(* Title: HOL/UNITY/Extend.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1998 University of Cambridge Extending of state setsExtending 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" *) section‹Extending State Sets› theory Extend imports Guar begin definition (*MOVE to Relation.thy?*) Restrict :: "[ 'a set, ('a*'b) set] => ('a*'b) set" where "Restrict A r = r ∩ (A × UNIV)" definition good_map :: "['a*'b => 'c] => bool" where "good_map h ⟷ surj h & (∀x y. fst (inv h (h (x,y))) = x)" (*Using the locale constant "f", this is f (h (x,y))) = x*) definition extend_set :: "['a*'b => 'c, 'a set] => 'c set" where "extend_set h A = h ` (A × UNIV)" definition project_set :: "['a*'b => 'c, 'c set] => 'a set" where "project_set h C = {x. ∃y. h(x,y) ∈ C}" definition extend_act :: "['a*'b => 'c, ('a*'a) set] => ('c*'c) set" where "extend_act h = (%act. ⋃(s,s') ∈ act. ⋃y. {(h(s,y), h(s',y))})" definition project_act :: "['a*'b => 'c, ('c*'c) set] => ('a*'a) set" where "project_act h act = {(x,x'). ∃y y'. (h(x,y), h(x',y')) ∈ act}" definition extend :: "['a*'b => 'c, 'a program] => 'c program" where "extend h F = mk_program (extend_set h (Init F), extend_act h ` Acts F, project_act h -` AllowedActs F)" definition (*Argument C allows weak safety laws to be projected*) project :: "['a*'b => 'c, 'c set, 'c program] => 'a program" where "project h C F = mk_program (project_set h (Init F), project_act h ` Restrict C ` Acts F, {act. Restrict (project_set h C) act ∈ project_act h ` Restrict C ` AllowedActs F})" locale Extend = fixes f :: "'c => 'a" and g :: "'c => 'b" and h :: "'a*'b => 'c" (*isomorphism between 'a * 'b and 'c *) and slice :: "['c set, 'b] => 'a set" assumes good_h: "good_map h" defines f_def: "f z == fst (inv h z)" and g_def: "g z == snd (inv h z)" and slice_def: "slice Z y == {x. h(x,y) ∈ Z}" (** These we prove OUTSIDE the locale. **) subsection‹Restrict› (*MOVE to Relation.thy?*) lemma Restrict_iff [iff]: "((x,y) ∈ Restrict A r) = ((x,y) ∈ r & x ∈ A)" by (unfold Restrict_def, blast) lemma Restrict_UNIV [simp]: "Restrict UNIV = id" apply (rule ext) apply (auto simp add: Restrict_def) done lemma Restrict_empty [simp]: "Restrict {} r = {}" by (auto simp add: Restrict_def) lemma Restrict_Int [simp]: "Restrict A (Restrict B r) = Restrict (A ∩ B) r" by (unfold Restrict_def, blast) lemma Restrict_triv: "Domain r ⊆ A ==> Restrict A r = r" by (unfold Restrict_def, auto) lemma Restrict_subset: "Restrict A r ⊆ r" by (unfold Restrict_def, auto) lemma Restrict_eq_mono: "[| A ⊆ B; Restrict B r = Restrict B s |] ==> Restrict A r = Restrict A s" by (unfold Restrict_def, blast) lemma Restrict_imageI: "[| s ∈ RR; Restrict A r = Restrict A s |] ==> Restrict A r ∈ Restrict A ` RR" by (unfold Restrict_def image_def, auto) lemma Domain_Restrict [simp]: "Domain (Restrict A r) = A ∩ Domain r" by blast lemma Image_Restrict [simp]: "(Restrict A r) `` B = r `` (A ∩ B)" by blast (*Possibly easier than reasoning about "inv h"*) lemma good_mapI: assumes surj_h: "surj h" and prem: "!! x x' y y'. h(x,y) = h(x',y') ==> x=x'" shows "good_map h" apply (simp add: good_map_def) apply (safe intro!: surj_h) apply (rule prem) apply (subst surjective_pairing [symmetric]) apply (subst surj_h [THEN surj_f_inv_f]) apply (rule refl) done lemma good_map_is_surj: "good_map h ==> surj h" by (unfold good_map_def, auto) (*A convenient way of finding a closed form for inv h*) lemma fst_inv_equalityI: assumes surj_h: "surj h" and prem: "!! x y. g (h(x,y)) = x" shows "fst (inv h z) = g z" by (metis UNIV_I