src/HOL/Integ/Equiv.ML
 changeset 1894 c2c8279d40f0 parent 1844 791e79473966 child 1978 e7df069acb74
```--- a/src/HOL/Integ/Equiv.ML	Mon Jul 29 18:31:39 1996 +0200
+++ b/src/HOL/Integ/Equiv.ML	Tue Jul 30 17:33:26 1996 +0200
@@ -10,18 +10,20 @@

open Equiv;

+Delrules [equalityI];
+
(*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***)

(** first half: equiv A r ==> converse(r) O r = r **)

goalw Equiv.thy [trans_def,sym_def,converse_def]
"!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
-by (fast_tac (comp_cs addSEs [converseD]) 1);
+by (fast_tac (!claset addSEs [converseD]) 1);
qed "sym_trans_comp_subset";

goalw Equiv.thy [refl_def]
"!!A r. refl A r ==> r <= converse(r) O r";
-by (fast_tac (rel_cs addIs [compI]) 1);
+by (fast_tac (!claset addIs [compI]) 1);
qed "refl_comp_subset";

goalw Equiv.thy [equiv_def]
@@ -36,9 +38,9 @@
"!!A r. [| converse(r) O r = r;  Domain(r) = A |] ==> equiv A r";
by (etac equalityE 1);
by (subgoal_tac "ALL x y. (x,y) : r --> (y,x) : r" 1);
-by (safe_tac set_cs);
+by (safe_tac (!claset));
qed "comp_equivI";

(** Equivalence classes **)
@@ -46,28 +48,28 @@
(*Lemma for the next result*)
goalw Equiv.thy [equiv_def,trans_def,sym_def]
"!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} <= r^^{b}";
-by (safe_tac rel_cs);
+by (safe_tac (!claset));
by (rtac ImageI 1);
-by (fast_tac rel_cs 2);
-by (fast_tac rel_cs 1);
+by (Fast_tac 2);
+by (Fast_tac 1);
qed "equiv_class_subset";

goal Equiv.thy "!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} = r^^{b}";
by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1));
by (rewrite_goals_tac [equiv_def,sym_def]);
-by (fast_tac rel_cs 1);
+by (Fast_tac 1);
qed "equiv_class_eq";

val prems = goalw Equiv.thy [equiv_def,refl_def]
"[| equiv A r;  a: A |] ==> a: r^^{a}";
by (cut_facts_tac prems 1);
-by (fast_tac rel_cs 1);
+by (Fast_tac 1);
qed "equiv_class_self";

(*Lemma for the next result*)
goalw Equiv.thy [equiv_def,refl_def]
"!!A r. [| equiv A r;  r^^{b} <= r^^{a};  b: A |] ==> (a,b): r";
-by (fast_tac rel_cs 1);
+by (Fast_tac 1);
qed "subset_equiv_class";

val prems = goal Equiv.thy
@@ -78,7 +80,7 @@
(*thus r^^{a} = r^^{b} as well*)
goalw Equiv.thy [equiv_def,trans_def,sym_def]
"!!A r. [| equiv A r;  x: (r^^{a} Int r^^{b}) |] ==> (a,b): r";
-by (fast_tac rel_cs 1);
+by (Fast_tac 1);
qed "equiv_class_nondisjoint";

val [major] = goalw Equiv.thy [equiv_def,refl_def]
@@ -88,7 +90,7 @@

goal Equiv.thy
"!!A r. equiv A r ==> ((x,y): r) = (r^^{x} = r^^{y} & x:A & y:A)";
-by (safe_tac rel_cs);
+by (safe_tac (!claset));
by ((rtac equiv_class_eq 1) THEN (assume_tac 1) THEN (assume_tac 1));
by ((rtac eq_equiv_class 3) THEN
(assume_tac 4) THEN (assume_tac 4) THEN (assume_tac 3));
@@ -100,7 +102,7 @@

goal Equiv.thy
"!!A r. [| equiv A r;  x: A;  y: A |] ==> (r^^{x} = r^^{y}) = ((x,y): r)";
-by (safe_tac rel_cs);
+by (safe_tac (!claset));
by ((rtac eq_equiv_class 1) THEN
(assume_tac 1) THEN (assume_tac 1) THEN (assume_tac 1));
by ((rtac equiv_class_eq 1) THEN
@@ -123,7 +125,7 @@
