(* Title: HOL/UNITY/ProgressSets.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 2003 University of Cambridge Progress Sets. From David Meier and Beverly Sanders, Composing Leads-to Properties Theoretical Computer Science 243:1-2 (2000), 339-361. David Meier, Progress Properties in Program Refinement and Parallel Composition Swiss Federal Institute of Technology Zurich (1997) *) section‹Progress Sets› theory ProgressSets imports Transformers begin subsection ‹Complete Lattices and the Operator @{term cl}› definition lattice :: "'a set set => bool" where ― ‹Meier calls them closure sets, but they are just complete lattices› "lattice L == (∀M. M ⊆ L --> ⋂M ∈ L) & (∀M. M ⊆ L --> ⋃M ∈ L)" definition cl :: "['a set set, 'a set] => 'a set" where ― ‹short for ``closure''› "cl L r == ⋂{x. x∈L & r ⊆ x}" lemma UNIV_in_lattice: "lattice L ==> UNIV ∈ L" by (force simp add: lattice_def) lemma empty_in_lattice: "lattice L ==> {} ∈ L" by (force simp add: lattice_def) lemma Union_in_lattice: "[|M ⊆ L; lattice L|] ==> ⋃M ∈ L" by (simp add: lattice_def) lemma Inter_in_lattice: "[|M ⊆ L; lattice L|] ==> ⋂M ∈ L" by (simp add: lattice_def) lemma UN_in_lattice: "[|lattice L; !!i. i∈I ==> r i ∈ L|] ==> (⋃i∈I. r i) ∈ L" apply (blast intro: Union_in_lattice) done lemma INT_in_lattice: "[|lattice L; !!i. i∈I ==> r i ∈ L|] ==> (⋂i∈I. r i) ∈ L" apply (blast intro: Inter_in_lattice) done lemma Un_in_lattice: "[|x∈L; y∈L; lattice L|] ==> x∪y ∈ L" using Union_in_lattice [of "{x, y}" L] by simp lemma Int_in_lattice: "[|x∈L; y∈L; lattice L|] ==> x∩y ∈ L" using Inter_in_lattice [of "{x, y}" L] by simp lemma lattice_stable: "lattice {X. F ∈ stable X}" by (simp add: lattice_def stable_def constrains_def, blast) text‹The next three results state that @{term "cl L r"} is the minimal element of @{term L} that includes @{term r}.› lemma cl_in_lattice: "lattice L ==> cl L r ∈ L" apply (simp add: lattice_def cl_def) apply (erule conjE) apply (drule spec, erule mp, blast) done lemma cl_least: "[|c∈L; r⊆c|] ==> cl L r ⊆ c" by (force simp add: cl_def) text‹The next three lemmas constitute assertion (4.61)› lemma cl_mono: "r ⊆ r' ==> cl L r ⊆ cl L r'" by (simp add: cl_def, blast) lemma subset_cl: "r ⊆ cl L r" by (simp add: cl_def le_Inf_iff) text‹A reformulation of @{thm subset_cl}› lemma clI: "x ∈ r ==> x ∈ cl L r" by (simp add: cl_def, blast) text‹A reformulation of @{thm cl_least}› lemma clD: "[|c ∈ cl L r; B ∈ L; r ⊆ B|] ==> c ∈ B" by (force simp add: cl_def) lemma cl_UN_subset: "(⋃i∈I. cl L (r i)) ⊆ cl L (⋃i∈I. r i)" by (simp add: cl_def, blast) lemma cl_Un: "lattice L ==> cl L (r∪s) = cl L r ∪ cl L s" apply (rule equalityI) prefer 2 apply (simp add: cl_def, blast) apply (rule cl_least) apply (blast intro: Un_in_lattice cl_in_lattice) apply (blast intro: subset_cl [THEN subsetD]) done lemma cl_UN: "lattice L ==> cl L (⋃i∈I. r i) = (⋃i∈I. cl L (r i))" apply (rule equalityI) prefer 2 apply (simp add: cl_def, blast) apply (rule cl_least) apply (blast intro: UN_in_lattice cl_in_lattice) apply (blast intro: subset_cl [THEN subsetD]) done