author | paulson |
Mon, 18 Oct 1999 15:18:24 +0200 | |
changeset 7878 | 43b03d412b82 |
parent 7689 | affe0c2fdfbf |
child 7947 | b999c1ab9327 |
permissions | -rw-r--r-- |
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(* Title: HOL/UNITY/Guar.ML |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1999 University of Cambridge |
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|
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Guarantees, etc. |
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|
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From Chandy and Sanders, "Reasoning About Program Composition" |
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*) |
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|
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(*** existential properties ***) |
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|
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Goalw [ex_prop_def] |
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"[| ex_prop X; finite GG |] ==> GG Int X ~= {} --> (JN G:GG. G) : X"; |
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by (etac finite_induct 1); |
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by (auto_tac (claset(), simpset() addsimps [Int_insert_left])); |
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qed_spec_mp "ex1"; |
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|
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Goalw [ex_prop_def] |
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"ALL GG. finite GG & GG Int X ~= {} --> (JN G:GG. G) : X ==> ex_prop X"; |
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by (Clarify_tac 1); |
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by (dres_inst_tac [("x", "{F,G}")] spec 1); |
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by Auto_tac; |
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qed "ex2"; |
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|
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(*Chandy & Sanders take this as a definition*) |
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Goal "ex_prop X = (ALL GG. finite GG & GG Int X ~= {} --> (JN G:GG. G) : X)"; |
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by (blast_tac (claset() addIs [ex1,ex2]) 1); |
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qed "ex_prop_finite"; |
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|
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(*Their "equivalent definition" given at the end of section 3*) |
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Goal "ex_prop X = (ALL G. G:X = (ALL H. G <= H --> H: X))"; |
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by Auto_tac; |
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by (rewrite_goals_tac [ex_prop_def, component_def]); |
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by (Blast_tac 1); |
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by Safe_tac; |
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by (stac Join_commute 2); |
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by (ALLGOALS Blast_tac); |
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qed "ex_prop_equiv"; |
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|
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|
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(*** universal properties ***) |
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|
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Goalw [uv_prop_def] |
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"[| uv_prop X; finite GG |] ==> GG <= X --> (JN G:GG. G) : X"; |
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by (etac finite_induct 1); |
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by (auto_tac (claset(), simpset() addsimps [Int_insert_left])); |
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qed_spec_mp "uv1"; |
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|
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Goalw [uv_prop_def] |
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"ALL GG. finite GG & GG <= X --> (JN G:GG. G) : X ==> uv_prop X"; |
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by (rtac conjI 1); |
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53 |
by (Clarify_tac 2); |
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by (dres_inst_tac [("x", "{F,G}")] spec 2); |
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by (dres_inst_tac [("x", "{}")] spec 1); |
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by Auto_tac; |
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qed "uv2"; |
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|
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(*Chandy & Sanders take this as a definition*) |
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Goal "uv_prop X = (ALL GG. finite GG & GG <= X --> (JN G:GG. G) : X)"; |
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by (blast_tac (claset() addIs [uv1,uv2]) 1); |
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qed "uv_prop_finite"; |
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|
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(*** guarantees ***) |
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|
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val prems = Goal |
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"(!!G. [| F Join G : X; Disjoint UNIV F G |] ==> F Join G : Y) \ |
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\ ==> F : X guarantees Y"; |
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by (simp_tac (simpset() addsimps [guar_def, component_eq]) 1); |
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by (blast_tac (claset() addIs prems) 1); |
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qed "guaranteesI"; |
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|
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Goalw [guar_def, component_def] |
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"[| F : X guarantees Y; F Join G : X |] ==> F Join G : Y"; |
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by (Blast_tac 1); |
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qed "guaranteesD"; |
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|
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(*This version of guaranteesD matches more easily in the conclusion*) |
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Goalw [guar_def] |
