(* Title: HOL/UNITY/Transformers.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2003 University of Cambridge
Predicate Transformers. From
David Meier and Beverly Sanders,
Composing Leads-to Properties
Theoretical Computer Science 243:1-2 (2000), 339-361.
David Meier,
Progress Properties in Program Refinement and Parallel Composition
Swiss Federal Institute of Technology Zurich (1997)
*)
header{*Predicate Transformers*}
theory Transformers imports Comp begin
subsection{*Defining the Predicate Transformers @{term wp},
@{term awp} and @{term wens}*}
definition wp :: "[('a*'a) set, 'a set] => 'a set" where
--{*Dijkstra's weakest-precondition operator (for an individual command)*}
"wp act B == - (act^-1 `` (-B))"
definition awp :: "['a program, 'a set] => 'a set" where
--{*Dijkstra's weakest-precondition operator (for a program)*}
"awp F B == (\<Inter>act \<in> Acts F. wp act B)"
definition wens :: "['a program, ('a*'a) set, 'a set] => 'a set" where
--{*The weakest-ensures transformer*}
"wens F act B == gfp(\<lambda>X. (wp act B \<inter> awp F (B \<union> X)) \<union> B)"
text{*The fundamental theorem for wp*}
theorem wp_iff: "(A <= wp act B) = (act `` A <= B)"
by (force simp add: wp_def)
text{*This lemma is a good deal more intuitive than the definition!*}
lemma in_wp_iff: "(a \<in> wp act B) = (\<forall>x. (a,x) \<in> act --> x \<in> B)"
by (simp add: wp_def, blast)
lemma Compl_Domain_subset_wp: "- (Domain act) \<subseteq> wp act B"
by (force simp add: wp_def)
lemma wp_empty [simp]: "wp act {} = - (Domain act)"
by (force simp add: wp_def)
text{*The identity relation is the skip action*}
lemma wp_Id [simp]: "wp Id B = B"
by (simp add: wp_def)
lemma wp_totalize_act:
"wp (totalize_act act) B = (wp act B \<inter> Domain act) \<union> (B - Domain act)"
by (simp add: wp_def totalize_act_def, blast)
lemma awp_subset: "(awp F A \<subseteq> A)"
by (force simp add: awp_def wp_def)
lemma awp_Int_eq: "awp F (A\<inter>B) = awp F A \<inter> awp F B"
by (simp add: awp_def wp_def, blast)
text{*The fundamental theorem for awp*}
theorem awp_iff_constrains: "(A <= awp F B) = (F \<in> A co B)"
by (simp add: awp_def constrains_def wp_iff INT_subset_iff)
lemma awp_iff_stable: "(A \<subseteq> awp F A) = (F \<in> stable A)"
by (simp add: awp_iff_constrains stable_def)
lemma stable_imp_awp_ident: "F \<in> stable A ==> awp F A = A"
apply (rule equalityI [OF awp_subset])
apply (simp add: awp_iff_stable)
done
lemma wp_mono: "(A \<subseteq> B) ==> wp act A \<subseteq> wp act B"
by (simp add: wp_def, blast)
lemma awp_mono: "(A \<subseteq> B) ==> awp F A \<subseteq> awp F B"
by (simp add: awp_def wp_def, blast)
lemma wens_unfold:
"wens F act B = (wp act B \<inter> awp F (B \<union> wens F act B)) \<union> B"
apply (simp add: wens_def)
apply (rule gfp_unfold)
apply (simp add: mono_def wp_def awp_def, blast)
done
lemma wens_Id [simp]: "wens F Id B = B"
by (simp add: wens_def gfp_def wp_def awp_def, blast)
text{*These two theorems justify the claim that @{term wens} returns the
weakest assertion satisfying the ensures property*}
lemma ensures_imp_wens: "F \<in> A ensures B ==> \<exists>act \<in> Acts F. A \<subseteq> wens F act B"
apply (simp add: wens_def ensures_def transient_def, clarify)
apply (rule rev_bexI, assumption)
apply (rule gfp_upperbound)
apply (simp add: constrains_def awp_def wp_def, blast)
done
lemma wens_ensures: "act \<in> Acts F ==> F \<in> (wens F act B) ensures B"
by (simp add: wens_def gfp_def constrains_def awp_def wp_def
ensures_def transient_def, blast)
text{*These two results constitute assertion (4.13) of the thesis*}
lemma wens_mono: "(A \<subseteq> B) ==> wens F act A \<subseteq> wens F act B"
apply (simp add: wens_def wp_def awp_def)
apply (rule gfp_mono, blast)
done
lemma wens_weakening: "B \<subseteq> wens F act B"
by (simp add: wens_def gfp_def, blast)
text{*Assertion (6), or 4.16 in the thesis*}
lemma subset_wens: "A-B \<subseteq> wp act B \<inter> awp F (B \<union> A) ==> A \<subseteq> wens F act B"
apply (simp add: wens_def wp_def awp_def)
apply (rule gfp_upperbound, blast)
done
text{*Assertion 4.17 in the thesis*}
lemma Diff_wens_constrains: "F \<in> (wens F act A - A) co wens F act A"
by (simp add: wens_def gfp_def wp_def awp_def constrains_def, blast)
--{*Proved instantly, yet remarkably fragile. If @{text Un_subset_iff}
is declared as an iff-rule, then it's almost impossible to prove.
