--- a/src/HOL/Relation.ML Tue Mar 03 15:11:26 1998 +0100
+++ b/src/HOL/Relation.ML Tue Mar 03 15:12:25 1998 +0100
@@ -8,11 +8,11 @@
(** Identity relation **)
-goalw Relation.thy [id_def] "(a,a) : id";
+goalw thy [id_def] "(a,a) : id";
by (Blast_tac 1);
qed "idI";
-val major::prems = goalw Relation.thy [id_def]
+val major::prems = goalw thy [id_def]
"[| p: id; !!x.[| p = (x,x) |] ==> P \
\ |] ==> P";
by (rtac (major RS CollectE) 1);
@@ -20,7 +20,7 @@
by (eresolve_tac prems 1);
qed "idE";
-goalw Relation.thy [id_def] "(a,b):id = (a=b)";
+goalw thy [id_def] "(a,b):id = (a=b)";
by (Blast_tac 1);
qed "pair_in_id_conv";
Addsimps [pair_in_id_conv];
@@ -28,13 +28,13 @@
(** Composition of two relations **)
-goalw Relation.thy [comp_def]
+goalw thy [comp_def]
"!!r s. [| (a,b):s; (b,c):r |] ==> (a,c) : r O s";
by (Blast_tac 1);
qed "compI";
(*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
-val prems = goalw Relation.thy [comp_def]
+val prems = goalw thy [comp_def]
"[| xz : r O s; \
\ !!x y z. [| xz = (x,z); (x,y):s; (y,z):r |] ==> P \
\ |] ==> P";
@@ -43,7 +43,7 @@
ORELSE ares_tac prems 1));
qed "compE";
-val prems = goal Relation.thy
+val prems = goal thy
"[| (a,c) : r O s; \
\ !!y. [| (a,y):s; (y,c):r |] ==> P \
\ |] ==> P";
@@ -54,45 +54,55 @@
AddIs [compI, idI];
AddSEs [compE, idE];
-goal Relation.thy "!!r s. [| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
+goal thy "R O id = R";
+by (Fast_tac 1);
+qed "R_O_id";
+
+goal thy "id O R = R";
+by (Fast_tac 1);
+qed "id_O_R";
+
+Addsimps [R_O_id,id_O_R];
+
+goal thy "!!r s. [| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
by (Blast_tac 1);
qed "comp_mono";
-goal Relation.thy
+goal thy
"!!r s. [| s <= A Times B; r <= B Times C |] ==> (r O s) <= A Times C";
by (Blast_tac 1);
qed "comp_subset_Sigma";
(** Natural deduction for trans(r) **)
-val prems = goalw Relation.thy [trans_def]
+val prems = goalw thy [trans_def]
"(!! x y z. [| (x,y):r; (y,z):r |] ==> (x,z):r) ==> trans(r)";
by (REPEAT (ares_tac (prems@[allI,impI]) 1));
qed "transI";
-goalw Relation.thy [trans_def]
+goalw thy [trans_def]
"!!r. [| trans(r); (a,b):r; (b,c):r |] ==> (a,c):r";
by (Blast_tac 1);
qed "transD";
(** Natural deduction for r^-1 **)
-goalw Relation.thy [inverse_def] "!!a b r. ((a,b): r^-1) = ((b,a):r)";
+goalw thy [inverse_def] "!!a b r. ((a,b): r^-1) = ((b,a):r)";
by (Simp_tac 1);
qed "inverse_iff";
AddIffs [inverse_iff];
-goalw Relation.thy [inverse_def] "!!a b r. (a,b):r ==> (b,a): r^-1";
+goalw thy [inverse_def] "!!a b r. (a,b):r ==> (b,a): r^-1";
by (Simp_tac 1);
qed "inverseI";
-goalw Relation.thy [inverse_def] "!!a b r. (a,b) : r^-1 ==> (b,a) : r";
+goalw thy [inverse_def] "!!a b r. (a,b) : r^-1 ==> (b,a) : r";
by (Blast_tac 1);
qed "inverseD";
(*More general than inverseD, as it "splits" the member of the relation*)
-qed_goalw "inverseE" Relation.thy [inverse_def]
+qed_goalw "inverseE" thy [inverse_def]
"[| yx : r^-1; \
\ !!x y. [| yx=(y,x); (x,y):r |] ==> P \
\ |] ==> P"
@@ -103,30 +113,30 @@
AddSEs [inverseE];
-goalw Relation.thy [inverse_def] "(r^-1)^-1 = r";
