theory Ipc_AI

lemma transfer_caps_loop_irq_handlers[wp]:
  "\<And>slots caps ep buffer n mi.
    \<lbrace>valid_irq_handlers and valid_objs and valid_mdb and K (distinct slots)
         and (\<lambda>s. \<forall>x \<in> set slots. real_cte_at x s \<and> cte_wp_at (\<lambda>cap. cap = cap.NullCap) x s)
         and transfer_caps_srcs caps\<rbrace>
      transfer_caps_loop ep buffer n caps slots mi
    \<lbrace>\<lambda>rv. valid_irq_handlers :: 'state_ext state \<Rightarrow> bool\<rbrace>" apply (rule hoare_pre) apply (rule transfer_caps_loop_presM[where vo=True and em=False and ex=False]) apply wp apply (clarsimp simp: cte_wp_at_caps_of_state) apply (clarsimp simp: is_derived_def split: if_split_asm) apply (simp add: cap_master_cap_irqs)+ apply (wp valid_irq_handlers_lift) apply (clarsimp simp:cte_wp_at_caps_of_state|intro conjI ballI)+ apply (drule(1) bspec,clarsimp) apply (frule(1) caps_of_state_valid) apply (fastforce simp:valid_cap_def) apply (drule(1) bspec) apply clarsimp done

(* Proof state trace *)
proof (prove)
goal (1 subgoal):
 1. \<And>slots caps ep buffer n mi. \<lbrace>valid_irq_handlers and valid_objs and valid_mdb and K (distinct slots) and (\<lambda>s. \<forall>x\<in>set slots. real_cte_at x s \<and> cte_wp_at (\<lambda>cap. cap = NullCap) x s) and (\<lambda>s. \<forall>x\<in>set caps. cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp = fst x) (snd x) s \<and> real_cte_at (snd x) s)\<rbrace> transfer_caps_loop ep buffer n caps slots mi \<lbrace>\<lambda>rv. valid_irq_handlers\<rbrace> proof (prove)
goal (2 subgoals):
 1. \<And>slots caps ep buffer n mi. \<lbrace>?Q slots caps ep buffer n mi\<rbrace> transfer_caps_loop ep buffer n caps slots mi \<lbrace>\<lambda>rv. valid_irq_handlers\<rbrace>
 2. \<And>slots caps ep buffer n mi s. (valid_irq_handlers and valid_objs and valid_mdb and K (distinct slots) and (\<lambda>s. \<forall>x\<in>set slots. real_cte_at x s \<and> cte_wp_at (\<lambda>cap. cap = NullCap) x s) and (\<lambda>s. \<forall>x\<in>set caps. cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp = fst x) (snd x) s \<and> real_cte_at (snd x) s)) s \<Longrightarrow> ?Q slots caps ep buffer n mi s proof (prove)
goal (3 subgoals):
 1. \<And>slots caps ep buffer n mi cap src dest. \<lbrace>\<lambda>s. valid_irq_handlers s \<and> (True \<longrightarrow> valid_objs s \<and> valid_mdb s \<and> real_cte_at dest s \<and> s \<turnstile> cap \<and> tcb_cap_valid cap dest s \<and> real_cte_at src s \<and> cte_wp_at (is_derived (cdt s) src cap) src s \<and> cap \<noteq> NullCap) \<and> (False \<longrightarrow> cte_wp_at ((=) NullCap) dest s) \<and> (False \<longrightarrow> ex_cte_cap_wp_to (appropriate_cte_cap cap) dest s)\<rbrace> cap_insert cap src dest \<lbrace>\<lambda>rv. valid_irq_handlers\<rbrace>
 2. \<And>slots caps ep buffer n mi b na. \<lbrace>valid_irq_handlers\<rbrace> set_extra_badge buffer b na \<lbrace>\<lambda>_. valid_irq_handlers\<rbrace>
