(* Title: HOL/UNITY/Alloc ID: $Id: Alloc.ML,v 1.3 2005/02/13 16:20:11 skalberg Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1998 University of Cambridge Specification of Chandy and Charpentier's Allocator *) (*Perhaps equalities.ML shouldn't add this in the first place!*) Delsimps [image_Collect]; AddIs [impOfSubs subset_preserves_o]; Addsimps [impOfSubs subset_preserves_o]; Addsimps [funPair_o_distrib]; Addsimps [Always_INT_distrib]; Delsimps [o_apply]; (*Eliminate the "o" operator*) val o_simp = simplify (simpset() addsimps [o_def]); (*For rewriting of specifications related by "guarantees"*) Addsimps [rename_image_constrains, rename_image_stable, rename_image_increasing, rename_image_invariant, rename_image_Constrains, rename_image_Stable, rename_image_Increasing, rename_image_Always, rename_image_leadsTo, rename_image_LeadsTo, rename_preserves, rename_image_preserves, lift_image_preserves, bij_image_INT, bij_is_inj RS image_Int, bij_image_Collect_eq]; (*Splits up conjunctions & intersections: like CONJUNCTS in the HOL system*) fun list_of_Int th = (list_of_Int (th RS conjunct1) @ list_of_Int (th RS conjunct2)) handle THM _ => (list_of_Int (th RS IntD1) @ list_of_Int (th RS IntD2)) handle THM _ => (list_of_Int (th RS INT_D)) handle THM _ => (list_of_Int (th RS bspec)) handle THM _ => [th]; (*Used just once, for Alloc_Increasing*) val lessThanBspec = lessThan_iff RS iffD2 RSN (2, bspec); fun normalize th = normalize (th RS spec handle THM _ => th RS lessThanBspec handle THM _ => th RS bspec handle THM _ => th RS (guarantees_INT_right_iff RS iffD1)) handle THM _ => th; (*** bijectivity of sysOfAlloc [MUST BE AUTOMATED] ***) val record_auto_tac = auto_tac (claset() addIs [ext] addSWrapper record_split_wrapper, simpset() addsimps [sysOfAlloc_def, sysOfClient_def, client_map_def, non_dummy_def, funPair_def, o_apply, Let_def]); Goalw [sysOfAlloc_def, Let_def] "inj sysOfAlloc"; by (rtac injI 1); by record_auto_tac; qed "inj_sysOfAlloc"; AddIffs [inj_sysOfAlloc]; (*We need the inverse; also having it simplifies the proof of surjectivity*) Goal "!!s. inv sysOfAlloc s = \ \ (| allocGiv = allocGiv s, \ \ allocAsk = allocAsk s, \ \ allocRel = allocRel s, \ \ allocState_d.dummy = (client s, dummy s) |)"; by (rtac (inj_sysOfAlloc RS inv_f_eq) 1); by record_auto_tac; qed "inv_sysOfAlloc_eq"; Addsimps [inv_sysOfAlloc_eq]; Goal "surj sysOfAlloc"; by (simp_tac (simpset() addsimps [surj_iff, expand_fun_eq, o_apply]) 1); by record_auto_tac; qed "surj_sysOfAlloc"; AddIffs [surj_sysOfAlloc]; Goal "bij sysOfAlloc"; by (blast_tac (claset() addIs [bijI]) 1); qed "bij_sysOfAlloc"; AddIffs [bij_sysOfAlloc]; (*** bijectivity of sysOfClient ***) Goalw [sysOfClient_def] "inj sysOfClient"; by (rtac injI 1); by record_auto_tac; qed "inj_sysOfClient"; AddIffs [inj_sysOfClient]; Goal "!!s. inv sysOfClient s = \ \ (client s, \ \ (| allocGiv = allocGiv s, \ \ allocAsk = allocAsk s, \ \ allocRel = allocRel s, \ \ allocState_d.dummy = systemState.dummy s|) )"; by (rtac (inj_sysOfClient RS inv_f_eq) 1); by record_auto_tac; qed "inv_sysOfClient_eq"; Addsimps [inv_sysOfClient_eq]; Goal "surj sysOfClient"; by (simp_tac (simpset() addsimps [surj_iff, expand_fun_eq, o_apply]) 1); by record_auto_tac; qed "surj_sysOfClient"; AddIffs [surj_sysOfClient]; Goal "bij sysOfClient"; by (blast_tac (claset() addIs [bijI]) 1); qed "bij_sysOfClient"; AddIffs [bij_sysOfClient]; (*** bijectivity of client_map ***) Goalw [inj_on_def] "inj client_map"; by record_auto_tac; qed "inj_client_map"; AddIffs [inj_client_map]; Goal "!!s. inv client_map s = \ \ (%(x,y).(|giv = giv x, ask = ask x, rel = rel x, \ \ clientState_d.dummy = y|)) s"; by (rtac (inj_client_map RS inv_f_eq) 1); by record_auto_tac; qed "inv_client_map_eq"; Addsimps [inv_client_map_eq]; Goal "surj client_map"; by (simp_tac (simpset() addsimps [surj_iff, expand_fun_eq, o_apply]) 1); by record_auto_tac; qed "surj_client_map"; AddIffs [surj_client_map]; Goal "bij client_map"; by (blast_tac (claset() addIs [bijI]) 1); qed "bij_client_map"; AddIffs [bij_client_map]; (** o-simprules for client_map **) Goalw [client_map_def] "fst o client_map = non_dummy"; by (rtac fst_o_funPair 1); qed "fst_o_client_map"; Addsimps (make_o_equivs fst_o_client_map); Goalw [client_map_def] "snd o client_map = clientState_d.dummy"; by (rtac snd_o_funPair 1); qed "snd_o_client_map"; Addsimps (make_o_equivs snd_o_client_map); (** o-simprules for sysOfAlloc [MUST BE AUTOMATED] **) Goal "client o sysOfAlloc = fst o allocState_d.dummy "; by record_auto_tac; qed "client_o_sysOfAlloc"; Addsimps (make_o_equivs client_o_sysOfAlloc); Goal "allocGiv o sysOfAlloc = allocGiv"; by record_auto_tac; qed "allocGiv_o_sysOfAlloc_eq"; Addsimps (make_o_equivs allocGiv_o_sysOfAlloc_eq); Goal "allocAsk o sysOfAlloc = allocAsk"; by record_auto_tac; qed "allocAsk_o_sysOfAlloc_eq"; Addsimps (make_o_equivs allocAsk_o_sysOfAlloc_eq); Goal "allocRel o sysOfAlloc = allocRel"; by record_auto_tac; qed "allocRel_o_sysOfAlloc_eq"; Addsimps (make_o_equivs allocRel_o_sysOfAlloc_eq); (** o-simprules for sysOfClient [MUST BE AUTOMATED] **) Goal "client o sysOfClient = fst"; by record_auto_tac; qed "client_o_sysOfClient"; Addsimps (make_o_equivs client_o_sysOfClient); Goal "allocGiv o sysOfClient = allocGiv o snd "; by record_auto_tac; qed "allocGiv_o_sysOfClient_eq"; Addsimps (make_o_equivs allocGiv_o_sysOfClient_eq); Goal "allocAsk o sysOfClient = allocAsk o snd "; by record_auto_tac; qed "allocAsk_o_sysOfClient_eq"; Addsimps (make_o_equivs allocAsk_o_sysOfClient_eq); Goal "allocRel o sysOfClient = allocRel o snd "; by record_auto_tac; qed "allocRel_o_sysOfClient_eq"; Addsimps (make_o_equivs allocRel_o_sysOfClient_eq); Goal "allocGiv o inv sysOfAlloc = allocGiv"; by (simp_tac (simpset() addsimps [o_def]) 1); qed "allocGiv_o_inv_sysOfAlloc_eq"; Addsimps (make_o_equivs allocGiv_o_inv_sysOfAlloc_eq); Goal "allocAsk o inv sysOfAlloc = allocAsk"; by (simp_tac (simpset() addsimps [o_def]) 1); qed "allocAsk_o_inv_sysOfAlloc_eq"; Addsimps (make_o_equivs allocAsk_o_inv_sysOfAlloc_eq); Goal "allocRel o inv sysOfAlloc = allocRel"; by (simp_tac (simpset() addsimps [o_def]) 1); qed "allocRel_o_inv_sysOfAlloc_eq"; Addsimps (make_o_equivs allocRel_o_inv_sysOfAlloc_eq); Goal "(rel o inv client_map o drop_map i o inv sysOfClient) = \ \ rel o sub i o client"; by (simp_tac (simpset() addsimps [o_def, drop_map_def]) 1); qed "rel_inv_client_map_drop_map"; Addsimps (make_o_equivs rel_inv_client_map_drop_map); Goal "(ask o inv client_map o drop_map i o inv sysOfClient) = \ \ ask o sub i o client"; by (simp_tac (simpset() addsimps [o_def, drop_map_def]) 1); qed "ask_inv_client_map_drop_map"; Addsimps (make_o_equivs ask_inv_client_map_drop_map); (** Open_locale "System"; val Alloc = thm "Alloc"; val Client = thm "Client"; val Network = thm "Network"; val System_def = thm "System_def"; CANNOT use bind_thm: it puts the theorem into standard form, in effect exporting it from the locale **) AddIffs [finite_lessThan]; (*Client : <unfolded specification> *) val client_spec_simps = [client_spec_def, client_increasing_def, client_bounded_def, client_progress_def, client_allowed_acts_def, client_preserves_def, guarantees_Int_right]; val [Client_Increasing_ask, Client_Increasing_rel, Client_Bounded, Client_Progress, Client_AllowedActs, Client_preserves_giv, Client_preserves_dummy] = Client |> simplify (simpset() addsimps client_spec_simps) |> list_of_Int; AddIffs [Client_Increasing_ask, Client_Increasing_rel, Client_Bounded, Client_preserves_giv, Client_preserves_dummy]; (*Network : <unfolded specification> *) val network_spec_simps = [network_spec_def, network_ask_def, network_giv_def, network_rel_def, network_allowed_acts_def, network_preserves_def, ball_conj_distrib]; val [Network_Ask, Network_Giv, Network_Rel, Network_AllowedActs, Network_preserves_allocGiv, Network_preserves_rel, Network_preserves_ask] = Network |> simplify (simpset() addsimps network_spec_simps) |> list_of_Int; AddIffs [Network_preserves_allocGiv]; Addsimps [Network_preserves_rel, Network_preserves_ask]; Addsimps [o_simp Network_preserves_rel, o_simp Network_preserves_ask]; (*Alloc : <unfolded specification> *) val alloc_spec_simps = [alloc_spec_def, alloc_increasing_def, alloc_safety_def, alloc_progress_def, alloc_allowed_acts_def, alloc_preserves_def]; val [Alloc_Increasing_0, Alloc_Safety, Alloc_Progress, Alloc_AllowedActs, Alloc_preserves_allocRel, Alloc_preserves_allocAsk, Alloc_preserves_dummy] = Alloc |> simplify (simpset() addsimps alloc_spec_simps) |> list_of_Int; (*Strip off the INT in the guarantees postcondition*) val Alloc_Increasing = normalize Alloc_Increasing_0; AddIffs [Alloc_preserves_allocRel, Alloc_preserves_allocAsk, Alloc_preserves_dummy]; (** Components lemmas [MUST BE AUTOMATED] **) Goal "Network Join \ \ ((rename sysOfClient \ \ (plam x: (lessThan Nclients). rename client_map Client)) Join \ \ rename sysOfAlloc Alloc) \ \ = System"; by (simp_tac (simpset() addsimps System_def::Join_ac) 1); qed "Network_component_System"; Goal "(rename sysOfClient \ \ (plam x: (lessThan Nclients). rename client_map Client)) Join \ \ (Network Join rename sysOfAlloc Alloc) = System"; by (simp_tac (simpset() addsimps System_def::Join_ac) 1); qed "Client_component_System"; Goal "rename sysOfAlloc Alloc Join \ \ ((rename sysOfClient (plam x: (lessThan Nclients). rename client_map Client)) Join \ \ Network) = System"; by (simp_tac (simpset() addsimps System_def::Join_ac) 1); qed "Alloc_component_System"; AddIffs [Client_component_System, Network_component_System, Alloc_component_System]; (** These preservation laws should be generated automatically **) Goal "Allowed Client = preserves rel Int preserves ask"; by (auto_tac (claset(), simpset() addsimps [Allowed_def, Client_AllowedActs, safety_prop_Acts_iff])); qed "Client_Allowed"; Addsimps [Client_Allowed]; Goal "Allowed Network = \ \ preserves allocRel Int \ \ (INT i: lessThan Nclients. preserves(giv o sub i o client))"; by (auto_tac (claset(), simpset() addsimps [Allowed_def, Network_AllowedActs, safety_prop_Acts_iff])); qed "Network_Allowed"; Addsimps [Network_Allowed]; Goal "Allowed Alloc = preserves allocGiv"; by (auto_tac (claset(), simpset() addsimps [Allowed_def, Alloc_AllowedActs, safety_prop_Acts_iff])); qed "Alloc_Allowed"; Addsimps [Alloc_Allowed]; Goal "OK I (%i. lift i (rename client_map Client))"; by (rtac OK_lift_I 1); by Auto_tac; by (dres_inst_tac [("w1", "rel")] (impOfSubs subset_preserves_o) 1); by (dres_inst_tac [("w1", "ask")] (impOfSubs subset_preserves_o) 2); by (auto_tac (claset(), simpset() addsimps [o_def, split_def])); qed "OK_lift_rename_Client"; Addsimps [OK_lift_rename_Client]; (*needed in rename_client_map_tac* (*The proofs of rename_Client_Increasing, rename_Client_Bounded and rename_Client_Progress are similar. All require copying out the original Client property. A forward proof can be constructed as follows: Client_Increasing_ask RS (bij_client_map RS rename_rename_guarantees_eq RS iffD2) RS (lift_lift_guarantees_eq RS iffD2) RS guarantees_PLam_I RS (bij_sysOfClient RS rename_rename_guarantees_eq RS iffD2) |> simplify (simpset() addsimps [lift_image_eq_rename, o_def, split_def, surj_rename RS surj_range]); However, the "preserves" property remains to be discharged, and the unfolding of "o" and "sub" complicates subsequent reasoning. The following tactic works for all three proofs, though it certainly looks ad-hoc! *) val rename_client_map_tac = EVERY [ simp_tac (simpset() addsimps [rename_guarantees_eq_rename_inv]) 1, rtac guarantees_PLam_I 1, assume_tac 2, (*preserves: routine reasoning*) asm_simp_tac (simpset() addsimps [lift_preserves_sub]) 2, (*the guarantee for "lift i (rename client_map Client)" *) asm_simp_tac (simpset() addsimps [lift_guarantees_eq_lift_inv, rename_guarantees_eq_rename_inv, bij_imp_bij_inv, surj_rename RS surj_range, inv_inv_eq]) 1, asm_simp_tac (simpset() addsimps [o_def, non_dummy_def, guarantees_Int_right]) 1]; (*Lifting Client_Increasing to systemState*) Goal "i : I \ \ ==> rename sysOfClient (plam x: I. rename client_map Client) : \ \ UNIV guarantees \ \ Increasing (ask o sub i o client) Int \ \ Increasing (rel o sub i o client)"; by rename_client_map_tac; qed "rename_Client_Increasing"; Goal "[| F : preserves w; i ~= j |] \ \ ==> F : preserves (sub i o fst o lift_map j o funPair v w)"; by (auto_tac (claset(), simpset() addsimps [lift_map_def, split_def, linorder_neq_iff, o_def])); by (ALLGOALS (dtac (impOfSubs subset_preserves_o))); by (auto_tac (claset(), simpset() addsimps [o_def])); qed "preserves_sub_fst_lift_map"; Goal "[| i < Nclients; j < Nclients |] \ \ ==> Client : preserves (giv o sub i o fst o lift_map j o client_map)"; by (case_tac "i=j" 1); by (asm_full_simp_tac (simpset() addsimps [o_def, non_dummy_def]) 1); by (dtac (Client_preserves_dummy RS preserves_sub_fst_lift_map) 1); by (ALLGOALS (dtac (impOfSubs subset_preserves_o))); by (asm_full_simp_tac (simpset() addsimps [o_def, client_map_def]) 1); qed "client_preserves_giv_oo_client_map"; Goal "rename sysOfClient (plam x: lessThan Nclients. rename client_map Client)\ \ ok Network"; by (auto_tac (claset(), simpset() addsimps [ok_iff_Allowed, client_preserves_giv_oo_client_map])); qed "rename_sysOfClient_ok_Network"; Goal "rename sysOfClient (plam x: lessThan Nclients. rename client_map Client)\ \ ok rename sysOfAlloc Alloc"; by (simp_tac (simpset() addsimps [ok_iff_Allowed]) 1); qed "rename_sysOfClient_ok_Alloc"; Goal "rename sysOfAlloc Alloc ok Network"; by (simp_tac (simpset() addsimps [ok_iff_Allowed]) 1); qed "rename_sysOfAlloc_ok_Network"; AddIffs [rename_sysOfClient_ok_Network, rename_sysOfClient_ok_Alloc, rename_sysOfAlloc_ok_Network]; (*The "ok" laws, re-oriented*) AddIffs [rename_sysOfClient_ok_Network RS ok_sym, rename_sysOfClient_ok_Alloc RS ok_sym, rename_sysOfAlloc_ok_Network RS ok_sym]; Goal "i < Nclients \ \ ==> System : Increasing (ask o sub i o client) Int \ \ Increasing (rel o sub i o client)"; by (rtac ([rename_Client_Increasing, Client_component_System] MRS component_guaranteesD) 1); by Auto_tac; qed "System_Increasing"; bind_thm ("rename_guarantees_sysOfAlloc_I", bij_sysOfAlloc RS rename_rename_guarantees_eq RS iffD2); (*Lifting Alloc_Increasing up to the level of systemState*) val rename_Alloc_Increasing = Alloc_Increasing RS rename_guarantees_sysOfAlloc_I |> simplify (simpset() addsimps [surj_rename RS surj_range, o_def]); Goalw [System_def] "i < Nclients ==> System : Increasing (sub i o allocGiv)"; by (simp_tac (simpset() addsimps [o_def]) 1); by (rtac (rename_Alloc_Increasing RS guarantees_Join_I1 RS guaranteesD) 1); by Auto_tac; qed "System_Increasing_allocGiv"; AddSIs (list_of_Int System_Increasing); (** Follows consequences. The "Always (INT ...) formulation expresses the general safety property and allows it to be combined using Always_Int_rule below. **) Goal "i < Nclients ==> System : ((sub i o allocRel) Fols (rel o sub i o client))"; by (auto_tac (claset() addSIs [Network_Rel RS component_guaranteesD], simpset())); qed "System_Follows_rel"; Goal "i < Nclients ==> System : ((sub i o allocAsk) Fols (ask o sub i o client))"; by (auto_tac (claset() addSIs [Network_Ask RS component_guaranteesD], simpset())); qed "System_Follows_ask"; Goal "i < Nclients ==> System : (giv o sub i