+goalw thy [inverse_def] "(r^-1)^-1 = r";
by (Blast_tac 1);
qed "inverse_inverse";
Addsimps [inverse_inverse];
-goal Relation.thy "(r O s)^-1 = s^-1 O r^-1";
+goal thy "(r O s)^-1 = s^-1 O r^-1";
by (Blast_tac 1);
qed "inverse_comp";
-goal Relation.thy "id^-1 = id";
+goal thy "id^-1 = id";
by (Blast_tac 1);
qed "inverse_id";
Addsimps [inverse_id];
(** Domain **)
-qed_goalw "Domain_iff" Relation.thy [Domain_def]
+qed_goalw "Domain_iff" thy [Domain_def]
"a: Domain(r) = (EX y. (a,y): r)"
(fn _=> [ (Blast_tac 1) ]);
-qed_goal "DomainI" Relation.thy "!!a b r. (a,b): r ==> a: Domain(r)"
+qed_goal "DomainI" thy "!!a b r. (a,b): r ==> a: Domain(r)"
(fn _ => [ (etac (exI RS (Domain_iff RS iffD2)) 1) ]);
-qed_goal "DomainE" Relation.thy
+qed_goal "DomainE" thy
"[| a : Domain(r); !!y. (a,y): r ==> P |] ==> P"
(fn prems=>
[ (rtac (Domain_iff RS iffD1 RS exE) 1),
@@ -142,10 +152,10 @@
(** Range **)
-qed_goalw "RangeI" Relation.thy [Range_def] "!!a b r.(a,b): r ==> b : Range(r)"
+qed_goalw "RangeI" thy [Range_def] "!!a b r.(a,b): r ==> b : Range(r)"
(fn _ => [ (etac (inverseI RS DomainI) 1) ]);
-qed_goalw "RangeE" Relation.thy [Range_def]
+qed_goalw "RangeE" thy [Range_def]
"[| b : Range(r); !!x. (x,b): r ==> P |] ==> P"
(fn major::prems=>
[ (rtac (major RS DomainE) 1),
@@ -162,20 +172,26 @@
(*** Image of a set under a relation ***)
-qed_goalw "Image_iff" Relation.thy [Image_def]
+qed_goalw "Image_iff" thy [Image_def]
"b : r^^A = (? x:A. (x,b):r)"
(fn _ => [ Blast_tac 1 ]);
-qed_goal "Image_singleton_iff" Relation.thy
+qed_goalw "Image_singleton" thy [Image_def]
+ "r^^{a} = {b. (a,b):r}"
+ (fn _ => [ Blast_tac 1 ]);
+
+qed_goal "Image_singleton_iff" thy
"(b : r^^{a}) = ((a,b):r)"
(fn _ => [ rtac (Image_iff RS trans) 1,
Blast_tac 1 ]);
-qed_goalw "ImageI" Relation.thy [Image_def]
+AddIffs [Image_singleton_iff];
+
+qed_goalw "ImageI" thy [Image_def]
"!!a b r. [| (a,b): r; a:A |] ==> b : r^^A"
(fn _ => [ (Blast_tac 1)]);
-qed_goalw "ImageE" Relation.thy [Image_def]
+qed_goalw "ImageE" thy [Image_def]
"[| b: r^^A; !!x.[| (x,b): r; x:A |] ==> P |] ==> P"
(fn major::prems=>
[ (rtac (major RS CollectE) 1),
@@ -187,7 +203,7 @@
AddSEs [ImageE];
-qed_goal "Image_empty" Relation.thy
+qed_goal "Image_empty" thy
"R^^{} = {}"
(fn _ => [ Blast_tac 1 ]);
@@ -199,27 +215,21 @@
Addsimps [Image_id];
-qed_goal "Image_Int_subset" Relation.thy
+qed_goal "Image_Int_subset" thy
"R ^^ (A Int B) <= R ^^ A Int R ^^ B"
(fn _ => [ Blast_tac 1 ]);
-qed_goal "Image_Un" Relation.thy
+qed_goal "Image_Un" thy
"R ^^ (A Un B) = R ^^ A Un R ^^ B"
(fn _ => [ Blast_tac 1 ]);
-qed_goal "Image_subset" Relation.thy
+qed_goal "Image_subset" thy
"!!A B r. r <= A Times B ==> r^^C <= B"
(fn _ =>
[ (rtac subsetI 1),
(REPEAT (eresolve_tac [asm_rl, ImageE, subsetD RS SigmaD2] 1)) ]);
-goal Relation.thy "R O id = R";
-by (Fast_tac 1);
-qed "R_O_id";
-
-goal Relation.thy "id O R = R";
-by (Fast_tac 1);
-qed "id_O_R";
-
-Addsimps [R_O_id,id_O_R];
+goal thy "f-``(r^-1 ^^ {x}) = (UN y: r^-1 ^^ {x}. f-``{y})";
+by (Blast_tac 1);
+qed "vimage_inverse_Image";