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theory Defs
imports "HOL-IMP.Sec_Type_Expr" "HOL-IMP.Def_Init_Small"
begin
inductive sec_type :: "nat \<Rightarrow> com \<Rightarrow> bool" ("(_/ \<turnstile> _)" [0,0] 50) where
Skip: "l \<turnstile> SKIP" |
Assign: "\<lbrakk> sec x \<ge> sec a; sec x \<ge> l \<rbrakk> \<Longrightarrow> l \<turnstile> x ::= a" |
Seq: "\<lbrakk> l \<turnstile> c\<^sub>1; l \<turnstile> c\<^sub>2 \<rbrakk> \<Longrightarrow> l \<turnstile> c\<^sub>1;;c\<^sub>2" |
If: "\<lbrakk> max (sec b) l \<turnstile> c\<^sub>1; max (sec b) l \<turnstile> c\<^sub>2 \<rbrakk> \<Longrightarrow> l \<turnstile> IF b THEN c\<^sub>1 ELSE c\<^sub>2" |
While: "max (sec b) l \<turnstile> c \<Longrightarrow> l \<turnstile> WHILE b DO c"
inductive_cases [elim!]: "l \<turnstile> x ::= a" "l \<turnstile> c\<^sub>1;;c\<^sub>2" "l \<turnstile> IF b THEN c\<^sub>1 ELSE c\<^sub>2" "l \<turnstile> WHILE b DO c"
lemma anti_mono: "\<lbrakk> l \<turnstile> c; l' \<le> l \<rbrakk> \<Longrightarrow> l' \<turnstile> c"
apply(induction arbitrary: l' rule: sec_type.induct)
apply (metis sec_type.intros(1))
apply (metis le_trans sec_type.intros(2))
apply (metis sec_type.intros(3))
apply (metis If le_refl sup_mono sup_nat_def)
apply (metis While le_refl sup_mono sup_nat_def)
done
inductive sec_type2 :: "com \<Rightarrow> level \<Rightarrow> bool" ("(\<turnstile> _ : _)" [0,0] 50) where
Skip2: "\<turnstile> SKIP : l" |
Assign2: "sec x \<ge> sec a \<Longrightarrow> \<turnstile> x ::= a : sec x" |
Seq2: "\<lbrakk> \<turnstile> c\<^sub>1 : l\<^sub>1; \<turnstile> c\<^sub>2 : l\<^sub>2 \<rbrakk> \<Longrightarrow> \<turnstile> c\<^sub>1;;c\<^sub>2 : min l\<^sub>1 l\<^sub>2 " |
If2: "\<lbrakk> sec b \<le> min l\<^sub>1 l\<^sub>2; \<turnstile> c\<^sub>1 : l\<^sub>1; \<turnstile> c\<^sub>2 : l\<^sub>2 \<rbrakk> \<Longrightarrow> \<turnstile> IF b THEN c\<^sub>1 ELSE c\<^sub>2 : min l\<^sub>1 l\<^sub>2" |
While2: "\<lbrakk> sec b \<le> l; \<turnstile> c : l \<rbrakk> \<Longrightarrow> \<turnstile> WHILE b DO c : l"
end
theory Submission imports Defs begin theorem bottom_up_impl_top_down: "\<turnstile> c : l \<Longrightarrow> l \<turnstile> c" sorry theorem top_down_impl_bottom_up: "l \<turnstile> c \<Longrightarrow> \<exists> l' \<ge> l. \<turnstile> c : l'" sorry end
theory Check imports Submission begin theorem bottom_up_impl_top_down: "\<turnstile> c : l \<Longrightarrow> l \<turnstile> c" by (rule Submission.bottom_up_impl_top_down) theorem top_down_impl_bottom_up: "l \<turnstile> c \<Longrightarrow> \<exists> l' \<ge> l. \<turnstile> c : l'" by (rule Submission.top_down_impl_bottom_up) end