 3. \<And>slots caps ep buffer n mi s. (valid_irq_handlers and valid_objs and valid_mdb and K (distinct slots) and (\<lambda>s. \<forall>x\<in>set slots. real_cte_at x s \<and> cte_wp_at (\<lambda>cap. cap = NullCap) x s) and (\<lambda>s. \<forall>x\<in>set caps. cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp = fst x) (snd x) s \<and> real_cte_at (snd x) s)) s \<Longrightarrow> valid_irq_handlers s \<and> (True \<longrightarrow> valid_objs s \<and> valid_mdb s \<and> distinct slots \<and> (\<forall>x\<in>set slots. cte_wp_at (\<lambda>cap. cap = NullCap) x s \<and> real_cte_at x s) \<and> (\<forall>x\<in>set caps. s \<turnstile> fst x \<and> cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp \<noteq> fst x \<longrightarrow> cp = masked_as_full (fst x) (fst x)) (snd x) s \<and> real_cte_at (snd x) s)) \<and> (False \<longrightarrow> (\<forall>x\<in>set slots. ex_cte_cap_wp_to is_cnode_cap x s)) proof (prove)
goal (3 subgoals):
 1. \<And>slots caps ep buffer n mi cap src dest s. valid_irq_handlers s \<and> (True \<longrightarrow> valid_objs s \<and> valid_mdb s \<and> real_cte_at dest s \<and> s \<turnstile> cap \<and> tcb_cap_valid cap dest s \<and> real_cte_at src s \<and> cte_wp_at (is_derived (cdt s) src cap) src s \<and> cap \<noteq> NullCap) \<and> (False \<longrightarrow> cte_wp_at ((=) NullCap) dest s) \<and> (False \<longrightarrow> ex_cte_cap_wp_to (appropriate_cte_cap cap) dest s) \<Longrightarrow> valid_irq_handlers s \<and> cte_wp_at (\<lambda>cap'. \<forall>irq\<in>cap_irqs cap - cap_irqs cap'. irq_issued irq s) src s
 2. \<And>slots caps ep buffer n mi b na. \<lbrace>valid_irq_handlers\<rbrace> set_extra_badge buffer b na \<lbrace>\<lambda>_. valid_irq_handlers\<rbrace>
 3. \<And>slots caps ep buffer n mi s. (valid_irq_handlers and valid_objs and valid_mdb and K (distinct slots) and (\<lambda>s. \<forall>x\<in>set slots. real_cte_at x s \<and> cte_wp_at (\<lambda>cap. cap = NullCap) x s) and (\<lambda>s. \<forall>x\<in>set caps. cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp = fst x) (snd x) s \<and> real_cte_at (snd x) s)) s \<Longrightarrow> valid_irq_handlers s \<and> (True \<longrightarrow> valid_objs s \<and> valid_mdb s \<and> distinct slots \<and> (\<forall>x\<in>set slots. cte_wp_at (\<lambda>cap. cap = NullCap) x s \<and> real_cte_at x s) \<and> (\<forall>x\<in>set caps. s \<turnstile> fst x \<and> cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp \<noteq> fst x \<longrightarrow> cp = masked_as_full (fst x) (fst x)) (snd x) s \<and> real_cte_at (snd x) s)) \<and> (False \<longrightarrow> (\<forall>x\<in>set slots. ex_cte_cap_wp_to is_cnode_cap x s)) proof (prove)
goal (3 subgoals):
 1. \<And>cap a b aa ba s capa irq. \<lbrakk>valid_irq_handlers s; valid_objs s; valid_mdb s; cap_table_at (length ba) aa s; s \<turnstile> cap; tcb_cap_valid cap (aa, ba) s; cap_table_at (length b) a s; cap \<noteq> NullCap; caps_of_state s (a, b) = Some capa; is_derived (cdt s) (a, b) cap capa; irq \<in> cap_irqs cap; irq \<notin> cap_irqs capa\<rbrakk> \<Longrightarrow> irq_issued irq s
 2. \<And>slots caps ep buffer n mi b na. \<lbrace>valid_irq_handlers\<rbrace> set_extra_badge buffer b na \<lbrace>\<lambda>_. valid_irq_handlers\<rbrace>
