Local mirror of The Unified Policy Framework (UPF) entry of the Archive of Formal Proofs (AFP). https://www.isa-afp.org/entries/UPF.shtml
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* HOL-TestGen --- theorem-prover based test case generation
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* 2009-2014 Université Paris-Sud, France
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(* $Id: SeqComposition.thy 10879 2014-10-26 11:35:31Z brucker $ *)
header{* Sequential Composition *}
theory
SeqComposition
imports
ElementaryPolicies
begin
text{*
Sequential composition is based on the idea that two policies are to be combined by applying
the second policy to the output of the first one. Again, there are four possibilities how the
decisions can be combined. *}
section {* Flattening *}
text{*
A key concept of sequential policy composition is the flattening of nested decisions. There are
four possibilities, and these possibilities will give the various flavours of policy composition.
*}
fun flat_orA :: "('\<alpha> decision) decision \<Rightarrow> ('\<alpha> decision)"
where "flat_orA(allow(allow y)) = allow y"
|"flat_orA(allow(deny y)) = allow y"
|"flat_orA(deny(allow y)) = allow y"
|"flat_orA(deny(deny y)) = deny y"
lemma flat_orA_deny[dest]:"flat_orA x = deny y \<Longrightarrow> x = deny(deny y)"
apply (case_tac "x")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
done
lemma flat_orA_allow[dest]: "flat_orA x = allow y \<Longrightarrow> x = allow(allow y)
\<or> x = allow(deny y)
\<or> x = deny(allow y)"
apply (case_tac "x")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
done
fun flat_orD :: "('\<alpha> decision) decision \<Rightarrow> ('\<alpha> decision)"
where "flat_orD(allow(allow y)) = allow y"
|"flat_orD(allow(deny y)) = deny y"
|"flat_orD(deny(allow y)) = deny y"
|"flat_orD(deny(deny y)) = deny y"
lemma flat_orD_allow[dest]: "flat_orD x = allow y \<Longrightarrow> x = allow(allow y)"
apply (case_tac "x")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
done
lemma flat_orD_deny[dest]: "flat_orD x = deny y \<Longrightarrow> x = deny(deny y)
\<or> x = allow(deny y)
\<or> x = deny(allow y)"
apply (case_tac "x")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
done
fun flat_1 :: "('\<alpha> decision) decision \<Rightarrow> ('\<alpha> decision)"
where "flat_1(allow(allow y)) = allow y"
|"flat_1(allow(deny y)) = allow y"
|"flat_1(deny(allow y)) = deny y"
|"flat_1(deny(deny y)) = deny y"
lemma flat_1_allow[dest]: "flat_1 x = allow y \<Longrightarrow> x = allow(allow y) \<or> x = allow(deny y)"
apply (case_tac "x")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
done
lemma flat_1_deny[dest]: "flat_1 x = deny y \<Longrightarrow> x = deny(deny y) \<or> x = deny(allow y)"
apply (case_tac "x")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
done
fun flat_2 :: "('\<alpha> decision) decision \<Rightarrow> ('\<alpha> decision)"
where "flat_2(allow(allow y)) = allow y"
|"flat_2(allow(deny y)) = deny y"
|"flat_2(deny(allow y)) = allow y"
|"flat_2(deny(deny y)) = deny y"
lemma flat_2_allow[dest]: "flat_2 x = allow y \<Longrightarrow> x = allow(allow y) \<or> x = deny(allow y)"
apply (case_tac "x")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
done
lemma flat_2_deny[dest]: "flat_2 x = deny y \<Longrightarrow> x = deny(deny y) \<or> x = allow(deny y)"
apply (case_tac "x")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
apply (case_tac "\<alpha>")
apply (simp_all)
done
section{* Policy Composition *}
text{*
The following definition allows to compose two policies. Denies and allows are transferred.
*}
fun lift :: "('\<alpha> \<mapsto> '\<beta>) \<Rightarrow> ('\<alpha> decision \<mapsto>'\<beta> decision)"
where "lift f (deny s) = (case f s of
\<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>deny y\<rfloor>
| \<bottom> \<Rightarrow> \<bottom>)"
| "lift f (allow s) = (case f s of
\<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>allow y\<rfloor>
| \<bottom> \<Rightarrow> \<bottom>)"
lemma lift_mt [simp]: "lift \<emptyset> = \<emptyset>"
apply (rule ext)
apply (case_tac "x")
apply (simp_all)
done
text{*
Since policies are maps, we inherit a composition on them. However, this results in nestings
of decisions---which must be flattened. As we now that there are four different forms of
flattening, we have four different forms of policy composition: *}
definition
comp_orA :: "['\<beta>\<mapsto>'\<gamma>, '\<alpha>\<mapsto>'\<beta>] \<Rightarrow> '\<alpha>\<mapsto>'\<gamma>" (infixl "o'_orA" 55) where
"p2 o_orA p1 \<equiv> (map_option flat_orA) o (lift p2 o_m p1)"
notation (xsymbols)
comp_orA (infixl "\<circ>\<^sub>\<or>\<^sub>A" 55)
lemma comp_orA_mt[simp]:"p \<circ>\<^sub>\<or>\<^sub>A \<emptyset> = \<emptyset>"
by(simp add: comp_orA_def)
lemma mt_comp_orA[simp]:"\<emptyset> \<circ>\<^sub>\<or>\<^sub>A p = \<emptyset>"
by(simp add: comp_orA_def)
definition
comp_orD :: "['\<beta>\<mapsto>'\<gamma>, '\<alpha>\<mapsto>'\<beta>] \<Rightarrow> '\<alpha>\<mapsto>'\<gamma>" (infixl "o'_orD" 55) where
"p2 o_orD p1 \<equiv> (map_option flat_orD) o (lift p2 o_m p1)"
notation (xsymbols)
comp_orD (infixl "\<circ>\<^sub>orD" 55)
lemma comp_orD_mt[simp]:"p o_orD \<emptyset> = \<emptyset>"
by(simp add: comp_orD_def)
lemma mt_comp_orD[simp]:"\<emptyset> o_orD p = \<emptyset>"
by(simp add: comp_orD_def)
definition
comp_1 :: "['\<beta>\<mapsto>'\<gamma>, '\<alpha>\<mapsto>'\<beta>] \<Rightarrow> '\<alpha>\<mapsto>'\<gamma>" (infixl "o'_1" 55) where
"p2 o_1 p1 \<equiv> (map_option flat_1) o (lift p2 o_m p1)"
notation (xsymbols)
comp_1 (infixl "\<circ>\<^sub>1" 55)
lemma comp_1_mt[simp]:"p \<circ>\<^sub>1 \<emptyset> = \<emptyset>"
by(simp add: comp_1_def)
lemma mt_comp_1[simp]:"\<emptyset> \<circ>\<^sub>1 p = \<emptyset>"
by(simp add: comp_1_def)
definition
comp_2 :: "['\<beta>\<mapsto>'\<gamma>, '\<alpha>\<mapsto>'\<beta>] \<Rightarrow> '\<alpha>\<mapsto>'\<gamma>" (infixl "o'_2" 55) where
"p2 o_2 p1 \<equiv> (map_option flat_2) o (lift p2 o_m p1)"
notation (xsymbols)
comp_2 (infixl "\<circ>\<^sub>2" 55)
lemma comp_2_mt[simp]:"p \<circ>\<^sub>2 \<emptyset> = \<emptyset>"
by(simp add: comp_2_def)
lemma mt_comp_2[simp]:"\<emptyset> \<circ>\<^sub>2 p = \<emptyset>"
by(simp add: comp_2_def)
end