author  paulson 
Fri, 22 Oct 1999 18:26:46 +0200  
changeset 7913  86be2946bb0b 
parent 7822  09aabe6d04b8 
child 8004  6273f58ea2c1 
permissions  rwrr 
1465  1 
(* Title: Relation.ML 
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ID: $Id$ 
1985
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3 
Authors: Lawrence C Paulson, Cambridge University Computer Laboratory 
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Copyright 1996 University of Cambridge 
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*) 
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(** Identity relation **) 
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5608  9 
Goalw [Id_def] "(a,a) : Id"; 
2891  10 
by (Blast_tac 1); 
5608  11 
qed "IdI"; 
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5608  13 
val major::prems = Goalw [Id_def] 
14 
"[ p: Id; !!x.[ p = (x,x) ] ==> P \ 

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\ ] ==> P"; 
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16 
by (rtac (major RS CollectE) 1); 
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17 
by (etac exE 1); 
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by (eresolve_tac prems 1); 
5608  19 
qed "IdE"; 
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20 

5608  21 
Goalw [Id_def] "(a,b):Id = (a=b)"; 
2891  22 
by (Blast_tac 1); 
5608  23 
qed "pair_in_Id_conv"; 
24 
Addsimps [pair_in_Id_conv]; 

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Goalw [refl_def] "reflexive Id"; 
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by Auto_tac; 
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28 
qed "reflexive_Id"; 
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29 

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(*A strange result, since Id is also symmetric.*) 
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Goalw [antisym_def] "antisym Id"; 
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by Auto_tac; 
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qed "antisym_Id"; 
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34 

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Goalw [trans_def] "trans Id"; 
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by Auto_tac; 
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qed "trans_Id"; 
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(** Diagonal relation: indentity restricted to some set **) 
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41 

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(*** Equality : the diagonal relation ***) 
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43 

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44 
Goalw [diag_def] "[ a=b; a:A ] ==> (a,b) : diag(A)"; 
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by (Blast_tac 1); 
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qed "diag_eqI"; 
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47 

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val diagI = refl RS diag_eqI > standard; 
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49 

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50 
(*The general elimination rule*) 
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val major::prems = Goalw [diag_def] 
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"[ c : diag(A); \ 
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\ !!x y. [ x:A; c = (x,x) ] ==> P \ 
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\ ] ==> P"; 
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55 
by (rtac (major RS UN_E) 1); 
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56 
by (REPEAT (eresolve_tac [asm_rl,singletonE] 1 ORELSE resolve_tac prems 1)); 
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57 
qed "diagE"; 
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58 

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59 
AddSIs [diagI]; 
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60 
AddSEs [diagE]; 
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61 

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Goal "((x,y) : diag A) = (x=y & x : A)"; 
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by (Blast_tac 1); 
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64 
qed "diag_iff"; 
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65 

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66 
Goal "diag(A) <= A Times A"; 
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by (Blast_tac 1); 
5995  68 
qed "diag_subset_Times"; 
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(** Composition of two relations **) 
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5069  74 
Goalw [comp_def] 
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"[ (a,b):s; (b,c):r ] ==> (a,c) : r O s"; 
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by (Blast_tac 1); 
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qed "compI"; 
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78 

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(*proof requires higherlevel assumptions or a delaying of hyp_subst_tac*) 
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val prems = Goalw [comp_def] 
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"[ xz : r O s; \ 
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82 
\ !!x y z. [ xz = (x,z); (x,y):s; (y,z):r ] ==> P \ 
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83 
\ ] ==> P"; 
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84 
by (cut_facts_tac prems 1); 
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85 
by (REPEAT (eresolve_tac [CollectE, splitE, exE, conjE] 1 
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86 
ORELSE ares_tac prems 1)); 
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qed "compE"; 
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5316  89 
val prems = Goal 
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"[ (a,c) : r O s; \ 
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91 
\ !!y. [ (a,y):s; (y,c):r ] ==> P \ 
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92 
\ ] ==> P"; 
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93 
by (rtac compE 1); 
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94 
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Pair_inject,ssubst] 1)); 
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95 
qed "compEpair"; 
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96 

