author | paulson |
Fri, 24 Jan 2003 18:13:59 +0100 | |
changeset 13786 | ab8f39f48a6f |
parent 11701 | 3d51fbf81c17 |
child 13796 | 19f50fa807ae |
permissions | -rw-r--r-- |
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(* Title: HOL/UNITY/Priority |
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ID: $Id$ |
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Author: Sidi O Ehmety, Cambridge University Computer Laboratory |
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Copyright 2001 University of Cambridge |
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5 |
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The priority system |
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|
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From Charpentier and Chandy, |
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Examples of Program Composition Illustrating the Use of Universal Properties |
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In J. Rolim (editor), Parallel and Distributed Processing, |
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Spriner LNCS 1586 (1999), pages 1215-1227. |
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*) |
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13 |
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Addsimps [Component_def RS def_prg_Init]; |
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program_defs_ref := [Component_def, system_def]; |
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Addsimps [highest_def, lowest_def]; |
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Addsimps [simp_of_act act_def]; |
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Addsimps (map simp_of_set [Highest_def, Lowest_def]); |
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19 |
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21 |
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22 |
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(**** Component correctness proofs ****) |
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(* neighbors is stable *) |
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Goal "Component i: stable {s. neighbors k s = n}"; |
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by (constrains_tac 1); |
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by Auto_tac; |
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qed "Component_neighbors_stable"; |
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(* property 4 *) |
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Goal |
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"Component i: {s. ((i,j):s) = b} Int (- Highest i) co {s. ((i,j):s)=b}"; |
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by (constrains_tac 1); |
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qed "Component_waits_priority"; |
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(* property 5: charpentier and Chandy mistakenly express it as |
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'transient Highest i'. Consider the case where i has neighbors *) |
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39 |
Goal |
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"Component i: {s. neighbors i s ~= {}} Int Highest i \ |
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\ ensures - Highest i"; |
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42 |
by (ensures_tac "act i" 1); |
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43 |
by (REPEAT(Fast_tac 1)); |
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qed "Component_yields_priority"; |
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45 |
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(* or better *) |
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Goal "Component i: Highest i ensures Lowest i"; |
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by (ensures_tac "act i" 1); |
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by (REPEAT(Fast_tac 1)); |
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qed "Component_yields_priority'"; |
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51 |
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(* property 6: Component doesn't introduce cycle *) |
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Goal "Component i: Highest i co Highest i Un Lowest i"; |
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54 |
by (constrains_tac 1); |
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55 |
by (Fast_tac 1); |
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qed "Component_well_behaves"; |
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57 |
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(* property 7: local axiom *) |
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Goal "Component i: stable {s. ALL j k. j~=i & k~=i--> ((j,k):s) = b j k}"; |
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60 |
by (constrains_tac 1); |
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qed "locality"; |
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62 |
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63 |
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(**** System properties ****) |
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(* property 8: strictly universal *) |
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66 |
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Goalw [Safety_def] |
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"system: stable Safety"; |
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69 |
by (rtac stable_INT 1); |
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70 |
by (constrains_tac 1); |
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71 |
by (Fast_tac 1); |
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qed "Safety"; |
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73 |
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(* property 13: universal *) |
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75 |
Goal |
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"system: {s. s = q} co {s. s=q} Un {s. EX i. derive i q s}"; |
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77 |
by (constrains_tac 1); |
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78 |
by (Blast_tac 1); |
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qed "p13"; |
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80 |
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81 |
(* property 14: the 'above set' of a Component that hasn't got |
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priority doesn't increase *) |
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83 |
Goal |
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84 |
"ALL j. system: -Highest i Int {s. j~:above i s} co {s. j~:above i s}"; |
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85 |
by (Clarify_tac 1); |
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86 |
by (cut_inst_tac [("i", "j")] reach_lemma 1); |
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87 |
by (constrains_tac 1); |
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88 |
by (auto_tac (claset(), simpset() addsimps [trancl_converse])); |
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89 |
qed "above_not_increase"; |
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90 |
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91 |
Goal |
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92 |
"system: -Highest i Int {s. above i s = x} co {s. above i s <= x}"; |
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93 |
by (cut_inst_tac [("i", "i")] above_not_increase 1); |
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94 |
by (asm_full_simp_tac (simpset() addsimps |
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[trancl_converse, constrains_def]) 1); |
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96 |
by (Blast_tac 1); |
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97 |
qed "above_not_increase'"; |
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98 |
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99 |