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"[| F : X guarantees Y; H : X; F <= H |] ==> H : Y"; |
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by (Blast_tac 1); |
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qed "component_guaranteesD"; |
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|
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(*This equation is more intuitive than the official definition*) |
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Goal "(F : X guarantees Y) = \ |
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\ (ALL G. F Join G : X & Disjoint UNIV F G --> F Join G : Y)"; |
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by (simp_tac (simpset() addsimps [guar_def, component_eq]) 1); |
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by (Blast_tac 1); |
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qed "guarantees_eq"; |
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|
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Goalw [guar_def] |
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"[| F: X guarantees X'; Y <= X; X' <= Y' |] ==> F: Y guarantees Y'"; |
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94 |
by (Blast_tac 1); |
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qed "guarantees_weaken"; |
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|
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Goalw [guar_def] |
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"[| F: X guarantees Y; F <= F' |] ==> F': X guarantees Y"; |
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99 |
by (blast_tac (claset() addIs [component_trans]) 1); |
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qed "guarantees_weaken_prog"; |
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101 |
|
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Goalw [guar_def] "X <= Y ==> X guarantees Y = UNIV"; |
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103 |
by (Blast_tac 1); |
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qed "subset_imp_guarantees_UNIV"; |
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|
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(*Equivalent to subset_imp_guarantees_UNIV but more intuitive*) |
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Goalw [guar_def] "X <= Y ==> F : X guarantees Y"; |
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108 |
by (Blast_tac 1); |
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109 |
qed "subset_imp_guarantees"; |
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110 |
|
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111 |
(*Remark at end of section 4.1*) |
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Goalw [guar_def] "ex_prop Y = (Y = UNIV guarantees Y)"; |
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113 |
by (simp_tac (simpset() addsimps [ex_prop_equiv]) 1); |
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114 |
by (blast_tac (claset() addEs [equalityE]) 1); |
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115 |
qed "ex_prop_equiv2"; |
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116 |
|
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117 |
(** Distributive laws. Re-orient to perform miniscoping **) |
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118 |
|
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119 |
Goalw [guar_def] |
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120 |
"(UN i:I. X i) guarantees Y = (INT i:I. X i guarantees Y)"; |
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121 |
by (Blast_tac 1); |
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122 |
qed "guarantees_UN_left"; |
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123 |
|
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124 |
Goalw [guar_def] |
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125 |
"(X Un Y) guarantees Z = (X guarantees Z) Int (Y guarantees Z)"; |
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126 |
by (Blast_tac 1); |
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127 |
qed "guarantees_Un_left"; |
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128 |
|
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129 |
Goalw [guar_def] |
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130 |
"X guarantees (INT i:I. Y i) = (INT i:I. X guarantees Y i)"; |
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131 |
by (Blast_tac 1); |
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132 |
qed "guarantees_INT_right"; |
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133 |
|
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134 |
Goalw [guar_def] |
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135 |
"Z guarantees (X Int Y) = (Z guarantees X) Int (Z guarantees Y)"; |
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136 |
by (Blast_tac 1); |
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137 |
qed "guarantees_Int_right"; |
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138 |
|
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139 |
Goalw [guar_def] "(X guarantees Y) = (UNIV guarantees (-X Un Y))"; |
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140 |
by (Blast_tac 1); |
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141 |
qed "shunting"; |
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142 |
|
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143 |
Goalw [guar_def] "(X guarantees Y) = -Y guarantees -X"; |
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144 |
by (Blast_tac 1); |
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145 |
qed "contrapositive"; |
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146 |
|
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147 |
(** The following two can be expressed using intersection and subset, which |
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148 |
is more faithful to the text but looks cryptic. |
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149 |
**) |
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150 |
|
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151 |
Goalw [guar_def] |
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152 |
"[| F : V guarantees X; F : (X Int Y) guarantees Z |]\ |
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153 |
\ ==> F : (V Int Y) guarantees Z"; |
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154 |
by (Blast_tac 1); |
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155 |
qed "combining1"; |
