(* Title: HOL/UNITY/SubstAx ID: $Id: SubstAx.thy,v 1.18 2005/06/17 14:13:10 haftmann Exp $ 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 constdefs Ensures :: "['a set, 'a set] => 'a program set" (infixl "Ensures" 60) "A Ensures B == {F. F ∈ (reachable F ∩ A) ensures B}" LeadsTo :: "['a set, 'a set] => 'a program set" (infixl "LeadsTo" 60) "A LeadsTo B == {F. F ∈ (reachable F ∩ A) leadsTo B}" syntax (xsymbols) "op LeadsTo" :: "['a set, 'a set] => 'a program set" (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: inv_image_def 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
lemma LeadsTo_eq_leadsTo:
A LeadsTo B = {F. F ∈ reachable F ∩ A leadsTo reachable F ∩ B}
lemma Always_LeadsTo_pre:
F ∈ Always INV ==> (F ∈ INV ∩ A LeadsTo A') = (F ∈ A LeadsTo A')
lemma Always_LeadsTo_post:
F ∈ Always INV ==> (F ∈ A LeadsTo INV ∩ A') = (F ∈ A LeadsTo A')
lemmas Always_LeadsToI:
[| F ∈ Always INV; F ∈ INV ∩ A LeadsTo A' |] ==> F ∈ A LeadsTo A'
lemmas Always_LeadsToI:
[| F ∈ Always INV; F ∈ INV ∩ A LeadsTo A' |] ==> F ∈ A LeadsTo A'
lemmas Always_LeadsToD:
[| F ∈ Always INV; F ∈ A LeadsTo A' |] ==> F ∈ A LeadsTo INV ∩ A'
lemmas Always_LeadsToD:
[| F ∈ Always INV; F ∈ A LeadsTo A' |] ==> F ∈ A LeadsTo INV ∩ A'
lemma leadsTo_imp_LeadsTo:
F ∈ A leadsTo B ==> F ∈ A LeadsTo B
lemma LeadsTo_Trans:
[| F ∈ A LeadsTo B; F ∈ B LeadsTo C |] ==> F ∈ A LeadsTo C
lemma LeadsTo_Union:
(!!A. A ∈ S ==> F ∈ A LeadsTo B) ==> F ∈ Union S LeadsTo B
lemma LeadsTo_UNIV:
F ∈ A LeadsTo UNIV
lemma LeadsTo_Un_duplicate:
F ∈ A LeadsTo A' ∪ A' ==> F ∈ A LeadsTo A'
lemma LeadsTo_Un_duplicate2:
F ∈ A LeadsTo A' ∪ C ∪ C ==> F ∈ A LeadsTo A' ∪ C
lemma LeadsTo_UN:
(!!i. i ∈ I ==> F ∈ A i LeadsTo B) ==> F ∈ (UN i:I. A i) LeadsTo B
lemma LeadsTo_Un:
[| F ∈ A LeadsTo C; F ∈ B LeadsTo C |] ==> F ∈ A ∪ B LeadsTo C
lemma single_LeadsTo_I:
(!!s. s ∈ A ==> F ∈ {s} LeadsTo B) ==> F ∈ A LeadsTo B
lemma subset_imp_LeadsTo:
A ⊆ B ==> F ∈ A LeadsTo B
lemmas empty_LeadsTo:
F ∈ {} LeadsTo B
lemmas empty_LeadsTo:
F ∈ {} LeadsTo B
lemma LeadsTo_weaken_R:
[| F ∈ A LeadsTo A'; A' ⊆ B' |] ==> F ∈ A LeadsTo B'
lemma LeadsTo_weaken_L:
[| F ∈ A LeadsTo A'; B ⊆ A |] ==> F ∈ B LeadsTo A'
lemma LeadsTo_weaken:
[| F ∈ A LeadsTo A'; B ⊆ A; A' ⊆ B' |] ==> F ∈ B LeadsTo B'
lemma Always_LeadsTo_weaken:
[| F ∈ Always C; F ∈ A LeadsTo A'; C ∩ B ⊆ A; C ∩ A' ⊆ B' |] ==> F ∈ B LeadsTo B'
lemma LeadsTo_Un_post:
F ∈ A LeadsTo B ==> F ∈ A ∪ B LeadsTo B
lemma LeadsTo_Trans_Un:
[| F ∈ A LeadsTo B; F ∈ B LeadsTo C |] ==> F ∈ A ∪ B LeadsTo C
lemma LeadsTo_Un_distrib:
(F ∈ A ∪ B LeadsTo C) = (F ∈ A LeadsTo C ∧ F ∈ B LeadsTo C)
lemma LeadsTo_UN_distrib:
(F ∈ (UN i:I. A i) LeadsTo B) = (∀i∈I. F ∈ A i LeadsTo B)
lemma LeadsTo_Union_distrib:
(F ∈ Union S LeadsTo B) = (∀A∈S. F ∈ A LeadsTo B)
lemma LeadsTo_Basis:
F ∈ A Ensures B ==> F ∈ A LeadsTo B
lemma EnsuresI:
[| F ∈ A - B Co A ∪ B; F ∈ transient (A - B) |] ==> F ∈ A Ensures B
lemma Always_LeadsTo_Basis:
[| F ∈ Always INV; F ∈ INV ∩ (A - A') Co A ∪ A'; F ∈ transient (INV ∩ (A - A')) |] ==> F ∈ A LeadsTo A'
lemma LeadsTo_Diff:
[| F ∈ A - B LeadsTo C; F ∈ A ∩ B LeadsTo C |] ==> F ∈ A LeadsTo C
lemma LeadsTo_UN_UN:
(!!i. i ∈ I ==> F ∈ A i LeadsTo A' i) ==> F ∈ (UN i:I. A i) LeadsTo (UN i:I. A' i)
lemma LeadsTo_UN_UN_noindex:
(!!i. F ∈ A i LeadsTo A' i) ==> F ∈ (UN i. A i) LeadsTo (UN i. A' i)
lemma all_LeadsTo_UN_UN:
∀i. F ∈ A i LeadsTo A' i ==> F ∈ (UN i. A i) LeadsTo (UN i. A' i)
lemma LeadsTo_Un_Un:
[| F ∈ A LeadsTo A'; F ∈ B LeadsTo B' |] ==> F ∈ A ∪ B LeadsTo A' ∪ B'
lemma LeadsTo_cancel2:
[| F ∈ A LeadsTo A' ∪ B; F ∈ B LeadsTo B' |] ==> F ∈ A LeadsTo A' ∪ B'
lemma LeadsTo_cancel_Diff2:
[| F ∈ A LeadsTo A' ∪ B; F ∈ B - A' LeadsTo B' |] ==> F ∈ A LeadsTo A' ∪ B'
lemma LeadsTo_cancel1:
[| F ∈ A LeadsTo B ∪ A'; F ∈ B LeadsTo B' |] ==> F ∈ A LeadsTo B' ∪ A'
lemma LeadsTo_cancel_Diff1:
[| F ∈ A LeadsTo B ∪ A'; F ∈ B - A' LeadsTo B' |] ==> F ∈ A LeadsTo B' ∪ A'
lemma LeadsTo_empty:
[| F ∈ A LeadsTo {}; all_total F |] ==> F ∈ Always (- A)
lemma PSP_Stable:
[| F ∈ A LeadsTo A'; F ∈ Stable B |] ==> F ∈ A ∩ B LeadsTo A' ∩ B
lemma PSP_Stable2:
[| F ∈ A LeadsTo A'; F ∈ Stable B |] ==> F ∈ B ∩ A LeadsTo B ∩ A'
lemma PSP:
[| F ∈ A LeadsTo A'; F ∈ B Co B' |] ==> F ∈ A ∩ B' LeadsTo A' ∩ B ∪ (B' - B)
lemma PSP2:
[| F ∈ A LeadsTo A'; F ∈ B Co B' |] ==> F ∈ B' ∩ A LeadsTo B ∩ A' ∪ (B' - B)
lemma PSP_Unless:
[| F ∈ A LeadsTo A'; F ∈ B Unless B' |] ==> F ∈ A ∩ B LeadsTo A' ∩ B ∪ B'
lemma Stable_transient_Always_LeadsTo:
[| F ∈ Stable A; F ∈ transient C; F ∈ Always (- A ∪ B ∪ C) |] ==> F ∈ A LeadsTo B
lemma LeadsTo_wf_induct:
[| wf r; ∀m. F ∈ A ∩ f -` {m} LeadsTo A ∩ f -` r^-1 `` {m} ∪ B |] ==> F ∈ A LeadsTo B
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
lemma LessThan_induct:
(!!m. F ∈ A ∩ f -` {m} LeadsTo A ∩ f -` {..<m} ∪ B) ==> F ∈ A LeadsTo B
lemma integ_0_le_induct:
[| F ∈ Always {s. 0 ≤ f s}; !!z. F ∈ A ∩ {s. f s = z} LeadsTo A ∩ {s. f s < z} ∪ B |] ==> F ∈ A LeadsTo B
lemma LessThan_bounded_induct:
∀m∈{l<..}. F ∈ A ∩ f -` {m} LeadsTo A ∩ f -` {..<m} ∪ B ==> F ∈ A LeadsTo A ∩ f -` {..l} ∪ B
lemma GreaterThan_bounded_induct:
∀m∈{..<l}. F ∈ A ∩ f -` {m} LeadsTo A ∩ f -` {m<..} ∪ B ==> F ∈ A LeadsTo A ∩ f -` {l..} ∪ B
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
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 ∈ (INT i:I. A i) LeadsTo (INT i:I. A' i) ∪ C
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 ∈ (INT i:I. A i) LeadsTo (INT i:I. A' i) ∪ C
lemma Stable_completion:
[| F ∈ A LeadsTo A'; F ∈ Stable A'; F ∈ B LeadsTo B'; F ∈ Stable B' |] ==> F ∈ A ∩ B LeadsTo A' ∩ B'
lemma Finite_stable_completion:
[| finite I; !!i. i ∈ I ==> F ∈ A i LeadsTo A' i; !!i. i ∈ I ==> F ∈ Stable (A' i) |] ==> F ∈ (INT i:I. A i) LeadsTo (INT i:I. A' i)