 3. \<And>slots caps ep buffer n mi s. (valid_irq_handlers and valid_objs and valid_mdb and K (distinct slots) and (\<lambda>s. \<forall>x\<in>set slots. real_cte_at x s \<and> cte_wp_at (\<lambda>cap. cap = NullCap) x s) and (\<lambda>s. \<forall>x\<in>set caps. cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp = fst x) (snd x) s \<and> real_cte_at (snd x) s)) s \<Longrightarrow> valid_irq_handlers s \<and> (True \<longrightarrow> valid_objs s \<and> valid_mdb s \<and> distinct slots \<and> (\<forall>x\<in>set slots. cte_wp_at (\<lambda>cap. cap = NullCap) x s \<and> real_cte_at x s) \<and> (\<forall>x\<in>set caps. s \<turnstile> fst x \<and> cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp \<noteq> fst x \<longrightarrow> cp = masked_as_full (fst x) (fst x)) (snd x) s \<and> real_cte_at (snd x) s)) \<and> (False \<longrightarrow> (\<forall>x\<in>set slots. ex_cte_cap_wp_to is_cnode_cap x s)) proof (prove)
goal (4 subgoals):
 1. \<And>cap a b aa ba s capa irq. \<lbrakk>irq \<in> cap_irqs cap; irq \<notin> cap_irqs capa; valid_irq_handlers s; valid_objs s; valid_mdb s; cap_table_at (length ba) aa s; s \<turnstile> cap; tcb_cap_valid cap (aa, ba) s; cap_table_at (length b) a s; cap \<noteq> NullCap; caps_of_state s (a, b) = Some capa; is_untyped_cap capa; \<not> is_zombie capa; cap \<noteq> IRQControlCap; cap_master_cap capa = cap_master_cap cap; descendants_of (a, b) (cdt s) = {}; is_master_reply_cap capa = is_reply_cap cap; is_derived_arch cap capa; \<not> is_reply_cap capa; \<not> is_master_reply_cap cap\<rbrakk> \<Longrightarrow> irq_issued irq s
 2. \<And>cap a b aa ba s capa irq. \<lbrakk>irq \<in> cap_irqs cap; irq \<notin> cap_irqs capa; valid_irq_handlers s; valid_objs s; valid_mdb s; cap_table_at (length ba) aa s; s \<turnstile> cap; tcb_cap_valid cap (aa, ba) s; cap_table_at (length b) a s; cap \<noteq> NullCap; caps_of_state s (a, b) = Some capa; \<not> is_untyped_cap capa; \<not> is_zombie capa; cap \<noteq> IRQControlCap; cap_master_cap capa = cap_master_cap cap; (cap_badge capa, cap_badge cap) \<in> capBadge_ordering False; is_master_reply_cap capa = is_reply_cap cap; is_derived_arch cap capa; \<not> is_reply_cap capa; \<not> is_master_reply_cap cap\<rbrakk> \<Longrightarrow> irq_issued irq s
 3. \<And>slots caps ep buffer n mi b na. \<lbrace>valid_irq_handlers\<rbrace> set_extra_badge buffer b na \<lbrace>\<lambda>_. valid_irq_handlers\<rbrace>
 4. \<And>slots caps ep buffer n mi s. (valid_irq_handlers and valid_objs and valid_mdb and K (distinct slots) and (\<lambda>s. \<forall>x\<in>set slots. real_cte_at x s \<and> cte_wp_at (\<lambda>cap. cap = NullCap) x s) and (\<lambda>s. \<forall>x\<in>set caps. cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp = fst x) (snd x) s \<and> real_cte_at (snd x) s)) s \<Longrightarrow> valid_irq_handlers s \<and> (True \<longrightarrow> valid_objs s \<and> valid_mdb s \<and> distinct slots \<and> (\<forall>x\<in>set slots. cte_wp_at (\<lambda>cap. cap = NullCap) x s \<and> real_cte_at x s) \<and> (\<forall>x\<in>set caps. s \<turnstile> fst x \<and> cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp \<noteq> fst x \<longrightarrow> cp = masked_as_full (fst x) (fst x)) (snd x) s \<and> real_cte_at (snd x) s)) \<and> (False \<longrightarrow> (\<forall>x\<in>set slots. ex_cte_cap_wp_to is_cnode_cap x s)) proof (prove)
