src/HOL/Relation.ML
changeset 1128 64b30e3cc6d4
child 1264 3eb91524b938
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Relation.ML	Fri May 26 18:11:47 1995 +0200
@@ -0,0 +1,173 @@
+(*  Title: 	Relation.ML
+    ID:         $Id$
+    Authors: 	Riccardo Mattolini, Dip. Sistemi e Informatica
+        	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1994 Universita' di Firenze
+    Copyright   1993  University of Cambridge
+*)
+
+val RSLIST = curry (op MRS);
+
+open Relation;
+
+(** Identity relation **)
+
+goalw Relation.thy [id_def] "(a,a) : id";  
+by (rtac CollectI 1);
+by (rtac exI 1);
+by (rtac refl 1);
+qed "idI";
+
+val major::prems = goalw Relation.thy [id_def]
+    "[| p: id;  !!x.[| p = (x,x) |] ==> P  \
+\    |] ==>  P";  
+by (rtac (major RS CollectE) 1);
+by (etac exE 1);
+by (eresolve_tac prems 1);
+qed "idE";
+
+goalw Relation.thy [id_def] "(a,b):id = (a=b)";
+by(fast_tac prod_cs 1);
+qed "pair_in_id_conv";
+
+
+(** Composition of two relations **)
+
+val prems = goalw Relation.thy [comp_def]
+    "[| (a,b):s; (b,c):r |] ==> (a,c) : r O s";
+by (fast_tac (set_cs addIs prems) 1);
+qed "compI";
+
+(*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
+val prems = goalw Relation.thy [comp_def]
+    "[| xz : r O s;  \
+\       !!x y z. [| xz = (x,z);  (x,y):s;  (y,z):r |] ==> P \
+\    |] ==> P";
+by (cut_facts_tac prems 1);
+by (REPEAT (eresolve_tac [CollectE, exE, conjE] 1 ORELSE ares_tac prems 1));
+qed "compE";
+
+val prems = goal Relation.thy
+    "[| (a,c) : r O s;  \
+\       !!y. [| (a,y):s;  (y,c):r |] ==> P \
+\    |] ==> P";
+by (rtac compE 1);
+by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Pair_inject,ssubst] 1));
+qed "compEpair";
+
+val comp_cs = prod_cs addIs [compI, idI] addSEs [compE, idE];
+
+goal Relation.thy "!!r s. [| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
+by (fast_tac comp_cs 1);
+qed "comp_mono";
+
+goal Relation.thy
+    "!!r s. [| s <= Sigma A (%x.B);  r <= Sigma B (%x.C) |] ==> \
+\           (r O s) <= Sigma A (%x.C)";
+by (fast_tac comp_cs 1);
+qed "comp_subset_Sigma";
+
+(** Natural deduction for trans(r) **)
+
+val prems = goalw Relation.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";
+
+val major::prems = goalw Relation.thy [trans_def]
+    "[| trans(r);  (a,b):r;  (b,c):r |] ==> (a,c):r";
+by (cut_facts_tac [major] 1);
+by (fast_tac (HOL_cs addIs prems) 1);
+qed "transD";
+
+(** Natural deduction for converse(r) **)
+
+goalw Relation.thy [converse_def] "!!a b r. (a,b):r ==> (b,a):converse(r)";
+by (simp_tac prod_ss 1);
+by (fast_tac set_cs 1);
+qed "converseI";
+
+goalw Relation.thy [converse_def] "!!a b r. (a,b) : converse(r) ==> (b,a) : r";
+by (fast_tac comp_cs 1);
+qed "converseD";
+
+qed_goalw "converseE" Relation.thy [converse_def]
+    "[| yx : converse(r);  \
+\       !!x y. [| yx=(y,x);  (x,y):r |] ==> P \
+\    |] ==> P"
+ (fn [major,minor]=>
+  [ (rtac (major RS CollectE) 1),
+    (REPEAT (eresolve_tac [bexE,exE, conjE, minor] 1)),
+    (hyp_subst_tac 1),
+    (assume_tac 1) ]);
+
+val converse_cs = comp_cs addSIs [converseI] 
+			  addSEs [converseD,converseE];
+
+(** Domain **)
+
+qed_goalw "Domain_iff" Relation.thy [Domain_def]
+    "a: Domain(r) = (EX y. (a,y): r)"
+ (fn _=> [ (fast_tac comp_cs 1) ]);
+
+qed_goal "DomainI" Relation.thy "!!a b r. (a,b): r ==> a: Domain(r)"
+ (fn _ => [ (etac (exI RS (Domain_iff RS iffD2)) 1) ]);
+
+qed_goal "DomainE" Relation.thy
+    "[| a : Domain(r);  !!y. (a,y): r ==> P |] ==> P"
+ (fn prems=>
+  [ (rtac (Domain_iff RS iffD1 RS exE) 1),
+    (REPEAT (ares_tac prems 1)) ]);
+
+(** Range **)
+
+qed_goalw "RangeI" Relation.thy [Range_def] "!!a b r.(a,b): r ==> b : Range(r)"
+ (fn _ => [ (etac (converseI RS DomainI) 1) ]);
+
+qed_goalw "RangeE" Relation.thy [Range_def]
+    "[| b : Range(r);  !!x. (x,b): r ==> P |] ==> P"
+ (fn major::prems=>
+  [ (rtac (major RS DomainE) 1),
+    (resolve_tac prems 1),
+    (etac converseD 1) ]);
+
+(*** Image of a set under a relation ***)
+
+qed_goalw "Image_iff" Relation.thy [Image_def]
+    "b : r^^A = (? x:A. (x,b):r)"
+ (fn _ => [ fast_tac (comp_cs addIs [RangeI]) 1 ]);
+
+qed_goal "Image_singleton_iff" Relation.thy
+    "(b : r^^{a}) = ((a,b):r)"
+ (fn _ => [ rtac (Image_iff RS trans) 1,
+	    fast_tac comp_cs 1 ]);
+
+qed_goalw "ImageI" Relation.thy [Image_def]
+    "!!a b r. [| (a,b): r;  a:A |] ==> b : r^^A"
+ (fn _ => [ (REPEAT (ares_tac [CollectI,RangeI,bexI] 1)),
+            (resolve_tac [conjI ] 1),
+            (resolve_tac [RangeI] 1),
+            (REPEAT (fast_tac set_cs 1))]);
+
+qed_goalw "ImageE" Relation.thy [Image_def]
+    "[| b: r^^A;  !!x.[| (x,b): r;  x:A |] ==> P |] ==> P"
+ (fn major::prems=>
+  [ (rtac (major RS CollectE) 1),
+    (safe_tac set_cs),
+    (etac RangeE 1),
+    (rtac (hd prems) 1),
+    (REPEAT (etac bexE 1 ORELSE ares_tac prems 1)) ]);
+
+qed_goal "Image_subset" Relation.thy
+    "!!A B r. r <= Sigma A (%x.B) ==> r^^C <= B"
+ (fn _ =>
+  [ (rtac subsetI 1),
+    (REPEAT (eresolve_tac [asm_rl, ImageE, subsetD RS SigmaD2] 1)) ]);
+
+val rel_cs = converse_cs addSIs [converseI] 
+                         addIs  [ImageI, DomainI, RangeI]
+                         addSEs [ImageE, DomainE, RangeE];
+
+val rel_eq_cs = rel_cs addSIs [equalityI];
+
+val rel_ss = prod_ss addsimps [pair_in_id_conv];