(* Title: HOL/UNITY/Extend.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
Extending of state sets
function f (forget) maps the extended state to the original state
function g (forgotten) maps the extended state to the "extending part"
*)
(** These we prove OUTSIDE the locale. **)
(*Possibly easier than reasoning about "inv h"*)
val [surj_h,prem] =
Goalw [good_map_def]
"[| surj h; !! x x' y y'. h(x,y) = h(x',y') ==> x=x' |] ==> good_map h";
by (safe_tac (claset() addSIs [surj_h]));
by (rtac prem 1);
by (stac (surjective_pairing RS sym) 1);
by (stac (surj_h RS surj_f_inv_f) 1);
by (rtac refl 1);
qed "good_mapI";
Goalw [good_map_def] "good_map h ==> surj h";
by Auto_tac;
qed "good_map_is_surj";
(*A convenient way of finding a closed form for inv h*)
val [surj,prem] = Goalw [inv_def]
"[| surj h; !! x y. g (h(x,y)) = x |] ==> fst (inv h z) = g z";
by (res_inst_tac [("y1", "z")] (surj RS surjD RS exE) 1);
by (rtac selectI2 1);
by (dres_inst_tac [("f", "g")] arg_cong 2);
by (auto_tac (claset(), simpset() addsimps [prem]));
qed "fst_inv_equalityI";
Open_locale "Extend";
val slice_def = thm "slice_def";
(*** Trivial properties of f, g, h ***)
val good_h = rewrite_rule [good_map_def] (thm "good_h");
val surj_h = good_h RS conjunct1;
val f_def = thm "f_def";
val g_def = thm "g_def";
Goal "f(h(x,y)) = x";
by (simp_tac (simpset() addsimps [f_def, good_h RS conjunct2]) 1);
qed "f_h_eq";
Addsimps [f_h_eq];
Goal "h(x,y) = h(x',y') ==> x=x'";
by (dres_inst_tac [("f", "fst o inv h")] arg_cong 1);
(*FIXME: If locales worked properly we could put just "f" above*)
by (full_simp_tac (simpset() addsimps [f_def, good_h RS conjunct2]) 1);
qed "h_inject1";
AddDs [h_inject1];
Goal "h(f z, g z) = z";
by (simp_tac (simpset() addsimps [f_def, g_def, surjective_pairing RS sym,
surj_h RS surj_f_inv_f]) 1);
qed "h_f_g_eq";
(** A sequence of proof steps borrowed from Provers/split_paired_all.ML **)
val cT = TFree ("'c", ["term"]);
(* "PROP P x == PROP P (h (f x, g x))" *)
val lemma1 = Thm.combination
(Thm.reflexive (cterm_of (sign_of thy) (Free ("P", cT --> propT))))
(Drule.unvarify (h_f_g_eq RS sym RS eq_reflection));
val prems = Goalw [lemma1] "(!!u y. PROP P (h (u, y))) ==> PROP P x";
by (resolve_tac prems 1);
val lemma2 = result();
val prems = Goal "(!!u y. PROP P (h (u, y))) ==> (!!z. PROP P z)";
by (rtac lemma2 1);
by (resolve_tac prems 1);
val lemma3 = result();
val prems = Goal "(!!z. PROP P z) ==> (!!u y. PROP P (h (u, y)))";
(*not needed for proof, but prevents generalization over h, 'c, etc.*)
by (cut_facts_tac [surj_h] 1);
by (resolve_tac prems 1);
val lemma4 = result();
val split_extended_all = Thm.equal_intr lemma4 lemma3;
(*** extend_set: basic properties ***)
Goalw [extend_set_def] "A<=B ==> extend_set h A <= extend_set h B";
by (Blast_tac 1);
qed "extend_set_mono";
Goalw [extend_set_def] "z : extend_set h A = (f z : A)";
by (force_tac (claset() addIs [h_f_g_eq RS sym], simpset()) 1);
qed "mem_extend_set_iff";
AddIffs [mem_extend_set_iff];
Goalw [extend_set_def]
"(extend_set h A <= extend_set h B) = (A <= B)";
by (Force_tac 1);
qed "extend_set_strict_mono";
AddIffs [extend_set_strict_mono];
Goal "{s. P (f s)} = extend_set h {s. P s}";
by Auto_tac;
qed "Collect_eq_extend_set";
Goal "extend_set h {x} = {s. f s = x}";
by Auto_tac;
