Theory SubstAx

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theory SubstAx
imports WFair Constrains

(*  Title:      HOL/UNITY/SubstAx
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge

Weak LeadsTo relation (restricted to the set of reachable states)
*)


header{*Weak Progress*}

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 (xsymbols)
LeadsTo (infixl " \<longmapsto>w " 60)



text{*Resembles the previous definition of LeadsTo*}
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{*Specialized laws for handling invariants*}

(** 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 ∈ Always C; F ∈ (C ∩ A) LeadsTo A' |] ==> F ∈ A LeadsTo A' *)
lemmas Always_LeadsToI = Always_LeadsTo_pre [THEN iffD1, standard]

(* [| F ∈ Always INV; F ∈ A LeadsTo A' |] ==> F ∈ A LeadsTo (INV ∩ A') *)
lemmas Always_LeadsToD = Always_LeadsTo_post [THEN iffD2, standard]


subsection{*Introduction rules: Basis, Trans, Union*}

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 ∈ (Union S) LeadsTo B"

apply (simp add: LeadsTo_def)
apply (subst Int_Union)
apply (blast intro: leadsTo_UN)
done


subsection{*Derived rules*}

lemma LeadsTo_UNIV [simp]: "F ∈ A LeadsTo UNIV"
by (simp add: LeadsTo_def)

text{*Useful with cancellation, disjunction*}
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 ∈ (\<Union>i ∈ I. A i) LeadsTo B"

apply (simp only: Union_image_eq [symmetric])
apply (blast intro: LeadsTo_Union)
done

text{*Binary union introduction rule*}
lemma LeadsTo_Un:
"[| F ∈ A LeadsTo C; F ∈ B LeadsTo C |] ==> F ∈ (A ∪ B) LeadsTo C"

apply (subst Un_eq_Union)
apply (blast intro: LeadsTo_Union)
done

text{*Lets us look at the starting state*}
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, standard, simp]

lemma LeadsTo_weaken_R [rule_format]:
"[| 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 [rule_format]:
"[| F ∈ A LeadsTo A'; B ⊆ A |]
==> F ∈ B LeadsTo A'"

apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_L)
done

lemma LeadsTo_weaken:
"[| F ∈ A LeadsTo A';
B ⊆ A; A' ⊆ B' |]
==> F ∈ 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 ∩ B ⊆ A; C ∩ A' ⊆ B' |]
==> F ∈ 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 ∈ (A ∪ 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 ∈ (\<Union>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 ∈ (Union 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 ∈ 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 ∈ (INV ∩ (A-A')) Co (A ∪ A');
F ∈ transient (INV ∩ (A-A')) |]
==> F ∈ A LeadsTo A'"

apply (rule Always_LeadsToI, assumption)
apply (blast intro: EnsuresI LeadsTo_Basis Always_ConstrainsD [THEN Constrains_weaken] transient_strengthen)
done

text{*Set difference: maybe combine with @{text leadsTo_weaken_L}??
This is the most useful form of the "disjunction" rule*}

lemma LeadsTo_Diff:
"[| F ∈ (A-B) LeadsTo C; F ∈ (A ∩ B) LeadsTo C |]
==> F ∈ A LeadsTo C"

by (blast intro: LeadsTo_Un LeadsTo_weaken)


lemma LeadsTo_UN_UN:
"(!! i. i ∈ I ==> F ∈ (A i) LeadsTo (A' i))
==> F ∈ (\<Union>i ∈ I. A i) LeadsTo (\<Union>i ∈ I. A' i)"

apply (simp only: Union_image_eq [symmetric])
apply (blast intro: LeadsTo_Union LeadsTo_weaken_R)
done


text{*Version with no index set*}
lemma LeadsTo_UN_UN_noindex:
"(!!i. F ∈ (A i) LeadsTo (A' i)) ==> F ∈ (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"

by (blast intro: LeadsTo_UN_UN)

text{*Version with no index set*}
lemma all_LeadsTo_UN_UN:
"∀i. F ∈ (A i) LeadsTo (A' i)
==> F ∈ (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"

by (blast intro: LeadsTo_UN_UN)

text{*Binary union version*}
lemma LeadsTo_Un_Un:
"[| F ∈ A LeadsTo A'; F ∈ B LeadsTo B' |]
==> F ∈ (A ∪ B) LeadsTo (A' ∪ 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 ∈ A LeadsTo (A' ∪ 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 ∈ A LeadsTo (A' ∪ 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 ∈ A LeadsTo (B' ∪ 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 ∈ A LeadsTo (B' ∪ A')"

