author | nipkow |
Sat, 27 Apr 1996 12:07:31 +0200 | |
changeset 1694 | 3452958f85a8 |
parent 1642 | 21db0cf9a1a4 |
child 1754 | 852093aeb0ab |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: Relation.ML |
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ID: $Id$ |
1465 | 3 |
Authors: Riccardo Mattolini, Dip. Sistemi e Informatica |
4 |
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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|
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val RSLIST = curry (op MRS); |
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|
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open Relation; |
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|
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(** Identity relation **) |
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|
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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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|
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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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|
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goalw Relation.thy [id_def] "(a,b):id = (a=b)"; |
1552 | 30 |
by (fast_tac prod_cs 1); |
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qed "pair_in_id_conv"; |
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Addsimps [pair_in_id_conv]; |
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|
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|
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(** Composition of two relations **) |
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|
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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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|
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(*proof requires higher-level 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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|
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val prems = goal Relation.thy |
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52 |
"[| (a,c) : r O s; \ |
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53 |
\ !!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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57 |
qed "compEpair"; |
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58 |
|
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val comp_cs = prod_cs addIs [compI, idI] addSEs [compE, idE]; |
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|
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goal Relation.thy "!!r s. [| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)"; |
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62 |
by (fast_tac comp_cs 1); |
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qed "comp_mono"; |
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64 |
|
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goal Relation.thy |
1642 | 66 |
"!!r s. [| s <= A Times B; r <= B Times C |] ==> \ |
67 |
\ (r O s) <= A Times C"; |
|
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by (fast_tac comp_cs 1); |
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qed "comp_subset_Sigma"; |
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|
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(** Natural deduction for trans(r) **) |
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|
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val prems = goalw Relation.thy [trans_def] |
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74 |
"(!! 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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|
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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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83 |
|
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84 |
(** Natural deduction for converse(r) **) |
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|
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goalw Relation.thy [converse_def] "!!a b r. (a,b):r ==> (b,a):converse(r)"; |
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87 |
by (Simp_tac 1); |
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qed "converseI"; |
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|
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goalw Relation.thy [converse_def] "!!a b r. (a,b) : converse(r) ==> (b,a) : r"; |
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91 |
by (fast_tac comp_cs 1); |
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92 |
qed "converseD"; |
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93 |
|
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qed_goalw "converseE" Relation.thy [converse_def] |
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95 |
"[| yx : converse(r); \ |
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96 |
\ !!x y. [| yx=(y,x); (x,y):r |] ==> P \ |
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97 |
\ |] ==> P" |
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98 |
(fn [major,minor]=> |
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99 |
[ (rtac (major RS CollectE) 1), |
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100 |
(REPEAT (eresolve_tac [splitE, bexE,exE, conjE, minor] 1)), |
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101 |
(assume_tac 1) ]); |
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102 |
|
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103 |
val converse_cs = comp_cs addSIs [converseI] |
1465 | 104 |
addSEs [converseD,converseE]; |
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105 |
|
1605 | 106 |
goalw Relation.thy [converse_def] "converse(converse R) = R"; |
107 |
by(fast_tac (prod_cs addSIs [equalityI]) 1); |
|
108 |
qed "converse_converse"; |
|
109 |
||
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110 |
(** Domain **) |
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111 |
|
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112 |
qed_goalw "Domain_iff" Relation.thy [Domain_def] |
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113 |
"a: Domain(r) = (EX y. (a,y): r)" |
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114 |
(fn _=> [ (fast_tac comp_cs 1) ]); |
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115 |
|
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116 |
qed_goal "DomainI" Relation.thy "!!a b r. (a,b): r ==> a: Domain(r)" |
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117 |
(fn _ => [ (etac (exI RS (Domain_iff RS iffD2)) 1) ]); |
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118 |
|
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119 |
qed_goal "DomainE" Relation.thy |
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120 |
"[| a : Domain(r); !!y. (a,y): r ==> P |] ==> P" |
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121 |
(fn prems=> |
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122 |
[ (rtac (Domain_iff RS iffD1 RS exE) 1), |
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123 |
(REPEAT (ares_tac prems 1)) ]); |
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124 |
|
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125 |
(** Range **) |
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126 |
|
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127 |
qed_goalw "RangeI" Relation.thy [Range_def] "!!a b r.(a,b): r ==> b : Range(r)" |
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128 |
(fn _ => [ (etac (converseI RS DomainI) 1) ]); |
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129 |
|
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Trancl is now based on Relation which used to be in Integ.