o client) Fols (sub i o allocGiv)"; by (auto_tac (claset() addSIs [Network_Giv RS component_guaranteesD, rename_Alloc_Increasing RS component_guaranteesD], simpset())); by (ALLGOALS (simp_tac (simpset() addsimps [o_def, non_dummy_def]))); by (auto_tac (claset() addSIs [rename_Alloc_Increasing RS component_guaranteesD], simpset())); qed "System_Follows_allocGiv"; Goal "System : Always (INT i: lessThan Nclients. \ \ {s. (giv o sub i o client) s <= (sub i o allocGiv) s})"; by Auto_tac; by (etac (System_Follows_allocGiv RS Follows_Bounded) 1); qed "Always_giv_le_allocGiv"; Goal "System : Always (INT i: lessThan Nclients. \ \ {s. (sub i o allocAsk) s <= (ask o sub i o client) s})"; by Auto_tac; by (etac (System_Follows_ask RS Follows_Bounded) 1); qed "Always_allocAsk_le_ask"; Goal "System : Always (INT i: lessThan Nclients. \ \ {s. (sub i o allocRel) s <= (rel o sub i o client) s})"; by (auto_tac (claset() addSIs [Follows_Bounded, System_Follows_rel], simpset())); qed "Always_allocRel_le_rel"; (*** Proof of the safety property (1) ***) (*safety (1), step 1 is System_Follows_rel*) (*safety (1), step 2*) (* i < Nclients ==> System : Increasing (sub i o allocRel) *) bind_thm ("System_Increasing_allocRel", System_Follows_rel RS Follows_Increasing1); (*Lifting Alloc_safety up to the level of systemState. Simplififying with o_def gets rid of the translations but it unfortunately gets rid of the other "o"s too.*) val rename_Alloc_Safety = Alloc_Safety RS rename_guarantees_sysOfAlloc_I |> simplify (simpset() addsimps [o_def]); (*safety (1), step 3*) Goal "System : Always {s. (∑i: lessThan Nclients. (tokens o sub i o allocGiv) s) \ \ <= NbT + (∑i: lessThan Nclients. (tokens o sub i o allocRel) s)}"; by (simp_tac (simpset() addsimps [o_apply]) 1); by (rtac (rename_Alloc_Safety RS component_guaranteesD) 1); by (auto_tac (claset(), simpset() addsimps [o_simp System_Increasing_allocRel])); qed "System_sum_bounded"; (** Follows reasoning **) Goal "System : Always (INT i: lessThan Nclients. \ \ {s. (tokens o giv o sub i o client) s \ \ <= (tokens o sub i o allocGiv) s})"; by (rtac (Always_giv_le_allocGiv RS Always_weaken) 1); by (auto_tac (claset() addIs [tokens_mono_prefix], simpset() addsimps [o_apply])); qed "Always_tokens_giv_le_allocGiv"; Goal "System : Always (INT i: lessThan Nclients. \ \ {s. (tokens o sub i o allocRel) s \ \ <= (tokens o rel o sub i o client) s})"; by (rtac (Always_allocRel_le_rel RS Always_weaken) 1); by (auto_tac (claset() addIs [tokens_mono_prefix], simpset() addsimps [o_apply])); qed "Always_tokens_allocRel_le_rel"; (*safety (1), step 4 (final result!) *) Goalw [system_safety_def] "System : system_safety"; by (rtac (Always_Int_rule [System_sum_bounded, Always_tokens_giv_le_allocGiv, Always_tokens_allocRel_le_rel] RS Always_weaken) 1); by Auto_tac; by (rtac (setsum_fun_mono RS order_trans) 1); by (dtac order_trans 2); by (rtac ([order_refl, setsum_fun_mono] MRS add_le_mono) 2); by (assume_tac 3); by Auto_tac; qed "System_safety"; (*** Proof of the progress property (2) ***) (*progress (2), step 1 is System_Follows_ask and System_Follows_rel*) (*progress (2), step 2; see also System_Increasing_allocRel*) (* i < Nclients ==> System : Increasing (sub i o allocAsk) *) bind_thm ("System_Increasing_allocAsk", System_Follows_ask RS Follows_Increasing1); (*progress (2), step 3: lifting "Client_Bounded" to systemState*) Goal "i : I \ \ ==> rename sysOfClient (plam x: I. rename client_map Client) : \ \ UNIV guarantees \ \ Always {s. ALL elt : set ((ask o sub i o client) s). elt <= NbT}"; by rename_client_map_tac; qed "rename_Client_Bounded"; Goal "i < Nclients \ \ ==> System : Always \ \ {s. ALL elt : set ((ask o sub i o client) s). elt <= NbT}"; by (rtac ([rename_Client_Bounded, Client_component_System] MRS component_guaranteesD) 1); by Auto_tac; qed "System_Bounded_ask"; Goal "{x. ALL y. P y --> Q x y} = (INT y: {y. P y}. {x. Q x y})"; by (Blast_tac 1); qed "Collect_all_imp_eq"; (*progress (2), step 4*) Goal "System : Always {s. ALL i<Nclients. \ \ ALL elt : set ((sub i o allocAsk) s). elt <= NbT}"; by (auto_tac (claset(), simpset() addsimps [Collect_all_imp_eq])); by (rtac (Always_Int_rule [Always_allocAsk_le_ask, System_Bounded_ask] RS Always_weaken) 1); by (auto_tac (claset() addDs [set_mono], simpset())); qed "System_Bounded_allocAsk"; (*progress (2), step 5 is System_Increasing_allocGiv*) (*progress (2), step 6*) (* i < Nclients ==> System : Increasing (giv o sub i o client) *) bind_thm ("System_Increasing_giv", System_Follows_allocGiv RS Follows_Increasing1); Goal "i: I \ \ ==> rename sysOfClient (plam x: I. rename client_map Client) \ \ : Increasing (giv o sub i o client) \ \ guarantees \ \ (INT h. {s. h <= (giv o sub i o client) s & \ \ h pfixGe (ask o sub i o client) s} \ \ LeadsTo {s. tokens h <= (tokens o rel o sub i o client) s})"; by rename_client_map_tac; by (asm_simp_tac (simpset() addsimps [o_simp Client_Progress]) 1); qed "rename_Client_Progress"; (*progress (2), step 7*) Goal "System : (INT i : (lessThan Nclients). \ \ INT h. {s. h <= (giv o sub i o client) s & \ \ h pfixGe (ask o sub i o client) s} \ \ LeadsTo {s. tokens h <= (tokens o rel o sub i o client) s})"; by (rtac INT_I 1); (*Couldn't have just used Auto_tac since the "INT h" must be kept*) by (rtac ([rename_Client_Progress, Client_component_System] MRS component_guaranteesD) 1); by (auto_tac (claset(), simpset() addsimps [System_Increasing_giv])); qed "System_Client_Progress"; (*Concludes System : {s. k <= (sub i o allocGiv) s} LeadsTo {s. (sub i o allocAsk) s <= (ask o sub i o client) s} Int {s. k <= (giv o sub i o client) s} *) val lemma = [System_Follows_ask RS Follows_Bounded, System_Follows_allocGiv RS Follows_LeadsTo] MRS Always_LeadsToD; (*A more complicated variant of the previous one*) val lemma2 = [lemma, System_Follows_ask RS Follows_Increasing1 RS IncreasingD] MRS PSP_Stable; Goal "i < Nclients \ \ ==> System : {s. h <= (sub i o allocGiv) s & \ \ h pfixGe (sub i o allocAsk) s} \ \ LeadsTo \ \ {s. h <= (giv o sub i o client) s & \ \ h pfixGe (ask o sub i o client) s}"; by (rtac single_LeadsTo_I 1); by (res_inst_tac [("k6", "h"), ("x2", "(sub i o allocAsk) s")] (lemma2 RS LeadsTo_weaken) 1); by Auto_tac; by (blast_tac (claset() addIs [trans_Ge RS trans_genPrefix RS transD, prefix_imp_pfixGe]) 1); val lemma3 = result(); (*progress (2), step 8: Client i's "release" action is visible system-wide*) Goal "i < Nclients \ \ ==> System : {s. h <= (sub i o allocGiv) s & \ \ h pfixGe (sub i o allocAsk) s} \ \ LeadsTo {s. tokens h <= (tokens o sub i o allocRel) s}"; by (rtac LeadsTo_Trans 1); by (dtac (System_Follows_rel RS impOfSubs (mono_tokens RS mono_Follows_o) RS Follows_LeadsTo) 2); by (asm_full_simp_tac (simpset() addsimps [o_assoc]) 2); by (rtac LeadsTo_Trans 1); by (cut_facts_tac [System_Client_Progress] 2); by (blast_tac (claset() addIs [LeadsTo_Basis]) 2); by (etac lemma3 1); qed "System_Alloc_Client_Progress"; (*Lifting Alloc_Progress up to the level of systemState*) val rename_Alloc_Progress = Alloc_Progress RS rename_guarantees_sysOfAlloc_I |> simplify (simpset() addsimps [o_def]); (*progress (2), step 9*) Goal "System : (INT i : (lessThan Nclients). \ \ INT h. {s. h <= (sub i o allocAsk) s} \ \ LeadsTo {s. h pfixLe (sub i o allocGiv) s})"; (*Can't use simpset(): the "INT h" must be kept*) by (simp_tac (HOL_ss addsimps [o_apply, sub_def]) 1); by (rtac (rename_Alloc_Progress RS component_guaranteesD) 1); by (auto_tac (claset(), simpset() addsimps [o_simp System_Increasing_allocRel, o_simp System_Increasing_allocAsk, o_simp System_Bounded_allocAsk, o_simp System_Alloc_Client_Progress])); qed "System_Alloc_Progress"; (*progress (2), step 10 (final result!) *) Goalw [system_progress_def] "System : system_progress"; by (cut_facts_tac [System_Alloc_Progress] 1); by (blast_tac (claset() addIs [LeadsTo_Trans, System_Follows_allocGiv RS Follows_LeadsTo_pfixLe, System_Follows_ask RS Follows_LeadsTo]) 1); qed "System_Progress"; (*Ultimate goal*) Goalw [system_spec_def] "System : system_spec"; by (blast_tac (claset() addIs [System_safety, System_Progress]) 1); qed "System_correct"; (** SOME lemmas no longer used **) Goal "non_dummy = (% (g,a,r). (| giv = g, ask = a, rel = r |)) o \ \ (funPair giv (funPair ask rel))"; by (rtac ext 1); by (auto_tac (claset(), simpset() addsimps [o_def, non_dummy_def])); qed "non_dummy_eq_o_funPair"; Goal "(preserves non_dummy) = \ \ (preserves rel Int preserves ask Int preserves giv)"; by (simp_tac (simpset() addsimps [non_dummy_eq_o_funPair]) 1); by Auto_tac; by (dres_inst_tac [("w1", "rel")] (impOfSubs subset_preserves_o) 1); by (dres_inst_tac [("w1", "ask")] (impOfSubs subset_preserves_o) 2); by (dres_inst_tac [("w1", "giv")] (impOfSubs subset_preserves_o) 3); by (auto_tac (claset(), simpset() addsimps [o_def])); qed "preserves_non_dummy_eq"; (*Could go to Extend.ML*) Goal "bij f ==> fst (inv (%(x, u). inv f x) z) = f z"; by (rtac fst_inv_equalityI 1); by (res_inst_tac [("f","%z. (f z, ?h z)")] surjI 1); by (asm_full_simp_tac (simpset() addsimps [bij_is_inj, inv_f_f]) 1); by (asm_full_simp_tac (simpset() addsimps [bij_is_surj, surj_f_inv_f]) 1); qed "bij_fst_inv_inv_eq";