author | paulson |
Fri, 18 Feb 2000 15:20:44 +0100 | |
changeset 8251 | 9be357df93d4 |
parent 8216 | e4b3192dfefa |
child 8311 | 6218522253e7 |
permissions | -rw-r--r-- |
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(* Title: HOL/UNITY/Client |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1998 University of Cambridge |
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Distributed Resource Management System: the Client |
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*) |
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Addsimps [Client_def RS def_prg_Init]; |
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program_defs_ref := [Client_def]; |
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Addsimps (map simp_of_act [rel_act_def, tok_act_def, ask_act_def]); |
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fun invariant_tac i = |
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rtac invariantI i THEN |
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constrains_tac (i+1); |
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(*Safety property 1: ask, rel are increasing*) |
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Goal "Client: UNIV guarantees[funPair rel ask] \ |
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\ Increasing ask Int Increasing rel"; |
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by (safe_tac (claset() addSIs [guaranteesI,increasing_imp_Increasing])); |
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by (auto_tac |
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(claset(), |
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simpset() addsimps [impOfSubs preserves_subset_increasing, |
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Join_increasing])); |
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by (auto_tac (claset(), simpset() addsimps [increasing_def])); |
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by (ALLGOALS constrains_tac); |
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by Auto_tac; |
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qed "increasing_ask_rel"; |
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Addsimps [nth_append, append_one_prefix]; |
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(*Safety property 2: the client never requests too many tokens. |
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With no Substitution Axiom, we must prove the two invariants |
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simultaneously. |
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*) |
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Goal "G : preserves (funPair ask tok) \ |
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\ ==> Client Join G : \ |
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\ Always ({s. tok s <= NbT} Int \ |
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\ {s. ALL elt : set (ask s). elt <= NbT})"; |
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by (rtac (invariantI RS stable_Join_Always2) 1); |
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by (fast_tac (claset() addSEs [impOfSubs preserves_subset_stable] |
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addSIs [stable_Int]) 3); |
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by (constrains_tac 2); |
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by (cut_inst_tac [("m", "tok s")] (NbT_pos RS mod_less_divisor) 2); |
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by Auto_tac; |
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qed "ask_bounded_lemma"; |
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(*export version, with no mention of tok in the postcondition, but |
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unfortunately tok must be declared local.*) |
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Goal "Client: UNIV guarantees[funPair ask tok] \ |
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\ Always {s. ALL elt : set (ask s). elt <= NbT}"; |
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by (rtac guaranteesI 1); |
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by (etac (ask_bounded_lemma RS Always_weaken) 1); |
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by (rtac Int_lower2 1); |
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qed "ask_bounded"; |
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(*** Towards proving the liveness property ***) |
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Goal "Client: stable {s. rel s <= giv s}"; |
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by (constrains_tac 1); |
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by Auto_tac; |
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qed "stable_rel_le_giv"; |
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Goal "[| Client Join G : Increasing giv; G : preserves rel |] \ |
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\ ==> Client Join G : Stable {s. rel s <= giv s}"; |
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by (rtac (stable_rel_le_giv RS Increasing_preserves_Stable) 1); |
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by Auto_tac; |
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qed "Join_Stable_rel_le_giv"; |
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Goal "[| Client Join G : Increasing giv; G : preserves rel |] \ |
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\ ==> Client Join G : Always {s. rel s <= giv s}"; |
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by (rtac AlwaysI 1); |
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by (rtac Join_Stable_rel_le_giv 2); |
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by Auto_tac; |
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qed "Join_Always_rel_le_giv"; |
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Goal "Client : transient {s. rel s = k & k<h & h <= giv s & h pfixGe ask s}"; |
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by (res_inst_tac [("act", "rel_act")] transientI 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [Domain_def, Client_def])); |
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by (blast_tac (claset() addIs [less_le_trans, prefix_length_le, |
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strict_prefix_length_less]) 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [prefix_def, genPrefix_iff_nth, Ge_def])); |
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by (blast_tac (claset() addIs [strict_prefix_length_less]) 1); |
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qed "transient_lemma"; |
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Goal "[| Client Join G : Increasing giv; G : preserves (funPair rel ask) |] \ |
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\ ==> Client Join G : {s. rel s = k & k<h & h <= giv s & h pfixGe ask s} \ |
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\ LeadsTo {s. k < rel s & rel s <= giv s & \ |
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\ h <= giv s & h pfixGe ask s}"; |
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by (rtac single_LeadsTo_I 1); |
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by (ftac (increasing_ask_rel RS guaranteesD) 1); |
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by Auto_tac; |