qed "extend_set_sing";
Goalw [extend_set_def, project_set_def]
"project_set h (extend_set h C) = C";
by Auto_tac;
qed "extend_set_inverse";
Addsimps [extend_set_inverse];
Goalw [extend_set_def, project_set_def]
"C <= extend_set h (project_set h C)";
by (auto_tac (claset(),
simpset() addsimps [split_extended_all]));
by (Blast_tac 1);
qed "extend_set_project_set";
Goal "inj (extend_set h)";
by (rtac inj_on_inverseI 1);
by (rtac extend_set_inverse 1);
qed "inj_extend_set";
Goalw [extend_set_def] "extend_set h UNIV = UNIV";
by (auto_tac (claset(),
simpset() addsimps [split_extended_all]));
qed "extend_set_UNIV_eq";
Addsimps [standard extend_set_UNIV_eq];
(*** project_set: basic properties ***)
(*project_set is simply image!*)
Goalw [project_set_def] "project_set h C = f `` C";
by (auto_tac (claset() addIs [f_h_eq RS sym],
simpset() addsimps [split_extended_all]));
qed "project_set_eq";
(*Converse appears to fail*)
Goalw [project_set_def] "!!z. z : C ==> f z : project_set h C";
by (auto_tac (claset(),
simpset() addsimps [split_extended_all]));
qed "project_set_I";
(*** More laws ***)
(*Because A and B could differ on the "other" part of the state,
cannot generalize result to
project_set h (A Int B) = project_set h A Int project_set h B
*)
Goalw [project_set_def]
"project_set h ((extend_set h A) Int B) = A Int (project_set h B)";
by Auto_tac;
qed "project_set_extend_set_Int";
Goal "extend_set h (A Un B) = extend_set h A Un extend_set h B";
by Auto_tac;
qed "extend_set_Un_distrib";
Goal "extend_set h (A Int B) = extend_set h A Int extend_set h B";
by Auto_tac;
qed "extend_set_Int_distrib";
Goal "extend_set h (INTER A B) = (INT x:A. extend_set h (B x))";
by Auto_tac;
qed "extend_set_INT_distrib";
Goal "extend_set h (A - B) = extend_set h A - extend_set h B";
by Auto_tac;
qed "extend_set_Diff_distrib";
Goal "extend_set h (Union A) = (UN X:A. extend_set h X)";
by (Blast_tac 1);
qed "extend_set_Union";
Goalw [extend_set_def] "(extend_set h A <= - extend_set h B) = (A <= - B)";
by Auto_tac;
qed "extend_set_subset_Compl_eq";
(*** extend_act ***)
Goalw [extend_act_def]
"((h(s,y), h(s',y)) : extend_act h act) = ((s, s') : act)";
by Auto_tac;
qed "mem_extend_act_iff";
AddIffs [mem_extend_act_iff];
(*Converse fails: (z,z') would include actions that changed the g-part*)
Goalw [extend_act_def]
"(z, z') : extend_act h act ==> (f z, f z') : act";
by Auto_tac;
qed "extend_act_D";
Goalw [extend_act_def, project_act_def, project_set_def]
"project_act h (Restrict C (extend_act h act)) = \
\ Restrict (project_set h C) act";
by (Blast_tac 1);
qed "project_act_extend_act_restrict";
Addsimps [project_act_extend_act_restrict];
Goalw [extend_act_def, project_act_def, project_set_def]
"(Restrict C (extend_act h act) = Restrict C (extend_act h act')) \
\ = (Restrict (project_set h C) act = Restrict (project_set h C) act')";
by Auto_tac;
by (ALLGOALS (blast_tac (claset() addSEs [equalityE])));
qed "Restrict_extend_act_eq_Restrict_project_act";
(*Premise is still undesirably strong, since Domain act can include
non-reachable states, but it seems necessary for this result.*)
Goal "Domain act <= project_set h C \
\ ==> project_act h (Restrict C (extend_act h act)) = act";
by (asm_simp_tac (simpset() addsimps [Restrict_triv]) 1);
qed "extend_act_inverse";
