src/ZF/EquivClass.thy
 changeset 23146 0bc590051d95 child 24892 c663e675e177
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/ZF/EquivClass.thy	Thu May 31 12:06:31 2007 +0200
1.3 @@ -0,0 +1,265 @@
1.4 +(*  Title:      ZF/EquivClass.thy
1.5 +    ID:         \$Id\$
1.6 +    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
1.7 +    Copyright   1994  University of Cambridge
1.8 +
1.9 +*)
1.10 +
1.12 +
1.13 +theory EquivClass imports Trancl Perm begin
1.14 +
1.15 +constdefs
1.16 +
1.17 +  quotient   :: "[i,i]=>i"    (infixl "'/'/" 90)  (*set of equiv classes*)
1.18 +      "A//r == {r``{x} . x:A}"
1.19 +
1.20 +  congruent  :: "[i,i=>i]=>o"
1.21 +      "congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)"
1.22 +
1.23 +  congruent2 :: "[i,i,[i,i]=>i]=>o"
1.24 +      "congruent2(r1,r2,b) == ALL y1 z1 y2 z2.
1.25 +           <y1,z1>:r1 --> <y2,z2>:r2 --> b(y1,y2) = b(z1,z2)"
1.26 +
1.27 +syntax
1.28 +  RESPECTS ::"[i=>i, i] => o"  (infixr "respects" 80)
1.29 +  RESPECTS2 ::"[i=>i, i] => o"  (infixr "respects2 " 80)
1.30 +    --{*Abbreviation for the common case where the relations are identical*}
1.31 +
1.32 +translations
1.33 +  "f respects r" == "congruent(r,f)"
1.34 +  "f respects2 r" => "congruent2(r,r,f)"
1.35 +
1.36 +subsection{*Suppes, Theorem 70:
1.37 +    @{term r} is an equiv relation iff @{term "converse(r) O r = r"}*}
1.38 +
1.39 +(** first half: equiv(A,r) ==> converse(r) O r = r **)
1.40 +
1.41 +lemma sym_trans_comp_subset:
1.42 +    "[| sym(r); trans(r) |] ==> converse(r) O r <= r"
1.43 +by (unfold trans_def sym_def, blast)
1.44 +
1.45 +lemma refl_comp_subset:
1.46 +    "[| refl(A,r); r <= A*A |] ==> r <= converse(r) O r"
1.47 +by (unfold refl_def, blast)
1.48 +
1.49 +lemma equiv_comp_eq:
1.50 +    "equiv(A,r) ==> converse(r) O r = r"
1.51 +apply (unfold equiv_def)
1.52 +apply (blast del: subsetI intro!: sym_trans_comp_subset refl_comp_subset)
1.53 +done
1.54 +
1.55 +(*second half*)
1.56 +lemma comp_equivI:
1.57 +    "[| converse(r) O r = r;  domain(r) = A |] ==> equiv(A,r)"
1.58 +apply (unfold equiv_def refl_def sym_def trans_def)
1.59 +apply (erule equalityE)
1.60 +apply (subgoal_tac "ALL x y. <x,y> : r --> <y,x> : r", blast+)
1.61 +done
1.62 +
1.63 +(** Equivalence classes **)
1.64 +
1.65 +(*Lemma for the next result*)
1.66 +lemma equiv_class_subset:
1.67 +    "[| sym(r);  trans(r);  <a,b>: r |] ==> r``{a} <= r``{b}"
1.68 +by (unfold trans_def sym_def, blast)
1.69 +
1.70 +lemma equiv_class_eq:
1.71 +    "[| equiv(A,r);  <a,b>: r |] ==> r``{a} = r``{b}"
1.72 +apply (unfold equiv_def)
1.73 +apply (safe del: subsetI intro!: equalityI equiv_class_subset)
1.74 +apply (unfold sym_def, blast)
1.75 +done
1.76 +
1.77 +lemma equiv_class_self:
1.78 +    "[| equiv(A,r);  a: A |] ==> a: r``{a}"
1.79 +by (unfold equiv_def refl_def, blast)
1.80 +
1.81 +(*Lemma for the next result*)
1.82 +lemma subset_equiv_class:
1.83 +    "[| equiv(A,r);  r``{b} <= r``{a};  b: A |] ==> <a,b>: r"
1.84 +by (unfold equiv_def refl_def, blast)
1.85 +
1.86 +lemma eq_equiv_class: "[| r``{a} = r``{b};  equiv(A,r);  b: A |] ==> <a,b>: r"
1.87 +by (assumption | rule equalityD2 subset_equiv_class)+
1.88 +
1.89 +(*thus r``{a} = r``{b} as well*)
1.90 +lemma equiv_class_nondisjoint:
1.91 +    "[| equiv(A,r);  x: (r``{a} Int r``{b}) |] ==> <a,b>: r"
1.92 +by (unfold equiv_def trans_def sym_def, blast)
1.93 +
1.94 +lemma equiv_type: "equiv(A,r) ==> r <= A*A"
1.95 +by (unfold equiv_def, blast)
1.96 +
1.97 +lemma equiv_class_eq_iff:
1.98 +     "equiv(A,r) ==> <x,y>: r <-> r``{x} = r``{y} & x:A & y:A"