lemma cl_Int_subset: "cl L (r∩s) ⊆ cl L r ∩ cl L s" by (simp add: cl_def, blast) lemma cl_idem [simp]: "cl L (cl L r) = cl L r" by (simp add: cl_def, blast) lemma cl_ident: "r∈L ==> cl L r = r" by (force simp add: cl_def) lemma cl_empty [simp]: "lattice L ==> cl L {} = {}" by (simp add: cl_ident empty_in_lattice) lemma cl_UNIV [simp]: "lattice L ==> cl L UNIV = UNIV" by (simp add: cl_ident UNIV_in_lattice) text‹Assertion (4.62)› lemma cl_ident_iff: "lattice L ==> (cl L r = r) = (r∈L)" apply (rule iffI) apply (erule subst) apply (erule cl_in_lattice) apply (erule cl_ident) done lemma cl_subset_in_lattice: "[|cl L r ⊆ r; lattice L|] ==> r∈L" by (simp add: cl_ident_iff [symmetric] equalityI subset_cl) subsection ‹Progress Sets and the Main Lemma› text‹A progress set satisfies certain closure conditions and is a simple way of including the set @{term "wens_set F B"}.› definition closed :: "['a program, 'a set, 'a set, 'a set set] => bool" where "closed F T B L == ∀M. ∀act ∈ Acts F. B⊆M & T∩M ∈ L --> T ∩ (B ∪ wp act M) ∈ L" definition progress_set :: "['a program, 'a set, 'a set] => 'a set set set" where "progress_set F T B == {L. lattice L & B ∈ L & T ∈ L & closed F T B L}" lemma closedD: "[|closed F T B L; act ∈ Acts F; B⊆M; T∩M ∈ L|] ==> T ∩ (B ∪ wp act M) ∈ L" by (simp add: closed_def) text‹Note: the formalization below replaces Meier's @{term q} by @{term B} and @{term m} by @{term X}.› text‹Part of the proof of the claim at the bottom of page 97. It's proved separately because the argument requires a generalization over all @{term "act ∈ Acts F"}.› lemma lattice_awp_lemma: assumes TXC: "T∩X ∈ C" ― ‹induction hypothesis in theorem below› and BsubX: "B ⊆ X" ― ‹holds in inductive step› and latt: "lattice C" and TC: "T ∈ C" and BC: "B ∈ C" and clos: "closed F T B C" shows "T ∩ (B ∪ awp F (X ∪ cl C (T∩r))) ∈ C" apply (simp del: INT_simps add: awp_def INT_extend_simps) apply (rule INT_in_lattice [OF latt]) apply (erule closedD [OF clos]) apply (simp add: subset_trans [OF BsubX Un_upper1]) apply (subgoal_tac "T ∩ (X ∪ cl C (T∩r)) = (T∩X) ∪ cl C (T∩r)") prefer 2 apply (blast intro: TC clD) apply (erule ssubst) apply (blast intro: Un_in_lattice latt cl_in_lattice TXC) done text‹Remainder of the proof of the claim at the bottom of page 97.› lemma lattice_lemma: assumes TXC: "T∩X ∈ C" ― ‹induction hypothesis in theorem below› and BsubX: "B ⊆ X" ― ‹holds in inductive step› and act: "act ∈ Acts F" and latt: "lattice C" and TC: "T ∈ C" and BC: "B ∈ C" and clos: "closed F T B C" shows "T ∩ (wp act X ∩ awp F (X ∪ cl C (T∩r)) ∪ X) ∈ C" apply (subgoal_tac "T ∩ (B ∪ wp act X) ∈ C") prefer 2 apply (simp add: closedD [OF clos] act BsubX TXC) apply (drule Int_in_lattice [OF _ lattice_awp_lemma [OF TXC BsubX latt TC BC clos, of r] latt]) apply (subgoal_tac "T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T∩r)))) = T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T∩r)))") prefer 2 apply blast apply simp apply (drule Un_in_lattice [OF _ TXC latt]) apply (subgoal_tac "T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T∩r))) ∪ T∩X = T ∩ (wp act X ∩ awp F (X ∪ cl C (T∩r)) ∪ X)") apply simp apply (blast intro: BsubX [THEN subsetD]) done text‹Induction step for the main lemma› lemma progress_induction_step: assumes TXC: "T∩X ∈ C" ― ‹induction hypothesis in theorem below› and act: "act ∈ Acts F" and Xwens: "X ∈ wens_set F B" and latt: "lattice C" and TC: "T ∈ C" and BC: "B ∈ C" and clos: "closed F T B C" and Fstable: "F ∈ stable T" shows "T ∩ wens F act X ∈ C" proof - from Xwens have BsubX: "B ⊆ X" by (rule wens_set_imp_subset) let ?r = "wens F act X" have "?r ⊆ (wp act X ∩ awp F (X∪?r)) ∪ X" by (simp add: wens_unfold [symmetric]) then have "T∩?r ⊆ T ∩ ((wp act X ∩ awp F (X∪?r)) ∪ X)" by blast then have "T∩?r ⊆ T ∩ ((wp act X ∩ awp F (T ∩ (X∪?r))) ∪ X)" by (simp add: awp_Int_eq Fstable stable_imp_awp_ident, blast) then have "T∩?r ⊆ T ∩ ((wp act X ∩ awp F (X ∪ cl C (T∩?r))) ∪ X)" by (blast intro: awp_mono [THEN [2] rev_subsetD] subset_cl [THEN subsetD]) then have "cl C (T∩?r) ⊆ cl C (T ∩ ((wp act X ∩ awp F (X ∪ cl C (T∩?r))) ∪ X))" by (rule cl_mono) then have "cl C (T∩?r) ⊆ T ∩ ((wp act X ∩ awp F (X ∪ cl C (T∩?r))) ∪ X)" by (simp add: cl_ident lattice_lemma [OF TXC BsubX act latt TC BC clos]) then have "cl C (T∩?r) ⊆ (wp act X ∩ awp F (X ∪ cl C (T∩?r))) ∪ X" by blast then have "cl C (T∩?r) ⊆ ?r" by (blast intro!: subset_wens) then have cl_subset: "cl C (T∩?r) ⊆ T∩?r" by (simp add: cl_ident TC subset_trans [OF cl_mono [OF Int_lower1]]) show ?thesis by (rule cl_subset_in_lattice [OF cl_subset latt]) qed text‹Proved on page 96 of Meier's thesis. The special case when @{term "T=UNIV"} states that every progress set for the program @{term F} and set @{term B} includes the set @{term "wens_set F B"}.› lemma progress_set_lemma: "[|C ∈ progress_set F T B; r ∈ wens_set F B; F ∈ stable T|] ==> T∩r ∈ C" apply (simp add: progress_set_def, clarify) apply (erule wens_set.induct) txt‹Base› apply (simp add: Int_in_lattice) txt‹The difficult @{term wens} case› apply (simp add: progress_induction_step) txt‹Disjunctive case› apply (subgoal_tac "(⋃U∈W. T ∩ U) ∈ C") apply simp apply (blast intro: UN_in_lattice) done subsection ‹The Progress Set Union Theorem› lemma closed_mono: assumes BB': "B ⊆ B'" and TBwp: "T ∩ (B ∪ wp act M) ∈ C" and B'C: "B' ∈ C" and TC: "T ∈ C" and latt: "lattice C" shows "T ∩ (B' ∪ wp act M) ∈ C" proof - from TBwp have "(T∩B) ∪ (T ∩ wp act M) ∈ C" by (simp add: Int_Un_distrib) then have TBBC: "(T∩B') ∪ ((T∩B) ∪ (T ∩ wp act M)) ∈ C" by (blast intro: Int_in_lattice Un_in_lattice TC B'C latt) show ?thesis by (rule eqelem_imp_iff [THEN iffD1, OF _ TBBC], blast intro: BB' [THEN subsetD]) qed lemma progress_set_mono: assumes BB': "B ⊆ B'" shows "[| B' ∈ C; C ∈ progress_set F T B|] ==> C ∈ progress_set F T B'" by (simp add: progress_set_def closed_def closed_mono [OF BB'] subset_trans [OF BB']) theorem progress_set_Union: assumes leadsTo: "F ∈ A leadsTo B'" and prog: "C ∈ progress_set F T B" and Fstable: "F ∈ stable T" and BB': "B ⊆ B'" and B'C: "B' ∈ C" and Gco: "!!X. X∈C ==> G ∈ X-B co X" shows "F⊔G ∈ T∩A leadsTo B'" apply (insert prog Fstable) apply (rule leadsTo_Join [OF leadsTo]) apply (force simp add: progress_set_def awp_iff_stable [symmetric]) apply (simp add: awp_iff_constrains) apply (drule progress_set_mono [OF BB' B'C]) apply (blast intro: progress_set_lemma Gco constrains_weaken_L BB' [THEN subsetD]) done subsection ‹Some Progress Sets› lemma UNIV_in_progress_set: "UNIV ∈ progress_set F T B" by (simp add: progress_set_def lattice_def closed_def) subsubsection ‹Lattices and Relations› text‹From Meier's thesis, section 4.5.3› definition relcl :: "'a set set => ('a * 'a) set" where ― ‹Derived relation from a lattice› "relcl L == {(x,y). y ∈ cl L {x}}" definition latticeof :: "('a * 'a) set => 'a set set" where ― ‹Derived lattice from a relation: the set of upwards-closed sets› "latticeof r == {X. ∀s t. s ∈ X & (s,t) ∈ r --> t ∈ X}" lemma relcl_refl: "(a,a) ∈ relcl L" by (simp add: relcl_def subset_cl [THEN subsetD]) lemma relcl_trans: "[| (a,b) ∈ relcl L; (b,c) ∈ relcl L; lattice L |] ==> (a,c) ∈ relcl L" apply (simp add: relcl_def) apply (blast intro: clD cl_in_lattice) done lemma refl_relcl: "lattice L ==> refl (relcl L)" by (simp add: refl_onI relcl_def subset_cl [THEN subsetD]) lemma trans_relcl: "lattice L ==> trans (relcl L)" by (blast intro: relcl_trans transI) lemma lattice_latticeof: "lattice (latticeof r)" by (auto simp add: lattice_def latticeof_def) lemma lattice_singletonI: "[|lattice L; !!s. s ∈ X ==> {s} ∈ L|] ==> X ∈ L" apply (cut_tac UN_singleton [of X]) apply (erule subst) apply (simp only: UN_in_lattice) done text‹Equation (4.71) of Meier's thesis. He gives no proof.› lemma cl_latticeof: "[|refl r; trans r|] ==> cl (latticeof r) X = {t. ∃s. s∈X & (s,t) ∈ r}" apply (rule equalityI) apply (rule cl_least) apply (simp (no_asm_use) add: latticeof_def trans_def, blast) apply (simp add: latticeof_def refl_on_def, blast) apply (simp add: latticeof_def, clarify) apply (unfold cl_def, blast) done text‹Related to (4.71).› lemma cl_eq_Collect_relcl: "lattice L ==> cl L X = {t. ∃s. s∈X & (s,t) ∈ relcl L}" apply (cut_tac UN_singleton [of X]) apply (erule subst) apply (force simp only: relcl_def cl_UN) done text‹Meier's theorem of section 4.5.3› theorem latticeof_relcl_eq: "lattice L ==> latticeof (relcl L) = L" apply (rule equalityI) prefer 2 apply (force simp add: latticeof_def relcl_def cl_def, clarify) apply (rename_tac X) apply (rule cl_subset_in_lattice) prefer 2 apply assumption apply (drule cl_ident_iff [OF lattice_latticeof, THEN iffD2]) apply (drule equalityD1) apply (rule subset_trans) prefer 2 apply assumption apply (thin_tac "_ ⊆ X") apply (cut_tac A=X in UN_singleton) apply (erule subst) apply (simp only: cl_UN lattice_latticeof cl_latticeof [OF refl_relcl trans_relcl]) apply (simp add: relcl_def) done theorem relcl_latticeof_eq: "[|refl r; trans r|] ==> relcl (latticeof r) = r" by (simp add: relcl_def cl_latticeof) subsubsection ‹Decoupling Theorems› definition decoupled :: "['a program, 'a program] => bool" where "decoupled F G == ∀act ∈ Acts F. ∀B. G ∈ stable B --> G ∈ stable (wp act B)" text‹Rao's Decoupling Theorem› lemma stableco: "F ∈ stable A ==> F ∈ A-B co A" by (simp add: stable_def constrains_def, blast) theorem decoupling: assumes leadsTo: "F ∈ A leadsTo B" and Gstable: "G ∈ stable B" and dec: "decoupled F G" shows "F⊔G ∈ A leadsTo B" proof - have prog: "{X. G ∈ stable X} ∈ progress_set F UNIV B" by (simp add: progress_set_def lattice_stable Gstable closed_def stable_Un [OF Gstable] dec [unfolded decoupled_def]) have "F⊔G ∈ (UNIV∩A) leadsTo B" by (rule progress_set_Union [OF leadsTo prog], simp_all add: Gstable stableco) thus ?thesis by simp qed text‹Rao's Weak Decoupling Theorem› theorem weak_decoupling: assumes leadsTo: "F ∈ A leadsTo B" and stable: "F⊔G ∈ stable B" and dec: "decoupled F (F⊔G)" shows "F⊔G ∈ A leadsTo B" proof - have prog: "{X. F⊔G ∈ stable X} ∈ progress_set F UNIV B" by (simp del: Join_stable add: progress_set_def lattice_stable stable closed_def stable_Un [OF stable] dec [unfolded decoupled_def]) have "F⊔G ∈ (UNIV∩A) leadsTo B" by (rule progress_set_Union [OF leadsTo prog], simp_all del: Join_stable add: stable, simp add: stableco) thus ?thesis by simp qed text‹The ``Decoupling via @{term G'} Union Theorem''› theorem decoupling_via_aux: assumes leadsTo: "F ∈ A leadsTo B" and prog: "{X. G' ∈ stable X} ∈ progress_set F UNIV B" and GG': "G ≤ G'" ― ‹Beware! This is the converse of the refinement relation!› shows "F⊔G ∈ A leadsTo B" proof - from prog have stable: "G' ∈ stable B" by (simp add: progress_set_def) have "F⊔G ∈ (UNIV∩A) leadsTo B" by (rule progress_set_Union [OF leadsTo prog], simp_all add: stable stableco component_stable [OF GG']) thus ?thesis by simp qed subsection‹Composition Theorems Based on Monotonicity and Commutativity› subsubsection‹Commutativity of @{term "cl L"} and assignment.› definition commutes :: "['a program, 'a set, 'a set, 'a set set] => bool" where "commutes F T B L == ∀M. ∀act ∈ Acts F. B ⊆ M --> cl L (T ∩ wp act M) ⊆ T ∩ (B ∪ wp act (cl L (T∩M)))" text‹From Meier's thesis, section 4.5.6› lemma commutativity1_lemma: assumes commutes: "commutes F T B L" and lattice: "lattice L" and BL: "B ∈ L" and TL: "T ∈ L" shows "closed F T B L" apply (simp add: closed_def, clarify) apply (rule ProgressSets.cl_subset_in_lattice [OF _ lattice]) apply (simp add: Int_Un_distrib cl_Un [OF lattice] cl_ident Int_in_lattice [OF TL BL lattice] Un_upper1) apply (subgoal_tac "cl L (T ∩ wp act M) ⊆ T ∩ (B ∪ wp act (cl L (T ∩ M)))") prefer 2 apply (cut_tac commutes, simp add: commutes_def) apply (erule subset_trans) apply (simp add: cl_ident) apply (blast intro: rev_subsetD [OF _ wp_mono]) done text‹Version packaged with @{thm progress_set_Union}› lemma commutativity1: assumes leadsTo: "F ∈ A leadsTo B" and lattice: "lattice L" and BL: "B ∈ L" and TL: "T ∈ L" and Fstable: "F ∈ stable T" and Gco: "!!X. X∈L ==> G ∈ X-B co X" and commutes: "commutes F T B L" shows "F⊔G ∈ T∩A leadsTo B" by (rule progress_set_Union [OF leadsTo _ Fstable subset_refl BL Gco], simp add: progress_set_def commutativity1_lemma commutes lattice