f_inv_into_f prod.collapse prem surj_h) subsection‹Trivial properties of f, g, h› context Extend begin lemma f_h_eq [simp]: "f(h(x,y)) = x" by (simp add: f_def good_h [unfolded good_map_def, THEN conjunct2]) lemma h_inject1 [dest]: "h(x,y) = h(x',y') ==> x=x'" apply (drule_tac f = f in arg_cong) apply (simp add: f_def good_h [unfolded good_map_def, THEN conjunct2]) done lemma h_f_g_equiv: "h(f z, g z) == z" by (simp add: f_def g_def good_h [unfolded good_map_def, THEN conjunct1, THEN surj_f_inv_f]) lemma h_f_g_eq: "h(f z, g z) = z" by (simp add: h_f_g_equiv) lemma split_extended_all: "(!!z. PROP P z) == (!!u y. PROP P (h (u, y)))" proof assume allP: "⋀z. PROP P z" fix u y show "PROP P (h (u, y))" by (rule allP) next assume allPh: "⋀u y. PROP P (h(u,y))" fix z have Phfgz: "PROP P (h (f z, g z))" by (rule allPh) show "PROP P z" by (rule Phfgz [unfolded h_f_g_equiv]) qed end subsection‹@{term extend_set}: basic properties› lemma project_set_iff [iff]: "(x ∈ project_set h C) = (∃y. h(x,y) ∈ C)" by (simp add: project_set_def) lemma extend_set_mono: "A ⊆ B ==> extend_set h A ⊆ extend_set h B" by (unfold extend_set_def, blast) context Extend begin lemma mem_extend_set_iff [iff]: "z ∈ extend_set h A = (f z ∈ A)" apply (unfold extend_set_def) apply (force intro: h_f_g_eq [symmetric]) done lemma extend_set_strict_mono [iff]: "(extend_set h A ⊆ extend_set h B) = (A ⊆ B)" by (unfold extend_set_def, force) lemma (in -) extend_set_empty [simp]: "extend_set h {} = {}" by (unfold extend_set_def, auto) lemma extend_set_eq_Collect: "extend_set h {s. P s} = {s. P(f s)}" by auto lemma extend_set_sing: "extend_set h {x} = {s. f s = x}" by auto lemma extend_set_inverse [simp]: "project_set h (extend_set h C) = C" by (unfold extend_set_def, auto) lemma extend_set_project_set: "C ⊆ extend_set h (project_set h C)" apply (unfold extend_set_def) apply (auto simp add: split_extended_all, blast) done lemma inj_extend_set: "inj (extend_set h)" apply (rule inj_on_inverseI) apply (rule extend_set_inverse) done lemma extend_set_UNIV_eq [simp]: "extend_set h UNIV = UNIV" apply (unfold extend_set_def) apply (auto simp add: split_extended_all) done subsection‹@{term project_set}: basic properties› (*project_set is simply image!*) lemma project_set_eq: "project_set h C = f ` C" by (auto intro: f_h_eq [symmetric] simp add: split_extended_all) (*Converse appears to fail*) lemma project_set_I: "!!z. z ∈ C ==> f z ∈ project_set h C" by (auto simp add: split_extended_all) subsection‹More laws› (*Because A and B could differ on the "other" part of the state, cannot generalize to project_set h (A ∩ B) = project_set h A ∩ project_set h B *) lemma project_set_extend_set_Int: "project_set h ((extend_set h A) ∩ B) = A ∩ (project_set h B)" by auto (*Unused, but interesting?*) lemma project_set_extend_set_Un: "project_set h ((extend_set h A) ∪ B) = A ∪ (project_set h B)" by auto lemma (in -) project_set_Int_subset: "project_set h (A ∩ B) ⊆ (project_set h A) ∩ (project_set h B)" by auto lemma extend_set_Un_distrib: "extend_set h (A ∪ B) = extend_set h A ∪ extend_set h B" by auto lemma extend_set_Int_distrib: "extend_set h (A ∩ B) = extend_set h A ∩ extend_set h B" by auto lemma extend_set_INT_distrib: "extend_set h (INTER A B) = (⋂x ∈ A. extend_set h (B x))" by auto lemma extend_set_Diff_distrib: "extend_set h (A - B) = extend_set h A - extend_set h B" by auto lemma extend_set_Union: "extend_set h (⋃A) = (⋃X ∈ A. extend_set h X)" by blast lemma extend_set_subset_Compl_eq: "(extend_set h A ⊆ - extend_set h B) = (A ⊆ - B)" by (auto simp: extend_set_def) subsection‹@{term extend_act}› (*Can't strengthen it to ((h(s,y), h(s',y')) ∈ extend_act h act) = ((s, s') ∈ act & y=y') because h doesn't have to be injective in the 2nd argument*) lemma mem_extend_act_iff [iff]: "((h(s,y), h(s',y)) ∈ extend_act h act) = ((s, s') ∈ act)" by (auto simp: extend_act_def) (*Converse fails: (z,z') would include actions that changed the g-part*) lemma extend_act_D: "(z, z') ∈ extend_act h act ==> (f z, f z') ∈ act" by (auto simp: extend_act_def) lemma extend_act_inverse [simp]: "project_act h (extend_act h act) = act" unfolding extend_act_def project_act_def by blast lemma project_act_extend_act_restrict [simp]: "project_act h (Restrict C (extend_act h act)) = Restrict (project_set h C) act" unfolding extend_act_def project_act_def by blast lemma subset_extend_act_D: "act' ⊆ extend_act h act ==> project_act h act' ⊆ act" unfolding extend_act_def project_act_def by force lemma inj_extend_act: "inj (extend_act h)" apply (rule inj_on_inverseI) apply (rule extend_act_inverse) done lemma extend_act_Image [simp]: "extend_act h act `` (extend_set h A) = extend_set h (act `` A)" unfolding extend_set_def extend_act_def by force lemma extend_act_strict_mono [iff]: "(extend_act h act' ⊆ extend_act h act) = (act'<=act)" by (auto simp: extend_act_def) lemma [iff]: "(extend_act h act = extend_act h act') = (act = act')" by (rule inj_extend_act [THEN inj_eq]) lemma (in -) Domain_extend_act: "Domain (extend_act h act) = extend_set h (Domain act)" unfolding extend_set_def extend_act_def by force lemma extend_act_Id [simp]: "extend_act h Id = Id" unfolding extend_act_def by (force intro: h_f_g_eq [symmetric]) lemma project_act_I: "!!z z'. (z, z') ∈ act ==> (f z, f z') ∈ project_act h act" unfolding project_act_def by (force simp add: split_extended_all) lemma project_act_Id [simp]: "project_act h Id = Id" unfolding project_act_def by force lemma Domain_project_act: "Domain (project_act h act) = project_set h (Domain act)" unfolding project_act_def by (force simp add: split_extended_all) subsection‹extend› text‹Basic properties› lemma (in -) Init_extend [simp]: "Init (extend h F) = extend_set h (Init F)" by (auto simp: extend_def) lemma (in -) Init_project [simp]: "Init (project h C F) = project_set h (Init F)" by (auto simp: project_def) lemma Acts_extend [simp]: "Acts (extend h F) = (extend_act h ` Acts F)" by (simp add: extend_def insert_Id_image_Acts) lemma AllowedActs_extend [simp]: "AllowedActs (extend h F) = project_act h -` AllowedActs F" by (simp add: extend_def insert_absorb) lemma (in -) Acts_project [simp]: "Acts(project h C F) = insert Id (project_act h ` Restrict C ` Acts F)" by (auto simp add: project_def image_iff) lemma AllowedActs_project [simp]: "AllowedActs(project h C F) = {act. Restrict (project_set h C) act ∈ project_act h ` Restrict C ` AllowedActs F}" apply (simp (no_asm) add: project_def image_iff) apply (subst insert_absorb) apply (auto intro!: bexI [of _ Id] simp add: project_act_def) done lemma Allowed_extend: "Allowed (extend h F) = project h UNIV -` Allowed F" by (auto simp add: Allowed_def) lemma extend_SKIP [simp]: "extend h SKIP = SKIP" apply (unfold SKIP_def) apply (rule program_equalityI, auto) done lemma (in -) project_set_UNIV [simp]: "project_set h UNIV = UNIV" by auto lemma (in -) project_set_Union: "project_set h (⋃A) = (⋃X ∈ A. project_set h X)" by blast (*Converse FAILS: the extended state contributing to project_set h C may not coincide with the one contributing to project_act h