theory structures_proofs

lemma (in kernel_all_substitute) cap_notification_cap_new_spec:
  "\<forall>s. \<Gamma> \<turnstile> {s}
       \<acute>ret__struct_cap_C :== PROC cap_notification_cap_new(\<acute>capNtfnBadge, \<acute>capNtfnCanReceive, \<acute>capNtfnCanSend, \<acute>capNtfnPtr)
       \<lbrace>cap_notification_cap_lift \<acute>ret__struct_cap_C = \<lparr>
          cap_notification_cap_CL.capNtfnBadge_CL = (\<^bsup>s\<^esup>capNtfnBadge___unsigned AND mask 28),
          cap_notification_cap_CL.capNtfnCanReceive_CL = (\<^bsup>s\<^esup>capNtfnCanReceive___unsigned AND mask 1),
          cap_notification_cap_CL.capNtfnCanSend_CL = (\<^bsup>s\<^esup>capNtfnCanSend___unsigned AND mask 1),
          cap_notification_cap_CL.capNtfnPtr_CL = (\<^bsup>s\<^esup>capNtfnPtr___unsigned AND NOT (mask 4)) \<rparr> \<and>
        cap_get_tag \<acute>ret__struct_cap_C = scast cap_notification_cap\<rbrace>" apply (rule allI, rule conseqPre, vcg) apply (clarsimp simp: guard_simps o_def mask_def shift_over_ao_dists) apply (rule context_conjI[THEN iffD1[OF conj_commute]],
         fastforce simp: cap_get_tag_eq_x cap_notification_cap_def
                         mask_def shift_over_ao_dists word_bw_assocs word_ao_dist) apply (simp add: cap_notification_cap_lift_def) apply (erule cap_lift_notification_cap[THEN subst[OF sym]]; simp?) apply ((intro conjI sign_extend_eq)?;
         (simp add: mask_def shift_over_ao_dists multi_shift_simps word_size
                    word_ao_dist word_bw_assocs word_and_max_simps cap_notification_cap_def))? done

(* Proof state trace *)
proof (prove)
goal (1 subgoal):
 1. \<forall>s. \<Gamma>\<turnstile> {s} Call cap_notification_cap_new_'proc \<lbrace>cap_notification_cap_lift \<acute>ret__struct_cap_C = \<lparr>capNtfnBadge_CL = \<^bsup>s\<^esup>capNtfnBadge && mask 28, capNtfnCanReceive_CL = \<^bsup>s\<^esup>capNtfnCanReceive && mask 1, capNtfnCanSend_CL = \<^bsup>s\<^esup>capNtfnCanSend && mask 1, capNtfnPtr_CL = \<^bsup>s\<^esup>capNtfnPtr && ~~ mask 4\<rparr> \<and> cap_get_tag \<acute>ret__struct_cap_C = SCAST(32 signed \<rightarrow> 32) cap_notification_cap\<rbrace> proof (prove)
goal (1 subgoal):
 1. \<And>s. {s} \<subseteq> \<lbrace>\<forall>v. 0 <s 2 \<and> 0 \<le>s 0 \<and> 0 <s 0x20 \<and> 0 \<le>s 0 \<and> 0 <s 0x20 \<and> 1 <s 2 \<and> 0 \<le>s 1 \<and> 0 \<le>s 0 \<and> 0 <s 0x20 \<and> 0 \<le>s 1 \<and> 1 <s 0x20 \<and> 0 \<le>s 4 \<and> 4 <s 0x20 \<and> cap_notification_cap_lift (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 1) (((SCAST(32 signed \<rightarrow> 32) 0 || (\<acute>capNtfnBadge && 0xFFFFFFF << unat 4)) || (\<acute>capNtfnCanReceive && 1 << unat 1)) || (\<acute>capNtfnCanSend && 1 << unat 0))) (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 0) ((SCAST(32 signed \<rightarrow> 32) 0 || (\<acute>capNtfnPtr && 0xFFFFFFF0 >> unat 0)) || (SCAST(32 signed \<rightarrow> 32) cap_notification_cap && 0xF << unat 0))) v)) = \<lparr>capNtfnBadge_CL = \<^bsup>s\<^esup>capNtfnBadge && mask 28, capNtfnCanReceive_CL = \<^bsup>s\<^esup>capNtfnCanReceive && mask 1, capNtfnCanSend_CL = \<^bsup>s\<^esup>capNtfnCanSend && mask 1, capNtfnPtr_CL = \<^bsup>s\<^esup>capNtfnPtr && ~~ mask 4\<rparr> \<and> cap_get_tag (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 1) (((SCAST(32 signed \<rightarrow> 32) 0 || (\<acute>capNtfnBadge && 0xFFFFFFF << unat 4)) || (\<acute>capNtfnCanReceive && 1 << unat 1)) || (\<acute>capNtfnCanSend && 1 << unat 0))) (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 0) ((SCAST(32 signed \<rightarrow> 32) 0 || (\<acute>capNtfnPtr && 0xFFFFFFF0 >> unat 0)) || (SCAST(32 signed \<rightarrow> 32) cap_notification_cap && 0xF << unat 0))) v)) = SCAST(32 signed \<rightarrow> 32) cap_notification_cap\<rbrace> proof (prove)