Goalw [extend_act_def, project_act_def]
"act' <= extend_act h act ==> project_act h act' <= act";
by (Force_tac 1);
qed "subset_extend_act_D";
Goal "inj (extend_act h)";
by (rtac inj_on_inverseI 1);
by (rtac extend_act_inverse 1);
by (force_tac (claset(), simpset() addsimps [project_set_def]) 1);
qed "inj_extend_act";
Goalw [extend_set_def, extend_act_def]
"extend_act h act ^^ (extend_set h A) = extend_set h (act ^^ A)";
by (Force_tac 1);
qed "extend_act_Image";
Addsimps [extend_act_Image];
Goalw [extend_act_def] "(extend_act h act' <= extend_act h act) = (act'<=act)";
by Auto_tac;
qed "extend_act_strict_mono";
AddIffs [extend_act_strict_mono, inj_extend_act RS inj_eq];
(*The latter theorem is (extend_act h act' = extend_act h act) = (act'=act) *)
Goalw [extend_set_def, extend_act_def]
"Domain (extend_act h act) = extend_set h (Domain act)";
by (Force_tac 1);
qed "Domain_extend_act";
Goalw [extend_act_def]
"extend_act h Id = Id";
by (force_tac (claset() addIs [h_f_g_eq RS sym], simpset()) 1);
qed "extend_act_Id";
Goalw [project_act_def]
"!!z z'. (z, z') : act ==> (f z, f z') : project_act h act";
by (force_tac (claset(),
simpset() addsimps [split_extended_all]) 1);
qed "project_act_I";
Goalw [project_set_def, project_act_def]
"UNIV <= project_set h C ==> project_act h (Restrict C Id) = Id";
by (Force_tac 1);
qed "project_act_Id";
Goalw [project_set_def, project_act_def]
"Domain (project_act h act) = project_set h (Domain act)";
by (force_tac (claset(),
simpset() addsimps [split_extended_all]) 1);
qed "Domain_project_act";
Addsimps [extend_act_Id, project_act_Id];
Goal "(extend_act h act = Id) = (act = Id)";
by Auto_tac;
by (rewtac extend_act_def);
by (ALLGOALS (blast_tac (claset() addEs [equalityE])));
qed "extend_act_Id_iff";
Goal "Id : extend_act h `` Acts F";
by (auto_tac (claset() addSIs [extend_act_Id RS sym],
simpset() addsimps [image_iff]));
qed "Id_mem_extend_act";
(**** extend ****)
(*** Basic properties ***)
Goalw [extend_def] "Init (extend h F) = extend_set h (Init F)";
by Auto_tac;
qed "Init_extend";
Goalw [project_def] "Init (project h C F) = project_set h (Init F)";
by Auto_tac;
qed "Init_project";
Goal "Acts (extend h F) = (extend_act h `` Acts F)";
by (auto_tac (claset() addSIs [extend_act_Id RS sym],
simpset() addsimps [extend_def, image_iff]));
qed "Acts_extend";
Goal "Acts (project h C F) = insert Id (project_act h `` Restrict C `` Acts F)";
by (auto_tac (claset(),
simpset() addsimps [project_def, image_iff]));
qed "Acts_project";
Addsimps [Init_extend, Init_project, Acts_extend, Acts_project];
Goalw [SKIP_def] "extend h SKIP = SKIP";
by (rtac program_equalityI 1);
by Auto_tac;
qed "extend_SKIP";
Goalw [project_set_def] "project_set h UNIV = UNIV";
by Auto_tac;
qed "project_set_UNIV";
Addsimps [project_set_UNIV];
Goal "project h C (extend h F) = \
\ mk_program (Init F, Restrict (project_set h C) `` Acts F)";
by (rtac program_equalityI 1);
by (asm_simp_tac (simpset() addsimps [image_eq_UN,
project_act_extend_act_restrict]) 2);
by (Simp_tac 1);
qed "project_extend_eq";
(*ALL act: Acts F. Domain act is MUCH TOO STRONG since Domain Id = UNIV!*)
Goal "UNIV <= project_set h C \
\ ==> project h C (extend h F) = F";
by (asm_simp_tac (simpset() addsimps [project_extend_eq, image_eq_UN,
subset_UNIV RS subset_trans RS Restrict_triv]) 1);
