author paulson Sat Sep 21 21:10:34 2002 +0200 (2002-09-21) changeset 13574 f9796358e66f parent 13573 37b22343c35a child 13575 ecb6ecd9af13
converted to Isar script
 src/ZF/Integ/EquivClass.ML file | annotate | diff | revisions src/ZF/Integ/EquivClass.thy file | annotate | diff | revisions src/ZF/IsaMakefile file | annotate | diff | revisions
```     1.1 --- a/src/ZF/Integ/EquivClass.ML	Fri Sep 20 11:49:38 2002 +0200
1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,246 +0,0 @@
1.4 -(*  Title:      ZF/EquivClass.ML
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 -Equivalence relations in Zermelo-Fraenkel Set Theory
1.10 -*)
1.11 -
1.12 -(*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***)
1.13 -
1.14 -(** first half: equiv(A,r) ==> converse(r) O r = r **)
1.15 -
1.16 -Goalw [trans_def,sym_def]
1.17 -    "[| sym(r); trans(r) |] ==> converse(r) O r <= r";
1.18 -by (Blast_tac 1);
1.19 -qed "sym_trans_comp_subset";
1.20 -
1.21 -Goalw [refl_def]
1.22 -    "[| refl(A,r); r <= A*A |] ==> r <= converse(r) O r";
1.23 -by (Blast_tac 1);
1.24 -qed "refl_comp_subset";
1.25 -
1.26 -Goalw [equiv_def]
1.27 -    "equiv(A,r) ==> converse(r) O r = r";
1.28 -by (blast_tac (subset_cs addSIs [sym_trans_comp_subset, refl_comp_subset]) 1);
1.29 -qed "equiv_comp_eq";
1.30 -
1.31 -(*second half*)
1.32 -Goalw [equiv_def,refl_def,sym_def,trans_def]
1.33 -    "[| converse(r) O r = r;  domain(r) = A |] ==> equiv(A,r)";
1.34 -by (etac equalityE 1);
1.35 -by (subgoal_tac "ALL x y. <x,y> : r --> <y,x> : r" 1);
1.36 -by (ALLGOALS Fast_tac);
1.37 -qed "comp_equivI";
1.38 -
1.39 -(** Equivalence classes **)
1.40 -
1.41 -(*Lemma for the next result*)
1.42 -Goalw [trans_def,sym_def]
1.43 -    "[| sym(r);  trans(r);  <a,b>: r |] ==> r``{a} <= r``{b}";
1.44 -by (Blast_tac 1);
1.45 -qed "equiv_class_subset";
1.46 -
1.47 -Goalw [equiv_def]
1.48 -    "[| equiv(A,r);  <a,b>: r |] ==> r``{a} = r``{b}";
1.49 -by (safe_tac (subset_cs addSIs [equalityI, equiv_class_subset]));
1.50 -by (rewtac sym_def);
1.51 -by (Blast_tac 1);
1.52 -qed "equiv_class_eq";
1.53 -
1.54 -Goalw [equiv_def,refl_def]
1.55 -    "[| equiv(A,r);  a: A |] ==> a: r``{a}";
1.56 -by (Blast_tac 1);
1.57 -qed "equiv_class_self";
1.58 -
1.59 -(*Lemma for the next result*)
1.60 -Goalw [equiv_def,refl_def]
1.61 -    "[| equiv(A,r);  r``{b} <= r``{a};  b: A |] ==> <a,b>: r";
1.62 -by (Blast_tac 1);
1.63 -qed "subset_equiv_class";
1.64 -
1.65 -Goal "[| r``{a} = r``{b};  equiv(A,r);  b: A |] ==> <a,b>: r";
1.66 -by (REPEAT (ares_tac[equalityD2, subset_equiv_class] 1));
1.67 -qed "eq_equiv_class";
1.68 -
1.69 -(*thus r``{a} = r``{b} as well*)
1.70 -Goalw [equiv_def,trans_def,sym_def]
1.71 -    "[| equiv(A,r);  x: (r``{a} Int r``{b}) |] ==> <a,b>: r";
1.72 -by (Blast_tac 1);
1.73 -qed "equiv_class_nondisjoint";
1.74 -
1.75 -Goalw [equiv_def] "equiv(A,r) ==> r <= A*A";
1.76 -by (safe_tac subset_cs);
1.77 -qed "equiv_type";
1.78 -
1.79 -Goal "equiv(A,r) ==> <x,y>: r <-> r``{x} = r``{y} & x:A & y:A";
1.80 -by (blast_tac (claset() addIs [eq_equiv_class, equiv_class_eq]
1.82 -qed "equiv_class_eq_iff";
1.83 -
1.84 -Goal "[| equiv(A,r);  x: A;  y: A |] ==> r``{x} = r``{y} <-> <x,y>: r";
1.85 -by (blast_tac (claset() addIs [eq_equiv_class, equiv_class_eq]
1.87 -qed "eq_equiv_class_iff";
1.88 -
1.89 -(*** Quotients ***)
1.90 -
1.91 -(** Introduction/elimination rules -- needed? **)
1.92 -
1.93 -Goalw [quotient_def] "x:A ==> r``{x}: A//r";
1.94 -by (etac RepFunI 1);
1.95 -qed "quotientI";
1.97 -
1.98 -val major::prems = Goalw [quotient_def]
1.99 -    "[| X: A//r;  !!x. [| X = r``{x};  x:A |] ==> P |]   \
1.100 -\    ==> P";
1.101 -by (rtac (major RS RepFunE) 1);
1.102 -by (eresolve_tac prems 1);
1.103 -by (assume_tac 1);
1.104 -qed "quotientE";
1.105 -
1.106 -Goalw [equiv_def,refl_def,quotient_def]
1.107 -    "equiv(A,r) ==> Union(A//r) = A";
1.108 -by (Blast_tac 1);
1.109 -qed "Union_quotient";
1.110 -
1.111 -Goalw [quotient_def]
1.112 -    "[| equiv(A,r);  X: A//r;  Y: A//r |] ==> X=Y | (X Int Y <= 0)";
1.113 -by (safe_tac (claset() addSIs [equiv_class_eq]));
1.114 -by (assume_tac 1);
