tuned proofs -- clarified flow of facts wrt. calculation;
(* Title: HOL/UNITY/ELT.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
leadsTo strengthened with a specification of the allowable sets transient parts
TRY INSTEAD (to get rid of the {} and to gain strong induction)
elt :: "['a set set, 'a program, 'a set] => ('a set) set"
inductive "elt CC F B"
intros
Weaken: "A <= B ==> A : elt CC F B"
ETrans: "[| F : A ensures A'; A-A' : CC; A' : elt CC F B |]
==> A : elt CC F B"
Union: "{A. A: S} : Pow (elt CC F B) ==> (Union S) : elt CC F B"
monos Pow_mono
*)
header{*Progress Under Allowable Sets*}
theory ELT imports Project begin
inductive_set
(*LEADS-TO constant for the inductive definition*)
elt :: "['a set set, 'a program] => ('a set * 'a set) set"
for CC :: "'a set set" and F :: "'a program"
where
Basis: "[| F : A ensures B; A-B : (insert {} CC) |] ==> (A,B) : elt CC F"
| Trans: "[| (A,B) : elt CC F; (B,C) : elt CC F |] ==> (A,C) : elt CC F"
| Union: "ALL A: S. (A,B) : elt CC F ==> (Union S, B) : elt CC F"
definition
(*the set of all sets determined by f alone*)
givenBy :: "['a => 'b] => 'a set set"
where "givenBy f = range (%B. f-` B)"
definition
(*visible version of the LEADS-TO relation*)
leadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
("(3_/ leadsTo[_]/ _)" [80,0,80] 80)
where "leadsETo A CC B = {F. (A,B) : elt CC F}"
definition
LeadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
("(3_/ LeadsTo[_]/ _)" [80,0,80] 80)
where "LeadsETo A CC B =
{F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) ` CC] B}"
(*** givenBy ***)
lemma givenBy_id [simp]: "givenBy id = UNIV"
by (unfold givenBy_def, auto)
lemma givenBy_eq_all: "(givenBy v) = {A. ALL x:A. ALL y. v x = v y --> y: A}"
apply (unfold givenBy_def, safe)
apply (rule_tac [2] x = "v ` ?u" in image_eqI, auto)
done
lemma givenByI: "(!!x y. [| x:A; v x = v y |] ==> y: A) ==> A: givenBy v"
by (subst givenBy_eq_all, blast)
lemma givenByD: "[| A: givenBy v; x:A; v x = v y |] ==> y: A"
by (unfold givenBy_def, auto)
lemma empty_mem_givenBy [iff]: "{} : givenBy v"
by (blast intro!: givenByI)
lemma givenBy_imp_eq_Collect: "A: givenBy v ==> EX P. A = {s. P(v s)}"
apply (rule_tac x = "%n. EX s. v s = n & s : A" in exI)
apply (simp (no_asm_use) add: givenBy_eq_all)
apply blast
done
lemma Collect_mem_givenBy: "{s. P(v s)} : givenBy v"
by (unfold givenBy_def, best)
lemma givenBy_eq_Collect: "givenBy v = {A. EX P. A = {s. P(v s)}}"
by (blast intro: Collect_mem_givenBy givenBy_imp_eq_Collect)
(*preserving v preserves properties given by v*)
lemma preserves_givenBy_imp_stable:
"[| F : preserves v; D : givenBy v |] ==> F : stable D"
by (force simp add: preserves_subset_stable [THEN subsetD] givenBy_eq_Collect)
lemma givenBy_o_subset: "givenBy (w o v) <= givenBy v"
apply (simp (no_asm) add: givenBy_eq_Collect)
apply best
done
lemma givenBy_DiffI:
"[| A : givenBy v; B : givenBy v |] ==> A-B : givenBy v"
apply (simp (no_asm_use) add: givenBy_eq_Collect)
apply safe
apply (rule_tac x = "%z. ?R z & ~ ?Q z" in exI)
unfolding set_diff_eq
apply auto
done
(** Standard leadsTo rules **)
lemma leadsETo_Basis [intro]:
"[| F: A ensures B; A-B: insert {} CC |] ==> F : A leadsTo[CC] B"
apply (unfold leadsETo_def)
apply (blast intro: elt.Basis)
done
lemma leadsETo_Trans:
"[| F : A leadsTo[CC] B; F : B leadsTo[CC] C |] ==> F : A leadsTo[CC] C"
apply (unfold leadsETo_def)
apply (blast intro: elt.Trans)
done
(*Useful with cancellation, disjunction*)
lemma leadsETo_Un_duplicate:
"F : A leadsTo[CC] (A' Un A') ==> F : A leadsTo[CC] A'"
by (simp add: Un_ac)
lemma leadsETo_Un_duplicate2:
"F : A leadsTo[CC] (A' Un C Un C) ==> F : A leadsTo[CC] (A' Un C)"
by (simp add: Un_ac)
(*The Union introduction rule as we should have liked to state it*)
lemma leadsETo_Union:
"(!!A. A : S ==> F : A leadsTo[CC] B) ==> F : (Union S) leadsTo[CC] B"
apply (unfold leadsETo_def)
apply (blast intro: elt.Union)
done
lemma leadsETo_UN:
"(!!i. i : I ==> F : (A i) leadsTo[CC] B)
==> F : (UN i:I. A i) leadsTo[CC] B"
apply (subst Union_image_eq [symmetric])
apply (blast intro: leadsETo_Union)
done
(*The INDUCTION rule as we should have liked to state it*)
lemma leadsETo_induct:
"[| F : za leadsTo[CC] zb;
!!A B. [| F : A ensures B; A-B : insert {} CC |] ==> P A B;
!!A B C. [| F : A leadsTo[CC] B; P A B; F : B leadsTo[CC] C; P B C |]
==> P A C;
!!B S. ALL A:S. F : A leadsTo[CC] B & P A B ==> P (Union S) B
|] ==> P za zb"
apply (unfold leadsETo_def)
apply (drule CollectD)
apply (erule elt.induct, blast+)
done
(** New facts involving leadsETo **)
lemma leadsETo_mono: "CC' <= CC ==> (A leadsTo[CC'] B) <= (A leadsTo[CC] B)"
apply safe
apply (erule leadsETo_induct)
prefer 3 apply (blast intro: leadsETo_Union)
prefer 2 apply (blast intro: leadsETo_Trans)
apply blast
done
lemma leadsETo_Trans_Un:
"[| F : A leadsTo[CC] B; F : B leadsTo[DD] C |]
==> F : A leadsTo[CC Un DD] C"
by (blast intro: leadsETo_mono [THEN subsetD] leadsETo_Trans)
lemma leadsETo_Union_Int:
"(!!A. A : S ==> F : (A Int C) leadsTo[CC] B)
==> F : (Union S Int C) leadsTo[CC] B"
apply (unfold leadsETo_def)
apply (simp only: Int_Union_Union)
apply (blast intro: elt.Union)
done
(*Binary union introduction rule*)
lemma leadsETo_Un:
"[| F : A leadsTo[CC] C; F : B leadsTo[CC] C |]
==> F : (A Un B) leadsTo[CC] C"
using leadsETo_Union [of "{A, B}" F CC C] by auto
lemma single_leadsETo_I:
"(!!x. x : A ==> F : {x} leadsTo[CC] B) ==> F : A leadsTo[CC] B"
by (subst UN_singleton [symmetric], rule leadsETo_UN, blast)
lemma subset_imp_leadsETo: "A<=B ==> F : A leadsTo[CC] B"
by (simp add: subset_imp_ensures [THEN leadsETo_Basis]
Diff_eq_empty_iff [THEN iffD2])
lemmas empty_leadsETo = empty_subsetI [THEN subset_imp_leadsETo, simp]
(** Weakening laws **)