qed "extend_inverse";
Addsimps [extend_inverse];
Goal "inj (extend h)";
by (rtac inj_on_inverseI 1);
by (rtac extend_inverse 1);
by (force_tac (claset(), simpset() addsimps [project_set_def]) 1);
qed "inj_extend";
Goal "extend h (F Join G) = extend h F Join extend h G";
by (rtac program_equalityI 1);
by (simp_tac (simpset() addsimps [image_Un]) 2);
by (simp_tac (simpset() addsimps [extend_set_Int_distrib]) 1);
qed "extend_Join";
Addsimps [extend_Join];
Goal "extend h (JOIN I F) = (JN i:I. extend h (F i))";
by (rtac program_equalityI 1);
by (simp_tac (simpset() addsimps [image_UN]) 2);
by (simp_tac (simpset() addsimps [extend_set_INT_distrib]) 1);
qed "extend_JN";
Addsimps [extend_JN];
(** These monotonicity results look natural but are UNUSED **)
Goal "F <= G ==> extend h F <= extend h G";
by (full_simp_tac (simpset() addsimps [component_eq_subset]) 1);
by Auto_tac;
qed "extend_mono";
Goal "F <= G ==> project h C F <= project h C G";
by (full_simp_tac
(simpset() addsimps [component_eq_subset, project_set_def]) 1);
by Auto_tac;
qed "project_mono";
(*** Safety: co, stable ***)
Goal "(extend h F : (extend_set h A) co (extend_set h B)) = \
\ (F : A co B)";
by (simp_tac (simpset() addsimps [constrains_def]) 1);
qed "extend_constrains";
Goal "(extend h F : stable (extend_set h A)) = (F : stable A)";
by (asm_simp_tac (simpset() addsimps [stable_def, extend_constrains]) 1);
qed "extend_stable";
Goal "(extend h F : invariant (extend_set h A)) = (F : invariant A)";
by (asm_simp_tac (simpset() addsimps [invariant_def, extend_stable]) 1);
qed "extend_invariant";
(*Converse fails: A and B may differ in their extra variables*)
Goal "extend h F : A co B ==> F : (project_set h A) co (project_set h B)";
by (auto_tac (claset(),
simpset() addsimps [constrains_def, project_set_def]));
by (Force_tac 1);
qed "extend_constrains_project_set";
(*** Diff, needed for localTo ***)
Goal "Restrict (extend_set h C) (extend_act h act) = \
\ extend_act h (Restrict C act)";
by (auto_tac (claset(),
simpset() addsimps [split_extended_all]));
by (auto_tac (claset(),
simpset() addsimps [extend_act_def]));
qed "Restrict_extend_set";
Goalw [Diff_def]
"(Diff (extend_set h C) (extend h G) (extend_act h `` acts)) = \
\ extend h (Diff C G acts)";
by (rtac program_equalityI 1);
by (Simp_tac 1);
by (simp_tac (simpset() addsimps [inj_extend_act RS image_set_diff]) 1);
by (simp_tac (simpset() addsimps [image_eq_UN,
Restrict_extend_set]) 1);
qed "Diff_extend_eq";
(*Opposite inclusion fails; this inclusion's more general than that of
Diff_extend_eq*)
Goal "Diff (project_set h C) G acts \
\ <= project h C (Diff C (extend h G) (extend_act h `` acts))";
by (simp_tac
(simpset() addsimps [component_eq_subset, Diff_def,image_UN,
image_image, image_Diff_image_eq,
Restrict_extend_act_eq_Restrict_project_act,
vimage_image_eq]) 1);
by (blast_tac (claset() addSEs [equalityE])1);
qed "Diff_project_set_component";
Goal "(Diff (extend_set h C) (extend h G) (extend_act h `` acts) \
\ : (extend_set h A) co (extend_set h B)) \
\ = (Diff C G acts : A co B)";
by (simp_tac (simpset() addsimps [Diff_extend_eq, extend_constrains]) 1);
qed "Diff_extend_constrains";
Goal "(Diff (extend_set h C) (extend h G) (extend_act h `` acts) \
\ : stable (extend_set h A)) \
\ = (Diff C G acts : stable A)";