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parents:
diff
changeset
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130 |
qed_goalw "RangeE" Relation.thy [Range_def] |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
131 |
"[| b : Range(r); !!x. (x,b): r ==> P |] ==> P" |
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
132 |
(fn major::prems=> |
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
133 |
[ (rtac (major RS DomainE) 1), |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
134 |
(resolve_tac prems 1), |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
135 |
(etac converseD 1) ]); |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
136 |
|
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
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changeset
|
137 |
(*** Image of a set under a relation ***) |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
138 |
|
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
139 |
qed_goalw "Image_iff" Relation.thy [Image_def] |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
140 |
"b : r^^A = (? x:A. (x,b):r)" |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
141 |
(fn _ => [ fast_tac (comp_cs addIs [RangeI]) 1 ]); |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
142 |
|
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
143 |
qed_goal "Image_singleton_iff" Relation.thy |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
144 |
"(b : r^^{a}) = ((a,b):r)" |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
145 |
(fn _ => [ rtac (Image_iff RS trans) 1, |
1465 | 146 |
fast_tac comp_cs 1 ]); |
1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
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parents:
diff
changeset
|
147 |
|
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
148 |
qed_goalw "ImageI" Relation.thy [Image_def] |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
149 |
"!!a b r. [| (a,b): r; a:A |] ==> b : r^^A" |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
150 |
(fn _ => [ (REPEAT (ares_tac [CollectI,RangeI,bexI] 1)), |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
151 |
(resolve_tac [conjI ] 1), |
1465 | 152 |
(rtac RangeI 1), |
1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
153 |
(REPEAT (fast_tac set_cs 1))]); |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
154 |
|
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
155 |
qed_goalw "ImageE" Relation.thy [Image_def] |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
156 |
"[| b: r^^A; !!x.[| (x,b): r; x:A |] ==> P |] ==> P" |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
157 |
(fn major::prems=> |
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
158 |
[ (rtac (major RS CollectE) 1), |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
159 |
(safe_tac set_cs), |
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
160 |
(etac RangeE 1), |
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
161 |
(rtac (hd prems) 1), |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
162 |
(REPEAT (etac bexE 1 ORELSE ares_tac prems 1)) ]); |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
163 |
|
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
164 |
qed_goal "Image_subset" Relation.thy |
1642 | 165 |
"!!A B r. r <= A Times B ==> r^^C <= B" |
1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
166 |
(fn _ => |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
167 |
[ (rtac subsetI 1), |
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Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
168 |
(REPEAT (eresolve_tac [asm_rl, ImageE, subsetD RS SigmaD2] 1)) ]); |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
169 |
|
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
170 |
val rel_cs = converse_cs addSIs [converseI] |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
171 |
addIs [ImageI, DomainI, RangeI] |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
172 |
addSEs [ImageE, DomainE, RangeE]; |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
173 |
|
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
174 |
val rel_eq_cs = rel_cs addSIs [equalityI]; |
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
175 |
|
1694 | 176 |
goal Relation.thy "R O id = R"; |
177 |
by(fast_tac (rel_cs addIs [set_ext] addbefore (split_all_tac 1)) 1); |
|
178 |
qed "R_O_id"; |
|
179 |
||
180 |
goal Relation.thy "id O R = R"; |
|
181 |
by(fast_tac (rel_cs addIs [set_ext] addbefore (split_all_tac 1)) 1); |
|
182 |
qed "id_O_R"; |
|
183 |
||
184 |
Addsimps [R_O_id,id_O_R]; |