<and> cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation \<and> cpspace_pde_array_relation (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cpspace_pte_array_relation (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) proof (prove)
goal (1 subgoal):
 1. \<lbrakk>cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation; cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation; cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)); cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)); cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation; cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation; cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation; carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr; carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr\<rbrakk> \<Longrightarrow> cmap_relation (map_to_tcbs (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation \<and> cmap_relation (map_to_eps (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (\<lambda>y. if y \<in> ep_Ptr ` set (new_cap_addrs n ptr ko) then Some (from_bytes (replicate (2 ^ objBitsKO ko) 0)) else cslift x y) ep_Ptr (cendpoint_relation (cslift x)) \<and> cmap_relation (map_to_ntfns (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)) \<and> cmap_relation (map_to_pdes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) pde_Ptr cpde_relation \<and> cmap_relation (map_to_ptes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) pte_Ptr cpte_relation \<and> cmap_relation (map_to_asidpools (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (cslift x) ap_Ptr casid_pool_relation \<and> cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation \<and> cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation \<and> carray_map_relation pdBits (map_to_pdes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (c_h_t_valid x) pd_Ptr \<and> carray_map_relation ptBits (map_to_ptes (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (c_h_t_valid x) pt_Ptr proof (prove)
goal (1 subgoal):
 1. \<lbrakk>cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation; cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation; cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)); cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)); cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation; cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation; cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation; carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr; carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr\<rbrakk> \<Longrightarrow> cmap_relation (map_to_eps (\<lambda>x. if x \<in> set (new_cap_addrs n ptr ko) then Some ko else ksPSpace \<sigma> x)) (\<lambda>y. if y \<in> ep_Ptr ` set (new_cap_addrs n ptr ko) then Some (from_bytes (replicate (2 ^ objBitsKO ko) 0)) else cslift x y) ep_Ptr (cendpoint_relation (cslift x)) proof (prove)
goal (1 subgoal):
 1. \<lbrakk>cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation; cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation; cmap_relation (map_to_eps (ksPSpace \<sigma>)) (cslift x) ep_Ptr (cendpoint_relation (cslift x)); cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)); cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation; cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation; cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation; carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr; carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr\<rbrakk> \<Longrightarrow> cmap_relation (\<lambda>x. if x \<in> set (new_cap_addrs n ptr (KOEndpoint makeObject)) then Some makeObject else map_to_eps (ksPSpace \<sigma>) x) (\<lambda>y. if y \<in> ep_Ptr ` set (new_cap_addrs n ptr (KOEndpoint makeObject)) then Some (from_bytes (replicate (2 ^ objBitsKO (KOEndpoint makeObject)) 0)) else cslift x y) ep_Ptr (cendpoint_relation (cslift x)) proof (prove)