by (simp_tac (simpset() addsimps [Diff_extend_constrains, stable_def]) 1);
qed "Diff_extend_stable";
(*UNUSED!!*)
Goal "Diff (extend_set h C) (extend h G) (extend_act h `` acts) : A co B \
\ ==> Diff C G acts : (project_set h A) co (project_set h B)";
by (asm_full_simp_tac (simpset() addsimps [Diff_extend_eq,
extend_constrains_project_set]) 1);
qed "Diff_extend_constrains_project_set";
Goalw [LOCALTO_def]
"(extend h F : (v o f) localTo[extend_set h C] extend h H) = \
\ (F : v localTo[C] H)";
by (Simp_tac 1);
(*A trick to strip both quantifiers*)
by (res_inst_tac [("f", "All")] (ext RS arg_cong) 1);
by (stac Collect_eq_extend_set 1);
by (simp_tac (simpset() addsimps [Diff_extend_stable]) 1);
qed "extend_localTo_extend_eq";
Goal "[| F : v localTo[C] H; C' <= extend_set h C |] \
\ ==> extend h F : (v o f) localTo[C'] extend h H";
by (blast_tac (claset() addDs [extend_localTo_extend_eq RS iffD2,
impOfSubs localTo_anti_mono]) 1);
qed "extend_localTo_extend_I";
(*USED?? opposite inclusion fails*)
Goal "Restrict C (extend_act h act) <= \
\ extend_act h (Restrict (project_set h C) act)";
by (auto_tac (claset(),
simpset() addsimps [split_extended_all]));
by (auto_tac (claset(),
simpset() addsimps [extend_act_def, project_set_def]));
qed "Restrict_extend_set_subset";
Goal "(extend_act h `` Acts F) - {Id} = extend_act h `` (Acts F - {Id})";
by (stac (extend_act_Id RS sym) 1);
by (simp_tac (simpset() addsimps [inj_extend_act RS image_set_diff]) 1);
qed "extend_act_image_Id_eq";
Goal "Disjoint C (extend h F) (extend h G) = Disjoint (project_set h C) F G";
by (simp_tac
(simpset() addsimps [Disjoint_def, disjoint_iff_not_equal,
image_Diff_image_eq,
Restrict_extend_act_eq_Restrict_project_act,
extend_act_Id_iff, vimage_def]) 1);
qed "Disjoint_extend_eq";
Addsimps [Disjoint_extend_eq];
(*** Weak safety primitives: Co, Stable ***)
Goal "p : reachable (extend h F) ==> f p : reachable F";
by (etac reachable.induct 1);
by (auto_tac
(claset() addIs reachable.intrs,
simpset() addsimps [extend_act_def, image_iff]));
qed "reachable_extend_f";
Goal "h(s,y) : reachable (extend h F) ==> s : reachable F";
by (force_tac (claset() addSDs [reachable_extend_f], simpset()) 1);
qed "h_reachable_extend";
Goalw [extend_set_def]
"reachable (extend h F) = extend_set h (reachable F)";
by (rtac equalityI 1);
by (force_tac (claset() addIs [h_f_g_eq RS sym]
addSDs [reachable_extend_f],
simpset()) 1);
by (Clarify_tac 1);
by (etac reachable.induct 1);
by (ALLGOALS (force_tac (claset() addIs reachable.intrs,
simpset())));
qed "reachable_extend_eq";
Goal "(extend h F : (extend_set h A) Co (extend_set h B)) = \
\ (F : A Co B)";
by (simp_tac
(simpset() addsimps [Constrains_def, reachable_extend_eq,
extend_constrains, extend_set_Int_distrib RS sym]) 1);
qed "extend_Constrains";
Goal "(extend h F : Stable (extend_set h A)) = (F : Stable A)";
by (simp_tac (simpset() addsimps [Stable_def, extend_Constrains]) 1);
qed "extend_Stable";
Goal "(extend h F : Always (extend_set h A)) = (F : Always A)";
by (asm_simp_tac (simpset() addsimps [Always_def, extend_Stable]) 1);
qed "extend_Always";
(*** Progress: transient, ensures ***)
Goal "(extend h F : transient (extend_set h A)) = (F : transient A)";
by (auto_tac (claset(),
simpset() addsimps [transient_def, extend_set_subset_Compl_eq,