BL TL) text‹Possibly move to Relation.thy, after @{term single_valued}› definition funof :: "[('a*'b)set, 'a] => 'b" where "funof r == (λx. THE y. (x,y) ∈ r)" lemma funof_eq: "[|single_valued r; (x,y) ∈ r|] ==> funof r x = y" by (simp add: funof_def single_valued_def, blast) lemma funof_Pair_in: "[|single_valued r; x ∈ Domain r|] ==> (x, funof r x) ∈ r" by (force simp add: funof_eq) lemma funof_in: "[|r``{x} ⊆ A; single_valued r; x ∈ Domain r|] ==> funof r x ∈ A" by (force simp add: funof_eq) lemma funof_imp_wp: "[|funof act t ∈ A; single_valued act|] ==> t ∈ wp act A" by (force simp add: in_wp_iff funof_eq) subsubsection‹Commutativity of Functions and Relation› text‹Thesis, page 109› (*FIXME: this proof is still an ungodly mess*) text‹From Meier's thesis, section 4.5.6› lemma commutativity2_lemma: assumes dcommutes: "⋀act s t. act ∈ Acts F ⟹ s ∈ T ⟹ (s, t) ∈ relcl L ⟹ s ∈ B | t ∈ B | (funof act s, funof act t) ∈ relcl L" and determ: "!!act. act ∈ Acts F ==> single_valued act" and total: "!!act. act ∈ Acts F ==> Domain act = UNIV" and lattice: "lattice L" and BL: "B ∈ L" and TL: "T ∈ L" and Fstable: "F ∈ stable T" shows "commutes F T B L" proof - { fix M and act and t assume 1: "B ⊆ M" "act ∈ Acts F" "t ∈ cl L (T ∩ wp act M)" then have "∃s. (s,t) ∈ relcl L ∧ s ∈ T ∩ wp act M" by (force simp add: cl_eq_Collect_relcl [OF lattice]) then obtain s where 2: "(s, t) ∈ relcl L" "s ∈ T" "s ∈ wp act M" by blast then have 3: "∀u∈L. s ∈ u --> t ∈ u" apply (intro ballI impI) apply (subst cl_ident [symmetric], assumption) apply (simp add: relcl_def) apply (blast intro: cl_mono [THEN [2] rev_subsetD]) done with 1 2 Fstable have 4: "funof act s ∈ T∩M" by (force intro!: funof_in simp add: wp_def stable_def constrains_def determ total) with 1 2 3 have 5: "s ∈ B | t ∈ B | (funof act s, funof act t) ∈ relcl L" by (intro dcommutes) assumption+ with 1 2 3 4 have "t ∈ B | funof act t ∈ cl L (T∩M)" by (simp add: relcl_def) (blast intro: BL cl_mono [THEN [2] rev_subsetD]) with 1 2 3 4 5 have "t ∈ B | t ∈ wp act (cl L (T∩M))" by (blast intro: funof_imp_wp determ) with 2 3 have "t ∈ T ∧ (t ∈ B ∨ t ∈ wp act (cl L (T ∩ M)))" by (blast intro: TL cl_mono [THEN [2] rev_subsetD]) then have"t ∈ T ∩ (B ∪ wp act (cl L (T ∩ M)))" by simp } then show "commutes F T B L" unfolding commutes_def by clarify qed text‹Version packaged with @{thm progress_set_Union}› lemma commutativity2: assumes leadsTo: "F ∈ A leadsTo B" and dcommutes: "∀act ∈ Acts F. ∀s ∈ T. ∀t. (s,t) ∈ relcl L --> s ∈ B | t ∈ B | (funof act s, funof act t) ∈ relcl L" and determ: "!!act. act ∈ Acts F ==> single_valued act" and total: "!!act. act ∈ Acts F ==> Domain act = UNIV" and lattice: "lattice L" and BL: "B ∈ L" and TL: "T ∈ L" and Fstable: "F ∈ stable T" and Gco: "!!X. X∈L ==> G ∈ X-B co X" shows "F⊔G ∈ T∩A leadsTo B" apply (rule commutativity1 [OF leadsTo lattice]) apply (simp_all add: Gco commutativity2_lemma dcommutes determ total lattice BL TL Fstable) done subsection ‹Monotonicity› text‹From Meier's thesis, section 4.5.7, page 110› (*to be continued?*) end