src/HOL/Integ/Equiv.ML
 changeset 1894 c2c8279d40f0 parent 1844 791e79473966 child 1978 e7df069acb74
```     1.1 --- a/src/HOL/Integ/Equiv.ML	Mon Jul 29 18:31:39 1996 +0200
1.2 +++ b/src/HOL/Integ/Equiv.ML	Tue Jul 30 17:33:26 1996 +0200
1.3 @@ -10,18 +10,20 @@
1.4
1.5  open Equiv;
1.6
1.7 +Delrules [equalityI];
1.8 +
1.9  (*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***)
1.10
1.11  (** first half: equiv A r ==> converse(r) O r = r **)
1.12
1.13  goalw Equiv.thy [trans_def,sym_def,converse_def]
1.14      "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
1.15 -by (fast_tac (comp_cs addSEs [converseD]) 1);
1.16 +by (fast_tac (!claset addSEs [converseD]) 1);
1.17  qed "sym_trans_comp_subset";
1.18
1.19  goalw Equiv.thy [refl_def]
1.20      "!!A r. refl A r ==> r <= converse(r) O r";
1.21 -by (fast_tac (rel_cs addIs [compI]) 1);
1.22 +by (fast_tac (!claset addIs [compI]) 1);
1.23  qed "refl_comp_subset";
1.24
1.25  goalw Equiv.thy [equiv_def]
1.26 @@ -36,9 +38,9 @@
1.27      "!!A r. [| converse(r) O r = r;  Domain(r) = A |] ==> equiv A r";
1.28  by (etac equalityE 1);
1.29  by (subgoal_tac "ALL x y. (x,y) : r --> (y,x) : r" 1);
1.30 -by (safe_tac set_cs);
1.33 +by (safe_tac (!claset));
1.36  qed "comp_equivI";
1.37
1.38  (** Equivalence classes **)
1.39 @@ -46,28 +48,28 @@
1.40  (*Lemma for the next result*)
1.41  goalw Equiv.thy [equiv_def,trans_def,sym_def]
1.42      "!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} <= r^^{b}";
1.43 -by (safe_tac rel_cs);
1.44 +by (safe_tac (!claset));
1.45  by (rtac ImageI 1);
1.46 -by (fast_tac rel_cs 2);
1.47 -by (fast_tac rel_cs 1);
1.48 +by (Fast_tac 2);
1.49 +by (Fast_tac 1);
1.50  qed "equiv_class_subset";
1.51
1.52  goal Equiv.thy "!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} = r^^{b}";
1.53  by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1));
1.54  by (rewrite_goals_tac [equiv_def,sym_def]);
1.55 -by (fast_tac rel_cs 1);
1.56 +by (Fast_tac 1);
1.57  qed "equiv_class_eq";
1.58
1.59  val prems = goalw Equiv.thy [equiv_def,refl_def]
1.60      "[| equiv A r;  a: A |] ==> a: r^^{a}";
1.61  by (cut_facts_tac prems 1);
1.62 -by (fast_tac rel_cs 1);
1.63 +by (Fast_tac 1);
1.64  qed "equiv_class_self";
1.65
1.66  (*Lemma for the next result*)
1.67  goalw Equiv.thy [equiv_def,refl_def]
1.68      "!!A r. [| equiv A r;  r^^{b} <= r^^{a};  b: A |] ==> (a,b): r";
1.69 -by (fast_tac rel_cs 1);
1.70 +by (Fast_tac 1);
1.71  qed "subset_equiv_class";
1.72
1.73  val prems = goal Equiv.thy
1.74 @@ -78,7 +80,7 @@
1.75  (*thus r^^{a} = r^^{b} as well*)
1.76  goalw Equiv.thy [equiv_def,trans_def,sym_def]
1.77      "!!A r. [| equiv A r;  x: (r^^{a} Int r^^{b}) |] ==> (a,b): r";
1.78 -by (fast_tac rel_cs 1);
1.79 +by (Fast_tac 1);
1.80  qed "equiv_class_nondisjoint";
1.81
1.82  val [major] = goalw Equiv.thy [equiv_def,refl_def]
1.83 @@ -88,7 +90,7 @@
1.84
1.85  goal Equiv.thy
1.86      "!!A r. equiv A r ==> ((x,y): r) = (r^^{x} = r^^{y} & x:A & y:A)";
1.87 -by (safe_tac rel_cs);
1.88 +by (safe_tac (!claset));
1.89  by ((rtac equiv_class_eq 1) THEN (assume_tac 1) THEN (assume_tac 1));
1.90  by ((rtac eq_equiv_class 3) THEN
1.91      (assume_tac 4) THEN (assume_tac 4) THEN (assume_tac 3));
1.92 @@ -100,7 +102,7 @@
1.93
1.94  goal Equiv.thy
1.95      "!!A r. [| equiv A r;  x: A;  y: A |] ==> (r^^{x} = r^^{y}) = ((x,y): r)";
1.96 -by (safe_tac rel_cs);
1.97 +by (safe_tac (!claset));
1.98  by ((rtac eq_equiv_class 1) THEN
1.99      (assume_tac 1) THEN (assume_tac 1) THEN (assume_tac 1));
1.100  by ((rtac equiv_class_eq 1) THEN
1.101 @@ -123,7 +125,7 @@
1.102  by (resolve_tac [major RS UN_E] 1);
1.103  by (rtac minor 1);
1.104  by (assume_tac 2);
1.105 -by (fast_tac rel_cs 1);
1.106 +by (Fast_tac 1);