goal (1 subgoal):
 1. \<lbrakk>cmap_relation (ctes_of \<sigma>) (cslift x) cte_Ptr ccte_relation; cmap_relation (map_to_tcbs (ksPSpace \<sigma>)) (cslift x) tcb_ptr_to_ctcb_ptr ctcb_relation; cmap_relation (map_to_ntfns (ksPSpace \<sigma>)) (cslift x) ntfn_Ptr (cnotification_relation (cslift x)); cmap_relation (map_to_pdes (ksPSpace \<sigma>)) (cslift x) pde_Ptr cpde_relation; cmap_relation (map_to_ptes (ksPSpace \<sigma>)) (cslift x) pte_Ptr cpte_relation; cmap_relation (map_to_asidpools (ksPSpace \<sigma>)) (cslift x) ap_Ptr casid_pool_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_C) cuser_user_data_relation; cmap_relation (\<lambda>p. map_option (\<lambda>_. byte_to_word_heap (underlying_memory (ksMachineState \<sigma>)) p) (map_to_user_data_device (ksPSpace \<sigma>) p)) (cslift x) PTR(user_data_device_C) cuser_user_data_device_relation; carray_map_relation pdBits (map_to_pdes (ksPSpace \<sigma>)) (c_h_t_valid x) pd_Ptr; carray_map_relation ptBits (map_to_ptes (ksPSpace \<sigma>)) (c_h_t_valid x) pt_Ptr\<rbrakk> \<Longrightarrow> cendpoint_relation (cslift x) makeObject (from_bytes (replicate (2 ^ objBitsKO (KOEndpoint makeObject)) 0)) proof (prove)
goal:
No subgoals! proof (state)
this:
cpspace_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))

goal (1 subgoal):
 1. \<And>\<sigma> x. (\<sigma>, x) \<in> rf_sr \<and> ptr \<noteq> 0 \<and> pspace_aligned' \<sigma> \<and> pspace_distinct' \<sigma> \<and> pspace_no_overlap' ptr sz \<sigma> \<and> (valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma> \<and> (region_is_bytes ptr (n * 2 ^ objBitsKO ko) x \<and> heap_list (hrs_mem (t_hrs_' (globals x))) (n * 2 ^ objBitsKO ko) ptr = replicate (n * 2 ^ objBitsKO ko) 0) \<and> range_cover ptr sz (objBitsKO ko) n \<and> {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {} \<Longrightarrow> (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
using this:
cpspace_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))

goal (1 subgoal):
 1. (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
using this:
cpspace_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))
(\<sigma>, x) \<in> rf_sr
region_is_bytes ptr (n * 2 ^ objBitsKO ko) x
{ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {}
(valid_objs' and untyped_ranges_zero' and caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko}) \<sigma>

goal (1 subgoal):
 1. (\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr proof (prove)
goal (1 subgoal):
 1. \<lbrakk>cpspace_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))); cpspace_relation (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>)) (t_hrs_' (globals x)) \<and> cready_queues_relation (cslift x) (ksReadyQueues_' (globals x)) (ksReadyQueues \<sigma>) \<and> zero_ranges_are_zero (gsUntypedZeroRanges \<sigma>) (t_hrs_' (globals x)) \<and> cbitmap_L1_relation (ksReadyQueuesL1Bitmap_' (globals x)) (ksReadyQueuesL1Bitmap \<sigma>) \<and> cbitmap_L2_relation (ksReadyQueuesL2Bitmap_' (globals x)) (ksReadyQueuesL2Bitmap \<sigma>) \<and> ksCurThread_' (globals x) = tcb_ptr_to_ctcb_ptr (ksCurThread \<sigma>) \<and> ksIdleThread_' (globals x) = tcb_ptr_to_ctcb_ptr (ksIdleThread \<sigma>) \<and> cinterrupt_relation (ksInterruptState \<sigma>) intStateIRQNode_array_Ptr (intStateIRQTable_' (globals x)) \<and> cscheduler_action_relation (ksSchedulerAction \<sigma>) (ksSchedulerAction_' (globals x)) \<and> carch_state_relation (ksArchState \<sigma>) (globals x) \<and> cmachine_state_relation (ksMachineState \<sigma>) (globals x) \<and> cte_array_relation \<sigma> (globals x) \<and> tcb_cte_array_relation \<sigma> (globals x) \<and> apsnd fst (ghost'state_' (globals x)) = (gsUserPages \<sigma>, gsCNodes \<sigma>) \<and> ghost_size_rel (ghost'state_' (globals x)) (gsMaxObjectSize \<sigma>) \<and> ksWorkUnitsCompleted_' (globals x) = ksWorkUnitsCompleted \<sigma> \<and> x \<Turnstile>\<^sub>c intStateIRQNode_array_Ptr \<and> {symbol_table ''intStateIRQNode''..