5608  97 
AddIs [compI, IdI]; 
98 
AddSEs [compE, IdE]; 

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99 

5608  100 
Goal "R O Id = R"; 
4673  101 
by (Fast_tac 1); 
5608  102 
qed "R_O_Id"; 
4673  103 

5608  104 
Goal "Id O R = R"; 
4673  105 
by (Fast_tac 1); 
5608  106 
qed "Id_O_R"; 
4673  107 

5608  108 
Addsimps [R_O_Id,Id_O_R]; 
4673  109 

5069  110 
Goal "(R O S) O T = R O (S O T)"; 
4830  111 
by (Blast_tac 1); 
112 
qed "O_assoc"; 

113 

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114 
Goal "[ r'<=r; s'<=s ] ==> (r' O s') <= (r O s)"; 
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by (Blast_tac 1); 
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qed "comp_mono"; 
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117 

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Goal "[ s <= A Times B; r <= B Times C ] ==> (r O s) <= A Times C"; 
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by (Blast_tac 1); 
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qed "comp_subset_Sigma"; 
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121 

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(** Natural deduction for refl(r) **) 
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123 

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val prems = Goalw [refl_def] 
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125 
"[ r <= A Times A; !! x. x:A ==> (x,x):r ] ==> refl A r"; 
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126 
by (REPEAT (ares_tac (prems@[ballI,conjI]) 1)); 
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127 
qed "reflI"; 
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128 

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129 
Goalw [refl_def] "[ refl A r; a:A ] ==> (a,a):r"; 
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by (Blast_tac 1); 
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131 
qed "reflD"; 
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132 

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133 
(** Natural deduction for antisym(r) **) 
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134 

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val prems = Goalw [antisym_def] 
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136 
"(!! x y. [ (x,y):r; (y,x):r ] ==> x=y) ==> antisym(r)"; 
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by (REPEAT (ares_tac (prems@[allI,impI]) 1)); 
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138 
qed "antisymI"; 
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139 

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140 
Goalw [antisym_def] "[ antisym(r); (a,b):r; (b,a):r ] ==> a=b"; 
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141 
by (Blast_tac 1); 
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142 
qed "antisymD"; 
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143 

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(** Natural deduction for trans(r) **) 
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145 

5316  146 
val prems = Goalw [trans_def] 
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147 
"(!! x y z. [ (x,y):r; (y,z):r ] ==> (x,z):r) ==> trans(r)"; 
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148 
by (REPEAT (ares_tac (prems@[allI,impI]) 1)); 
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149 
qed "transI"; 
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150 

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151 
Goalw [trans_def] "[ trans(r); (a,b):r; (b,c):r ] ==> (a,c):r"; 
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by (Blast_tac 1); 
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qed "transD"; 
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154 

3439  155 
(** Natural deduction for r^1 **) 
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156 

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157 
Goalw [converse_def] "((a,b): r^1) = ((b,a):r)"; 
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158 
by (Simp_tac 1); 
4746  159 
qed "converse_iff"; 
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160 

4746  161 
AddIffs [converse_iff]; 
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162 

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163 
Goalw [converse_def] "(a,b):r ==> (b,a): r^1"; 
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164 
by (Simp_tac 1); 
4746  165 
qed "converseI"; 
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166 

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167 
Goalw [converse_def] "(a,b) : r^1 ==> (b,a) : r"; 
2891  168 
by (Blast_tac 1); 
4746  169 
qed "converseD"; 
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170 

4746  171 
(*More general than converseD, as it "splits" the member of the relation*) 
7031  172 

173 
val [major,minor] = Goalw [converse_def] 

3439  174 
"[ yx : r^1; \ 
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175 
\ !!x y. [ yx=(y,x); (x,y):r ] ==> P \ 
7031  176 
\ ] ==> P"; 
177 
by (rtac (major RS CollectE) 1); 