One proof is via @{text meson} after expanding all definitions, but it's
slow!*}
text{*Assertion (7): 4.18 in the thesis. NOTE that many of these results
hold for an arbitrary action. We often do not require @{term "act \<in> Acts F"}*}
lemma stable_wens: "F \<in> stable A ==> F \<in> stable (wens F act A)"
apply (simp add: stable_def)
apply (drule constrains_Un [OF Diff_wens_constrains [of F act A]])
apply (simp add: Un_Int_distrib2 Compl_partition2)
apply (erule constrains_weaken, blast)
apply (simp add: wens_weakening)
done
text{*Assertion 4.20 in the thesis.*}
lemma wens_Int_eq_lemma:
"[|T-B \<subseteq> awp F T; act \<in> Acts F|]
==> T \<inter> wens F act B \<subseteq> wens F act (T\<inter>B)"
apply (rule subset_wens)
apply (rule_tac P="\<lambda>x. ?f x \<subseteq> ?b" in ssubst [OF wens_unfold])
apply (simp add: wp_def awp_def, blast)
done
text{*Assertion (8): 4.21 in the thesis. Here we indeed require
@{term "act \<in> Acts F"}*}
lemma wens_Int_eq:
"[|T-B \<subseteq> awp F T; act \<in> Acts F|]
==> T \<inter> wens F act B = T \<inter> wens F act (T\<inter>B)"
apply (rule equalityI)
apply (simp_all add: Int_lower1)
apply (rule wens_Int_eq_lemma, assumption+)
apply (rule subset_trans [OF _ wens_mono [of "T\<inter>B" B]], auto)
done
subsection{*Defining the Weakest Ensures Set*}
inductive_set
wens_set :: "['a program, 'a set] => 'a set set"
for F :: "'a program" and B :: "'a set"
where
Basis: "B \<in> wens_set F B"
| Wens: "[|X \<in> wens_set F B; act \<in> Acts F|] ==> wens F act X \<in> wens_set F B"
| Union: "W \<noteq> {} ==> \<forall>U \<in> W. U \<in> wens_set F B ==> \<Union>W \<in> wens_set F B"
lemma wens_set_imp_co: "A \<in> wens_set F B ==> F \<in> (A-B) co A"
apply (erule wens_set.induct)
apply (simp add: constrains_def)
apply (drule_tac act1=act and A1=X
in constrains_Un [OF Diff_wens_constrains])
apply (erule constrains_weaken, blast)
apply (simp add: wens_weakening)
apply (rule constrains_weaken)
apply (rule_tac I=W and A="\<lambda>v. v-B" and A'="\<lambda>v. v" in constrains_UN, blast+)
done
lemma wens_set_imp_leadsTo: "A \<in> wens_set F B ==> F \<in> A leadsTo B"
apply (erule wens_set.induct)
apply (rule leadsTo_refl)
apply (blast intro: wens_ensures leadsTo_Trans)
apply (blast intro: leadsTo_Union)
done
lemma leadsTo_imp_wens_set: "F \<in> A leadsTo B ==> \<exists>C \<in> wens_set F B. A \<subseteq> C"
apply (erule leadsTo_induct_pre)
apply (blast dest!: ensures_imp_wens intro: wens_set.Basis wens_set.Wens)
apply (clarify, drule ensures_weaken_R, assumption)
apply (blast dest!: ensures_imp_wens intro: wens_set.Wens)
apply (case_tac "S={}")
apply (simp, blast intro: wens_set.Basis)
apply (clarsimp dest!: bchoice simp: ball_conj_distrib Bex_def)
apply (rule_tac x = "\<Union>{Z. \<exists>U\<in>S. Z = f U}" in exI)
apply (blast intro: wens_set.Union)
done
text{*Assertion (9): 4.27 in the thesis.*}
lemma leadsTo_iff_wens_set: "(F \<in> A leadsTo B) = (\<exists>C \<in> wens_set F B. A \<subseteq> C)"
by (blast intro: leadsTo_imp_wens_set leadsTo_weaken_L wens_set_imp_leadsTo)
text{*This is the result that requires the definition of @{term wens_set} to
require @{term W} to be non-empty in the Unio case, for otherwise we should
always have @{term "{} \<in> wens_set F B"}.*}
lemma wens_set_imp_subset: "A \<in> wens_set F B ==> B \<subseteq> A"
apply (erule wens_set.induct)
apply (blast intro: wens_weakening [THEN subsetD])+
done