act*) lemma (in -) project_act_Restrict_subset: "project_act h (Restrict C act) ⊆ Restrict (project_set h C) (project_act h act)" by (auto simp add: project_act_def) lemma project_act_Restrict_Id_eq: "project_act h (Restrict C Id) = Restrict (project_set h C) Id" by (auto simp add: project_act_def) lemma project_extend_eq: "project h C (extend h F) = mk_program (Init F, Restrict (project_set h C) ` Acts F, {act. Restrict (project_set h C) act ∈ project_act h ` Restrict C ` (project_act h -` AllowedActs F)})" apply (rule program_equalityI) apply simp apply (simp add: image_image) apply (simp add: project_def) done lemma extend_inverse [simp]: "project h UNIV (extend h F) = F" apply (simp (no_asm_simp) add: project_extend_eq subset_UNIV [THEN subset_trans, THEN Restrict_triv]) apply (rule program_equalityI) apply (simp_all (no_asm)) apply (subst insert_absorb) apply (simp (no_asm) add: bexI [of _ Id]) apply auto apply (simp add: image_def) using project_act_Id apply blast apply (simp add: image_def) apply (rename_tac "act") apply (rule_tac x = "extend_act h act" in exI) apply simp done lemma inj_extend: "inj (extend h)" apply (rule inj_on_inverseI) apply (rule extend_inverse) done lemma extend_Join [simp]: "extend h (F⊔G) = extend h F⊔extend h G" apply (rule program_equalityI) apply (simp (no_asm) add: extend_set_Int_distrib) apply (simp add: image_Un, auto) done lemma extend_JN [simp]: "extend h (JOIN I F) = (⨆i ∈ I. extend h (F i))" apply (rule program_equalityI) apply (simp (no_asm) add: extend_set_INT_distrib) apply (simp add: image_UN, auto) done (** These monotonicity results look natural but are UNUSED **) lemma extend_mono: "F ≤ G ==> extend h F ≤ extend h G" by (force simp add: component_eq_subset) lemma project_mono: "F ≤ G ==> project h C F ≤ project h C G" by (simp add: component_eq_subset, blast) lemma all_total_extend: "all_total F ==> all_total (extend h F)" by (simp add: all_total_def Domain_extend_act) subsection‹Safety: co, stable› lemma extend_constrains: "(extend h F ∈ (extend_set h A) co (extend_set h B)) = (F ∈ A co B)" by (simp add: constrains_def) lemma extend_stable: "(extend h F ∈ stable (extend_set h A)) = (F ∈ stable A)" by (simp add: stable_def extend_constrains) lemma extend_invariant: "(extend h F ∈ invariant (extend_set h A)) = (F ∈ invariant A)" by (simp add: invariant_def extend_stable) (*Projects the state predicates in the property satisfied by extend h F. Converse fails: A and B may differ in their extra variables*) lemma extend_constrains_project_set: "extend h F ∈ A co B ==> F ∈ (project_set h A) co (project_set h B)" by (auto simp add: constrains_def, force) lemma extend_stable_project_set: "extend h F ∈ stable A ==> F ∈ stable (project_set h A)" by (simp add: stable_def extend_constrains_project_set) subsection‹Weak safety primitives: Co, Stable› lemma reachable_extend_f: "p ∈ reachable (extend h F) ==> f p ∈ reachable F" by (induct set: reachable) (auto intro: reachable.intros simp add: extend_act_def image_iff) lemma h_reachable_extend: "h(s,y) ∈ reachable (extend h F) ==> s ∈ reachable F" by (force dest!: reachable_extend_f) lemma reachable_extend_eq: "reachable (extend h F) = extend_set h (reachable F)" apply (unfold extend_set_def) apply (rule equalityI) apply (force intro: h_f_g_eq [symmetric] dest!: reachable_extend_f, clarify) apply (erule reachable.induct) apply (force intro: reachable.intros)+ done lemma extend_Constrains: "(extend h F ∈ (extend_set h A) Co (extend_set h B)) = (F ∈ A Co B)" by (simp add: Constrains_def