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by (rtac (transient_lemma RS Join_transient_I1 RS transient_imp_leadsTo RS |
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leadsTo_imp_LeadsTo RS PSP_Stable RS LeadsTo_weaken) 1); |
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by (rtac (Stable_Int RS Stable_Int RS Stable_Int) 1); |
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by (eres_inst_tac [("f", "giv"), ("x", "giv s")] IncreasingD 1); |
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by (eres_inst_tac [("f", "ask"), ("x", "ask s")] IncreasingD 1); |
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by (eres_inst_tac [("f", "rel"), ("x", "rel s")] IncreasingD 1); |
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by (etac Join_Stable_rel_le_giv 1); |
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by (Blast_tac 1); |
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by (blast_tac (claset() addIs [sym, order_less_le RS iffD2, |
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order_trans, prefix_imp_pfixGe, |
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pfixGe_trans]) 2); |
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by (blast_tac (claset() addIs [order_less_imp_le, order_trans]) 1); |
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qed "induct_lemma"; |
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Goal "[| Client Join G : Increasing giv; G : preserves (funPair rel ask) |] \ |
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\ ==> Client Join G : {s. rel s < h & h <= giv s & h pfixGe ask s} \ |
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\ LeadsTo {s. h <= rel s}"; |
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by (res_inst_tac [("f", "%s. size h - size (rel s)")] LessThan_induct 1); |
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by (auto_tac (claset(), simpset() addsimps [vimage_def])); |
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by (rtac single_LeadsTo_I 1); |
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by (rtac (induct_lemma RS LeadsTo_weaken) 1); |
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by Auto_tac; |
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by (blast_tac (claset() addIs [order_less_le RS iffD2] |
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addDs [common_prefix_linear]) 1); |
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by (REPEAT (dtac strict_prefix_length_less 1)); |
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by (arith_tac 1); |
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qed "rel_progress_lemma"; |
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Goal "[| Client Join G : Increasing giv; G : preserves (funPair rel ask) |] \ |
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\ ==> Client Join G : {s. h <= giv s & h pfixGe ask s} \ |
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\ LeadsTo {s. h <= rel s}"; |
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by (rtac (Join_Always_rel_le_giv RS Always_LeadsToI) 1); |
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by (rtac ([rel_progress_lemma, subset_refl RS subset_imp_LeadsTo] MRS |
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LeadsTo_Un RS LeadsTo_weaken_L) 3); |
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by Auto_tac; |
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by (blast_tac (claset() addIs [order_less_le RS iffD2] |
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addDs [common_prefix_linear]) 1); |
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qed "client_progress_lemma"; |
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(*Progress property: all tokens that are given will be released*) |
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Goal "Client : \ |
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\ Increasing giv guarantees[funPair rel ask] \ |
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\ (INT h. {s. h <= giv s & h pfixGe ask s} LeadsTo {s. h <= rel s})"; |
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by (rtac guaranteesI 1); |
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by (Clarify_tac 1); |
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by (blast_tac (claset() addIs [client_progress_lemma]) 1); |
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qed "client_progress"; |
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Goal "Client : preserves extra"; |
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by (rewtac preserves_def); |
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by Auto_tac; |
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by (constrains_tac 1); |
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by Auto_tac; |
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qed "client_preserves_extra"; |
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Goal "bij client_map"; |
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by (auto_tac (claset() addSWrapper record_split_wrapper, |
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simpset() addsimps [client_map_def, non_extra_def, bij_def, |
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inj_on_def, surj_def])); |
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by (res_inst_tac |
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[("x", "(|giv=?a, ask=?b, rel=?c, tok=?d, extra=?e|)")] exI 1); |
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by (Force_tac 1); |
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qed "bij_client_map"; |
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Goal "rename client_map Client : preserves snd"; |
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by (asm_simp_tac (simpset() addsimps [bij_client_map RS rename_preserves]) 1); |
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by (asm_simp_tac (simpset() addsimps [client_map_def]) 1); |
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by (rtac client_preserves_extra 1); |
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qed "client_preserves_extra"; |
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(** Obsolete lemmas from first version of the Client **) |
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Goal "Client: stable {s. size (rel s) <= size (giv s)}"; |
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by (constrains_tac 1); |
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by Auto_tac; |
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qed "stable_size_rel_le_giv"; |
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(*clients return the right number of tokens*) |
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Goal "Client : Increasing giv guarantees[rel] Always {s. rel s <= giv s}"; |
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by (rtac guaranteesI 1); |
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by (rtac AlwaysI 1); |
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by (Force_tac 1); |
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by (blast_tac (claset() addIs [Increasing_preserves_Stable, |
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stable_rel_le_giv]) 1); |
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qed "ok_guar_rel_prefix_giv"; |