subsection{*Properties Involving Program Union*}
text{*Assertion (4.30) of thesis, reoriented*}
lemma awp_Join_eq: "awp (F\<squnion>G) B = awp F B \<inter> awp G B"
by (simp add: awp_def wp_def, blast)
lemma wens_subset: "wens F act B - B \<subseteq> wp act B \<inter> awp F (B \<union> wens F act B)"
by (subst wens_unfold, fast)
text{*Assertion (4.31)*}
lemma subset_wens_Join:
"[|A = T \<inter> wens F act B; T-B \<subseteq> awp F T; A-B \<subseteq> awp G (A \<union> B)|]
==> A \<subseteq> wens (F\<squnion>G) act B"
apply (subgoal_tac "(T \<inter> wens F act B) - B \<subseteq>
wp act B \<inter> awp F (B \<union> wens F act B) \<inter> awp F T")
apply (rule subset_wens)
apply (simp add: awp_Join_eq awp_Int_eq Un_commute)
apply (simp add: awp_def wp_def, blast)
apply (insert wens_subset [of F act B], blast)
done
text{*Assertion (4.32)*}
lemma wens_Join_subset: "wens (F\<squnion>G) act B \<subseteq> wens F act B"
apply (simp add: wens_def)
apply (rule gfp_mono)
apply (auto simp add: awp_Join_eq)
done
text{*Lemma, because the inductive step is just too messy.*}
lemma wens_Union_inductive_step:
assumes awpF: "T-B \<subseteq> awp F T"
and awpG: "!!X. X \<in> wens_set F B ==> (T\<inter>X) - B \<subseteq> awp G (T\<inter>X)"
shows "[|X \<in> wens_set F B; act \<in> Acts F; Y \<subseteq> X; T\<inter>X = T\<inter>Y|]
==> wens (F\<squnion>G) act Y \<subseteq> wens F act X \<and>
T \<inter> wens F act X = T \<inter> wens (F\<squnion>G) act Y"
apply (subgoal_tac "wens (F\<squnion>G) act Y \<subseteq> wens F act X")
prefer 2
apply (blast dest: wens_mono intro: wens_Join_subset [THEN subsetD], simp)
apply (rule equalityI)
prefer 2 apply blast
apply (simp add: Int_lower1)
apply (frule wens_set_imp_subset)
apply (subgoal_tac "T-X \<subseteq> awp F T")
prefer 2 apply (blast intro: awpF [THEN subsetD])
apply (rule_tac B = "wens (F\<squnion>G) act (T\<inter>X)" in subset_trans)
prefer 2 apply (blast intro!: wens_mono)
apply (subst wens_Int_eq, assumption+)
apply (rule subset_wens_Join [of _ T], simp, blast)
apply (subgoal_tac "T \<inter> wens F act (T\<inter>X) \<union> T\<inter>X = T \<inter> wens F act X")
prefer 2
apply (subst wens_Int_eq [symmetric], assumption+)
apply (blast intro: wens_weakening [THEN subsetD], simp)
apply (blast intro: awpG [THEN subsetD] wens_set.Wens)
done
theorem wens_Union:
assumes awpF: "T-B \<subseteq> awp F T"
and awpG: "!!X. X \<in> wens_set F B ==> (T\<inter>X) - B \<subseteq> awp G (T\<inter>X)"
and major: "X \<in> wens_set F B"
shows "\<exists>Y \<in> wens_set (F\<squnion>G) B. Y \<subseteq> X & T\<inter>X = T\<inter>Y"
apply (rule wens_set.induct [OF major])
txt{*Basis: trivial*}
apply (blast intro: wens_set.Basis)
txt{*Inductive step*}
apply clarify
apply (rule_tac x = "wens (F\<squnion>G) act Y" in rev_bexI)
apply (force intro: wens_set.Wens)
apply (simp add: wens_Union_inductive_step [OF awpF awpG])
txt{*Union: by Axiom of Choice*}
apply (simp add: ball_conj_distrib Bex_def)
apply (clarify dest!: bchoice)
apply (rule_tac x = "\<Union>{Z. \<exists>U\<in>W. Z = f U}" in exI)
apply (blast intro: wens_set.Union)
done
theorem leadsTo_Join:
assumes leadsTo: "F \<in> A leadsTo B"
and awpF: "T-B \<subseteq> awp F T"
and awpG: "!!X. X \<in> wens_set F B ==> (T\<inter>X) - B \<subseteq> awp G (T\<inter>X)"
shows "F\<squnion>G \<in> T\<inter>A leadsTo B"
apply (rule leadsTo [THEN leadsTo_imp_wens_set, THEN bexE])
apply (rule wens_Union [THEN bexE])
apply (rule awpF)