goal (1 subgoal):
 1. \<And>s v. cap_notification_cap_lift (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x 0 (capNtfnPtr_' s && 0xFFFFFFF0 || SCAST(32 signed \<rightarrow> 32) cap_notification_cap && 0xF)) (Suc 0) (((capNtfnBadge_' s << 4) && 0xFFFFFFF0 || (capNtfnCanReceive_' s << Suc 0) && 2) || capNtfnCanSend_' s && 1)) v) = \<lparr>capNtfnBadge_CL = capNtfnBadge_' s && 0xFFFFFFF, capNtfnCanReceive_CL = capNtfnCanReceive_' s && 1, capNtfnCanSend_CL = capNtfnCanSend_' s && 1, capNtfnPtr_CL = capNtfnPtr_' s && 0xFFFFFFF0\<rparr> \<and> cap_get_tag (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x 0 (capNtfnPtr_' s && 0xFFFFFFF0 || SCAST(32 signed \<rightarrow> 32) cap_notification_cap && 0xF)) (Suc 0) (((capNtfnBadge_' s << 4) && 0xFFFFFFF0 || (capNtfnCanReceive_' s << Suc 0) && 2) || capNtfnCanSend_' s && 1)) v) = SCAST(32 signed \<rightarrow> 32) cap_notification_cap proof (prove)
goal (1 subgoal):
 1. \<And>s v. cap_get_tag (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x 0 (capNtfnPtr_' s && 0xFFFFFFF0 || SCAST(32 signed \<rightarrow> 32) cap_notification_cap && 0xF)) (Suc 0) (((capNtfnBadge_' s << 4) && 0xFFFFFFF0 || (capNtfnCanReceive_' s << Suc 0) && 2) || capNtfnCanSend_' s && 1)) v) = SCAST(32 signed \<rightarrow> 32) cap_notification_cap \<Longrightarrow> cap_notification_cap_lift (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x 0 (capNtfnPtr_' s && 0xFFFFFFF0 || SCAST(32 signed \<rightarrow> 32) cap_notification_cap && 0xF)) (Suc 0) (((capNtfnBadge_' s << 4) && 0xFFFFFFF0 || (capNtfnCanReceive_' s << Suc 0) && 2) || capNtfnCanSend_' s && 1)) v) = \<lparr>capNtfnBadge_CL = capNtfnBadge_' s && 0xFFFFFFF, capNtfnCanReceive_CL = capNtfnCanReceive_' s && 1, capNtfnCanSend_CL = capNtfnCanSend_' s && 1, capNtfnPtr_CL = capNtfnPtr_' s && 0xFFFFFFF0\<rparr> proof (prove)
goal (1 subgoal):
 1. \<And>s v. cap_get_tag (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x 0 (capNtfnPtr_' s && 0xFFFFFFF0 || SCAST(32 signed \<rightarrow> 32) cap_notification_cap && 0xF)) (Suc 0) (((capNtfnBadge_' s << 4) && 0xFFFFFFF0 || (capNtfnCanReceive_' s << Suc 0) && 2) || capNtfnCanSend_' s && 1)) v) = SCAST(32 signed \<rightarrow> 32) cap_notification_cap \<Longrightarrow> (case cap_lift (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x 0 (capNtfnPtr_' s && 0xFFFFFFF0 || SCAST(32 signed \<rightarrow> 32) cap_notification_cap && 0xF)) (Suc 0) (((capNtfnBadge_' s << 4) && 0xFFFFFFF0 || (capNtfnCanReceive_' s << Suc 0) && 2) || capNtfnCanSend_' s && 1)) v) of Some (Cap_notification_cap rec) \<Rightarrow> rec) = \<lparr>capNtfnBadge_CL = capNtfnBadge_' s && 0xFFFFFFF, capNtfnCanReceive_CL = capNtfnCanReceive_' s && 1, capNtfnCanSend_CL = capNtfnCanSend_' s && 1, capNtfnPtr_CL = capNtfnPtr_' s && 0xFFFFFFF0\<rparr> proof (prove)
goal (1 subgoal):
 1. \<And>s. (((capNtfnBadge_' s << 4) && 0xFFFFFFF0 || (capNtfnCanReceive_' s << Suc 0) && 2) || capNtfnCanSend_' s && 1 >> 4) && mask 28 = capNtfnBadge_' s && 0xFFFFFFF \<and> (((capNtfnBadge_' s << 4) && 0xFFFFFFF0 || (capNtfnCanReceive_' s << Suc 0) && 2) || capNtfnCanSend_' s && 1 >> Suc 0) && 1 = capNtfnCanReceive_' s && 1 \<and> (((capNtfnBadge_' s << 4) && 0xFFFFFFF0 || (capNtfnCanReceive_' s << Suc 0) && 2) || capNtfnCanSend_' s && 1) && 1 = capNtfnCanSend_' s && 1 \<and> (capNtfnPtr_' s && 0xFFFFFFF0 || SCAST(32 signed \<rightarrow> 32) cap_notification_cap && 0xF) && ~~ mask 4 = capNtfnPtr_' s && 0xFFFFFFF0 proof (prove)
goal:
No subgoals!