theory structures_proofs

lemma (in kernel_all_substitute) cap_untyped_cap_get_capFreeIndex_spec:
  "\<forall>s. \<Gamma> \<turnstile> \<lbrace>s. cap_get_tag \<acute>cap = scast cap_untyped_cap\<rbrace>
       \<acute>ret__unsigned :== PROC cap_untyped_cap_get_capFreeIndex(\<acute>cap)
       \<lbrace>\<acute>ret__unsigned = cap_untyped_cap_CL.capFreeIndex_CL (cap_untyped_cap_lift \<^bsup>s\<^esup>cap)\<rbrace>" apply(rule allI, rule conseqPre, vcg) apply (clarsimp simp:guard_simps) apply (simp add:cap_untyped_cap_lift_def) apply (subst cap_lift_untyped_cap) apply (simp add: o_def
                    cap_get_tag_def
                    cap_untyped_cap_def
                    mask_def word_size shift_over_ao_dists) apply (subst cap_lift_untyped_cap, simp)? apply (simp add: o_def
                   cap_get_tag_def
                   cap_untyped_cap_def
                   mask_def word_size shift_over_ao_dists multi_shift_simps
                   word_bw_assocs word_oa_dist word_and_max_simps ucast_def
                   sign_extend_def' nth_is_and_neq_0) 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_get_capFreeIndex_'proc \<lbrace>\<acute>ret__unsigned = capFreeIndex_CL (cap_untyped_cap_lift \<^bsup>s\<^esup>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>\<forall>v. 0 \<le>s 6 \<and> 6 <s 0x20 \<and> 1 <s 2 \<and> 0 \<le>s 1 \<and> ((if False then 1 else 0) \<noteq> 0 \<longrightarrow> 0 \<le>s 0x1F \<and> 0x1F <s 0x20 \<and> ((if (cap_C.words_C \<acute>cap.[unat 1] && 0xFFFFFFC0 >> unat 6) && (1 << unat 0x1F) \<noteq> 0 then 1 else 0) \<noteq> 0 \<longrightarrow> (cap_C.words_C \<acute>cap.[unat 1] && 0xFFFFFFC0 >> unat 6) || SCAST(32 signed \<rightarrow> 32) 0 = capFreeIndex_CL (cap_untyped_cap_lift \<^bsup>s\<^esup>cap)) \<and> ((if (cap_C.words_C \<acute>cap.[unat 1] && 0xFFFFFFC0 >> unat 6) && (1 << unat 0x1F) \<noteq> 0 then 1 else 0) = 0 \<longrightarrow> cap_C.words_C \<acute>cap.[unat 1] && 0xFFFFFFC0 >> unat 6 = capFreeIndex_CL (cap_untyped_cap_lift \<^bsup>s\<^esup>cap))) \<and> ((if False then 1 else 0) = 0 \<longrightarrow> cap_C.words_C \<acute>cap.[unat 1] && 0xFFFFFFC0 >> unat 6 = capFreeIndex_CL (cap_untyped_cap_lift \<^bsup>s\<^esup>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_C.words_C (cap_' x).[Suc 0] && 0xFFFFFFC0 >> 6 = capFreeIndex_CL (cap_untyped_cap_lift (cap_' x)) proof (prove)
goal (1 subgoal):
 1. \<And>x. cap_get_tag (cap_' x) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap \<Longrightarrow> cap_C.words_C (cap_' x).[Suc 0] && 0xFFFFFFC0 >> 6 = capFreeIndex_CL (case cap_lift (cap_' x) of Some (Cap_untyped_cap rec) \<Rightarrow> rec) proof (prove)
goal (2 subgoals):
 1. \<And>x. cap_get_tag (cap_' x) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap \<Longrightarrow> cap_get_tag (cap_' x) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap
 2. \<And>x. cap_get_tag (cap_' x) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap \<Longrightarrow> cap_C.words_C (cap_' x).[Suc 0] && 0xFFFFFFC0 >> 6 = capFreeIndex_CL (case Some (Cap_untyped_cap \<lparr>capFreeIndex_CL = (cap_C.words_C (cap_' x).[1] >> 6) && mask 26, capIsDevice_CL = (cap_C.words_C (cap_' x).[1] >> 5) && mask 1, capBlockSize_CL = (cap_C.words_C (cap_' x).[1] >> 0) && mask 5, capPtr_CL = (cap_C.words_C (cap_' x).[0] << 0) && ~~ mask 4\<rparr>) of Some (Cap_untyped_cap rec) \<Rightarrow> rec) proof (prove)
goal (1 subgoal):
 1. \<And>x. cap_get_tag (cap_' x) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap \<Longrightarrow> cap_C.words_C (cap_' x).[Suc 0] && 0xFFFFFFC0 >> 6 = capFreeIndex_CL (case Some (Cap_untyped_cap \<lparr>capFreeIndex_CL = (cap_C.words_C (cap_' x).[1] >> 6) && mask 26, capIsDevice_CL = (cap_C.words_C (cap_' x).[1] >> 5) && mask 1, capBlockSize_CL = (cap_C.words_C (cap_' x).[1] >> 0) && mask 5, capPtr_CL = (cap_C.words_C (cap_' x).[0] << 0) && ~~ mask 4\<rparr>) of Some (Cap_untyped_cap rec) \<Rightarrow> rec) proof (prove)
goal (1 subgoal):
 1. \<And>x. cap_get_tag (cap_' x) = SCAST(32 signed \<rightarrow> 32) cap_untyped_cap \<Longrightarrow> cap_C.words_C (cap_' x).[Suc 0] && 0xFFFFFFC0 >> 6 = capFreeIndex_CL (case Some (Cap_untyped_cap \<lparr>capFreeIndex_CL = (cap_C.words_C (cap_' x).[1] >> 6) && mask 26, capIsDevice_CL = (cap_C.words_C (cap_' x).[1] >> 5) && mask 1, capBlockSize_CL = (cap_C.words_C (cap_' x).[1] >> 0) && mask 5, capPtr_CL = (cap_C.words_C (cap_' x).[0] << 0) && ~~ mask 4\<rparr>) of Some (Cap_untyped_cap rec) \<Rightarrow> rec) proof (prove)
goal:
No subgoals!