Domain_extend_act]));
qed "extend_transient";
Goal "(extend h F : (extend_set h A) ensures (extend_set h B)) = \
\ (F : A ensures B)";
by (simp_tac
(simpset() addsimps [ensures_def, extend_constrains, extend_transient,
extend_set_Un_distrib RS sym,
extend_set_Diff_distrib RS sym]) 1);
qed "extend_ensures";
Goal "F : A leadsTo B \
\ ==> extend h F : (extend_set h A) leadsTo (extend_set h B)";
by (etac leadsTo_induct 1);
by (asm_simp_tac (simpset() addsimps [leadsTo_UN, extend_set_Union]) 3);
by (blast_tac (claset() addIs [leadsTo_Trans]) 2);
by (asm_simp_tac (simpset() addsimps [leadsTo_Basis, extend_ensures]) 1);
qed "leadsTo_imp_extend_leadsTo";
(*** Proving the converse takes some doing! ***)
Goalw [slice_def] "slice (Union S) y = (UN x:S. slice x y)";
by Auto_tac;
qed "slice_Union";
Goalw [slice_def] "slice (extend_set h A) y = A";
by Auto_tac;
qed "slice_extend_set";
Goalw [slice_def, project_set_def] "project_set h A = (UN y. slice A y)";
by Auto_tac;
qed "project_set_is_UN_slice";
Goalw [slice_def, transient_def]
"extend h F : transient A ==> F : transient (slice A y)";
by Auto_tac;
by (rtac bexI 1);
by Auto_tac;
by (force_tac (claset(), simpset() addsimps [extend_act_def]) 1);
qed "extend_transient_slice";
Goal "extend h F : A ensures B ==> F : (slice A y) ensures (project_set h B)";
by (full_simp_tac
(simpset() addsimps [ensures_def, extend_constrains, extend_transient,
project_set_eq, image_Un RS sym,
extend_set_Un_distrib RS sym,
extend_set_Diff_distrib RS sym]) 1);
by Safe_tac;
by (full_simp_tac (simpset() addsimps [constrains_def, extend_act_def,
extend_set_def]) 1);
by (Clarify_tac 1);
by (ball_tac 1);
by (full_simp_tac (simpset() addsimps [slice_def, image_iff, Image_iff]) 1);
by (force_tac (claset() addSIs [h_f_g_eq RS sym], simpset()) 1);
(*transient*)
by (dtac extend_transient_slice 1);
by (etac transient_strengthen 1);
by (force_tac (claset() addIs [f_h_eq RS sym],
simpset() addsimps [slice_def]) 1);
qed "extend_ensures_slice";
Goal "ALL y. F : (slice B y) leadsTo CU ==> F : (project_set h B) leadsTo CU";
by (simp_tac (simpset() addsimps [project_set_is_UN_slice]) 1);
by (blast_tac (claset() addIs [leadsTo_UN]) 1);
qed "leadsTo_slice_image";
Goal "extend h F : AU leadsTo BU \
\ ==> ALL y. F : (slice AU y) leadsTo (project_set h BU)";
by (etac leadsTo_induct 1);
by (full_simp_tac (simpset() addsimps [slice_Union]) 3);
by (blast_tac (claset() addIs [leadsTo_UN]) 3);
by (blast_tac (claset() addIs [leadsTo_slice_image, leadsTo_Trans]) 2);
by (blast_tac (claset() addIs [extend_ensures_slice, leadsTo_Basis]) 1);
qed_spec_mp "extend_leadsTo_slice";
Goal "(extend h F : (extend_set h A) leadsTo (extend_set h B)) = \
\ (F : A leadsTo B)";
by Safe_tac;
by (etac leadsTo_imp_extend_leadsTo 2);
by (dtac extend_leadsTo_slice 1);
by (full_simp_tac (simpset() addsimps [slice_extend_set]) 1);
qed "extend_leadsto";
Goal "(extend h F : (extend_set h A) LeadsTo (extend_set h B)) = \
\ (F : A LeadsTo B)";
by (simp_tac
(simpset() addsimps [LeadsTo_def, reachable_extend_eq,
extend_leadsto, extend_set_Int_distrib RS sym]) 1);
qed "extend_LeadsTo";
Close_locale "Extend";
(*Close_locale should do this!
Delsimps [f_h_eq, extend_set_inverse, f_image_extend_set, extend_act_inverse,
extend_act_Image];
Delrules [make_elim h_inject1];
*)