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156 |
|
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157 |
Goalw [guar_def] |
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158 |
"[| F : V guarantees (X Un Y); F : Y guarantees Z |]\ |
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159 |
\ ==> F : V guarantees (X Un Z)"; |
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160 |
by (Blast_tac 1); |
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161 |
qed "combining2"; |
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162 |
|
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163 |
(** The following two follow Chandy-Sanders, but the use of object-quantifiers |
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164 |
does not suit Isabelle... **) |
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165 |
|
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166 |
(*Premise should be (!!i. i: I ==> F: X guarantees Y i) *) |
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167 |
Goalw [guar_def] |
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168 |
"ALL i:I. F : X guarantees (Y i) ==> F : X guarantees (INT i:I. Y i)"; |
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169 |
by (Blast_tac 1); |
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170 |
qed "all_guarantees"; |
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171 |
|
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172 |
(*Premises should be [| F: X guarantees Y i; i: I |] *) |
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173 |
Goalw [guar_def] |
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174 |
"EX i:I. F : X guarantees (Y i) ==> F : X guarantees (UN i:I. Y i)"; |
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175 |
by (Blast_tac 1); |
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176 |
qed "ex_guarantees"; |
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177 |
|
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178 |
(*** Additional guarantees laws, by lcp ***) |
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179 |
|
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180 |
Goalw [guar_def] |
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181 |
"[| F: U guarantees V; G: X guarantees Y |] \ |
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182 |
\ ==> F Join G: (U Int X) guarantees (V Int Y)"; |
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183 |
by (simp_tac (simpset() addsimps [Join_component_iff]) 1); |
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184 |
by (Blast_tac 1); |
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185 |
qed "guarantees_Join_Int"; |
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186 |
|
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187 |
Goalw [guar_def] |
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188 |
"[| F: U guarantees V; G: X guarantees Y |] \ |
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189 |
\ ==> F Join G: (U Un X) guarantees (V Un Y)"; |
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190 |
by (simp_tac (simpset() addsimps [Join_component_iff]) 1); |
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191 |
by (Blast_tac 1); |
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192 |
qed "guarantees_Join_Un"; |
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193 |
|
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194 |
Goal "((JOIN I F) <= H) = (ALL i: I. F i <= H)"; |
7537 | 195 |
by (simp_tac (simpset() addsimps [component_eq_subset]) 1); |
7400
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196 |
by (Blast_tac 1); |
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197 |
qed "JN_component_iff"; |
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198 |
|
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199 |
Goalw [guar_def] |
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200 |
"[| ALL i:I. F i : X i guarantees Y i |] \ |
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201 |
\ ==> (JOIN I F) : (INTER I X) guarantees (INTER I Y)"; |
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202 |
by (simp_tac (simpset() addsimps [JN_component_iff]) 1); |
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203 |
by (Blast_tac 1); |
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204 |
bind_thm ("guarantees_JN_INT", ballI RS result()); |
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205 |
|
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206 |
Goalw [guar_def] |
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207 |
"[| ALL i:I. F i : X i guarantees Y i |] \ |
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208 |
\ ==> (JOIN I F) : (UNION I X) guarantees (UNION I Y)"; |
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209 |
by (simp_tac (simpset() addsimps [JN_component_iff]) 1); |
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210 |
by (Blast_tac 1); |
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211 |
bind_thm ("guarantees_JN_UN", ballI RS result()); |
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212 |
|
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213 |
|
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214 |
(*** guarantees laws for breaking down the program, by lcp ***) |
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215 |
|
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216 |
Goalw [guar_def] |
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217 |
"F: X guarantees Y | G: X guarantees Y ==> F Join G: X guarantees Y"; |
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218 |
by (simp_tac (simpset() addsimps [Join_component_iff]) 1); |
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219 |
by (Blast_tac 1); |
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220 |
qed "guarantees_Join_I"; |
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221 |
|
7689 | 222 |
bind_thm ("guarantees_Join_I1", disjI1 RS guarantees_Join_I); |
223 |
bind_thm ("guarantees_Join_I2", disjI2 RS guarantees_Join_I); |
|
224 |
||
7400
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225 |
Goalw [guar_def] |
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226 |
"[| i : I; F i: X guarantees Y |] ==> (JN i:I. (F i)) : X guarantees Y"; |
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227 |
by (simp_tac (simpset() addsimps [JN_component_iff]) 1); |
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228 |
by (Blast_tac 1); |
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229 |
