--- a/src/ZF/Integ/EquivClass.thy Fri Sep 03 22:40:57 2004 +0200
+++ b/src/ZF/Integ/EquivClass.thy Mon Sep 06 15:57:58 2004 +0200
@@ -17,9 +17,18 @@
congruent :: "[i,i=>i]=>o"
"congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)"
- congruent2 :: "[i,[i,i]=>i]=>o"
- "congruent2(r,b) == ALL y1 z1 y2 z2.
- <y1,z1>:r --> <y2,z2>:r --> b(y1,y2) = b(z1,z2)"
+ congruent2 :: "[i,i,[i,i]=>i]=>o"
+ "congruent2(r1,r2,b) == ALL y1 z1 y2 z2.
+ <y1,z1>:r1 --> <y2,z2>:r2 --> b(y1,y2) = b(z1,z2)"
+
+syntax
+ RESPECTS ::"[i=>i, i] => o" (infixr "respects" 80)
+ RESPECTS2 ::"[i=>i, i] => o" (infixr "respects2 " 80)
+ --{*Abbreviation for the common case where the relations are identical*}
+
+translations
+ "f respects r" == "congruent(r,f)"
+ "f respects2 r" => "congruent2(r,r,f)"
subsection{*Suppes, Theorem 70:
@{term r} is an equiv relation iff @{term "converse(r) O r = r"}*}
@@ -28,19 +37,16 @@
lemma sym_trans_comp_subset:
"[| sym(r); trans(r) |] ==> converse(r) O r <= r"
-apply (unfold trans_def sym_def, blast)
-done
+by (unfold trans_def sym_def, blast)
lemma refl_comp_subset:
"[| refl(A,r); r <= A*A |] ==> r <= converse(r) O r"
-apply (unfold refl_def, blast)
-done
+by (unfold refl_def, blast)
lemma equiv_comp_eq:
"equiv(A,r) ==> converse(r) O r = r"
apply (unfold equiv_def)
-apply (blast del: subsetI
- intro!: sym_trans_comp_subset refl_comp_subset)
+apply (blast del: subsetI intro!: sym_trans_comp_subset refl_comp_subset)
done
(*second half*)
@@ -124,7 +130,7 @@
(*Conversion rule*)
lemma UN_equiv_class:
- "[| equiv(A,r); congruent(r,b); a: A |] ==> (UN x:r``{a}. b(x)) = b(a)"
+ "[| equiv(A,r); b respects r; a: A |] ==> (UN x:r``{a}. b(x)) = b(a)"
apply (subgoal_tac "\<forall>x \<in> r``{a}. b(x) = b(a)")
apply simp
apply (blast intro: equiv_class_self)
@@ -133,7 +139,7 @@
(*type checking of UN x:r``{a}. b(x) *)
lemma UN_equiv_class_type:
- "[| equiv(A,r); congruent(r,b); X: A//r; !!x. x : A ==> b(x) : B |]
+ "[| equiv(A,r); b respects r; X: A//r; !!x. x : A ==> b(x) : B |]
==> (UN x:X. b(x)) : B"
apply (unfold quotient_def, safe)
apply (simp (no_asm_simp) add: UN_equiv_class)
@@ -143,7 +149,7 @@
major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
*)
lemma UN_equiv_class_inject:
- "[| equiv(A,r); congruent(r,b);
+ "[| equiv(A,r); b respects r;
(UN x:X. b(x))=(UN y:Y. b(y)); X: A//r; Y: A//r;
!!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |]
==> X=Y"
@@ -156,28 +162,28 @@
subsection{*Defining Binary Operations upon Equivalence Classes*}
lemma congruent2_implies_congruent:
- "[| equiv(A,r); congruent2(r,b); a: A |] ==> congruent(r,b(a))"
+ "[| equiv(A,r1); congruent2(r1,r2,b); a: A |] ==> congruent(r2,b(a))"
by (unfold congruent_def congruent2_def equiv_def refl_def, blast)
lemma congruent2_implies_congruent_UN:
- "[| equiv(A,r); congruent2(r,b); a: A |] ==>
- congruent(r, %x1. UN x2:r``{a}. b(x1,x2))"
+ "[| equiv(A1,r1); equiv(A2,r2); congruent2(r1,r2,b); a: A2 |] ==>
+ congruent(r1, %x1. \<Union>x2 \<in> r2``{a}. b(x1,x2))"
apply (unfold congruent_def, safe)
apply (frule equiv_type [THEN subsetD], assumption)
apply clarify
-apply (simp add: UN_equiv_class [of A r] congruent2_implies_congruent)
+apply (simp add: UN_equiv_class congruent2_implies_congruent)
apply (unfold congruent2_def equiv_def refl_def, blast)
done
lemma UN_equiv_class2:
- "[| equiv(A,r); congruent2(r,b); a1: A; a2: A |]
- ==> (UN x1:r``{a1}. UN x2:r``{a2}. b(x1,x2)) = b(a1,a2)"
-by (simp add: UN_equiv_class [of A r] congruent2_implies_congruent
+ "[| equiv(A1,r1); equiv(A2,r2); congruent2(r1,r2,b); a1: A1; a2: A2 |]
+ ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. b(x1,x2)) = b(a1,a2)"
+by (simp add: UN_equiv_class congruent2_implies_congruent
congruent2_implies_congruent_UN)
(*type checking*)
lemma UN_equiv_class_type2:
- "[| equiv(A,r); congruent2(r,b);
+ "[| equiv(A,r); b respects2 r;
X1: A//r; X2: A//r;
!!x1 x2. [| x1: A; x2: A |] ==> b(x1,x2) : B
|] ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B"
@@ -190,10 +196,10 @@
(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
than the direct proof*)
lemma congruent2I:
- "[| equiv(A,r);
- !! y z w. [| w: A; <y,z> : r |] ==> b(y,w) = b(z,w);
- !! y z w. [| w: A; <y,z> : r |] ==> b(w,y) = b(w,z)
- |] ==> congruent2(r,b)"
+ "[| equiv(A1,r1); equiv(A2,r2);
+ !! y z w. [| w \<in> A2; <y,z> \<in> r1 |] ==> b(y,w) = b(z,w);
+ !! y z w. [| w \<in> A1; <y,z> \<in> r2 |] ==> b(w,y) = b(w,z)
+ |] ==> congruent2(r1,r2,b)"
apply (unfold congruent2_def equiv_def refl_def, safe)
apply (blast intro: trans)
done
@@ -202,9 +208,9 @@
assumes equivA: "equiv(A,r)"
and commute: "!! y z. [| y: A; z: A |] ==> b(y,z) = b(z,y)"
and congt: "!! y z w. [| w: A; <y,z>: r |] ==> b(w,y) = b(w,z)"
- shows "congruent2(r,b)"
+ shows "b respects2 r"
apply (insert equivA [THEN equiv_type, THEN subsetD])
-apply (rule congruent2I [OF equivA])
+apply (rule congruent2I [OF equivA equivA])
apply (rule commute [THEN trans])
apply (rule_tac [3] commute [THEN trans, symmetric])
apply (rule_tac [5] sym)