reachable_extend_eq extend_constrains extend_set_Int_distrib [symmetric]) lemma extend_Stable: "(extend h F ∈ Stable (extend_set h A)) = (F ∈ Stable A)" by (simp add: Stable_def extend_Constrains) lemma extend_Always: "(extend h F ∈ Always (extend_set h A)) = (F ∈ Always A)" by (simp add: Always_def extend_Stable) (** Safety and "project" **) (** projection: monotonicity for safety **) lemma (in -) project_act_mono: "D ⊆ C ==> project_act h (Restrict D act) ⊆ project_act h (Restrict C act)" by (auto simp add: project_act_def) lemma project_constrains_mono: "[| D ⊆ C; project h C F ∈ A co B |] ==> project h D F ∈ A co B" apply (auto simp add: constrains_def) apply (drule project_act_mono, blast) done lemma project_stable_mono: "[| D ⊆ C; project h C F ∈ stable A |] ==> project h D F ∈ stable A" by (simp add: stable_def project_constrains_mono) (*Key lemma used in several proofs about project and co*) lemma project_constrains: "(project h C F ∈ A co B) = (F ∈ (C ∩ extend_set h A) co (extend_set h B) & A ⊆ B)" apply (unfold constrains_def) apply (auto intro!: project_act_I simp add: ball_Un) apply (force intro!: project_act_I dest!: subsetD) (*the <== direction*) apply (unfold project_act_def) apply (force dest!: subsetD) done lemma project_stable: "(project h UNIV F ∈ stable A) = (F ∈ stable (extend_set h A))" by (simp add: stable_def project_constrains) lemma project_stable_I: "F ∈ stable (extend_set h A) ==> project h C F ∈ stable A" apply (drule project_stable [THEN iffD2]) apply (blast intro: project_stable_mono) done lemma Int_extend_set_lemma: "A ∩ extend_set h ((project_set h A) ∩ B) = A ∩ extend_set h B" by (auto simp add: split_extended_all) (*Strange (look at occurrences of C) but used in leadsETo proofs*) lemma project_constrains_project_set: "G ∈ C co B ==> project h C G ∈ project_set h C co project_set h B" by (simp add: constrains_def project_def project_act_def, blast) lemma project_stable_project_set: "G ∈ stable C ==> project h C G ∈ stable (project_set h C)" by (simp add: stable_def project_constrains_project_set) subsection‹Progress: transient, ensures› lemma extend_transient: "(extend h F ∈ transient (extend_set h A)) = (F ∈ transient A)" by (auto simp add: transient_def extend_set_subset_Compl_eq Domain_extend_act) lemma extend_ensures: "(extend h F ∈ (extend_set h A) ensures (extend_set h B)) = (F ∈ A ensures B)" by (simp add: ensures_def extend_constrains extend_transient extend_set_Un_distrib [symmetric] extend_set_Diff_distrib [symmetric]) lemma leadsTo_imp_extend_leadsTo: "F ∈ A leadsTo B ==> extend h F ∈ (extend_set h A) leadsTo (extend_set h B)" apply (erule leadsTo_induct) apply (simp add: leadsTo_Basis extend_ensures) apply (blast intro: leadsTo_Trans) apply (simp add: leadsTo_UN extend_set_Union) done subsection‹Proving the converse takes some doing!› lemma slice_iff [iff]: "(x ∈ slice C y) = (h(x,y) ∈ C)" by (simp add: slice_def) lemma slice_Union: "slice (⋃S) y = (⋃x ∈ S. slice x y)" by auto lemma slice_extend_set: "slice (extend_set h A) y = A" by auto lemma project_set_is_UN_slice: "project_set h A = (⋃y. slice A y)" by auto lemma extend_transient_slice: "extend h F ∈ transient A ==> F ∈ transient (slice A y)" by (auto simp: transient_def) (*Converse?