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100 |
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101 |
(* p15: universal property: all Components well behave *) |
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102 |
Goal "ALL i. system: Highest i co Highest i Un Lowest i"; |
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103 |
by (Clarify_tac 1); |
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104 |
by (constrains_tac 1); |
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105 |
by Auto_tac; |
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106 |
qed "system_well_behaves"; |
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107 |
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108 |
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109 |
Goal "Acyclic = (INT i. {s. i~:above i s})"; |
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110 |
by (auto_tac (claset(), simpset() |
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111 |
addsimps [Acyclic_def, acyclic_def, trancl_converse])); |
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112 |
qed "Acyclic_eq"; |
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113 |
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114 |
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115 |
val lemma = [above_not_increase RS spec, |
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116 |
system_well_behaves RS spec] MRS constrains_Un; |
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117 |
Goal |
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118 |
"system: stable Acyclic"; |
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119 |
by (auto_tac (claset() addSIs [stable_INT, stableI, |
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120 |
lemma RS constrains_weaken], |
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121 |
simpset() addsimps [Acyclic_eq, |
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122 |
image0_r_iff_image0_trancl,trancl_converse])); |
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123 |
qed "Acyclic_stable"; |
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124 |
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125 |
|
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126 |
Goalw [Acyclic_def, Maximal_def] |
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127 |
"Acyclic <= Maximal"; |
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128 |
by (Clarify_tac 1); |
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|
129 |
by (dtac above_lemma_b 1); |
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130 |
by Auto_tac; |
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131 |
qed "Acyclic_subset_Maximal"; |
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132 |
|
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133 |
(* property 17: original one is an invariant *) |
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134 |
Goal |
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135 |
"system: stable (Acyclic Int Maximal)"; |
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|
136 |
by (simp_tac (simpset() addsimps |
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137 |
[Acyclic_subset_Maximal RS Int_absorb2, Acyclic_stable]) 1); |
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|
138 |
qed "Acyclic_Maximal_stable"; |
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|
139 |
|
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diff
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|
140 |
|
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|
141 |
(* propert 5: existential property *) |
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142 |
|
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parents:
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143 |
Goal "system: Highest i leadsTo Lowest i"; |
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parents:
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|
144 |
by (ensures_tac "act i" 1); |
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paulson
parents:
diff
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|
145 |
by (auto_tac (claset(), simpset() addsimps [Component_def])); |
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|
146 |
qed "Highest_leadsTo_Lowest"; |
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|
147 |
|
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148 |
(* a lowest i can never be in any abover set *) |
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149 |
Goal "Lowest i <= (INT k. {s. i~:above k s})"; |
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150 |
by (auto_tac (claset(), |
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151 |
simpset() addsimps [image0_r_iff_image0_trancl, trancl_converse])); |
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|
152 |
qed "Lowest_above_subset"; |
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diff
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|
153 |
|
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154 |
(* property 18: a simpler proof than the original, one which uses psp *) |
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155 |
Goal "system: Highest i leadsTo (INT k. {s. i~:above k s})"; |
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paulson
parents:
diff
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|
156 |
by (rtac leadsTo_weaken_R 1); |
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paulson
parents:
diff
changeset
|
157 |
by (rtac Lowest_above_subset 2); |
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paulson
parents:
diff
changeset
|
158 |
by (rtac Highest_leadsTo_Lowest 1); |
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paulson
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diff
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|
159 |
qed "Highest_escapes_above"; |
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|
160 |
|
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diff
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|
161 |
Goal |
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162 |
"system: Highest j Int {s. j:above i s} leadsTo {s. j~:above i s}"; |
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paulson
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diff
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|
163 |
by (blast_tac (claset() addIs |
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paulson
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diff
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|
164 |
[[Highest_escapes_above, Int_lower1, INT_lower] MRS leadsTo_weaken]) 1); |
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|
165 |
qed "Highest_escapes_above'"; |
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|
166 |
|
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167 |
(*** The main result: above set decreases ***) |
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168 |
(* The original proof of the following formula was wrong *) |
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169 |
val above_decreases_lemma = |
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|
170 |
[Highest_escapes_above', above_not_increase'] MRS psp RS leadsTo_weaken; |
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|
171 |
|
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parents:
diff
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|
172 |
Goal "Highest i = {s. above i s ={}}"; |
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paulson
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|
173 |
by (auto_tac (claset(), |
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|
174 |
simpset() addsimps [image0_trancl_iff_image0_r])); |
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|
175 |
qed "Highest_iff_above0"; |
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diff
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|
176 |
|
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parents:
diff
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|
177 |
|
