theory structures_proofs

lemma (in kernel_all_substitute) cap_reply_cap_new_spec:
  "\<forall>s. \<Gamma> \<turnstile> {s}
       \<acute>ret__struct_cap_C :== PROC cap_reply_cap_new(\<acute>capReplyCanGrant, \<acute>capReplyMaster, \<acute>capTCBPtr)
       \<lbrace>cap_reply_cap_lift \<acute>ret__struct_cap_C = \<lparr>
          cap_reply_cap_CL.capReplyCanGrant_CL = (\<^bsup>s\<^esup>capReplyCanGrant___unsigned AND mask 1),
          cap_reply_cap_CL.capReplyMaster_CL = (\<^bsup>s\<^esup>capReplyMaster___unsigned AND mask 1),
          cap_reply_cap_CL.capTCBPtr_CL = (\<^bsup>s\<^esup>capTCBPtr___unsigned AND NOT (mask 6)) \<rparr> \<and>
        cap_get_tag \<acute>ret__struct_cap_C = scast cap_reply_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_reply_cap_def
                         mask_def shift_over_ao_dists word_bw_assocs word_ao_dist) apply (simp add: cap_reply_cap_lift_def) apply (erule cap_lift_reply_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_reply_cap_def))? done

(* Proof state trace *)
proof (prove)
goal (1 subgoal):
 1. \<forall>s. \<Gamma>\<turnstile> {s} Call cap_reply_cap_new_'proc \<lbrace>cap_reply_cap_lift \<acute>ret__struct_cap_C = \<lparr>capReplyCanGrant_CL = \<^bsup>s\<^esup>capReplyCanGrant && mask 1, capReplyMaster_CL = \<^bsup>s\<^esup>capReplyMaster && mask 1, capTCBPtr_CL = \<^bsup>s\<^esup>capTCBPtr && ~~ mask 6\<rparr> \<and> cap_get_tag \<acute>ret__struct_cap_C = SCAST(32 signed \<rightarrow> 32) cap_reply_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> 0 \<le>s 4 \<and> 4 <s 0x20 \<and> 0 \<le>s 5 \<and> 5 <s 0x20 \<and> 1 <s 2 \<and> 0 \<le>s 1 \<and> cap_reply_cap_lift (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 1) (SCAST(32 signed \<rightarrow> 32) 0)) (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 0) ((((SCAST(32 signed \<rightarrow> 32) 0 || (\<acute>capReplyCanGrant && 1 << unat 5)) || (\<acute>capReplyMaster && 1 << unat 4)) || (\<acute>capTCBPtr && 0xFFFFFFC0 >> unat 0)) || (SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF << unat 0))) v)) = \<lparr>capReplyCanGrant_CL = \<^bsup>s\<^esup>capReplyCanGrant && mask 1, capReplyMaster_CL = \<^bsup>s\<^esup>capReplyMaster && mask 1, capTCBPtr_CL = \<^bsup>s\<^esup>capTCBPtr && ~~ mask 6\<rparr> \<and> cap_get_tag (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 1) (SCAST(32 signed \<rightarrow> 32) 0)) (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 0) ((((SCAST(32 signed \<rightarrow> 32) 0 || (\<acute>capReplyCanGrant && 1 << unat 5)) || (\<acute>capReplyMaster && 1 << unat 4)) || (\<acute>capTCBPtr && 0xFFFFFFC0 >> unat 0)) || (SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF << unat 0))) v)) = SCAST(32 signed \<rightarrow> 32) cap_reply_cap\<rbrace> proof (prove)
goal (1 subgoal):
 1. \<And>s v. cap_reply_cap_lift (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x (Suc 0) 0) 0 ((((capReplyCanGrant_' s << 5) && 0x20 || (capReplyMaster_' s << 4) && 0x10) || capTCBPtr_' s && 0xFFFFFFC0) || SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF)) v) = \<lparr>capReplyCanGrant_CL = capReplyCanGrant_' s && 1, capReplyMaster_CL = capReplyMaster_' s && 1, capTCBPtr_CL = capTCBPtr_' s && 0xFFFFFFC0\<rparr> \<and> cap_get_tag (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x (Suc 0) 0) 0 ((((capReplyCanGrant_' s << 5) && 0x20 || (capReplyMaster_' s << 4) && 0x10) || capTCBPtr_' s && 0xFFFFFFC0) || SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF)) v) = SCAST(32 signed \<rightarrow> 32) cap_reply_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 (Suc 0) 0) 0 ((((capReplyCanGrant_' s << 5) && 0x20 || (capReplyMaster_' s << 4) && 0x10) || capTCBPtr_' s && 0xFFFFFFC0) || SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF)) v) = SCAST(32 signed \<rightarrow> 32) cap_reply_cap \<Longrightarrow> cap_reply_cap_lift (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x (Suc 0) 0) 0 ((((capReplyCanGrant_' s << 5) && 0x20 || (capReplyMaster_' s << 4) && 0x10) || capTCBPtr_' s && 0xFFFFFFC0) || SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF)) v) = \<lparr>capReplyCanGrant_CL = capReplyCanGrant_' s && 1, capReplyMaster_CL = capReplyMaster_' s && 1, capTCBPtr_CL = capTCBPtr_' s && 0xFFFFFFC0\<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 (Suc 0) 0) 0 ((((capReplyCanGrant_' s << 5) && 0x20 || (capReplyMaster_' s << 4) && 0x10) || capTCBPtr_' s && 0xFFFFFFC0) || SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF)) v) = SCAST(32 signed \<rightarrow> 32) cap_reply_cap \<Longrightarrow> (case cap_lift (cap_C.words_C_update (\<lambda>x. Arrays.update (Arrays.update x (Suc 0) 0) 0 ((((capReplyCanGrant_' s << 5) && 0x20 || (capReplyMaster_' s << 4) && 0x10) || capTCBPtr_' s && 0xFFFFFFC0) || SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF)) v) of Some (Cap_reply_cap rec) \<Rightarrow> rec) = \<lparr>capReplyCanGrant_CL = capReplyCanGrant_' s && 1, capReplyMaster_CL = capReplyMaster_' s && 1, capTCBPtr_CL = capTCBPtr_' s && 0xFFFFFFC0\<rparr> proof (prove)
goal (1 subgoal):
 1. \<And>s. ((((capReplyCanGrant_' s << 5) && 0x20 || (capReplyMaster_' s << 4) && 0x10) || capTCBPtr_' s && 0xFFFFFFC0) || SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF >> 5) && 1 = capReplyCanGrant_' s && 1 \<and> ((((capReplyCanGrant_' s << 5) && 0x20 || (capReplyMaster_' s << 4) && 0x10) || capTCBPtr_' s && 0xFFFFFFC0) || SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF >> 4) && 1 = capReplyMaster_' s && 1 \<and> ((((capReplyCanGrant_' s << 5) && 0x20 || (capReplyMaster_' s << 4) && 0x10) || capTCBPtr_' s && 0xFFFFFFC0) || SCAST(32 signed \<rightarrow> 32) cap_reply_cap && 0xF) && ~~ mask 6 = capTCBPtr_' s && 0xFFFFFFC0 proof (prove)
goal:
No subgoals!