lemma leadsETo_weaken_R:
"[| F : A leadsTo[CC] A'; A'<=B' |] ==> F : A leadsTo[CC] B'"
by (blast intro: subset_imp_leadsETo leadsETo_Trans)
lemma leadsETo_weaken_L:
"[| F : A leadsTo[CC] A'; B<=A |] ==> F : B leadsTo[CC] A'"
by (blast intro: leadsETo_Trans subset_imp_leadsETo)
(*Distributes over binary unions*)
lemma leadsETo_Un_distrib:
"F : (A Un B) leadsTo[CC] C =
(F : A leadsTo[CC] C & F : B leadsTo[CC] C)"
by (blast intro: leadsETo_Un leadsETo_weaken_L)
lemma leadsETo_UN_distrib:
"F : (UN i:I. A i) leadsTo[CC] B =
(ALL i : I. F : (A i) leadsTo[CC] B)"
by (blast intro: leadsETo_UN leadsETo_weaken_L)
lemma leadsETo_Union_distrib:
"F : (Union S) leadsTo[CC] B = (ALL A : S. F : A leadsTo[CC] B)"
by (blast intro: leadsETo_Union leadsETo_weaken_L)
lemma leadsETo_weaken:
"[| F : A leadsTo[CC'] A'; B<=A; A'<=B'; CC' <= CC |]
==> F : B leadsTo[CC] B'"
apply (drule leadsETo_mono [THEN subsetD], assumption)
apply (blast del: subsetCE
intro: leadsETo_weaken_R leadsETo_weaken_L leadsETo_Trans)
done
lemma leadsETo_givenBy:
"[| F : A leadsTo[CC] A'; CC <= givenBy v |]
==> F : A leadsTo[givenBy v] A'"
by (blast intro: leadsETo_weaken)
(*Set difference*)
lemma leadsETo_Diff:
"[| F : (A-B) leadsTo[CC] C; F : B leadsTo[CC] C |]
==> F : A leadsTo[CC] C"
by (blast intro: leadsETo_Un leadsETo_weaken)
(*Binary union version*)
lemma leadsETo_Un_Un:
"[| F : A leadsTo[CC] A'; F : B leadsTo[CC] B' |]
==> F : (A Un B) leadsTo[CC] (A' Un B')"
by (blast intro: leadsETo_Un leadsETo_weaken_R)
(** The cancellation law **)
lemma leadsETo_cancel2:
"[| F : A leadsTo[CC] (A' Un B); F : B leadsTo[CC] B' |]
==> F : A leadsTo[CC] (A' Un B')"
by (blast intro: leadsETo_Un_Un subset_imp_leadsETo leadsETo_Trans)
lemma leadsETo_cancel1:
"[| F : A leadsTo[CC] (B Un A'); F : B leadsTo[CC] B' |]
==> F : A leadsTo[CC] (B' Un A')"
apply (simp add: Un_commute)
apply (blast intro!: leadsETo_cancel2)
done
lemma leadsETo_cancel_Diff1:
"[| F : A leadsTo[CC] (B Un A'); F : (B-A') leadsTo[CC] B' |]
==> F : A leadsTo[CC] (B' Un A')"
apply (rule leadsETo_cancel1)
prefer 2 apply assumption
apply simp_all
done
(** PSP: Progress-Safety-Progress **)
(*Special case of PSP: Misra's "stable conjunction"*)
lemma e_psp_stable:
"[| F : A leadsTo[CC] A'; F : stable B; ALL C:CC. C Int B : CC |]
==> F : (A Int B) leadsTo[CC] (A' Int B)"
apply (unfold stable_def)
apply (erule leadsETo_induct)
prefer 3 apply (blast intro: leadsETo_Union_Int)
prefer 2 apply (blast intro: leadsETo_Trans)
apply (rule leadsETo_Basis)
prefer 2 apply (force simp add: Diff_Int_distrib2 [symmetric])
apply (simp add: ensures_def Diff_Int_distrib2 [symmetric]
Int_Un_distrib2 [symmetric])
apply (blast intro: transient_strengthen constrains_Int)
done
lemma e_psp_stable2:
"[| F : A leadsTo[CC] A'; F : stable B; ALL C:CC. C Int B : CC |]
==> F : (B Int A) leadsTo[CC] (B Int A')"