178 
by (REPEAT (eresolve_tac [splitE, bexE,exE, conjE, minor] 1)); 

179 
by (assume_tac 1); 

180 
qed "converseE"; 

4746  181 
AddSEs [converseE]; 
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182 

5069  183 
Goalw [converse_def] "(r^1)^1 = r"; 
2891  184 
by (Blast_tac 1); 
4746  185 
qed "converse_converse"; 
186 
Addsimps [converse_converse]; 

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187 

5069  188 
Goal "(r O s)^1 = s^1 O r^1"; 
4423  189 
by (Blast_tac 1); 
4746  190 
qed "converse_comp"; 
1605  191 

5608  192 
Goal "Id^1 = Id"; 
4644  193 
by (Blast_tac 1); 
5608  194 
qed "converse_Id"; 
195 
Addsimps [converse_Id]; 

4644  196 

5995  197 
Goal "(diag A) ^1 = diag A"; 
198 
by (Blast_tac 1); 

199 
qed "converse_diag"; 

200 
Addsimps [converse_diag]; 

201 

7083  202 
Goalw [refl_def] "refl A r ==> refl A (converse r)"; 
203 
by (Blast_tac 1); 

204 
qed "refl_converse"; 

205 

206 
Goalw [antisym_def] "antisym (converse r) = antisym r"; 

207 
by (Blast_tac 1); 

208 
qed "antisym_converse"; 

209 

210 
Goalw [trans_def] "trans (converse r) = trans r"; 

211 
by (Blast_tac 1); 

212 
qed "trans_converse"; 

213 

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214 
(** Domain **) 
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215 

5811  216 
Goalw [Domain_def] "a: Domain(r) = (EX y. (a,y): r)"; 
217 
by (Blast_tac 1); 

218 
qed "Domain_iff"; 

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219 

7007  220 
Goal "(a,b): r ==> a: Domain(r)"; 
221 
by (etac (exI RS (Domain_iff RS iffD2)) 1) ; 

222 
qed "DomainI"; 

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223 

7007  224 
val prems= Goal "[ a : Domain(r); !!y. (a,y): r ==> P ] ==> P"; 
225 
by (rtac (Domain_iff RS iffD1 RS exE) 1); 

226 
by (REPEAT (ares_tac prems 1)) ; 

227 
qed "DomainE"; 

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228 

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229 
AddIs [DomainI]; 
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230 
AddSEs [DomainE]; 
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231 

5608  232 
Goal "Domain Id = UNIV"; 
4644  233 
by (Blast_tac 1); 
5608  234 
qed "Domain_Id"; 
235 
Addsimps [Domain_Id]; 

4644  236 

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237 
Goal "Domain (diag A) = A"; 
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238 
by Auto_tac; 
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239 
qed "Domain_diag"; 
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240 
Addsimps [Domain_diag]; 
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241 

5811  242 
Goal "Domain(A Un B) = Domain(A) Un Domain(B)"; 
243 
by (Blast_tac 1); 

244 
qed "Domain_Un_eq"; 

245 

246 
Goal "Domain(A Int B) <= Domain(A) Int Domain(B)"; 

247 
by (Blast_tac 1); 

248 
qed "Domain_Int_subset"; 

249 

250 
Goal "Domain(A)  Domain(B) <= Domain(A  B)"; 

251 
by (Blast_tac 1); 

252 
qed "Domain_Diff_subset"; 

253 

6005  254 
Goal "Domain (Union S) = (UN A:S. Domain A)"; 
255 
by (Blast_tac 1); 

256 
qed "Domain_Union"; 

257 

7822  258 
Goal "r <= s ==> Domain r <= Domain s"; 
259 
by (Blast_tac 1); 

260 
qed "Domain_mono"; 

261 

5811  262 

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263 
(** Range **) 
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264 

5811  265 
Goalw [Domain_def, Range_def] "a: Range(r) = (EX y. (y,a): r)"; 
266 
by (Blast_tac 1); 

267 
qed "Range_iff"; 