goal (2 subgoals):
 1. \<And>buffer b na. \<lbrace>valid_irq_handlers\<rbrace> set_extra_badge buffer b na \<lbrace>\<lambda>_. valid_irq_handlers\<rbrace>
 2. \<And>slots caps ep buffer n mi s. (valid_irq_handlers and valid_objs and valid_mdb and K (distinct slots) and (\<lambda>s. \<forall>x\<in>set slots. real_cte_at x s \<and> cte_wp_at (\<lambda>cap. cap = NullCap) x s) and (\<lambda>s. \<forall>x\<in>set caps. cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp = fst x) (snd x) s \<and> real_cte_at (snd x) s)) s \<Longrightarrow> valid_irq_handlers s \<and> (True \<longrightarrow> valid_objs s \<and> valid_mdb s \<and> distinct slots \<and> (\<forall>x\<in>set slots. cte_wp_at (\<lambda>cap. cap = NullCap) x s \<and> real_cte_at x s) \<and> (\<forall>x\<in>set caps. s \<turnstile> fst x \<and> cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp \<noteq> fst x \<longrightarrow> cp = masked_as_full (fst x) (fst x)) (snd x) s \<and> real_cte_at (snd x) s)) \<and> (False \<longrightarrow> (\<forall>x\<in>set slots. ex_cte_cap_wp_to is_cnode_cap x s)) proof (prove)
goal (1 subgoal):
 1. \<And>slots caps ep buffer n mi s. (valid_irq_handlers and valid_objs and valid_mdb and K (distinct slots) and (\<lambda>s. \<forall>x\<in>set slots. real_cte_at x s \<and> cte_wp_at (\<lambda>cap. cap = NullCap) x s) and (\<lambda>s. \<forall>x\<in>set caps. cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp = fst x) (snd x) s \<and> real_cte_at (snd x) s)) s \<Longrightarrow> valid_irq_handlers s \<and> (True \<longrightarrow> valid_objs s \<and> valid_mdb s \<and> distinct slots \<and> (\<forall>x\<in>set slots. cte_wp_at (\<lambda>cap. cap = NullCap) x s \<and> real_cte_at x s) \<and> (\<forall>x\<in>set caps. s \<turnstile> fst x \<and> cte_wp_at (\<lambda>cp. fst x \<noteq> NullCap \<longrightarrow> cp \<noteq> fst x \<longrightarrow> cp = masked_as_full (fst x) (fst x)) (snd x) s \<and> real_cte_at (snd x) s)) \<and> (False \<longrightarrow> (\<forall>x\<in>set slots. ex_cte_cap_wp_to is_cnode_cap x s)) proof (prove)
goal (2 subgoals):
 1. \<And>slots caps s a aa b. \<lbrakk>(a, aa, b) \<in> set caps; valid_irq_handlers s; valid_objs s; valid_mdb s; distinct slots; \<forall>x\<in>set slots. real_cte_at x s \<and> caps_of_state s x = Some NullCap; \<forall>x\<in>set caps. (\<exists>cap. caps_of_state s (snd x) = Some cap \<and> (fst x \<noteq> NullCap \<longrightarrow> cap = fst x)) \<and> real_cte_at (snd x) s\<rbrakk> \<Longrightarrow> s \<turnstile> a
 2. \<And>slots caps s a aa b. \<lbrakk>(a, aa, b) \<in> set caps; valid_irq_handlers s; valid_objs s; valid_mdb s; distinct slots; \<forall>x\<in>set slots. real_cte_at x s \<and> caps_of_state s x = Some NullCap; \<forall>x\<in>set caps. (\<exists>cap. caps_of_state s (snd x) = Some cap \<and> (fst x \<noteq> NullCap \<longrightarrow> cap = fst x)) \<and> real_cte_at (snd x) s\<rbrakk> \<Longrightarrow> \<exists>cap. caps_of_state s (aa, b) = Some cap \<and> (a \<noteq> NullCap \<longrightarrow> cap \<noteq> a \<longrightarrow> cap = masked_as_full a a) proof (prove)
goal (2 subgoals):