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|
178 |
Goal |
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179 |
"system: (UN j. {s. above i s = x} Int {s. j:above i s} Int Highest j) \ |
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180 |
\ leadsTo {s. above i s < x}"; |
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paulson
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diff
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|
181 |
by (rtac leadsTo_UN 1); |
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paulson
parents:
diff
changeset
|
182 |
by (rtac single_leadsTo_I 1); |
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paulson
parents:
diff
changeset
|
183 |
by (Clarify_tac 1); |
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paulson
parents:
diff
changeset
|
184 |
by (res_inst_tac [("x2", "above i x")] above_decreases_lemma 1); |
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paulson
parents:
diff
changeset
|
185 |
by (ALLGOALS(full_simp_tac (simpset() delsimps [Highest_def] |
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paulson
parents:
diff
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|
186 |
addsimps [Highest_iff_above0]))); |
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paulson
parents:
diff
changeset
|
187 |
by (REPEAT(Blast_tac 1)); |
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paulson
parents:
diff
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|
188 |
qed "above_decreases"; |
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diff
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|
189 |
|
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|
190 |
(** Just a massage of conditions to have the desired form ***) |
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191 |
Goalw [Maximal_def, Maximal'_def, Highest_def] |
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|
192 |
"Maximal = Maximal'"; |
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paulson
parents:
diff
changeset
|
193 |
by (Blast_tac 1); |
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|
194 |
qed "Maximal_eq_Maximal'"; |
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diff
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|
195 |
|
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|
196 |
Goal "x~={} ==> \ |
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|
197 |
\ Acyclic Int {s. above i s = x} <= \ |
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|
198 |
\ (UN j. {s. above i s = x} Int {s. j:above i s} Int Highest j)"; |
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paulson
parents:
diff
changeset
|
199 |
by (res_inst_tac [("B", "Maximal' Int {s. above i s = x}")] subset_trans 1); |
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paulson
parents:
diff
changeset
|
200 |
by (simp_tac (simpset() addsimps [Maximal_eq_Maximal' RS sym]) 1); |
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paulson
parents:
diff
changeset
|
201 |
by (blast_tac (claset() addIs [Acyclic_subset_Maximal RS subsetD]) 1); |
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paulson
parents:
diff
changeset
|
202 |
by (simp_tac (simpset() delsimps [above_def] addsimps [Maximal'_def, Highest_iff_above0]) 1); |
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paulson
parents:
diff
changeset
|
203 |
by (Blast_tac 1); |
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paulson
parents:
diff
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|
204 |
qed "Acyclic_subset"; |
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paulson
parents:
diff
changeset
|
205 |
|
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|
206 |
val above_decreases' = [above_decreases, Acyclic_subset] MRS leadsTo_weaken_L; |
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|
207 |
val above_decreases_psp = [above_decreases', Acyclic_stable] MRS psp_stable; |
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paulson
parents:
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changeset
|
208 |
|
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parents:
diff
changeset
|
209 |
Goal |
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|
210 |
"x~={}==> \ |
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paulson
parents:
diff
changeset
|
211 |
\ system: Acyclic Int {s. above i s = x} leadsTo Acyclic Int {s. above i s < x}"; |
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paulson
parents:
diff
changeset
|
212 |
by (etac (above_decreases_psp RS leadsTo_weaken) 1); |
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paulson
parents:
diff
changeset
|
213 |
by (Blast_tac 1); |
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paulson
parents:
diff
changeset
|
214 |
by Auto_tac; |
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paulson
parents:
diff
changeset
|
215 |
qed "above_decreases_psp'"; |
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paulson
parents:
diff
changeset
|
216 |
|
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paulson
parents:
diff
changeset
|
217 |
|
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|
218 |
val finite_psubset_induct = wf_finite_psubset RS leadsTo_wf_induct; |
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|
219 |
val leadsTo_weaken_L' = rotate_prems 1 leadsTo_weaken_L; |
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paulson
parents:
diff
changeset
|
220 |
|
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paulson
parents:
diff
changeset
|
221 |
|
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paulson
parents:
diff
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|
222 |
Goal "system: Acyclic leadsTo Highest i"; |
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paulson
parents:
diff
changeset
|
223 |
by (res_inst_tac [("f", "%s. above i s")] finite_psubset_induct 1); |
11701
3d51fbf81c17
sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents:
11194
diff
changeset
|
224 |
by (asm_simp_tac (simpset() delsimps [Highest_def, thm "above_def"] |
11194
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|
225 |
addsimps [Highest_iff_above0, |
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paulson
parents:
diff
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|
226 |
vimage_def, finite_psubset_def]) 1); |
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paulson
parents:
diff
changeset
|
227 |
by (Clarify_tac 1); |
ea13ff5a26d1
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paulson
parents:
diff
changeset
|
228 |
by (case_tac "m={}" 1); |
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paulson
parents:
diff
changeset
|
229 |
by (rtac (Int_lower2 RS leadsTo_weaken_L') 1); |
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paulson
parents:
diff
changeset
|
230 |
by (force_tac (claset(), simpset() addsimps [leadsTo_refl]) 1); |
ea13ff5a26d1
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paulson
parents:
diff
changeset
|
231 |
by (res_inst_tac [("A'", "Acyclic Int {x. above i x < m}")] |
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paulson
parents:
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|
232 |
leadsTo_weaken_R 1); |
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paulson
parents:
diff
changeset
|
233 |
by (REPEAT(blast_tac (claset() addIs [above_decreases_psp']) 1)); |
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paulson
parents:
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changeset
|
234 |
qed "Progress"; |
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paulson
parents:
diff
changeset
|
235 |
|
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paulson
parents:
diff
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|
236 |
(* We have proved all (relevant) theorems given in the paper *) |
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|
237 |
(* We didn't assume any thing about the relation r *) |
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|
238 |
(* It is not necessary that r be a priority relation as assumed in the original proof *) |
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|
239 |
(* It suffices that we start from a state which is finite and acyclic *) |