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(* Title: HOL/UNITY/SubstAx.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Weak LeadsTo relation (restricted to the set of reachable states)
*)
section\<open>Weak Progress\<close>
theory SubstAx imports WFair Constrains begin
definition Ensures :: "['a set, 'a set] => 'a program set" (infixl "Ensures" 60) where
"A Ensures B == {F. F \ (reachable F \ A) ensures B}"
definition LeadsTo :: "['a set, 'a set] => 'a program set" (infixl "LeadsTo" 60) where
"A LeadsTo B == {F. F \ (reachable F \ A) leadsTo B}"
notation LeadsTo (infixl "\w" 60)
text\<open>Resembles the previous definition of LeadsTo\<close>
lemma LeadsTo_eq_leadsTo:
"A LeadsTo B = {F. F \ (reachable F \ A) leadsTo (reachable F \ B)}"
apply (unfold LeadsTo_def)
apply (blast dest: psp_stable2 intro: leadsTo_weaken)
done
subsection\<open>Specialized laws for handling invariants\<close>
(** Conjoining an Always property **)
lemma Always_LeadsTo_pre:
"F \ Always INV ==> (F \ (INV \ A) LeadsTo A') = (F \ A LeadsTo A')"
by (simp add: LeadsTo_def Always_eq_includes_reachable Int_absorb2
Int_assoc [symmetric])
lemma Always_LeadsTo_post:
"F \ Always INV ==> (F \ A LeadsTo (INV \ A')) = (F \ A LeadsTo A')"
by (simp add: LeadsTo_eq_leadsTo Always_eq_includes_reachable Int_absorb2
Int_assoc [symmetric])
(* [| F \<in> Always C; F \<in> (C \<inter> A) LeadsTo A' |] ==> F \<in> A LeadsTo A' *)
lemmas Always_LeadsToI = Always_LeadsTo_pre [THEN iffD1]
(* [| F \<in> Always INV; F \<in> A LeadsTo A' |] ==> F \<in> A LeadsTo (INV \<inter> A') *)
lemmas Always_LeadsToD = Always_LeadsTo_post [THEN iffD2]
subsection\<open>Introduction rules: Basis, Trans, Union\<close>
lemma leadsTo_imp_LeadsTo: "F \ A leadsTo B ==> F \ A LeadsTo B"
apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_L)
done
lemma LeadsTo_Trans:
"[| F \ A LeadsTo B; F \ B LeadsTo C |] ==> F \ A LeadsTo C"
apply (simp add: LeadsTo_eq_leadsTo)
apply (blast intro: leadsTo_Trans)
done
lemma LeadsTo_Union:
"(!!A. A \ S ==> F \ A LeadsTo B) ==> F \ (\S) LeadsTo B"
apply (simp add: LeadsTo_def)
apply (subst Int_Union)
apply (blast intro: leadsTo_UN)
done
subsection\<open>Derived rules\<close>
lemma LeadsTo_UNIV [simp]: "F \ A LeadsTo UNIV"
by (simp add: LeadsTo_def)
text\<open>Useful with cancellation, disjunction\<close>
lemma LeadsTo_Un_duplicate:
"F \ A LeadsTo (A' \ A') ==> F \ A LeadsTo A'"
by (simp add: Un_ac)
lemma LeadsTo_Un_duplicate2:
"F \ A LeadsTo (A' \ C \ C) ==> F \ A LeadsTo (A' \ C)"
by (simp add: Un_ac)
lemma LeadsTo_UN:
"(!!i. i \ I ==> F \ (A i) LeadsTo B) ==> F \ (\i \ I. A i) LeadsTo B"
apply (blast intro: LeadsTo_Union)
done
text\<open>Binary union introduction rule\<close>
lemma LeadsTo_Un:
"[| F \ A LeadsTo C; F \ B LeadsTo C |] ==> F \ (A \ B) LeadsTo C"
using LeadsTo_UN [of "{A, B}" F id C] by auto
text\<open>Lets us look at the starting state\<close>
lemma single_LeadsTo_I:
"(!!s. s \ A ==> F \ {s} LeadsTo B) ==> F \ A LeadsTo B"
by (subst UN_singleton [symmetric], rule LeadsTo_UN, blast)
lemma subset_imp_LeadsTo: "A \ B ==> F \ A LeadsTo B"
apply (simp add: LeadsTo_def)
apply (blast intro: subset_imp_leadsTo)
done
lemmas empty_LeadsTo = empty_subsetI [THEN subset_imp_LeadsTo, simp]
lemma LeadsTo_weaken_R:
"[| F \ A LeadsTo A'; A' \ B' |] ==> F \ A LeadsTo B'"
apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_R)
done
lemma LeadsTo_weaken_L:
"[| F \ A LeadsTo A'; B \ A |]
==> F \<in> B LeadsTo A'"
apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_L)
done
lemma LeadsTo_weaken:
"[| F \ A LeadsTo A';
B \<subseteq> A; A' \<subseteq> B' |]
==> F \<in> B LeadsTo B'"
by (blast intro: LeadsTo_weaken_R LeadsTo_weaken_L LeadsTo_Trans)
lemma Always_LeadsTo_weaken:
"[| F \ Always C; F \ A LeadsTo A';
C \<inter> B \<subseteq> A; C \<inter> A' \<subseteq> B' |]
==> F \<in> B LeadsTo B'"
by (blast dest: Always_LeadsToI intro: LeadsTo_weaken intro: Always_LeadsToD)
(** Two theorems for "proof lattices" **)
lemma LeadsTo_Un_post: "F \ A LeadsTo B ==> F \ (A \ B) LeadsTo B"
by (blast intro: LeadsTo_Un subset_imp_LeadsTo)
lemma LeadsTo_Trans_Un:
"[| F \ A LeadsTo B; F \ B LeadsTo C |]
==> F \<in> (A \<union> B) LeadsTo C"
by (blast intro: LeadsTo_Un subset_imp_LeadsTo LeadsTo_weaken_L LeadsTo_Trans)
(** Distributive laws **)
lemma LeadsTo_Un_distrib:
"(F \ (A \ B) LeadsTo C) = (F \ A LeadsTo C & F \ B LeadsTo C)"
by (blast intro: LeadsTo_Un LeadsTo_weaken_L)
lemma LeadsTo_UN_distrib:
"(F \ (\i \ I. A i) LeadsTo B) = (\i \ I. F \ (A i) LeadsTo B)"
by (blast intro: LeadsTo_UN LeadsTo_weaken_L)
lemma LeadsTo_Union_distrib:
"(F \ (\S) LeadsTo B) = (\A \ S. F \ A LeadsTo B)"
by (blast intro: LeadsTo_Union LeadsTo_weaken_L)
(** More rules using the premise "Always INV" **)
lemma LeadsTo_Basis: "F \ A Ensures B ==> F \ A LeadsTo B"
by (simp add: Ensures_def LeadsTo_def leadsTo_Basis)
lemma EnsuresI:
"[| F \ (A-B) Co (A \ B); F \ transient (A-B) |]
==> F \<in> A Ensures B"
apply (simp add: Ensures_def Constrains_eq_constrains)
apply (blast intro: ensuresI constrains_weaken transient_strengthen)
done
lemma Always_LeadsTo_Basis:
"[| F \ Always INV;
F \<in> (INV \<inter> (A-A')) Co (A \<union> A');
F \<in> transient (INV \<inter> (A-A')) |]
==> F \<in> A LeadsTo A'"
apply (rule Always_LeadsToI, assumption)
apply (blast intro: EnsuresI LeadsTo_Basis Always_ConstrainsD [THEN Constrains_weaken] transient_strengthen)
done
text\<open>Set difference: maybe combine with \<open>leadsTo_weaken_L\<close>??