apply (erule awpG, assumption)
apply (blast intro: wens_set_imp_leadsTo [THEN leadsTo_weaken_L])
done
subsection {*The Set @{term "wens_set F B"} for a Single-Assignment Program*}
text{*Thesis Section 4.3.3*}
text{*We start by proving laws about single-assignment programs*}
lemma awp_single_eq [simp]:
"awp (mk_program (init, {act}, allowed)) B = B \<inter> wp act B"
by (force simp add: awp_def wp_def)
lemma wp_Un_subset: "wp act A \<union> wp act B \<subseteq> wp act (A \<union> B)"
by (force simp add: wp_def)
lemma wp_Un_eq: "single_valued act ==> wp act (A \<union> B) = wp act A \<union> wp act B"
apply (rule equalityI)
apply (force simp add: wp_def single_valued_def)
apply (rule wp_Un_subset)
done
lemma wp_UN_subset: "(\<Union>i\<in>I. wp act (A i)) \<subseteq> wp act (\<Union>i\<in>I. A i)"
by (force simp add: wp_def)
lemma wp_UN_eq:
"[|single_valued act; I\<noteq>{}|]
==> wp act (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. wp act (A i))"
apply (rule equalityI)
prefer 2 apply (rule wp_UN_subset)
apply (simp add: wp_def Image_INT_eq)
done
lemma wens_single_eq:
"wens (mk_program (init, {act}, allowed)) act B = B \<union> wp act B"
by (simp add: wens_def gfp_def wp_def, blast)
text{*Next, we express the @{term "wens_set"} for single-assignment programs*}
definition wens_single_finite :: "[('a*'a) set, 'a set, nat] => 'a set" where
"wens_single_finite act B k == \<Union>i \<in> atMost k. (wp act ^^ i) B"
definition wens_single :: "[('a*'a) set, 'a set] => 'a set" where
"wens_single act B == \<Union>i. (wp act ^^ i) B"
lemma wens_single_Un_eq:
"single_valued act
==> wens_single act B \<union> wp act (wens_single act B) = wens_single act B"
apply (rule equalityI)
apply (simp_all add: Un_upper1)
apply (simp add: wens_single_def wp_UN_eq, clarify)
apply (rule_tac a="Suc xa" in UN_I, auto)
done
lemma atMost_nat_nonempty: "atMost (k::nat) \<noteq> {}"
by force
lemma wens_single_finite_0 [simp]: "wens_single_finite act B 0 = B"
by (simp add: wens_single_finite_def)
lemma wens_single_finite_Suc:
"single_valued act
==> wens_single_finite act B (Suc k) =
wens_single_finite act B k \<union> wp act (wens_single_finite act B k)"
apply (simp add: wens_single_finite_def image_def
wp_UN_eq [OF _ atMost_nat_nonempty])
apply (force elim!: le_SucE)
done
lemma wens_single_finite_Suc_eq_wens:
"single_valued act
==> wens_single_finite act B (Suc k) =
wens (mk_program (init, {act}, allowed)) act
(wens_single_finite act B k)"
by (simp add: wens_single_finite_Suc wens_single_eq)
lemma def_wens_single_finite_Suc_eq_wens:
"[|F = mk_program (init, {act}, allowed); single_valued act|]
==> wens_single_finite act B (Suc k) =
wens F act (wens_single_finite act B k)"
by (simp add: wens_single_finite_Suc_eq_wens)
lemma wens_single_finite_Un_eq:
"single_valued act
==> wens_single_finite act B k \<union> wp act (wens_single_finite act B k)
\<in> range (wens_single_finite act B)"
by (simp add: wens_single_finite_Suc [symmetric])
lemma wens_single_eq_Union:
"wens_single act B = \<Union>range (wens_single_finite act B)"
by (simp add: wens_single_finite_def wens_single_def, blast)
lemma wens_single_finite_eq_Union:
"wens_single_finite act B n = (\<Union>k\<in>atMost n. wens_single_finite act B k)"
apply (auto simp add: wens_single_finite_def)
apply (blast intro: le_trans)
done
lemma wens_single_finite_mono:
"m \<le> n ==> wens_single_finite act B m \<subseteq> wens_single_finite act B n"
