author  paulson 
Wed, 06 Mar 1996 12:52:11 +0100  
changeset 1552  6f71b5d46700 
parent 1465  5d7a7e439cec 
child 1605  248e1e125ca0 
permissions  rwrr 
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(* Title: Relation.ML 
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ID: $Id$ 
1465  3 
Authors: Riccardo Mattolini, Dip. Sistemi e Informatica 
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Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1994 Universita' di Firenze 
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Copyright 1993 University of Cambridge 
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*) 
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val RSLIST = curry (op MRS); 
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open Relation; 
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(** Identity relation **) 
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goalw Relation.thy [id_def] "(a,a) : id"; 
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by (rtac CollectI 1); 
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by (rtac exI 1); 
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by (rtac refl 1); 
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qed "idI"; 
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val major::prems = goalw Relation.thy [id_def] 
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"[ p: id; !!x.[ p = (x,x) ] ==> P \ 
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\ ] ==> P"; 
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by (rtac (major RS CollectE) 1); 
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by (etac exE 1); 
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by (eresolve_tac prems 1); 
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qed "idE"; 
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goalw Relation.thy [id_def] "(a,b):id = (a=b)"; 
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by (fast_tac prod_cs 1); 
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qed "pair_in_id_conv"; 
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(** Composition of two relations **) 
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val prems = goalw Relation.thy [comp_def] 
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"[ (a,b):s; (b,c):r ] ==> (a,c) : r O s"; 
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by (fast_tac (prod_cs addIs prems) 1); 
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qed "compI"; 
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(*proof requires higherlevel assumptions or a delaying of hyp_subst_tac*) 
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val prems = goalw Relation.thy [comp_def] 
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"[ xz : r O s; \ 
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\ !!x y z. [ xz = (x,z); (x,y):s; (y,z):r ] ==> P \ 
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\ ] ==> P"; 
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by (cut_facts_tac prems 1); 
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by (REPEAT (eresolve_tac [CollectE, splitE, exE, conjE] 1 ORELSE ares_tac prems 1)); 
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qed "compE"; 
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val prems = goal Relation.thy 
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"[ (a,c) : r O s; \ 
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\ !!y. [ (a,y):s; (y,c):r ] ==> P \ 
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\ ] ==> P"; 
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by (rtac compE 1); 
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by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Pair_inject,ssubst] 1)); 
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qed "compEpair"; 
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val comp_cs = prod_cs addIs [compI, idI] addSEs [compE, idE]; 
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goal Relation.thy "!!r s. [ r'<=r; s'<=s ] ==> (r' O s') <= (r O s)"; 
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by (fast_tac comp_cs 1); 
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qed "comp_mono"; 
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goal Relation.thy 
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"!!r s. [ s <= Sigma A (%x.B); r <= Sigma B (%x.C) ] ==> \ 
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\ (r O s) <= Sigma A (%x.C)"; 
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by (fast_tac comp_cs 1); 
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qed "comp_subset_Sigma"; 
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(** Natural deduction for trans(r) **) 
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val prems = goalw Relation.thy [trans_def] 
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"(!! x y z. [ (x,y):r; (y,z):r ] ==> (x,z):r) ==> trans(r)"; 
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by (REPEAT (ares_tac (prems@[allI,impI]) 1)); 
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qed "transI"; 
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val major::prems = goalw Relation.thy [trans_def] 
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"[ trans(r); (a,b):r; (b,c):r ] ==> (a,c):r"; 
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by (cut_facts_tac [major] 1); 
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by (fast_tac (HOL_cs addIs prems) 1); 
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qed "transD"; 
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(** Natural deduction for converse(r) **) 
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goalw Relation.thy [converse_def] "!!a b r. (a,b):r ==> (b,a):converse(r)"; 
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by (Simp_tac 1); 
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qed "converseI"; 
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goalw Relation.thy [converse_def] "!!a b r. (a,b) : converse(r) ==> (b,a) : r"; 
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by (fast_tac comp_cs 1); 
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qed "converseD"; 
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qed_goalw "converseE" Relation.thy [converse_def] 
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"[ yx : converse(r); \ 
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\ !!x y. [ yx=(y,x); (x,y):r ] ==> P \ 
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\ ] ==> P" 
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(fn [major,minor]=> 
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[ (rtac (major RS CollectE) 1), 
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(REPEAT (eresolve_tac [splitE, bexE,exE, conjE, minor] 1)), 
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(assume_tac 1) ]); 
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val converse_cs = comp_cs addSIs [converseI] 
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addSEs [converseD,converseE]; 
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(** Domain **) 
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qed_goalw "Domain_iff" Relation.thy [Domain_def] 
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"a: Domain(r) = (EX y. (a,y): r)" 
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(fn _=> [ (fast_tac comp_cs 1) ]); 
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qed_goal "DomainI" Relation.thy "!!a b r. (a,b): r ==> a: Domain(r)" 
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(fn _ => [ (etac (exI RS (Domain_iff RS iffD2)) 1) ]); 
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qed_goal "DomainE" Relation.thy 
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"[ a : Domain(r); !!y. (a,y): r ==> P ] ==> P" 
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(fn prems=> 
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[ (rtac (Domain_iff RS iffD1 RS exE) 1), 
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(REPEAT (ares_tac prems 1)) ]); 
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(** Range **) 
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qed_goalw "RangeI" Relation.thy [Range_def] "!!a b r.(a,b): r ==> b : Range(r)" 
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(fn _ => [ (etac (converseI RS DomainI) 1) ]); 
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124 