by (simp (no_asm_simp) add: e_psp_stable Int_ac)
lemma e_psp:
"[| F : A leadsTo[CC] A'; F : B co B';
ALL C:CC. C Int B Int B' : CC |]
==> F : (A Int B') leadsTo[CC] ((A' Int B) Un (B' - B))"
apply (erule leadsETo_induct)
prefer 3 apply (blast intro: leadsETo_Union_Int)
(*Transitivity case has a delicate argument involving "cancellation"*)
apply (rule_tac [2] leadsETo_Un_duplicate2)
apply (erule_tac [2] leadsETo_cancel_Diff1)
prefer 2
apply (simp add: Int_Diff Diff_triv)
apply (blast intro: leadsETo_weaken_L dest: constrains_imp_subset)
(*Basis case*)
apply (rule leadsETo_Basis)
apply (blast intro: psp_ensures)
apply (subgoal_tac "A Int B' - (Ba Int B Un (B' - B)) = (A - Ba) Int B Int B'")
apply auto
done
lemma e_psp2:
"[| F : A leadsTo[CC] A'; F : B co B';
ALL C:CC. C Int B Int B' : CC |]
==> F : (B' Int A) leadsTo[CC] ((B Int A') Un (B' - B))"
by (simp add: e_psp Int_ac)
(*** Special properties involving the parameter [CC] ***)
(*??IS THIS NEEDED?? or is it just an example of what's provable??*)
lemma gen_leadsETo_imp_Join_leadsETo:
"[| F: (A leadsTo[givenBy v] B); G : preserves v;
F\<squnion>G : stable C |]
==> F\<squnion>G : ((C Int A) leadsTo[(%D. C Int D) ` givenBy v] B)"
apply (erule leadsETo_induct)
prefer 3
apply (subst Int_Union)
apply (blast intro: leadsETo_UN)
prefer 2
apply (blast intro: e_psp_stable2 [THEN leadsETo_weaken_L] leadsETo_Trans)
apply (rule leadsETo_Basis)
apply (auto simp add: Diff_eq_empty_iff [THEN iffD2]
Int_Diff ensures_def givenBy_eq_Collect)
prefer 3 apply (blast intro: transient_strengthen)
apply (drule_tac [2] P1 = P in preserves_subset_stable [THEN subsetD])
apply (drule_tac P1 = P in preserves_subset_stable [THEN subsetD])
apply (unfold stable_def)
apply (blast intro: constrains_Int [THEN constrains_weaken])+
done
(**** Relationship with traditional "leadsTo", strong & weak ****)
(** strong **)
lemma leadsETo_subset_leadsTo: "(A leadsTo[CC] B) <= (A leadsTo B)"
apply safe
apply (erule leadsETo_induct)
prefer 3 apply (blast intro: leadsTo_Union)
prefer 2 apply (blast intro: leadsTo_Trans, blast)
done
lemma leadsETo_UNIV_eq_leadsTo: "(A leadsTo[UNIV] B) = (A leadsTo B)"
apply safe
apply (erule leadsETo_subset_leadsTo [THEN subsetD])
(*right-to-left case*)
apply (erule leadsTo_induct)
prefer 3 apply (blast intro: leadsETo_Union)
prefer 2 apply (blast intro: leadsETo_Trans, blast)
done
(**** weak ****)
lemma LeadsETo_eq_leadsETo:
"A LeadsTo[CC] B =
{F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) ` CC]
(reachable F Int B)}"
apply (unfold LeadsETo_def)
apply (blast dest: e_psp_stable2 intro: leadsETo_weaken)
done
(*** Introduction rules: Basis, Trans, Union ***)
lemma LeadsETo_Trans:
"[| F : A LeadsTo[CC] B; F : B LeadsTo[CC] C |]
==> F : A LeadsTo[CC] C"
apply (simp add: LeadsETo_eq_leadsETo)
apply (blast intro: leadsETo_Trans)
done
lemma LeadsETo_Union:
"(!!A. A : S ==> F : A LeadsTo[CC] B) ==> F : (Union S) LeadsTo[CC] B"
apply (simp add: LeadsETo_def)
apply (subst Int_Union)
apply (blast intro: leadsETo_UN)
done
lemma LeadsETo_UN:
"(!!i. i : I ==> F : (A i) LeadsTo[CC] B)
==> F : (UN i:I. A i) LeadsTo[CC] B"
apply (simp only: Union_image_eq [symmetric])
apply (blast intro: LeadsETo_Union)
done
(*Binary union introduction rule*)
lemma LeadsETo_Un:
"[| F : A LeadsTo[CC] C; F : B LeadsTo[CC] C |]
==> F : (A Un B) LeadsTo[CC] C"
using LeadsETo_Union [of "{A, B}" F CC C] by auto
(*Lets us look at the starting state*)
lemma single_LeadsETo_I:
"(!!s. s : A ==> F : {s} LeadsTo[CC] B) ==> F : A LeadsTo[CC] B"
by (subst UN_singleton [symmetric], rule LeadsETo_UN, blast)
lemma subset_imp_LeadsETo:
"A <= B ==> F : A LeadsTo[CC] B"
apply (simp (no_asm) add: LeadsETo_def)
apply (blast intro: subset_imp_leadsETo)
done
lemmas empty_LeadsETo = empty_subsetI [THEN subset_imp_LeadsETo]
lemma LeadsETo_weaken_R:
"[| F : A LeadsTo[CC] A'; A' <= B' |] ==> F : A LeadsTo[CC] B'"
apply (simp add: LeadsETo_def)
apply (blast intro: leadsETo_weaken_R)
done
lemma LeadsETo_weaken_L:
"[| F : A LeadsTo[CC] A'; B <= A |] ==> F : B LeadsTo[CC] A'"
apply (simp add: LeadsETo_def)
apply (blast intro: leadsETo_weaken_L)
done
lemma LeadsETo_weaken:
"[| F : A LeadsTo[CC'] A';
B <= A; A' <= B'; CC' <= CC |]
==> F : B LeadsTo[CC] B'"
apply (simp (no_asm_use) add: LeadsETo_def)
apply (blast intro: leadsETo_weaken)
done
lemma LeadsETo_subset_LeadsTo: "(A LeadsTo[CC] B) <= (A LeadsTo B)"
apply (unfold LeadsETo_def LeadsTo_def)
apply (blast intro: leadsETo_subset_leadsTo [THEN subsetD])
done
(*Postcondition can be strengthened to (reachable F Int B) *)
lemma reachable_ensures:
"F : A ensures B ==> F : (reachable F Int A) ensures B"
apply (rule stable_ensures_Int [THEN ensures_weaken_R], auto)
done
lemma lel_lemma:
"F : A leadsTo B ==> F : (reachable F Int A) leadsTo[Pow(reachable F)] B"
apply (erule leadsTo_induct)
apply (blast intro: reachable_ensures)
apply (blast dest: e_psp_stable2 intro: leadsETo_Trans leadsETo_weaken_L)
apply (subst Int_Union)
apply (blast intro: leadsETo_UN)
done
lemma LeadsETo_UNIV_eq_LeadsTo: "(A LeadsTo[UNIV] B) = (A LeadsTo B)"
apply safe
apply (erule LeadsETo_subset_LeadsTo [THEN subsetD])
(*right-to-left case*)
apply (unfold LeadsETo_def LeadsTo_def)
apply (blast intro: lel_lemma [THEN leadsETo_weaken])
done
(**** EXTEND/PROJECT PROPERTIES ****)
context Extend
begin
lemma givenBy_o_eq_extend_set:
"givenBy (v o f) = extend_set h ` (givenBy v)"
apply (simp add: givenBy_eq_Collect)
apply (rule equalityI, best)
apply blast
done
lemma givenBy_eq_extend_set: "givenBy f = range (extend_set h)"
by (simp add: givenBy_eq_Collect, best)
lemma extend_set_givenBy_I:
"D : givenBy v ==> extend_set h D : givenBy (v o f)"
apply (simp (no_asm_use) add: givenBy_eq_all, blast)
done
lemma leadsETo_imp_extend_leadsETo:
"F : A leadsTo[CC] B
==> extend h F : (extend_set h A) leadsTo[extend_set h ` CC]
(extend_set h B)"
apply (erule leadsETo_induct)
apply (force intro: subset_imp_ensures
simp add: extend_ensures extend_set_Diff_distrib [symmetric])
apply (blast intro: leadsETo_Trans)
apply (simp add: leadsETo_UN extend_set_Union)
done
(*This version's stronger in the "ensures" precondition
BUT there's no ensures_weaken_L*)
lemma Join_project_ensures_strong:
"[| project h C G ~: transient (project_set h C Int (A-B)) |
project_set h C Int (A - B) = {};
extend h F\<squnion>G : stable C;
F\<squnion>project h C G : (project_set h C Int A) ensures B |]
==> extend h F\<squnion>G : (C Int extend_set h A) ensures (extend_set h B)"
apply (subst Int_extend_set_lemma [symmetric])
apply (rule Join_project_ensures)
apply (auto simp add: Int_Diff)
done
(*NOT WORKING. MODIFY AS IN Project.thy
lemma pld_lemma:
"[| extend h F\<squnion>G : stable C;
F\<squnion>project h C G : (project_set h C Int A) leadsTo[(%D. project_set h C Int D)`givenBy v] B;
G : preserves (v o f) |]
==> extend h F\<squnion>G :
(C Int extend_set h (project_set h C Int A))
leadsTo[(%D. C Int extend_set h D)`givenBy v] (extend_set h B)"
apply (erule leadsETo_induct)
prefer 3
apply (simp del: UN_simps add: Int_UN_distrib leadsETo_UN extend_set_Union)
prefer 2
apply (blast intro: e_psp_stable2 [THEN leadsETo_weaken_L] leadsETo_Trans)
txt{*Base case is hard*}
apply auto
apply (force intro: leadsETo_Basis subset_imp_ensures)
apply (rule leadsETo_Basis)
prefer 2
apply (simp add: Int_Diff Int_extend_set_lemma extend_set_Diff_distrib [symmetric])
apply (rule Join_project_ensures_strong)
apply (auto intro: project_stable_project_set simp add: Int_left_absorb)
apply (simp (no_asm_simp) add: stable_ensures_Int [THEN ensures_weaken_R] Int_lower2 project_stable_project_set extend_stable_project_set)
done
lemma project_leadsETo_D_lemma:
"[| extend h F\<squnion>G : stable C;
F\<squnion>project h C G :
(project_set h C Int A)
leadsTo[(%D. project_set h C Int D)`givenBy v] B;
G : preserves (v o f) |]
==> extend h F\<squnion>G : (C Int extend_set h A)
leadsTo[(%D. C Int extend_set h D)`givenBy v] (extend_set h B)"
apply (rule pld_lemma [THEN leadsETo_weaken])
apply (auto simp add: split_extended_all)
done
lemma project_leadsETo_D:
"[| F\<squnion>project h UNIV G : A leadsTo[givenBy v] B;
G : preserves (v o f) |]
==> extend h F\<squnion>G : (extend_set h A)
leadsTo[givenBy (v o f)] (extend_set h B)"
apply (cut_tac project_leadsETo_D_lemma [of _ _ UNIV], auto)
apply (erule leadsETo_givenBy)
apply (rule givenBy_o_eq_extend_set [THEN equalityD2])
done
lemma project_LeadsETo_D:
"[| F\<squnion>project h (reachable (extend h F\<squnion>G)) G
: A LeadsTo[givenBy v] B;
G : preserves (v o f) |]
==> extend h F\<squnion>G :
(extend_set h A) LeadsTo[givenBy (v o f)] (extend_set h B)"