This is the most useful form of the "disjunction" rule\<close>
lemma LeadsTo_Diff:
"[| F \ (A-B) LeadsTo C; F \ (A \ B) LeadsTo C |]
==> F \<in> A LeadsTo C"
by (blast intro: LeadsTo_Un LeadsTo_weaken)
lemma LeadsTo_UN_UN:
"(!! i. i \ I ==> F \ (A i) LeadsTo (A' i))
==> F \<in> (\<Union>i \<in> I. A i) LeadsTo (\<Union>i \<in> I. A' i)"
apply (blast intro: LeadsTo_Union LeadsTo_weaken_R)
done
text\<open>Version with no index set\<close>
lemma LeadsTo_UN_UN_noindex:
"(!!i. F \ (A i) LeadsTo (A' i)) ==> F \ (\i. A i) LeadsTo (\i. A' i)"
by (blast intro: LeadsTo_UN_UN)
text\<open>Version with no index set\<close>
lemma all_LeadsTo_UN_UN:
"\i. F \ (A i) LeadsTo (A' i)
==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
by (blast intro: LeadsTo_UN_UN)
text\<open>Binary union version\<close>
lemma LeadsTo_Un_Un:
"[| F \ A LeadsTo A'; F \ B LeadsTo B' |]
==> F \<in> (A \<union> B) LeadsTo (A' \<union> B')"
by (blast intro: LeadsTo_Un LeadsTo_weaken_R)
(** The cancellation law **)
lemma LeadsTo_cancel2:
"[| F \ A LeadsTo (A' \ B); F \ B LeadsTo B' |]
==> F \<in> A LeadsTo (A' \<union> B')"
by (blast intro: LeadsTo_Un_Un subset_imp_LeadsTo LeadsTo_Trans)
lemma LeadsTo_cancel_Diff2:
"[| F \ A LeadsTo (A' \ B); F \ (B-A') LeadsTo B' |]
==> F \<in> A LeadsTo (A' \<union> B')"
apply (rule LeadsTo_cancel2)
prefer 2 apply assumption
apply (simp_all (no_asm_simp))
done
lemma LeadsTo_cancel1:
"[| F \ A LeadsTo (B \ A'); F \ B LeadsTo B' |]
==> F \<in> A LeadsTo (B' \<union> A')"
apply (simp add: Un_commute)
apply (blast intro!: LeadsTo_cancel2)
done
lemma LeadsTo_cancel_Diff1:
"[| F \ A LeadsTo (B \ A'); F \ (B-A') LeadsTo B' |]
==> F \<in> A LeadsTo (B' \<union> A')"
apply (rule LeadsTo_cancel1)
prefer 2 apply assumption
apply (simp_all (no_asm_simp))
done
text\<open>The impossibility law\<close>
text\<open>The set "A" may be non-empty, but it contains no reachable states\<close>
lemma LeadsTo_empty: "[|F \ A LeadsTo {}; all_total F|] ==> F \ Always (-A)"
apply (simp add: LeadsTo_def Always_eq_includes_reachable)
apply (drule leadsTo_empty, auto)
done
subsection\<open>PSP: Progress-Safety-Progress\<close>
text\<open>Special case of PSP: Misra's "stable conjunction"\<close>
lemma PSP_Stable:
"[| F \ A LeadsTo A'; F \ Stable B |]
==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B)"
apply (simp add: LeadsTo_eq_leadsTo Stable_eq_stable)
apply (drule psp_stable, assumption)
apply (simp add: Int_ac)
done
lemma PSP_Stable2:
"[| F \ A LeadsTo A'; F \ Stable B |]
==> F \<in> (B \<inter> A) LeadsTo (B \<inter> A')"
by (simp add: PSP_Stable Int_ac)
lemma PSP:
"[| F \ A LeadsTo A'; F \ B Co B' |]
==> F \<in> (A \<inter> B') LeadsTo ((A' \<inter> B) \<union> (B' - B))"
apply (simp add: LeadsTo_def Constrains_eq_constrains)
apply (blast dest: psp intro: leadsTo_weaken)
done
lemma PSP2:
"[| F \ A LeadsTo A'; F \ B Co B' |]
==> F \<in> (B' \<inter> A) LeadsTo ((B \<inter> A') \<union> (B' - B))"
by (simp add: PSP Int_ac)
lemma PSP_Unless:
"[| F \ A LeadsTo A'; F \ B Unless B' |]
==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B) \<union> B')"
apply (unfold Unless_def)
apply (drule PSP, assumption)
apply (blast intro: LeadsTo_Diff LeadsTo_weaken subset_imp_LeadsTo)
done
lemma Stable_transient_Always_LeadsTo:
"[| F \ Stable A; F \ transient C;
F \<in> Always (-A \<union> B \<union> C) |] ==> F \<in> A LeadsTo B"
apply (erule Always_LeadsTo_weaken)
apply (rule LeadsTo_Diff)
prefer 2
apply (erule
transient_imp_leadsTo [THEN leadsTo_imp_LeadsTo, THEN PSP_Stable2])
apply (blast intro: subset_imp_LeadsTo)+
done
subsection\<open>Induction rules\<close>
(** Meta or object quantifier ????? **)
lemma LeadsTo_wf_induct:
"[| wf r;
\<forall>m. F \<in> (A \<inter> f-`{m}) LeadsTo