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qed_goalw "RangeE" Relation.thy [Range_def] 
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"[ b : Range(r); !!x. (x,b): r ==> P ] ==> P" 
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(fn major::prems=> 
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[ (rtac (major RS DomainE) 1), 
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(resolve_tac prems 1), 
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(etac converseD 1) ]); 
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(*** Image of a set under a relation ***) 
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qed_goalw "Image_iff" Relation.thy [Image_def] 
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"b : r^^A = (? x:A. (x,b):r)" 
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136 
(fn _ => [ fast_tac (comp_cs addIs [RangeI]) 1 ]); 
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137 

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138 
qed_goal "Image_singleton_iff" Relation.thy 
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139 
"(b : r^^{a}) = ((a,b):r)" 
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140 
(fn _ => [ rtac (Image_iff RS trans) 1, 
1465  141 
fast_tac comp_cs 1 ]); 
1128
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142 

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143 
qed_goalw "ImageI" Relation.thy [Image_def] 
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144 
"!!a b r. [ (a,b): r; a:A ] ==> b : r^^A" 
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145 
(fn _ => [ (REPEAT (ares_tac [CollectI,RangeI,bexI] 1)), 
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146 
(resolve_tac [conjI ] 1), 
1465  147 
(rtac RangeI 1), 
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148 
(REPEAT (fast_tac set_cs 1))]); 
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149 

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150 
qed_goalw "ImageE" Relation.thy [Image_def] 
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151 
"[ b: r^^A; !!x.[ (x,b): r; x:A ] ==> P ] ==> P" 
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152 
(fn major::prems=> 
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153 
[ (rtac (major RS CollectE) 1), 
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154 
(safe_tac set_cs), 
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155 
(etac RangeE 1), 
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156 
(rtac (hd prems) 1), 
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157 
(REPEAT (etac bexE 1 ORELSE ares_tac prems 1)) ]); 
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158 

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159 
qed_goal "Image_subset" Relation.thy 
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160 
"!!A B r. r <= Sigma A (%x.B) ==> r^^C <= B" 
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161 
(fn _ => 
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162 
[ (rtac subsetI 1), 
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163 
(REPEAT (eresolve_tac [asm_rl, ImageE, subsetD RS SigmaD2] 1)) ]); 
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164 

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165 
val rel_cs = converse_cs addSIs [converseI] 
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166 
addIs [ImageI, DomainI, RangeI] 
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167 
addSEs [ImageE, DomainE, RangeE]; 
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168 

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169 
val rel_eq_cs = rel_cs addSIs [equalityI]; 
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170 

1264
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added local simpsets; removed IOA from 'make test'
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171 
Addsimps [pair_in_id_conv]; 