 1. \<And>slots caps s a aa b cap. \<lbrakk>(a, aa, b) \<in> set caps; valid_irq_handlers s; valid_objs s; valid_mdb s; distinct slots; \<forall>x\<in>set slots. real_cte_at x s \<and> caps_of_state s x = Some NullCap; cap_table_at (length b) aa s; caps_of_state s (aa, b) = Some cap; a \<noteq> NullCap \<longrightarrow> cap = a\<rbrakk> \<Longrightarrow> s \<turnstile> a
 2. \<And>slots caps s a aa b. \<lbrakk>(a, aa, b) \<in> set caps; valid_irq_handlers s; valid_objs s; valid_mdb s; distinct slots; \<forall>x\<in>set slots. real_cte_at x s \<and> caps_of_state s x = Some NullCap; \<forall>x\<in>set caps. (\<exists>cap. caps_of_state s (snd x) = Some cap \<and> (fst x \<noteq> NullCap \<longrightarrow> cap = fst x)) \<and> real_cte_at (snd x) s\<rbrakk> \<Longrightarrow> \<exists>cap. caps_of_state s (aa, b) = Some cap \<and> (a \<noteq> NullCap \<longrightarrow> cap \<noteq> a \<longrightarrow> cap = masked_as_full a a) proof (prove)
goal (2 subgoals):
 1. \<And>slots caps s a aa b cap. \<lbrakk>(a, aa, b) \<in> set caps; valid_irq_handlers s; valid_objs s; valid_mdb s; distinct slots; \<forall>x\<in>set slots. real_cte_at x s \<and> caps_of_state s x = Some NullCap; cap_table_at (length b) aa s; caps_of_state s (aa, b) = Some cap; a \<noteq> NullCap \<longrightarrow> cap = a; s \<turnstile> cap\<rbrakk> \<Longrightarrow> s \<turnstile> a
 2. \<And>slots caps s a aa b. \<lbrakk>(a, aa, b) \<in> set caps; valid_irq_handlers s; valid_objs s; valid_mdb s; distinct slots; \<forall>x\<in>set slots. real_cte_at x s \<and> caps_of_state s x = Some NullCap; \<forall>x\<in>set caps. (\<exists>cap. caps_of_state s (snd x) = Some cap \<and> (fst x \<noteq> NullCap \<longrightarrow> cap = fst x)) \<and> real_cte_at (snd x) s\<rbrakk> \<Longrightarrow> \<exists>cap. caps_of_state s (aa, b) = Some cap \<and> (a \<noteq> NullCap \<longrightarrow> cap \<noteq> a \<longrightarrow> cap = masked_as_full a a) proof (prove)
goal (1 subgoal):
 1. \<And>slots caps s a aa b. \<lbrakk>(a, aa, b) \<in> set caps; valid_irq_handlers s; valid_objs s; valid_mdb s; distinct slots; \<forall>x\<in>set slots. real_cte_at x s \<and> caps_of_state s x = Some NullCap; \<forall>x\<in>set caps. (\<exists>cap. caps_of_state s (snd x) = Some cap \<and> (fst x \<noteq> NullCap \<longrightarrow> cap = fst x)) \<and> real_cte_at (snd x) s\<rbrakk> \<Longrightarrow> \<exists>cap. caps_of_state s (aa, b) = Some cap \<and> (a \<noteq> NullCap \<longrightarrow> cap \<noteq> a \<longrightarrow> cap = masked_as_full a a) proof (prove)
goal (1 subgoal):
 1. \<And>slots caps s a aa b. \<lbrakk>(a, aa, b) \<in> set caps; valid_irq_handlers s; valid_objs s; valid_mdb s; distinct slots; \<forall>x\<in>set slots. real_cte_at x s \<and> caps_of_state s x = Some NullCap; (\<exists>cap. caps_of_state s (snd (a, aa, b)) = Some cap \<and> (fst (a, aa, b) \<noteq> NullCap \<longrightarrow> cap = fst (a, aa, b))) \<and> real_cte_at (snd (a, aa, b)) s\<rbrakk> \<Longrightarrow> \<exists>cap. caps_of_state s (aa, b) = Some cap \<and> (a \<noteq> NullCap \<longrightarrow> cap \<noteq> a \<longrightarrow> cap = masked_as_full a a) proof (prove)
goal:
No subgoals!