*) lemma extend_constrains_slice: "extend h F ∈ A co B ==> F ∈ (slice A y) co (slice B y)" by (auto simp add: constrains_def) lemma extend_ensures_slice: "extend h F ∈ A ensures B ==> F ∈ (slice A y) ensures (project_set h B)" apply (auto simp add: ensures_def extend_constrains extend_transient) apply (erule_tac [2] extend_transient_slice [THEN transient_strengthen]) apply (erule extend_constrains_slice [THEN constrains_weaken], auto) done lemma leadsTo_slice_project_set: "∀y. F ∈ (slice B y) leadsTo CU ==> F ∈ (project_set h B) leadsTo CU" apply (simp add: project_set_is_UN_slice) apply (blast intro: leadsTo_UN) done lemma extend_leadsTo_slice [rule_format]: "extend h F ∈ AU leadsTo BU ==> ∀y. F ∈ (slice AU y) leadsTo (project_set h BU)" apply (erule leadsTo_induct) apply (blast intro: extend_ensures_slice) apply (blast intro: leadsTo_slice_project_set leadsTo_Trans) apply (simp add: leadsTo_UN slice_Union) done lemma extend_leadsTo: "(extend h F ∈ (extend_set h A) leadsTo (extend_set h B)) = (F ∈ A leadsTo B)" apply safe apply (erule_tac [2] leadsTo_imp_extend_leadsTo) apply (drule extend_leadsTo_slice) apply (simp add: slice_extend_set) done lemma extend_LeadsTo: "(extend h F ∈ (extend_set h A) LeadsTo (extend_set h B)) = (F ∈ A LeadsTo B)" by (simp add: LeadsTo_def reachable_extend_eq extend_leadsTo extend_set_Int_distrib [symmetric]) subsection‹preserves› lemma project_preserves_I: "G ∈ preserves (v o f) ==> project h C G ∈ preserves v" by (auto simp add: preserves_def project_stable_I extend_set_eq_Collect) (*to preserve f is to preserve the whole original state*) lemma project_preserves_id_I: "G ∈ preserves f ==> project h C G ∈ preserves id" by (simp add: project_preserves_I) lemma extend_preserves: "(extend h G ∈ preserves (v o f)) = (G ∈ preserves v)" by (auto simp add: preserves_def extend_stable [symmetric] extend_set_eq_Collect) lemma inj_extend_preserves: "inj h ==> (extend h G ∈ preserves g)" by (auto simp add: preserves_def extend_def extend_act_def stable_def constrains_def g_def) subsection‹Guarantees› lemma project_extend_Join: "project h UNIV ((extend h F)⊔G) = F⊔(project h UNIV G)" apply (rule program_equalityI) apply (auto simp add: project_set_extend_set_Int image_iff) apply (metis Un_iff extend_act_inverse image_iff) apply (metis Un_iff extend_act_inverse image_iff) done lemma extend_Join_eq_extend_D: "(extend h F)⊔G = extend h H ==> H = F⊔(project h UNIV G)" apply (drule_tac f = "project h UNIV" in arg_cong) apply (simp add: project_extend_Join) done (** Strong precondition and postcondition; only useful when the old and new state sets are in bijection **) lemma ok_extend_imp_ok_project: "extend h F ok G ==> F ok project h UNIV G" apply (auto simp add: ok_def) apply (drule subsetD) apply (auto intro!: rev_image_eqI) done lemma ok_extend_iff: "(extend h F ok extend h G) = (F ok G)" apply (simp add: ok_def, safe) apply force+ done lemma OK_extend_iff: "OK I (%i. extend h (F i)) = (OK I F)" apply (unfold OK_def, safe) apply (drule_tac x = i in bspec) apply (drule_tac [2] x = j in bspec) apply force+ done lemma guarantees_imp_extend_guarantees: "F ∈ X guarantees Y ==> extend h F ∈ (extend h ` X) guarantees (extend h ` Y)" apply (rule guaranteesI, clarify) apply (blast dest: ok_extend_imp_ok_project extend_Join_eq_extend_D guaranteesD) done lemma extend_guarantees_imp_guarantees: "extend h F ∈ (extend h ` X) guarantees (extend h ` Y) ==> F ∈ X guarantees Y" apply (auto simp add: guar_def) apply (drule_tac x = "extend h G" in spec) apply (simp del: extend_Join add: extend_Join [symmetric] ok_extend_iff inj_extend [THEN inj_image_mem_iff]) done lemma extend_guarantees_eq: "(extend h F ∈ (extend h ` X) guarantees (extend h ` Y)) = (F ∈ X guarantees Y)" by (blast intro: guarantees_imp_extend_guarantees extend_guarantees_imp_guarantees) end end