apply (cut_tac subset_refl [THEN stable_reachable, THEN project_leadsETo_D_lemma])
apply (auto simp add: LeadsETo_def)
apply (erule leadsETo_mono [THEN [2] rev_subsetD])
apply (blast intro: extend_set_givenBy_I)
apply (simp add: project_set_reachable_extend_eq [symmetric])
done
lemma extending_leadsETo:
"(ALL G. extend h F ok G --> G : preserves (v o f))
==> extending (%G. UNIV) h F
(extend_set h A leadsTo[givenBy (v o f)] extend_set h B)
(A leadsTo[givenBy v] B)"
apply (unfold extending_def)
apply (auto simp add: project_leadsETo_D)
done
lemma extending_LeadsETo:
"(ALL G. extend h F ok G --> G : preserves (v o f))
==> extending (%G. reachable (extend h F\<squnion>G)) h F
(extend_set h A LeadsTo[givenBy (v o f)] extend_set h B)
(A LeadsTo[givenBy v] B)"
apply (unfold extending_def)
apply (blast intro: project_LeadsETo_D)
done
*)
(*** leadsETo in the precondition ***)
(*Lemma for the Trans case*)
lemma pli_lemma:
"[| extend h F\<squnion>G : stable C;
F\<squnion>project h C G
: project_set h C Int project_set h A leadsTo project_set h B |]
==> F\<squnion>project h C G
: project_set h C Int project_set h A leadsTo
project_set h C Int project_set h B"
apply (rule psp_stable2 [THEN leadsTo_weaken_L])
apply (auto simp add: project_stable_project_set extend_stable_project_set)
done
lemma project_leadsETo_I_lemma:
"[| extend h F\<squnion>G : stable C;
extend h F\<squnion>G :
(C Int A) leadsTo[(%D. C Int D)`givenBy f] B |]
==> F\<squnion>project h C G
: (project_set h C Int project_set h (C Int A)) leadsTo (project_set h B)"
apply (erule leadsETo_induct)
prefer 3
apply (simp only: Int_UN_distrib project_set_Union)
apply (blast intro: leadsTo_UN)
prefer 2 apply (blast intro: leadsTo_Trans pli_lemma)
apply (simp add: givenBy_eq_extend_set)
apply (rule leadsTo_Basis)
apply (blast intro: ensures_extend_set_imp_project_ensures)
done
lemma project_leadsETo_I:
"extend h F\<squnion>G : (extend_set h A) leadsTo[givenBy f] (extend_set h B)
==> F\<squnion>project h UNIV G : A leadsTo B"
apply (rule project_leadsETo_I_lemma [THEN leadsTo_weaken], auto)
done
lemma project_LeadsETo_I:
"extend h F\<squnion>G : (extend_set h A) LeadsTo[givenBy f] (extend_set h B)
==> F\<squnion>project h (reachable (extend h F\<squnion>G)) G
: A LeadsTo B"
apply (simp (no_asm_use) add: LeadsTo_def LeadsETo_def)
apply (rule project_leadsETo_I_lemma [THEN leadsTo_weaken])
apply (auto simp add: project_set_reachable_extend_eq [symmetric])
done
lemma projecting_leadsTo:
"projecting (%G. UNIV) h F
(extend_set h A leadsTo[givenBy f] extend_set h B)
(A leadsTo B)"
apply (unfold projecting_def)
apply (force dest: project_leadsETo_I)
done
lemma projecting_LeadsTo:
"projecting (%G. reachable (extend h F\<squnion>G)) h F
(extend_set h A LeadsTo[givenBy f] extend_set h B)
(A LeadsTo B)"
apply (unfold projecting_def)
apply (force dest: project_LeadsETo_I)
done
end
end