+256 * size_of TYPE(cte_C)} \<subseteq> kernel_data_refs \<and> x \<Turnstile>\<^sub>c armKSGlobalPD_Ptr \<and> {symbol_table ''armKSGlobalPD''..+16384} \<subseteq> kernel_data_refs \<and> htd_safe domain (hrs_htd (t_hrs_' (globals x))) \<and> - domain \<subseteq> kernel_data_refs \<and> globals_list_distinct (- kernel_data_refs) symbol_table kernel_all_global_addresses.global_data_list \<and> cdom_schedule_relation (kernel_state.ksDomSchedule \<sigma>) kernel_all_global_addresses.ksDomSchedule \<and> ksDomScheduleIdx_' (globals x) = word_of_nat (ksDomScheduleIdx \<sigma>) \<and> ksCurDomain_' (globals x) = UCAST(8 \<rightarrow> 32) (ksCurDomain \<sigma>) \<and> ksDomainTime_' (globals x) = ksDomainTime \<sigma>; region_is_bytes ptr (n * 2 ^ objBitsKO ko) x; {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {}; valid_objs' \<sigma> \<and> untyped_ranges_zero' \<sigma> \<and> caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko} \<sigma>\<rbrakk> \<Longrightarrow> zero_ranges_are_zero (gsUntypedZeroRanges \<sigma>) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> carch_state_relation (ksArchState \<sigma>) (globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>) \<and> cmachine_state_relation (ksMachineState \<sigma>) (globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>) \<and> cvariable_array_map_relation (gsCNodes \<sigma>) ((^) 2) cte_Ptr (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) \<and> cvariable_array_map_relation (map_to_tcbs (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (\<lambda>x. 5) cte_Ptr (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) \<and> hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<Turnstile>\<^sub>t intStateIRQNode_array_Ptr \<and> hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<Turnstile>\<^sub>t armKSGlobalPD_Ptr \<and> htd_safe domain (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) proof (prove)
goal (1 subgoal):
 1. \<lbrakk>cpspace_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))); cpspace_relation (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>)) (t_hrs_' (globals x)) \<and> cready_queues_relation (cslift x) (ksReadyQueues_' (globals x)) (ksReadyQueues \<sigma>) \<and> zero_ranges_are_zero (gsUntypedZeroRanges \<sigma>) (t_hrs_' (globals x)) \<and> cbitmap_L1_relation (ksReadyQueuesL1Bitmap_' (globals x)) (ksReadyQueuesL1Bitmap \<sigma>) \<and> cbitmap_L2_relation (ksReadyQueuesL2Bitmap_' (globals x)) (ksReadyQueuesL2Bitmap \<sigma>) \<and> ksCurThread_' (globals x) = tcb_ptr_to_ctcb_ptr (ksCurThread \<sigma>) \<and> ksIdleThread_' (globals x) = tcb_ptr_to_ctcb_ptr (ksIdleThread \<sigma>) \<and> cinterrupt_relation (ksInterruptState \<sigma>) intStateIRQNode_array_Ptr (intStateIRQTable_' (globals x)) \<and> cscheduler_action_relation (ksSchedulerAction \<sigma>) (ksSchedulerAction_' (globals x)) \<and> armKSNextASID_' (globals x) = armKSNextASID (ksArchState \<sigma>) \<and> armKSKernelVSpace (ksArchState \<sigma>) = armKSKernelVSpace_C \<and> array_relation ((=) \<circ> option_to_0) 0xFF (armKSHWASIDTable (ksArchState \<sigma>)) (armKSHWASIDTable_' (globals x)) \<and> array_relation ((=) \<circ> option_to_ptr) (2 ^ local.asid_high_bits - 1) (armKSASIDTable (ksArchState \<sigma>)) (armKSASIDTable_' (globals x)) \<and> asid_map_pd_to_hwasids (armKSASIDMap (ksArchState \<sigma>)) = set_option \<circ> (pde_stored_asid \<circ>\<^sub>m cslift x \<circ>\<^sub>m pd_pointer_to_asid_slot) \<and> carch_globals (ksArchState \<sigma>) \<and> irq_masks (ksMachineState \<sigma>) = irq_masks (phantom_machine_state_' (globals x)) \<and> irq_state (ksMachineState \<sigma>) = irq_state (phantom_machine_state_' (globals x)) \<and> device_state (ksMachineState \<sigma>) = device_state (phantom_machine_state_' (globals x)) \<and> exclusive_state (ksMachineState \<sigma>) = exclusive_state (phantom_machine_state_' (globals x)) \<and> machine_state_rest (ksMachineState \<sigma>) = machine_state_rest (phantom_machine_state_' (globals x)) \<and> cte_array_relation \<sigma> (globals x) \<and> tcb_cte_array_relation \<sigma> (globals x) \<and> apsnd fst (ghost'state_' (globals x)) = (gsUserPages \<sigma>, gsCNodes \<sigma>) \<and> ghost_size_rel (ghost'state_' (globals x)) (gsMaxObjectSize \<sigma>) \<and> ksWorkUnitsCompleted_' (globals x) = ksWorkUnitsCompleted \<sigma> \<and> x \<Turnstile>\<^sub>c intStateIRQNode_array_Ptr \<and> {symbol_table ''intStateIRQNode''..+256 * size_of TYPE(cte_C)} \<subseteq> kernel_data_refs \<and> x \<Turnstile>\<^sub>c armKSGlobalPD_Ptr \<and> {symbol_table ''armKSGlobalPD''..+16384} \<subseteq> kernel_data_refs \<and> htd_safe domain (hrs_htd (t_hrs_' (globals x))) \<and> - domain \<subseteq> kernel_data_refs \<and> globals_list_distinct (- kernel_data_refs) symbol_table kernel_all_global_addresses.global_data_list \<and> cdom_schedule_relation (kernel_state.ksDomSchedule \<sigma>) kernel_all_global_addresses.ksDomSchedule \<and> ksDomScheduleIdx_' (globals x) = word_of_nat (ksDomScheduleIdx \<sigma>) \<and> ksCurDomain_' (globals x) = UCAST(8 \<rightarrow> 32) (ksCurDomain \<sigma>) \<and> ksDomainTime_' (globals x) = ksDomainTime \<sigma>; region_is_bytes ptr (n * 2 ^ objBitsKO ko) x; {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {}; valid_objs' \<sigma> \<and> untyped_ranges_zero' \<sigma> \<and> caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko} \<sigma>\<rbrakk> \<Longrightarrow> zero_ranges_are_zero (gsUntypedZeroRanges \<sigma>) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> set_option \<circ> (pde_stored_asid \<circ>\<^sub>m cslift x \<circ>\<^sub>m pd_pointer_to_asid_slot) = set_option \<circ> (pde_stored_asid \<circ>\<^sub>m clift (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<circ>\<^sub>m pd_pointer_to_asid_slot) \<and> cvariable_array_map_relation (gsCNodes \<sigma>) ((^) 2) cte_Ptr (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) \<and> cvariable_array_map_relation (map_to_tcbs (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (\<lambda>x. 5) cte_Ptr (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) \<and> hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<Turnstile>\<^sub>t intStateIRQNode_array_Ptr \<and> hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<Turnstile>\<^sub>t armKSGlobalPD_Ptr \<and> htd_safe domain (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) proof (prove)
goal (1 subgoal):
 1. \<lbrakk>cpspace_relation (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)) (underlying_memory (ksMachineState \<sigma>)) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))); cpspace_relation (ksPSpace \<sigma>) (underlying_memory (ksMachineState \<sigma>)) (t_hrs_' (globals x)) \<and> cready_queues_relation (cslift x) (ksReadyQueues_' (globals x)) (ksReadyQueues \<sigma>) \<and> zero_ranges_are_zero (gsUntypedZeroRanges \<sigma>) (t_hrs_' (globals x)) \<and> cbitmap_L1_relation (ksReadyQueuesL1Bitmap_' (globals x)) (ksReadyQueuesL1Bitmap \<sigma>) \<and> cbitmap_L2_relation (ksReadyQueuesL2Bitmap_' (globals x)) (ksReadyQueuesL2Bitmap \<sigma>) \<and> ksCurThread_' (globals x) = tcb_ptr_to_ctcb_ptr (ksCurThread \<sigma>) \<and> ksIdleThread_' (globals x) = tcb_ptr_to_ctcb_ptr (ksIdleThread \<sigma>) \<and> cinterrupt_relation (ksInterruptState \<sigma>) intStateIRQNode_array_Ptr (intStateIRQTable_' (globals x)) \<and> cscheduler_action_relation (ksSchedulerAction \<sigma>) (ksSchedulerAction_' (globals x)) \<and> armKSNextASID_' (globals x) = armKSNextASID (ksArchState \<sigma>) \<and> armKSKernelVSpace (ksArchState \<sigma>) = armKSKernelVSpace_C \<and> array_relation ((=) \<circ> option_to_0) 0xFF (armKSHWASIDTable (ksArchState \<sigma>)) (armKSHWASIDTable_' (globals x)) \<and> array_relation ((=) \<circ> option_to_ptr) (2 ^ local.asid_high_bits - 1) (armKSASIDTable (ksArchState \<sigma>)) (armKSASIDTable_' (globals x)) \<and> asid_map_pd_to_hwasids (armKSASIDMap (ksArchState \<sigma>)) = set_option \<circ> (pde_stored_asid \<circ>\<^sub>m cslift x \<circ>\<^sub>m pd_pointer_to_asid_slot) \<and> carch_globals (ksArchState \<sigma>) \<and> irq_masks (ksMachineState \<sigma>) = irq_masks (phantom_machine_state_' (globals x)) \<and> irq_state (ksMachineState \<sigma>) = irq_state (phantom_machine_state_' (globals x)) \<and> device_state (ksMachineState \<sigma>) = device_state (phantom_machine_state_' (globals x)) \<and> exclusive_state (ksMachineState \<sigma>) = exclusive_state (phantom_machine_state_' (globals x)) \<and> machine_state_rest (ksMachineState \<sigma>) = machine_state_rest (phantom_machine_state_' (globals x)) \<and> cte_array_relation \<sigma> (globals x) \<and> tcb_cte_array_relation \<sigma> (globals x) \<and> apsnd fst (ghost'state_' (globals x)) = (gsUserPages \<sigma>, gsCNodes \<sigma>) \<and> ghost_size_rel (ghost'state_' (globals x)) (gsMaxObjectSize \<sigma>) \<and> ksWorkUnitsCompleted_' (globals x) = ksWorkUnitsCompleted \<sigma> \<and> (\<exists>v. cslift x intStateIRQNode_array_Ptr = Some v) \<and> {symbol_table ''intStateIRQNode''..+256 * size_of TYPE(cte_C)} \<subseteq> kernel_data_refs \<and> (\<exists>v. cslift x armKSGlobalPD_Ptr = Some v) \<and> {symbol_table ''armKSGlobalPD''..+16384} \<subseteq> kernel_data_refs \<and> htd_safe domain (hrs_htd (t_hrs_' (globals x))) \<and> - domain \<subseteq> kernel_data_refs \<and> globals_list_distinct (- kernel_data_refs) symbol_table kernel_all_global_addresses.global_data_list \<and> cdom_schedule_relation (kernel_state.ksDomSchedule \<sigma>) kernel_all_global_addresses.ksDomSchedule \<and> ksDomScheduleIdx_' (globals x) = word_of_nat (ksDomScheduleIdx \<sigma>) \<and> ksCurDomain_' (globals x) = UCAST(8 \<rightarrow> 32) (ksCurDomain \<sigma>) \<and> ksDomainTime_' (globals x) = ksDomainTime \<sigma>; region_is_bytes ptr (n * 2 ^ objBitsKO ko) x; {ptr..+n * 2 ^ objBitsKO ko} \<inter> kernel_data_refs = {}; valid_objs' \<sigma> \<and> untyped_ranges_zero' \<sigma> \<and> caps_overlap_reserved' {ptr..+n * 2 ^ objBitsKO ko} \<sigma>\<rbrakk> \<Longrightarrow> zero_ranges_are_zero (gsUntypedZeroRanges \<sigma>) (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))) \<and> cvariable_array_map_relation (gsCNodes \<sigma>) ((^) 2) cte_Ptr (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) \<and> cvariable_array_map_relation (map_to_tcbs (foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>))) (\<lambda>x. 5) cte_Ptr (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) \<and> htd_safe domain (hrs_htd (hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x)))) proof (prove)
goal:
No subgoals! proof (state)
this:
(\<sigma>\<lparr>ksPSpace := foldr (\<lambda>addr. data_map_insert addr ko) (new_cap_addrs n ptr ko) (ksPSpace \<sigma>)\<rparr>, x\<lparr>globals := globals x\<lparr>t_hrs_' := hrs_htd_update (ptr_retyps_gen n (ep_Ptr ptr) False) (t_hrs_' (globals x))\<rparr>\<rparr>) \<in> rf_sr

goal:
No subgoals!