268 

7031  269 
Goalw [Range_def] "(a,b): r ==> b : Range(r)"; 
270 
by (etac (converseI RS DomainI) 1); 

271 
qed "RangeI"; 

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272 

7031  273 
val major::prems = Goalw [Range_def] 
274 
"[ b : Range(r); !!x. (x,b): r ==> P ] ==> P"; 

275 
by (rtac (major RS DomainE) 1); 

276 
by (resolve_tac prems 1); 

277 
by (etac converseD 1) ; 

278 
qed "RangeE"; 

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279 

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280 
AddIs [RangeI]; 
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281 
AddSEs [RangeE]; 
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282 

5608  283 
Goal "Range Id = UNIV"; 
4644  284 
by (Blast_tac 1); 
5608  285 
qed "Range_Id"; 
286 
Addsimps [Range_Id]; 

4644  287 

5995  288 
Goal "Range (diag A) = A"; 
289 
by Auto_tac; 

290 
qed "Range_diag"; 

291 
Addsimps [Range_diag]; 

292 

5811  293 
Goal "Range(A Un B) = Range(A) Un Range(B)"; 
294 
by (Blast_tac 1); 

295 
qed "Range_Un_eq"; 

296 

297 
Goal "Range(A Int B) <= Range(A) Int Range(B)"; 

298 
by (Blast_tac 1); 

299 
qed "Range_Int_subset"; 

300 

301 
Goal "Range(A)  Range(B) <= Range(A  B)"; 

302 
by (Blast_tac 1); 

303 
qed "Range_Diff_subset"; 

304 

6005  305 
Goal "Range (Union S) = (UN A:S. Range A)"; 
306 
by (Blast_tac 1); 

307 
qed "Range_Union"; 

308 

5811  309 

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310 
(*** Image of a set under a relation ***) 
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311 

5649  312 
overload_1st_set "Relation.op ^^"; 
5335  313 

7031  314 
Goalw [Image_def] "b : r^^A = (? x:A. (x,b):r)"; 
315 
by (Blast_tac 1); 

316 
qed "Image_iff"; 

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317 

7031  318 
Goalw [Image_def] "r^^{a} = {b. (a,b):r}"; 
319 
by (Blast_tac 1); 

320 
qed "Image_singleton"; 

4673  321 

7031  322 
Goal "(b : r^^{a}) = ((a,b):r)"; 
7007  323 
by (rtac (Image_iff RS trans) 1); 
324 
by (Blast_tac 1); 

325 
qed "Image_singleton_iff"; 

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326 

4673  327 
AddIffs [Image_singleton_iff]; 
328 

7007  329 
Goalw [Image_def] "[ (a,b): r; a:A ] ==> b : r^^A"; 
330 
by (Blast_tac 1); 

331 
qed "ImageI"; 

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332 

7031  333 
val major::prems = Goalw [Image_def] 
334 
"[ b: r^^A; !!x.[ (x,b): r; x:A ] ==> P ] ==> P"; 

335 
by (rtac (major RS CollectE) 1); 

336 
by (Clarify_tac 1); 

337 
by (rtac (hd prems) 1); 

338 
by (REPEAT (etac bexE 1 ORELSE ares_tac prems 1)) ; 

339 
qed "ImageE"; 

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340 

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341 
AddIs [ImageI]; 
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342 
AddSEs [ImageE]; 
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343 

4593  344 

7031  345 
Goal "R^^{} = {}"; 
7007  346 
by (Blast_tac 1); 
347 
qed "Image_empty"; 

4593  348 

349 
Addsimps [Image_empty]; 

350 

5608  351 
Goal "Id ^^ A = A"; 
4601  352 
by (Blast_tac 1); 
5608  353 
qed "Image_Id"; 
4601  354 

5998  355 
Goal "diag A ^^ B = A Int B"; 
5995  356 
by (Blast_tac 1); 
357 
qed "Image_diag"; 