((A \<inter> f-`(r\<inverse> `` {m})) \<union> B) |]
==> F \<in> A LeadsTo B"
apply (simp add: LeadsTo_eq_leadsTo)
apply (erule leadsTo_wf_induct)
apply (blast intro: leadsTo_weaken)
done
lemma Bounded_induct:
"[| wf r;
\<forall>m \<in> I. F \<in> (A \<inter> f-`{m}) LeadsTo
((A \<inter> f-`(r\<inverse> `` {m})) \<union> B) |]
==> F \<in> A LeadsTo ((A - (f-`I)) \<union> B)"
apply (erule LeadsTo_wf_induct, safe)
apply (case_tac "m \ I")
apply (blast intro: LeadsTo_weaken)
apply (blast intro: subset_imp_LeadsTo)
done
lemma LessThan_induct:
"(!!m::nat. F \ (A \ f-`{m}) LeadsTo ((A \ f-`(lessThan m)) \ B))
==> F \<in> A LeadsTo B"
by (rule wf_less_than [THEN LeadsTo_wf_induct], auto)
text\<open>Integer version. Could generalize from 0 to any lower bound\<close>
lemma integ_0_le_induct:
"[| F \ Always {s. (0::int) \ f s};
!! z. F \<in> (A \<inter> {s. f s = z}) LeadsTo
((A \<inter> {s. f s < z}) \<union> B) |]
==> F \<in> A LeadsTo B"
apply (rule_tac f = "nat o f" in LessThan_induct)
apply (simp add: vimage_def)
apply (rule Always_LeadsTo_weaken, assumption+)
apply (auto simp add: nat_eq_iff nat_less_iff)
done
lemma LessThan_bounded_induct:
"!!l::nat. \m \ greaterThan l.
F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B)
==> F \<in> A LeadsTo ((A \<inter> (f-`(atMost l))) \<union> B)"
apply (simp only: Diff_eq [symmetric] vimage_Compl
Compl_greaterThan [symmetric])
apply (rule wf_less_than [THEN Bounded_induct], simp)
done
lemma GreaterThan_bounded_induct:
"!!l::nat. \m \ lessThan l.
F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(greaterThan m)) \<union> B)
==> F \<in> A LeadsTo ((A \<inter> (f-`(atLeast l))) \<union> B)"
apply (rule_tac f = f and f1 = "%k. l - k"
in wf_less_than [THEN wf_inv_image, THEN LeadsTo_wf_induct])
apply (simp add: Image_singleton, clarify)
apply (case_tac "m)
apply (blast intro: LeadsTo_weaken_R diff_less_mono2)
apply (blast intro: not_le_imp_less subset_imp_LeadsTo)
done
subsection\<open>Completion: Binary and General Finite versions\<close>
lemma Completion:
"[| F \ A LeadsTo (A' \ C); F \ A' Co (A' \ C);
F \<in> B LeadsTo (B' \<union> C); F \<in> B' Co (B' \<union> C) |]
==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B') \<union> C)"
apply (simp add: LeadsTo_eq_leadsTo Constrains_eq_constrains Int_Un_distrib)
apply (blast intro: completion leadsTo_weaken)
done
lemma Finite_completion_lemma:
"finite I
==> (\<forall>i \<in> I. F \<in> (A i) LeadsTo (A' i \<union> C)) -->
(\<forall>i \<in> I. F \<in> (A' i) Co (A' i \<union> C)) -->
F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
apply (erule finite_induct, auto)
apply (rule Completion)
prefer 4
apply (simp only: INT_simps [symmetric])
apply (rule Constrains_INT, auto)
done
lemma Finite_completion:
"[| finite I;
!!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i \<union> C);
!!i. i \<in> I ==> F \<in> (A' i) Co (A' i \<union> C) |]
==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
by (blast intro: Finite_completion_lemma [THEN mp, THEN mp])
lemma Stable_completion:
"[| F \ A LeadsTo A'; F \ Stable A';
F \<in> B LeadsTo B'; F \<in> Stable B' |]
==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B')"
apply (unfold Stable_def)
apply (rule_tac C1 = "{}" in Completion [THEN LeadsTo_weaken_R])
apply (force+)
done
lemma Finite_stable_completion:
"[| finite I;
!!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i);
!!i. i \<in> I ==> F \<in> Stable (A' i) |]
==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo (\<Inter>i \<in> I. A' i)"
apply (unfold Stable_def)
apply (rule_tac C1 = "{}" in Finite_completion [THEN LeadsTo_weaken_R])
apply (simp_all, blast+)
done
end
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