120 lines
3.6 KiB
Plaintext
120 lines
3.6 KiB
Plaintext
(*
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* Copyright 2014, NICTA
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*
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* This software may be distributed and modified according to the terms of
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* the BSD 2-Clause license. Note that NO WARRANTY is provided.
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* See "LICENSE_BSD2.txt" for details.
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*
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* @TAG(NICTA_BSD)
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*)
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(* Author: Gerwin Klein, 2012
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Maintainers: Gerwin Klein <kleing at cse.unsw.edu.au>
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Rafal Kolanski <rafal.kolanski at nicta.com.au>
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*)
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header "Equivalence between Separation Algebra Formulations"
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theory Sep_Eq
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imports
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Separation_Algebra
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Separation_Algebra_Alt
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begin
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text {*
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In this theory we show that our total formulation of separation algebra is
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equivalent in strength to Calcagno et al's original partial one.
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This theory is not intended to be included in own developments.
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*}
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no_notation map_add (infixl "++" 100)
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section "Total implies Partial"
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definition add2 :: "'a::sep_algebra => 'a => 'a option" where
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"add2 x y \<equiv> if x ## y then Some (x + y) else None"
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lemma add2_zero: "add2 x 0 = Some x"
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by (simp add: add2_def)
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lemma add2_comm: "add2 x y = add2 y x"
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by (auto simp: add2_def sep_add_commute sep_disj_commute)
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lemma add2_assoc:
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"lift2 add2 a (lift2 add2 b c) = lift2 add2 (lift2 add2 a b) c"
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by (auto simp: add2_def lift2_def sep_add_assoc
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dest: sep_disj_addD sep_disj_addI3
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sep_add_disjD sep_disj_addI2 sep_disj_commuteI
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split: option.splits)
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interpretation total_partial: sep_algebra_alt 0 add2
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by (unfold_locales) (auto intro: add2_zero add2_comm add2_assoc)
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section "Partial implies Total"
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definition
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sep_add' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a :: sep_algebra_alt" where
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"sep_add' x y \<equiv> if disjoint x y then the (add x y) else undefined"
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lemma sep_disj_zero':
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"disjoint x 0"
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by simp
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lemma sep_disj_commuteI':
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"disjoint x y \<Longrightarrow> disjoint y x"
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by (clarsimp simp: disjoint_def add_comm)
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lemma sep_add_zero':
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"sep_add' x 0 = x"
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by (simp add: sep_add'_def)
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lemma sep_add_commute':
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"disjoint x y \<Longrightarrow> sep_add' x y = sep_add' y x"
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by (clarsimp simp: sep_add'_def disjoint_def add_comm)
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lemma sep_add_assoc':
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"\<lbrakk> disjoint x y; disjoint y z; disjoint x z \<rbrakk> \<Longrightarrow>
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sep_add' (sep_add' x y) z = sep_add' x (sep_add' y z)"
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using add_assoc [of "Some x" "Some y" "Some z"]
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by (clarsimp simp: disjoint_def sep_add'_def lift2_def
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split: option.splits)
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lemma sep_disj_addD1':
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"disjoint x (sep_add' y z) \<Longrightarrow> disjoint y z \<Longrightarrow> disjoint x y"
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proof (clarsimp simp: disjoint_def sep_add'_def)
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fix a assume a: "y \<oplus> z = Some a"
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fix b assume b: "x \<oplus> a = Some b"
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with a have "Some x ++ (Some y ++ Some z) = Some b" by (simp add: lift2_def)
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hence "(Some x ++ Some y) ++ Some z = Some b" by (simp add: add_assoc)
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thus "\<exists>b. x \<oplus> y = Some b" by (simp add: lift2_def split: option.splits)
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qed
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lemma sep_disj_addI1':
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"disjoint x (sep_add' y z) \<Longrightarrow> disjoint y z \<Longrightarrow> disjoint (sep_add' x y) z"
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apply (clarsimp simp: disjoint_def sep_add'_def)
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apply (rule conjI)
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apply clarsimp
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apply (frule lift_to_add2, assumption)
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apply (simp add: add_assoc)
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apply (clarsimp simp: lift2_def add_comm)
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apply clarsimp
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apply (frule lift_to_add2, assumption)
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apply (simp add: add_assoc)
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apply (clarsimp simp: lift2_def)
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done
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interpretation partial_total: sep_algebra sep_add' 0 disjoint
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apply (unfold_locales)
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apply (rule sep_disj_zero')
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apply (erule sep_disj_commuteI')
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apply (rule sep_add_zero')
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apply (erule sep_add_commute')
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apply (erule (2) sep_add_assoc')
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apply (erule (1) sep_disj_addD1')
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apply (erule (1) sep_disj_addI1')
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done
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end
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