1.99 +by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type)
1.100 +
1.101 +lemma eq_equiv_class_iff:
1.102 +     "[| equiv(A,r);  x: A;  y: A |] ==> r``{x} = r``{y} <-> <x,y>: r"
1.103 +by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type)
1.104 +
1.105 +(*** Quotients ***)
1.106 +
1.107 +(** Introduction/elimination rules -- needed? **)
1.108 +
1.109 +lemma quotientI [TC]: "x:A ==> r``{x}: A//r"
1.110 +apply (unfold quotient_def)
1.111 +apply (erule RepFunI)
1.112 +done
1.113 +
1.114 +lemma quotientE:
1.115 +    "[| X: A//r;  !!x. [| X = r``{x};  x:A |] ==> P |] ==> P"
1.116 +by (unfold quotient_def, blast)
1.117 +
1.118 +lemma Union_quotient:
1.119 +    "equiv(A,r) ==> Union(A//r) = A"
1.120 +by (unfold equiv_def refl_def quotient_def, blast)
1.121 +
1.122 +lemma quotient_disj:
1.123 +    "[| equiv(A,r);  X: A//r;  Y: A//r |] ==> X=Y | (X Int Y <= 0)"
1.124 +apply (unfold quotient_def)
1.125 +apply (safe intro!: equiv_class_eq, assumption)
1.126 +apply (unfold equiv_def trans_def sym_def, blast)
1.127 +done
1.128 +
1.129 +subsection{*Defining Unary Operations upon Equivalence Classes*}
1.130 +
1.131 +(** Could have a locale with the premises equiv(A,r)  and  congruent(r,b)
1.132 +**)
1.133 +
1.134 +(*Conversion rule*)
1.135 +lemma UN_equiv_class:
1.136 +    "[| equiv(A,r);  b respects r;  a: A |] ==> (UN x:r``{a}. b(x)) = b(a)"
1.137 +apply (subgoal_tac "\<forall>x \<in> r``{a}. b(x) = b(a)")
1.138 + apply simp
1.139 + apply (blast intro: equiv_class_self)
1.140 +apply (unfold equiv_def sym_def congruent_def, blast)
1.141 +done
1.142 +
1.143 +(*type checking of  UN x:r``{a}. b(x) *)
1.144 +lemma UN_equiv_class_type:
1.145 +    "[| equiv(A,r);  b respects r;  X: A//r;  !!x.  x : A ==> b(x) : B |]
1.146 +     ==> (UN x:X. b(x)) : B"
1.147 +apply (unfold quotient_def, safe)
1.148 +apply (simp (no_asm_simp) add: UN_equiv_class)
1.149 +done
1.150 +
1.151 +(*Sufficient conditions for injectiveness.  Could weaken premises!
1.152 +  major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
1.153 +*)
1.154 +lemma UN_equiv_class_inject:
1.155 +    "[| equiv(A,r);   b respects r;
1.156 +        (UN x:X. b(x))=(UN y:Y. b(y));  X: A//r;  Y: A//r;
1.157 +        !!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |]
1.158 +     ==> X=Y"
1.159 +apply (unfold quotient_def, safe)
1.160 +apply (rule equiv_class_eq, assumption)
1.161 +apply (simp add: UN_equiv_class [of A r b])
1.162 +done
1.163 +
1.164 +
1.165 +subsection{*Defining Binary Operations upon Equivalence Classes*}
1.166 +
1.167 +lemma congruent2_implies_congruent:
1.168 +    "[| equiv(A,r1);  congruent2(r1,r2,b);  a: A |] ==> congruent(r2,b(a))"
1.169 +by (unfold congruent_def congruent2_def equiv_def refl_def, blast)
1.170 +
1.171 +lemma congruent2_implies_congruent_UN:
1.172 +    "[| equiv(A1,r1);  equiv(A2,r2);  congruent2(r1,r2,b);  a: A2 |] ==>
1.173 +     congruent(r1, %x1. \<Union>x2 \<in> r2``{a}. b(x1,x2))"
1.174 +apply (unfold congruent_def, safe)
1.175 +apply (frule equiv_type [THEN subsetD], assumption)
1.176 +apply clarify
1.177 +apply (simp add: UN_equiv_class congruent2_implies_congruent)
1.178 +apply (unfold congruent2_def equiv_def refl_def, blast)
1.179 +done
1.180 +
1.181 +lemma UN_equiv_class2:
1.182 +    "[| equiv(A1,r1);  equiv(A2,r2);  congruent2(r1,r2,b);  a1: A1;  a2: A2 |]
1.183 +     ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. b(x1,x2)) = b(a1,a2)"
1.184 +by (simp add: UN_equiv_class congruent2_implies_congruent
1.185 +              congruent2_implies_congruent_UN)
1.186 +
1.187 +(*type checking*)
1.188 +lemma UN_equiv_class_type2:
1.189 +    "[| equiv(A,r);  b respects2 r;
1.190 +        X1: A//r;  X2: A//r;
1.191 +        !!x1 x2.  [| x1: A; x2: A |] ==> b(x1,x2) : B
1.192 +     |] ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B"
1.193 +apply (unfold quotient_def, safe)
1.194 +apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
1.195 +                    congruent2_implies_congruent quotientI)
1.196 +done
1.197 +
1.198 +
1.199 +(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
1.200 +  than the direct proof*)
1.201 +lemma congruent2I:
1.202 +    "[|  equiv(A1,r1);  equiv(A2,r2);
1.203 +        !! y z w. [| w \<in> A2;  <y,z> \<in> r1 |] ==> b(y,w) = b(z,w);
1.204 +        !! y z w. [| w \<in> A1;  <y,z> \<in> r2 |] ==> b(w,y) = b(w,z)
1.205 +     |] ==> congruent2(r1,r2,b)"
1.206 +apply (unfold congruent2_def equiv_def refl_def, safe)
1.207 +apply (blast intro: trans)
1.208 +done
1.209 +
1.210 +lemma congruent2_commuteI:
1.211 + assumes equivA: "equiv(A,r)"
1.212 +     and commute: "!! y z. [| y: A;  z: A |] ==> b(y,z) = b(z,y)"
1.213 +     and congt:   "!! y z w. [| w: A;  <y,z>: r |] ==> b(w,y) = b(w,z)"
1.214 + shows "b respects2 r"
1.215 +apply (insert equivA [THEN equiv_type, THEN subsetD])
1.216 +apply (rule congruent2I [OF equivA equivA])
1.217 +apply (rule commute [THEN trans])
1.218 +apply (rule_tac [3] commute [THEN trans, symmetric])
1.219 +apply (rule_tac [5] sym)
1.220 +apply (blast intro: congt)+
1.221 +done
1.222 +
1.223 +(*Obsolete?*)
1.224 +lemma congruent_commuteI:
1.225 +    "[| equiv(A,r);  Z: A//r;
1.226 +        !!w. [| w: A |] ==> congruent(r, %z. b(w,z));
1.227 +        !!x y. [| x: A;  y: A |] ==> b(y,x) = b(x,y)
1.228 +     |] ==> congruent(r, %w. UN z: Z. b(w,z))"
1.229 +apply (simp (no_asm) add: congruent_def)
1.230 +apply (safe elim!: quotientE)
1.231 +apply (frule equiv_type [THEN subsetD], assumption)
1.232 +apply (simp add: UN_equiv_class [of A r])
1.233 +apply (simp add: congruent_def)
1.234 +done
1.235 +
1.236 +ML
1.237 +{*
1.238 +val sym_trans_comp_subset = thm "sym_trans_comp_subset";
1.239 +val refl_comp_subset = thm "refl_comp_subset";
1.240 +val equiv_comp_eq = thm "equiv_comp_eq";
1.241 +val comp_equivI = thm "comp_equivI";
1.242 +val equiv_class_subset = thm "equiv_class_subset";
1.243 +val equiv_class_eq = thm "equiv_class_eq";
1.244 +val equiv_class_self = thm "equiv_class_self";
1.245 +val subset_equiv_class = thm "subset_equiv_class";
1.246 +val eq_equiv_class = thm "eq_equiv_class";
1.247 +val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";
1.248 +val equiv_type = thm "equiv_type";
1.249 +val equiv_class_eq_iff = thm "equiv_class_eq_iff";
1.250 +val eq_equiv_class_iff = thm "eq_equiv_class_iff";
1.251 +val quotientI = thm "quotientI";
1.252 +val quotientE = thm "quotientE";
1.253 +val Union_quotient = thm "Union_quotient";
1.254 +val quotient_disj = thm "quotient_disj";
1.255 +val UN_equiv_class = thm "UN_equiv_class";
1.256 +val UN_equiv_class_type = thm "UN_equiv_class_type";
1.257 +val UN_equiv_class_inject = thm "UN_equiv_class_inject";
1.258 +val congruent2_implies_congruent = thm "congruent2_implies_congruent";
1.259 +val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";
1.260 +val congruent_commuteI = thm "congruent_commuteI";
1.261 +val UN_equiv_class2 = thm "UN_equiv_class2";
1.262 +val UN_equiv_class_type2 = thm "UN_equiv_class_type2";
1.263 +val congruent2I = thm "congruent2I";
1.264 +val congruent2_commuteI = thm "congruent2_commuteI";
1.265 +val congruent_commuteI = thm "congruent_commuteI";
1.266 +*}
1.267 +
1.268 +end
```