theory CSpaceInv_AI

lemma set_cap_live[wp]:
  "\<lbrace>\<lambda>s. P (obj_at live p' s)\<rbrace>
     set_cap cap p \<lbrace>\<lambda>rv s. P (obj_at live p' s)\<rbrace>" apply (simp add: set_cap_def split_def set_object_def) apply (wp get_object_wp | wpc)+ by (fastforce simp: obj_at_def live_def)

(* Proof state trace *)
proof (prove)
goal (1 subgoal):
 1. \<lbrace>\<lambda>s. P (obj_at live p' s)\<rbrace> set_cap cap p \<lbrace>\<lambda>rv s. P (obj_at live p' s)\<rbrace> proof (prove)
goal (1 subgoal):
 1. \<lbrace>\<lambda>s. P (obj_at live p' s)\<rbrace> do obj <- get_object (fst p);
obj' <- case obj of CNode sz cn \<Rightarrow> if snd p \<in> dom cn \<and> well_formed_cnode_n sz cn then return $ CNode sz $ cn(snd p \<mapsto> cap) else fail | TCB tcb \<Rightarrow> if snd p = tcb_cnode_index 0 then return $ TCB $ tcb\<lparr>tcb_ctable := cap\<rparr> else if snd p = tcb_cnode_index 1 then return $ TCB $ tcb\<lparr>tcb_vtable := cap\<rparr> else if snd p = tcb_cnode_index 2 then return $ TCB $ tcb\<lparr>tcb_reply := cap\<rparr> else if snd p = tcb_cnode_index 3 then return $ TCB $ tcb\<lparr>tcb_caller := cap\<rparr> else if snd p = tcb_cnode_index 4 then return $ TCB $ tcb\<lparr>tcb_ipcframe := cap\<rparr> else fail | _ \<Rightarrow> fail;
kobj <- get_object (fst p);
y <- assert (a_type kobj = a_type obj');
s <- get;
put (s\<lparr>kheap := (kheap s)(fst p \<mapsto> obj')\<rparr>)
od \<lbrace>\<lambda>rv s. P (obj_at live p' s)\<rbrace> proof (prove)
goal (1 subgoal):
 1. \<And>s. P (obj_at live p' s) \<Longrightarrow> \<forall>ko. ko_at ko (fst p) s \<longrightarrow> (\<forall>x xa. ko = CNode x xa \<longrightarrow> (snd p \<in> dom xa \<and> well_formed_cnode_n x xa \<longrightarrow> (\<forall>ko. ko_at ko (fst p) s \<longrightarrow> a_type ko = a_type (CNode x $ xa(snd p \<mapsto> cap)) \<longrightarrow> P (obj_at live p' (s\<lparr>kheap := (kheap s)(fst p \<mapsto> CNode x $ xa(snd p \<mapsto> cap))\<rparr>)))) \<and> (\<not> (snd p \<in> dom xa \<and> well_formed_cnode_n x xa) \<longrightarrow> True)) \<and> (\<forall>x. ko = TCB x \<longrightarrow> (snd p = tcb_cnode_index 0 \<longrightarrow> (\<forall>ko. ko_at ko (fst p) s \<longrightarrow> a_type ko = a_type (TCB $ x\<lparr>tcb_ctable := cap\<rparr>) \<longrightarrow> P (obj_at live p' (s\<lparr>kheap := (kheap s)(fst p \<mapsto> TCB $ x\<lparr>tcb_ctable := cap\<rparr>)\<rparr>)))) \<and> (snd p \<noteq> tcb_cnode_index 0 \<longrightarrow> (snd p = tcb_cnode_index 1 \<longrightarrow> (\<forall>ko. ko_at ko (fst p) s \<longrightarrow> a_type ko = a_type (TCB $ x\<lparr>tcb_vtable := cap\<rparr>) \<longrightarrow> P (obj_at live p' (s\<lparr>kheap := (kheap s)(fst p \<mapsto> TCB $ x\<lparr>tcb_vtable := cap\<rparr>)\<rparr>)))) \<and> (snd p \<noteq> tcb_cnode_index 1 \<longrightarrow> (snd p = tcb_cnode_index 2 \<longrightarrow> (\<forall>ko. ko_at ko (fst p) s \<longrightarrow> a_type ko = a_type (TCB $ x\<lparr>tcb_reply := cap\<rparr>) \<longrightarrow> P (obj_at live p' (s\<lparr>kheap := (kheap s)(fst p \<mapsto> TCB $ x\<lparr>tcb_reply := cap\<rparr>)\<rparr>)))) \<and> (snd p \<noteq> tcb_cnode_index 2 \<longrightarrow> (snd p = tcb_cnode_index 3 \<longrightarrow> (\<forall>ko. ko_at ko (fst p) s \<longrightarrow> a_type ko = a_type (TCB $ x\<lparr>tcb_caller := cap\<rparr>) \<longrightarrow> P (obj_at live p' (s\<lparr>kheap := (kheap s)(fst p \<mapsto> TCB $ x\<lparr>tcb_caller := cap\<rparr>)\<rparr>)))) \<and> (snd p \<noteq> tcb_cnode_index 3 \<longrightarrow> (snd p = tcb_cnode_index 4 \<longrightarrow> (\<forall>ko. ko_at ko (fst p) s \<longrightarrow> a_type ko = a_type (TCB $ x\<lparr>tcb_ipcframe := cap\<rparr>) \<longrightarrow> P (obj_at live p' (s\<lparr>kheap := (kheap s)(fst p \<mapsto> TCB $ x\<lparr>tcb_ipcframe := cap\<rparr>)\<rparr>)))) \<and> (snd p \<noteq> tcb_cnode_index 4 \<longrightarrow> True)))))) \<and> (\<forall>x. ko = Endpoint x \<longrightarrow> True) \<and> (\<forall>x. ko = Notification x \<longrightarrow> True) \<and> (\<forall>x. ko = ArchObj x \<longrightarrow> True)