theory structures_proofs

lemma (in kernel_all_substitute) cap_untyped_cap_set_capFreeIndex_spec:
  "\<forall>s. \<Gamma> \<turnstile> \<lbrace>s. cap_get_tag \<acute>cap = scast cap_untyped_cap\<rbrace>
       \<acute>ret__struct_cap_C :== PROC cap_untyped_cap_set_capFreeIndex(\<acute>cap, \<acute>v32)
       \<lbrace>cap_untyped_cap_lift \<acute>ret__struct_cap_C = cap_untyped_cap_lift \<^bsup>s\<^esup>cap \<lparr> cap_untyped_cap_CL.capFreeIndex_CL :=  (\<^bsup>s\<^esup>v32 AND mask 26)\<rparr> \<and>
        cap_get_tag \<acute>ret__struct_cap_C = scast cap_untyped_cap\<rbrace>" apply (rule allI, rule conseqPre, vcg) apply clarsimp apply (rule context_conjI[THEN iffD1[OF conj_commute]],
         fastforce simp: cap_get_tag_eq_x cap_lift_def cap_tag_defs
                         mask_def shift_over_ao_dists multi_shift_simps word_size
                         word_ao_dist word_bw_assocs) apply (simp add: cap_untyped_cap_lift_def cap_lift_def cap_tag_defs) 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))? done

(* Proof state trace *)
proof (prove)
goal (1 subgoal):
 1. \<forall>s. \<Gamma>\<turnstile> \<lbrace>s. cap_get_tag \<acute>cap = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap\<rbrace> Call cap_untyped_cap_set_capFreeIndex_'proc \<lbrace>cap_untyped_cap_lift \<acute>ret__struct_cap_C = cap_untyped_cap_lift \<^bsup>s\<^esup>cap\<lparr>capFreeIndex_CL := \<^bsup>s\<^esup>v32 && mask 26\<rparr> \<and> cap_get_tag \<acute>ret__struct_cap_C = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap\<rbrace> proof (prove)
goal (1 subgoal):
 1. \<And>s. \<lbrace>s. cap_get_tag \<acute>cap = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap\<rbrace> \<subseteq> \<lbrace>1 <s 2 \<and> 0 \<le>s 1 \<and> 1 <s 2 \<and> 0 \<le>s 1 \<and> 1 <s 2 \<and> 0 \<le>s 1 \<and> 0 \<le>s 6 \<and> 6 <s 0x20 \<and> 1 <s 2 \<and> 0 \<le>s 1 \<and> cap_untyped_cap_lift (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 1) (v.[unat 1] || (\<acute>v32 << unat 6) && 0xFFFFFFC0)) (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 1) (v.[unat 1] && ~~ 0xFFFFFFC0)) \<acute>cap)) = cap_untyped_cap_lift \<^bsup>s\<^esup>cap\<lparr>capFreeIndex_CL := \<^bsup>s\<^esup>v32 && mask 26\<rparr> \<and> cap_get_tag (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 1) (v.[unat 1] || (\<acute>v32 << unat 6) && 0xFFFFFFC0)) (cap_C.words_C_update (\<lambda>v. Arrays.update v (unat 1) (v.[unat 1] && ~~ 0xFFFFFFC0)) \<acute>cap)) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap\<rbrace> proof (prove)
goal (1 subgoal):
 1. \<And>x. cap_get_tag (cap_' x) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap \<Longrightarrow> cap_untyped_cap_lift (cap_C.words_C_update ((\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] || (v32_' x << 6) && 0xFFFFFFC0)) \<circ> (\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] && 0x3F))) (cap_' x)) = cap_untyped_cap_lift (cap_' x)\<lparr>capFreeIndex_CL := v32_' x && mask 26\<rparr> \<and> cap_get_tag (cap_C.words_C_update ((\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] || (v32_' x << 6) && 0xFFFFFFC0)) \<circ> (\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] && 0x3F))) (cap_' x)) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap proof (prove)
goal (1 subgoal):
 1. \<And>x. \<lbrakk>cap_get_tag (cap_' x) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap; cap_get_tag (cap_C.words_C_update ((\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] || (v32_' x << 6) && 0xFFFFFFC0)) \<circ> (\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] && 0x3F))) (cap_' x)) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap\<rbrakk> \<Longrightarrow> cap_untyped_cap_lift (cap_C.words_C_update ((\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] || (v32_' x << 6) && 0xFFFFFFC0)) \<circ> (\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] && 0x3F))) (cap_' x)) = cap_untyped_cap_lift (cap_' x)\<lparr>capFreeIndex_CL := v32_' x && mask 26\<rparr> proof (prove)
goal (1 subgoal):
 1. \<And>x. \<lbrakk>cap_get_tag (cap_' x) = 2; cap_get_tag (cap_C.words_C_update ((\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] || (v32_' x << 6) && 0xFFFFFFC0)) \<circ> (\<lambda>v. Arrays.update v (Suc 0) (v.[Suc 0] && 0x3F))) (cap_' x)) = 2\<rbrakk> \<Longrightarrow> (cap_C.words_C (cap_' x).[Suc 0] && 0x3F || (v32_' x << 6) && 0xFFFFFFC0 >> 6) && mask 26 = v32_' x && mask 26 \<and> (cap_C.words_C (cap_' x).[Suc 0] && 0x3F || (v32_' x << 6) && 0xFFFFFFC0 >> 5) && 1 = (cap_C.words_C (cap_' x).[Suc 0] >> 5) && 1 \<and> (cap_C.words_C (cap_' x).[Suc 0] && 0x3F || (v32_' x << 6) && 0xFFFFFFC0) && mask 5 = cap_C.words_C (cap_' x).[Suc 0] && mask 5 proof (prove)
goal:
No subgoals!