by (force simp add: wens_single_finite_eq_Union [of act B n])
lemma wens_single_finite_subset_wens_single:
"wens_single_finite act B k \<subseteq> wens_single act B"
by (simp add: wens_single_eq_Union, blast)
lemma subset_wens_single_finite:
"[|W \<subseteq> wens_single_finite act B ` (atMost k); single_valued act; W\<noteq>{}|]
==> \<exists>m. \<Union>W = wens_single_finite act B m"
apply (induct k)
apply (rule_tac x=0 in exI, simp, blast)
apply (auto simp add: atMost_Suc)
apply (case_tac "wens_single_finite act B (Suc k) \<in> W")
prefer 2 apply blast
apply (drule_tac x="Suc k" in spec)
apply (erule notE, rule equalityI)
prefer 2 apply blast
apply (subst wens_single_finite_eq_Union)
apply (simp add: atMost_Suc, blast)
done
text{*lemma for Union case*}
lemma Union_eq_wens_single:
"\<lbrakk>\<forall>k. \<not> W \<subseteq> wens_single_finite act B ` {..k};
W \<subseteq> insert (wens_single act B)
(range (wens_single_finite act B))\<rbrakk>
\<Longrightarrow> \<Union>W = wens_single act B"
apply (cases "wens_single act B \<in> W")
apply (blast dest: wens_single_finite_subset_wens_single [THEN subsetD])
apply (simp add: wens_single_eq_Union)
apply (rule equalityI, blast)
apply (simp add: UN_subset_iff, clarify)
apply (subgoal_tac "\<exists>y\<in>W. \<exists>n. y = wens_single_finite act B n & i\<le>n")
apply (blast intro: wens_single_finite_mono [THEN subsetD])
apply (drule_tac x=i in spec)
apply (force simp add: atMost_def)
done
lemma wens_set_subset_single:
"single_valued act
==> wens_set (mk_program (init, {act}, allowed)) B \<subseteq>
insert (wens_single act B) (range (wens_single_finite act B))"
apply (rule subsetI)
apply (erule wens_set.induct)
txt{*Basis*}
apply (fastforce simp add: wens_single_finite_def)
txt{*Wens inductive step*}
apply (case_tac "acta = Id", simp)
apply (simp add: wens_single_eq)
apply (elim disjE)
apply (simp add: wens_single_Un_eq)
apply (force simp add: wens_single_finite_Un_eq)
txt{*Union inductive step*}
apply (case_tac "\<exists>k. W \<subseteq> wens_single_finite act B ` (atMost k)")
apply (blast dest!: subset_wens_single_finite, simp)
apply (rule disjI1 [OF Union_eq_wens_single], blast+)
done
lemma wens_single_finite_in_wens_set:
"single_valued act \<Longrightarrow>
wens_single_finite act B k
\<in> wens_set (mk_program (init, {act}, allowed)) B"
apply (induct_tac k)
apply (simp add: wens_single_finite_def wens_set.Basis)
apply (simp add: wens_set.Wens
wens_single_finite_Suc_eq_wens [of act B _ init allowed])
done
lemma single_subset_wens_set:
"single_valued act
==> insert (wens_single act B) (range (wens_single_finite act B)) \<subseteq>
wens_set (mk_program (init, {act}, allowed)) B"
apply (simp add: SUP_def image_def wens_single_eq_Union)
apply (blast intro: wens_set.Union wens_single_finite_in_wens_set)
done
text{*Theorem (4.29)*}
theorem wens_set_single_eq:
"[|F = mk_program (init, {act}, allowed); single_valued act|]
==> wens_set F B =
insert (wens_single act B) (range (wens_single_finite act B))"
apply (rule equalityI)
apply (simp add: wens_set_subset_single)
apply (erule ssubst, erule single_subset_wens_set)
done
text{*Generalizing Misra's Fixed Point Union Theorem (4.41)*}
lemma fp_leadsTo_Join:
"[|T-B \<subseteq> awp F T; T-B \<subseteq> FP G; F \<in> A leadsTo B|] ==> F\<squnion>G \<in> T\<inter>A leadsTo B"
apply (rule leadsTo_Join, assumption, blast)
apply (simp add: FP_def awp_iff_constrains stable_def constrains_def, blast)
done
end