author | nipkow |
Mon, 13 May 2002 15:27:28 +0200 | |
changeset 13145 | 59bc43b51aa2 |
parent 11194 | ea13ff5a26d1 |
child 13797 | baefae13ad37 |
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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5 |
|
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Distributed Resource Management System: the Client |
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*) |
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|
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Addsimps [Client_def RS def_prg_Init, |
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Client_def RS def_prg_AllowedActs]; |
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program_defs_ref := [Client_def]; |
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|
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Addsimps (map simp_of_act [rel_act_def, tok_act_def, ask_act_def]); |
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14 |
|
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Goal "(Client ok G) = \ |
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\ (G : preserves rel & G : preserves ask & G : preserves tok &\ |
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\ Client : Allowed G)"; |
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18 |
by (auto_tac (claset(), |
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simpset() addsimps [ok_iff_Allowed, Client_def RS def_prg_Allowed])); |
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qed "Client_ok_iff"; |
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AddIffs [Client_ok_iff]; |
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|
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23 |
|
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(*Safety property 1: ask, rel are increasing*) |
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Goal "Client: UNIV guarantees Increasing ask Int Increasing rel"; |
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by (auto_tac |
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(claset() addSIs [increasing_imp_Increasing], |
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simpset() addsimps [guar_def, impOfSubs preserves_subset_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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36 |
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37 |
(*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 "Client ok G \ |
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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 Auto_tac; |
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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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|
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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 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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|
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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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|
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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 (force_tac (claset() addIs [AlwaysI, Join_Stable_rel_le_giv], simpset()) 1); |
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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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83 |
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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87 |
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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91 |
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92 |
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Goal "[| Client Join G : Increasing giv; Client ok G |] \ |
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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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98 |
by (ftac (increasing_ask_rel RS guaranteesD) 1); |
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99 |
by Auto_tac; |
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100 |
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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102 |
by (rtac (Stable_Int RS Stable_Int RS Stable_Int) 1); |
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103 |
by (eres_inst_tac [("f", "giv"), ("x", "giv s")] IncreasingD 1); |
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104 |
by (eres_inst_tac [("f", "ask"), ("x", "ask s")] IncreasingD 1); |
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105 |
by (eres_inst_tac [("f", "rel"), ("x", "rel s")] IncreasingD 1); |
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106 |
by (etac Join_Stable_rel_le_giv 1); |
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107 |
by (Blast_tac 1); |
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108 |
by (blast_tac (claset() addIs [sym, order_less_le RS iffD2, |
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109 |
order_trans, prefix_imp_pfixGe, |
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110 |
pfixGe_trans]) 2); |
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111 |
by (blast_tac (claset() addIs [order_less_imp_le, order_trans]) 1); |
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112 |
qed "induct_lemma"; |
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113 |
|
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114 |
|
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115 |
Goal "[| Client Join G : Increasing giv; Client ok G |] \ |
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116 |
\ ==> Client Join G : {s. rel s < h & h <= giv s & h pfixGe ask s} \ |
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117 |
\ LeadsTo {s. h <= rel s}"; |
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118 |
by (res_inst_tac [("f", "%s. size h - size (rel s)")] LessThan_induct 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
119 |
by (auto_tac (claset(), simpset() addsimps [vimage_def])); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
120 |
by (rtac single_LeadsTo_I 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
121 |
by (rtac (induct_lemma RS LeadsTo_weaken) 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
122 |
by Auto_tac; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
123 |
by (blast_tac (claset() addIs [order_less_le RS iffD2] |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
124 |
addDs [common_prefix_linear]) 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
125 |
by (REPEAT (dtac strict_prefix_length_less 1)); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
126 |
by (arith_tac 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
127 |
qed "rel_progress_lemma"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
128 |
|
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
129 |
|
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paulson
parents:
diff
changeset
|
130 |
Goal "[| Client Join G : Increasing giv; Client ok G |] \ |
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reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
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|
131 |
\ ==> Client Join G : {s. h <= giv s & h pfixGe ask s} \ |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
132 |
\ LeadsTo {s. h <= rel s}"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
133 |
by (rtac (Join_Always_rel_le_giv RS Always_LeadsToI) 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
134 |
by (rtac ([rel_progress_lemma, subset_refl RS subset_imp_LeadsTo] MRS |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
135 |
LeadsTo_Un RS LeadsTo_weaken_L) 3); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
136 |
by Auto_tac; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
137 |
by (blast_tac (claset() addIs [order_less_le RS iffD2] |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
138 |
addDs [common_prefix_linear]) 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
139 |
qed "client_progress_lemma"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
140 |
|
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
141 |
(*Progress property: all tokens that are given will be released*) |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
142 |
Goal "Client : \ |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
143 |
\ Increasing giv guarantees \ |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
144 |
\ (INT h. {s. h <= giv s & h pfixGe ask s} LeadsTo {s. h <= rel s})"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
145 |
by (rtac guaranteesI 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
146 |
by (Clarify_tac 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
147 |
by (blast_tac (claset() addIs [client_progress_lemma]) 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
148 |
qed "client_progress"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
149 |
|
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
150 |
(*This shows that the Client won't alter other variables in any state |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
151 |
that it is combined with*) |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
152 |
Goal "Client : preserves dummy"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
153 |
by (rewtac preserves_def); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
154 |
by Auto_tac; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
155 |
by (constrains_tac 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
156 |
by Auto_tac; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
157 |
qed "client_preserves_dummy"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
158 |
|
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
159 |
|
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
160 |
(** Obsolete lemmas from first version of the Client **) |
ea13ff5a26d1
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paulson
parents:
diff
changeset
|
161 |
|
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
162 |
Goal "Client: stable {s. size (rel s) <= size (giv s)}"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
163 |
by (constrains_tac 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
164 |
by Auto_tac; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
165 |
qed "stable_size_rel_le_giv"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
166 |
|
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
167 |
(*clients return the right number of tokens*) |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
168 |
Goal "Client : Increasing giv guarantees Always {s. rel s <= giv s}"; |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
169 |
by (rtac guaranteesI 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
170 |
by (rtac AlwaysI 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
171 |
by (Force_tac 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
172 |
by (blast_tac (claset() addIs [Increasing_preserves_Stable, |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
173 |
stable_rel_le_giv]) 1); |
ea13ff5a26d1
reorganization of HOL/UNITY, moving examples to subdirectories Simple and Comp
paulson
parents:
diff
changeset
|
174 |
qed "ok_guar_rel_prefix_giv"; |