358 

359 
Addsimps [Image_Id, Image_diag]; 

4601  360 

7007  361 
Goal "R ^^ (A Int B) <= R ^^ A Int R ^^ B"; 
362 
by (Blast_tac 1); 

363 
qed "Image_Int_subset"; 

4593  364 

7007  365 
Goal "R ^^ (A Un B) = R ^^ A Un R ^^ B"; 
366 
by (Blast_tac 1); 

367 
qed "Image_Un"; 

4593  368 

7007  369 
Goal "r <= A Times B ==> r^^C <= B"; 
370 
by (rtac subsetI 1); 

371 
by (REPEAT (eresolve_tac [asm_rl, ImageE, subsetD RS SigmaD2] 1)) ; 

372 
qed "Image_subset"; 

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373 

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374 
(*NOT suitable for rewriting*) 
5069  375 
Goal "r^^B = (UN y: B. r^^{y})"; 
4673  376 
by (Blast_tac 1); 
4733
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377 
qed "Image_eq_UN"; 
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378 

7913  379 
Goal "[ r'<=r; A'<=A ] ==> (r' ^^ A') <= (r ^^ A)"; 
380 
by (Blast_tac 1); 

381 
qed "Image_mono"; 

382 

383 
Goal "(r ^^ (UNION A B)) = (UN x:A.(r ^^ (B x)))"; 

384 
by (Blast_tac 1); 

385 
qed "Image_UN"; 

386 

387 
(*Converse inclusion fails*) 

388 
Goal "(r ^^ (INTER A B)) <= (INT x:A.(r ^^ (B x)))"; 

389 
by (Blast_tac 1); 

390 
qed "Image_INT_subset"; 

391 

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392 

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393 
section "Univalent"; 
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diff
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394 

7031  395 
Goalw [Univalent_def] 
396 
"!x y. (x,y):r > (!z. (x,z):r > y=z) ==> Univalent r"; 

397 
by (assume_tac 1); 

398 
qed "UnivalentI"; 

4760
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changeset

399 

7031  400 
Goalw [Univalent_def] 
401 
"[ Univalent r; (x,y):r; (x,z):r] ==> y=z"; 

402 
by Auto_tac; 

403 
qed "UnivalentD"; 

5231  404 

405 

406 
(** Graphs of partial functions **) 

407 

408 
Goal "Domain{(x,y). y = f x & P x} = {x. P x}"; 

409 
by (Blast_tac 1); 

410 
qed "Domain_partial_func"; 

411 

412 
Goal "Range{(x,y). y = f x & P x} = f``{x. P x}"; 

413 
by (Blast_tac 1); 

414 
qed "Range_partial_func"; 

415 

7014
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Added some definitions and theorems needed for the
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7007
diff
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416 

11ee650edcd2
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diff
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417 
(** Composition of function and relation **) 
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418 

11ee650edcd2
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diff
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419 
Goalw [fun_rel_comp_def] "A <= B ==> fun_rel_comp f A <= fun_rel_comp f B"; 
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420 
by (Fast_tac 1); 
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421 
qed "fun_rel_comp_mono"; 
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422 

11ee650edcd2
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423 
Goalw [fun_rel_comp_def] "! x. ?! y. (f x, y) : R ==> ?! g. g : fun_rel_comp f R"; 
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424 
by (res_inst_tac [("a","%x. @y. (f x, y) : R")] ex1I 1); 
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425 
by (rtac CollectI 1); 
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426 
by (rtac allI 1); 
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diff
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427 
by (etac allE 1); 
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diff
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428 
by (rtac (select_eq_Ex RS iffD2) 1); 
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diff
changeset

429 
by (etac ex1_implies_ex 1); 
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7007
diff
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430 
by (rtac ext 1); 
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diff
changeset

431 
by (etac CollectE 1); 
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diff
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432 
by (REPEAT (etac allE 1)); 
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433 
by (rtac (select1_equality RS sym) 1); 
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diff
changeset

434 
by (atac 1); 
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diff
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435 
by (atac 1); 
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diff
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436 
qed "fun_rel_comp_unique"; 