apply (rule LeadsTo_cancel1)
prefer 2 apply assumption
apply (simp_all (no_asm_simp))
done


text{*The impossibility law*}

text{*The set "A" may be non-empty, but it contains no reachable states*}
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{*PSP: Progress-Safety-Progress*}

text{*Special case of PSP: Misra's "stable conjunction"*}
lemma PSP_Stable:
"[| F ∈ A LeadsTo A'; F ∈ Stable B |]
==> F ∈ (A ∩ B) LeadsTo (A' ∩ 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 ∈ (B ∩ A) LeadsTo (B ∩ A')"

by (simp add: PSP_Stable Int_ac)

lemma PSP:
"[| F ∈ A LeadsTo A'; F ∈ B Co B' |]
==> F ∈ (A ∩ B') LeadsTo ((A' ∩ B) ∪ (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 ∈ (B' ∩ A) LeadsTo ((B ∩ A') ∪ (B' - B))"

by (simp add: PSP Int_ac)

lemma PSP_Unless:
"[| F ∈ A LeadsTo A'; F ∈ B Unless B' |]
==> F ∈ (A ∩ B) LeadsTo ((A' ∩ B) ∪ 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 ∈ Always (-A ∪ B ∪ C) |] ==> F ∈ 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{*Induction rules*}

(** Meta or object quantifier ????? **)
lemma LeadsTo_wf_induct:
"[| wf r;
∀m. F ∈ (A ∩ f-`{m}) LeadsTo
((A ∩ f-`(r^-1 `` {m})) ∪ B) |]
==> F ∈ 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;
∀m ∈ I. F ∈ (A ∩ f-`{m}) LeadsTo
((A ∩ f-`(r^-1 `` {m})) ∪ B) |]
==> F ∈ A LeadsTo ((A - (f-`I)) ∪ 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 ∈ A LeadsTo B"

by (rule wf_less_than [THEN LeadsTo_wf_induct], auto)

text{*Integer version. Could generalize from 0 to any lower bound*}
lemma integ_0_le_induct:
"[| F ∈ Always {s. (0::int) ≤ f s};
!! z. F ∈ (A ∩ {s. f s = z}) LeadsTo
((A ∩ {s. f s < z}) ∪ B) |]
==> F ∈ 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 ∈ (A ∩ f-`{m}) LeadsTo ((A ∩ f-`(lessThan m)) ∪ B)
==> F ∈ A LeadsTo ((A ∩ (f-`(atMost l))) ∪ 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 ∈ (A ∩ f-`{m}) LeadsTo ((A ∩ f-`(greaterThan m)) ∪ B)
==> F ∈ A LeadsTo ((A ∩ (f-`(atLeast l))) ∪ 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<l")
apply (blast intro: LeadsTo_weaken_R diff_less_mono2)
apply (blast intro: not_leE subset_imp_LeadsTo)
done


subsection{*Completion: Binary and General Finite versions*}

lemma Completion:
"[| F ∈ A LeadsTo (A' ∪ C); F ∈ A' Co (A' ∪ C);
F ∈ B LeadsTo (B' ∪ C); F ∈ B' Co (B' ∪ C) |]
==> F ∈ (A ∩ B) LeadsTo ((A' ∩ B') ∪ 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
==> (∀i ∈ I. F ∈ (A i) LeadsTo (A' i ∪ C)) -->
(∀i ∈ I. F ∈ (A' i) Co (A' i ∪ C)) -->
F ∈ (\<Inter>i ∈ I. A i) LeadsTo ((\<Inter>i ∈ I. A' i) ∪ 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 ∈ I ==> F ∈ (A i) LeadsTo (A' i ∪ C);
!!i. i ∈ I ==> F ∈ (A' i) Co (A' i ∪ C) |]
==> F ∈ (\<Inter>i ∈ I. A i) LeadsTo ((\<Inter>i ∈ I. A' i) ∪ C)"

by (blast intro: Finite_completion_lemma [THEN mp, THEN mp])

lemma Stable_completion:
"[| F ∈ A LeadsTo A'; F ∈ Stable A';
F ∈ B LeadsTo B'; F ∈ Stable B' |]
==> F ∈ (A ∩ B) LeadsTo (A' ∩ 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 ∈ I ==> F ∈ (A i) LeadsTo (A' i);
!!i. i ∈ I ==> F ∈ Stable (A' i) |]
==> F ∈ (\<Inter>i ∈ I. A i) LeadsTo (\<Inter>i ∈ I. A' i)"

apply (unfold Stable_def)
apply (rule_tac C1 = "{}" in Finite_completion [THEN LeadsTo_weaken_R])
apply (simp_all, blast+)
done

end