qed "guarantees_JN_I"; |
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230 |
|
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231 |
|
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232 |
(*** well-definedness ***) |
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233 |
|
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234 |
Goalw [welldef_def] "F Join G: welldef ==> F: welldef"; |
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235 |
by Auto_tac; |
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qed "Join_welldef_D1"; |
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237 |
|
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Goalw [welldef_def] "F Join G: welldef ==> G: welldef"; |
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239 |
by Auto_tac; |
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240 |
qed "Join_welldef_D2"; |
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241 |
|
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242 |
(*** refinement ***) |
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243 |
|
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244 |
Goalw [refines_def] "F refines F wrt X"; |
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245 |
by (Blast_tac 1); |
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246 |
qed "refines_refl"; |
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247 |
|
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248 |
Goalw [refines_def] |
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"[| H refines G wrt X; G refines F wrt X |] ==> H refines F wrt X"; |
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250 |
by Auto_tac; |
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251 |
qed "refines_trans"; |
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252 |
|
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253 |
Goalw [strict_ex_prop_def] |
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254 |
"strict_ex_prop X \ |
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\ ==> (ALL H. F Join H : X --> G Join H : X) = (F:X --> G:X)"; |
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256 |
by (Blast_tac 1); |
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257 |
qed "strict_ex_refine_lemma"; |
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258 |
|
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259 |
Goalw [strict_ex_prop_def] |
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260 |
"strict_ex_prop X \ |
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\ ==> (ALL H. F Join H : welldef & F Join H : X --> G Join H : X) = \ |
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\ (F: welldef Int X --> G:X)"; |
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263 |
by Safe_tac; |
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|
264 |
by (eres_inst_tac [("x","SKIP"), ("P", "%H. ?PP H --> ?RR H")] allE 1); |
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265 |
by (auto_tac (claset() addDs [Join_welldef_D1, Join_welldef_D2], simpset())); |
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266 |
qed "strict_ex_refine_lemma_v"; |
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|
267 |
|
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268 |
Goal "[| strict_ex_prop X; \ |
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|
269 |
\ ALL H. F Join H : welldef Int X --> G Join H : welldef |] \ |
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\ ==> (G refines F wrt X) = (G iso_refines F wrt X)"; |
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271 |
by (res_inst_tac [("x","SKIP")] allE 1 |
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272 |
THEN assume_tac 1); |
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|
273 |
by (asm_full_simp_tac |
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274 |
(simpset() addsimps [refines_def, iso_refines_def, |
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|
275 |
strict_ex_refine_lemma_v]) 1); |
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276 |
qed "ex_refinement_thm"; |
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|
277 |
|
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|
278 |
|
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279 |
Goalw [strict_uv_prop_def] |
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|
280 |
"strict_uv_prop X \ |
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|
281 |
\ ==> (ALL H. F Join H : X --> G Join H : X) = (F:X --> G:X)"; |
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|
282 |
by (Blast_tac 1); |
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|
283 |
qed "strict_uv_refine_lemma"; |
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|
284 |
|
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|
285 |
Goalw [strict_uv_prop_def] |
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|
286 |
"strict_uv_prop X \ |
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|
287 |
\ ==> (ALL H. F Join H : welldef & F Join H : X --> G Join H : X) = \ |
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|
288 |
\ (F: welldef Int X --> G:X)"; |
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|
289 |
by Safe_tac; |
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|
290 |
by (eres_inst_tac [("x","SKIP"), ("P", "%H. ?PP H --> ?RR H")] allE 1); |
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|
291 |
by (auto_tac (claset() addDs [Join_welldef_D1, Join_welldef_D2], |
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|
292 |
simpset())); |
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|
293 |
qed "strict_uv_refine_lemma_v"; |
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|
294 |
|
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|
295 |
Goal "[| strict_uv_prop X; \ |
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|
296 |
\ ALL H. F Join H : welldef Int X --> G Join H : welldef |] \ |
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|
297 |
\ ==> (G refines F wrt X) = (G iso_refines F wrt X)"; |
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|
298 |
by (res_inst_tac [("x","SKIP")] allE 1 |
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|
299 |
THEN assume_tac 1); |
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|
300 |
by (asm_full_simp_tac (simpset() addsimps [refines_def, iso_refines_def, |
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|
301 |
strict_uv_refine_lemma_v]) 1); |
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|
302 |
qed "uv_refinement_thm"; |