1.107  qed "quotientE";
1.108
1.109  (** Not needed by Theory Integ --> bypassed **)
1.110 @@ -136,10 +138,10 @@
1.111  (** Not needed by Theory Integ --> bypassed **)
1.112  (*goalw Equiv.thy [quotient_def]
1.113      "!!A r. [| equiv A r;  X: A/r;  Y: A/r |] ==> X=Y | (X Int Y <= 0)";
1.114 -by (safe_tac (ZF_cs addSIs [equiv_class_eq]));
1.115 +by (safe_tac (!claset addSIs [equiv_class_eq]));
1.116  by (assume_tac 1);
1.117  by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
1.118 -by (fast_tac ZF_cs 1);
1.119 +by (Fast_tac 1);
1.120  qed "quotient_disj";
1.121  **)
1.122
1.123 @@ -147,7 +149,7 @@
1.124
1.125  (* theorem needed to prove UN_equiv_class *)
1.126  goal Set.thy "!!A. [| a:A; ! y:A. b(y)=b(a) |] ==> (UN y:A. b(y))=b(a)";
1.127 -by (fast_tac (!claset addSEs [equalityE]) 1);
1.129  qed "UN_singleton_lemma";
1.130  val UN_singleton = ballI RSN (2,UN_singleton_lemma);
1.131
1.132 @@ -165,7 +167,7 @@
1.133  by (assume_tac 1);
1.134  by (assume_tac 1);
1.135  by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
1.136 -by (fast_tac rel_cs 1);
1.137 +by (Fast_tac 1);
1.138  qed "UN_equiv_class";
1.139
1.140  (*Resolve th against the "local" premises*)
1.141 @@ -177,7 +179,7 @@
1.142  \       !!x.  x : A ==> b(x) : B |]     \
1.143  \    ==> (UN x:X. b(x)) : B";
1.144  by (cut_facts_tac prems 1);
1.145 -by (safe_tac rel_cs);
1.146 +by (safe_tac (!claset));
1.147  by (rtac (localize UN_equiv_class RS ssubst) 1);
1.148  by (REPEAT (ares_tac prems 1));
1.149  qed "UN_equiv_class_type";
1.150 @@ -191,7 +193,7 @@
1.151  \       !!x y. [| x:A; y:A; b(x)=b(y) |] ==> (x,y):r |]         \
1.152  \    ==> X=Y";
1.153  by (cut_facts_tac prems 1);
1.154 -by (safe_tac rel_cs);
1.155 +by (safe_tac ((!claset) delrules [equalityI]));
1.156  by (rtac (equivA RS equiv_class_eq) 1);
1.157  by (REPEAT (ares_tac prems 1));
1.158  by (etac box_equals 1);
1.159 @@ -204,20 +206,20 @@
1.160
1.161  goalw Equiv.thy [congruent_def,congruent2_def,equiv_def,refl_def]
1.162      "!!A r. [| equiv A r;  congruent2 r b;  a: A |] ==> congruent r (b a)";
1.163 -by (fast_tac rel_cs 1);
1.164 +by (Fast_tac 1);
1.165  qed "congruent2_implies_congruent";
1.166
1.167  val equivA::prems = goalw Equiv.thy [congruent_def]
1.168      "[| equiv A r;  congruent2 r b;  a: A |] ==> \
1.169  \    congruent r (%x1. UN x2:r^^{a}. b x1 x2)";
1.170  by (cut_facts_tac (equivA::prems) 1);
1.171 -by (safe_tac rel_cs);
1.172 +by (safe_tac (!claset));
1.173  by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
1.174  by (assume_tac 1);
1.175  by (asm_simp_tac (!simpset addsimps [equivA RS UN_equiv_class,
1.176                                       congruent2_implies_congruent]) 1);
1.177  by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
1.178 -by (fast_tac rel_cs 1);
1.179 +by (Fast_tac 1);
1.180  qed "congruent2_implies_congruent_UN";
1.181
1.182  val prems as equivA::_ = goal Equiv.thy
1.183 @@ -236,7 +238,7 @@
1.184  \       !!x1 x2.  [| x1: A; x2: A |] ==> b x1 x2 : B |]    \
1.185  \    ==> (UN x1:X1. UN x2:X2. b x1 x2) : B";
1.186  by (cut_facts_tac prems 1);
1.187 -by (safe_tac rel_cs);
1.188 +by (safe_tac (!claset));
1.189  by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
1.190                               congruent2_implies_congruent_UN,
1.191                               congruent2_implies_congruent, quotientI]) 1));
1.192 @@ -251,7 +253,7 @@
1.193  \       !! y z w. [| w: A;  (y,z) : r |] ==> b w y = b w z       \
1.194  \    |] ==> congruent2 r b";
1.195  by (cut_facts_tac prems 1);
1.196 -by (safe_tac rel_cs);
1.197 +by (safe_tac (!claset));
1.198  by (rtac trans 1);
1.199  by (REPEAT (ares_tac prems 1
1.200       ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));
```