by (resolve_tac [major RS UN_E] 1);
by (rtac minor 1);
by (assume_tac 2);
-by (fast_tac rel_cs 1);
+by (Fast_tac 1);
qed "quotientE";

(** Not needed by Theory Integ --> bypassed **)
@@ -136,10 +138,10 @@
(** Not needed by Theory Integ --> bypassed **)
(*goalw Equiv.thy [quotient_def]
"!!A r. [| equiv A r;  X: A/r;  Y: A/r |] ==> X=Y | (X Int Y <= 0)";
by (assume_tac 1);
by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
-by (fast_tac ZF_cs 1);
+by (Fast_tac 1);
qed "quotient_disj";
**)

@@ -147,7 +149,7 @@

(* theorem needed to prove UN_equiv_class *)
goal Set.thy "!!A. [| a:A; ! y:A. b(y)=b(a) |] ==> (UN y:A. b(y))=b(a)";
-by (fast_tac (!claset addSEs [equalityE]) 1);
qed "UN_singleton_lemma";
val UN_singleton = ballI RSN (2,UN_singleton_lemma);

@@ -165,7 +167,7 @@
by (assume_tac 1);
by (assume_tac 1);
by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
-by (fast_tac rel_cs 1);
+by (Fast_tac 1);
qed "UN_equiv_class";

(*Resolve th against the "local" premises*)
@@ -177,7 +179,7 @@
\       !!x.  x : A ==> b(x) : B |]     \
\    ==> (UN x:X. b(x)) : B";
by (cut_facts_tac prems 1);
-by (safe_tac rel_cs);
+by (safe_tac (!claset));
by (rtac (localize UN_equiv_class RS ssubst) 1);
by (REPEAT (ares_tac prems 1));
qed "UN_equiv_class_type";
@@ -191,7 +193,7 @@
\       !!x y. [| x:A; y:A; b(x)=b(y) |] ==> (x,y):r |]         \
\    ==> X=Y";
by (cut_facts_tac prems 1);
-by (safe_tac rel_cs);
+by (safe_tac ((!claset) delrules [equalityI]));
by (rtac (equivA RS equiv_class_eq) 1);
by (REPEAT (ares_tac prems 1));
by (etac box_equals 1);
@@ -204,20 +206,20 @@

goalw Equiv.thy [congruent_def,congruent2_def,equiv_def,refl_def]
"!!A r. [| equiv A r;  congruent2 r b;  a: A |] ==> congruent r (b a)";
-by (fast_tac rel_cs 1);
+by (Fast_tac 1);
qed "congruent2_implies_congruent";

val equivA::prems = goalw Equiv.thy [congruent_def]
"[| equiv A r;  congruent2 r b;  a: A |] ==> \
\    congruent r (%x1. UN x2:r^^{a}. b x1 x2)";
by (cut_facts_tac (equivA::prems) 1);
-by (safe_tac rel_cs);
+by (safe_tac (!claset));
by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
by (assume_tac 1);
by (asm_simp_tac (!simpset addsimps [equivA RS UN_equiv_class,
congruent2_implies_congruent]) 1);
by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
-by (fast_tac rel_cs 1);
+by (Fast_tac 1);
qed "congruent2_implies_congruent_UN";

val prems as equivA::_ = goal Equiv.thy
@@ -236,7 +238,7 @@
\       !!x1 x2.  [| x1: A; x2: A |] ==> b x1 x2 : B |]    \
\    ==> (UN x1:X1. UN x2:X2. b x1 x2) : B";
by (cut_facts_tac prems 1);
-by (safe_tac rel_cs);
+by (safe_tac (!claset));
by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
congruent2_implies_congruent_UN,
congruent2_implies_congruent, quotientI]) 1));
@@ -251,7 +253,7 @@
\       !! y z w. [| w: A;  (y,z) : r |] ==> b w y = b w z       \
\    |] ==> congruent2 r b";
by (cut_facts_tac prems 1);
-by (safe_tac rel_cs);
+by (safe_tac (!claset));
by (rtac trans 1);
by (REPEAT (ares_tac prems 1
ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));```