1.115 -by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
1.116 -by (Blast_tac 1);
1.117 -qed "quotient_disj";
1.118 -
1.119 -(**** Defining unary operations upon equivalence classes ****)
1.120 -
1.121 -(** These proofs really require as local premises
1.122 -     equiv(A,r);  congruent(r,b)
1.123 -**)
1.124 -
1.125 -(*Conversion rule*)
1.126 -val prems as [equivA,bcong,_] = goal (the_context ())
1.127 -    "[| equiv(A,r);  congruent(r,b);  a: A |] ==> (UN x:r``{a}. b(x)) = b(a)";
1.128 -by (cut_facts_tac prems 1);
1.129 -by (rtac ([refl RS UN_cong, UN_constant] MRS trans) 1);
1.130 -by (etac equiv_class_self 2);
1.131 -by (assume_tac 2);
1.132 -by (rewrite_goals_tac [equiv_def,sym_def,congruent_def]);
1.133 -by (Blast_tac 1);
1.134 -qed "UN_equiv_class";
1.135 -
1.136 -(*type checking of  UN x:r``{a}. b(x) *)
1.137 -val prems = Goalw [quotient_def]
1.138 -    "[| equiv(A,r);  congruent(r,b);  X: A//r;   \
1.139 -\       !!x.  x : A ==> b(x) : B |]     \
1.140 -\    ==> (UN x:X. b(x)) : B";
1.141 -by (cut_facts_tac prems 1);
1.142 -by Safe_tac;
1.143 -by (asm_simp_tac (simpset() addsimps UN_equiv_class::prems) 1);
1.144 -qed "UN_equiv_class_type";
1.145 -
1.146 -(*Sufficient conditions for injectiveness.  Could weaken premises!
1.147 -  major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
1.148 -*)
1.149 -val prems = Goalw [quotient_def]
1.150 -    "[| equiv(A,r);   congruent(r,b);  \
1.151 -\       (UN x:X. b(x))=(UN y:Y. b(y));  X: A//r;  Y: A//r;  \
1.152 -\       !!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |]         \
1.153 -\    ==> X=Y";
1.154 -by (cut_facts_tac prems 1);
1.155 -by Safe_tac;
1.156 -by (rtac equiv_class_eq 1);
1.157 -by (REPEAT (ares_tac prems 1));
1.158 -by (etac box_equals 1);
1.159 -by (REPEAT (ares_tac [UN_equiv_class] 1));
1.160 -qed "UN_equiv_class_inject";
1.161 -
1.162 -
1.163 -(**** Defining binary operations upon equivalence classes ****)
1.164 -
1.165 -
1.166 -Goalw [congruent_def,congruent2_def,equiv_def,refl_def]
1.167 -    "[| equiv(A,r);  congruent2(r,b);  a: A |] ==> congruent(r,b(a))";
1.168 -by (Blast_tac 1);
1.169 -qed "congruent2_implies_congruent";
1.170 -
1.171 -val equivA::prems = goalw (the_context ()) [congruent_def]
1.172 -    "[| equiv(A,r);  congruent2(r,b);  a: A |] ==> \
1.173 -\    congruent(r, %x1. UN x2:r``{a}. b(x1,x2))";
1.174 -by (cut_facts_tac (equivA::prems) 1);
1.175 -by Safe_tac;
1.176 -by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
1.177 -by (assume_tac 1);
1.178 -by (asm_simp_tac (simpset() addsimps [equivA RS UN_equiv_class,
1.179 -                                     congruent2_implies_congruent]) 1);
1.180 -by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
1.181 -by (Blast_tac 1);
1.182 -qed "congruent2_implies_congruent_UN";
1.183 -
1.184 -val prems as equivA::_ = goal (the_context ())
1.185 -    "[| equiv(A,r);  congruent2(r,b);  a1: A;  a2: A |]  \
1.186 -\    ==> (UN x1:r``{a1}. UN x2:r``{a2}. b(x1,x2)) = b(a1,a2)";
1.187 -by (cut_facts_tac prems 1);
1.188 -by (asm_simp_tac (simpset() addsimps [equivA RS UN_equiv_class,
1.189 -                                     congruent2_implies_congruent,
1.190 -                                     congruent2_implies_congruent_UN]) 1);
1.191 -qed "UN_equiv_class2";
1.192 -
1.193 -(*type checking*)
1.194 -val prems = Goalw [quotient_def]
1.195 -    "[| equiv(A,r);  congruent2(r,b);                   \
1.196 -\       X1: A//r;  X2: A//r;                            \
1.197 -\       !!x1 x2.  [| x1: A; x2: A |] ==> b(x1,x2) : B   \
1.198 -\    |] ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B";
1.199 -by (cut_facts_tac prems 1);
1.200 -by Safe_tac;
1.201 -by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
1.202 -                             congruent2_implies_congruent_UN,
1.203 -                             congruent2_implies_congruent, quotientI]) 1));
1.204 -qed "UN_equiv_class_type2";
1.205 -
1.206 -
1.207 -(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
1.208 -  than the direct proof*)
1.209 -val prems = Goalw [congruent2_def,equiv_def,refl_def]
1.210 -    "[| equiv(A,r);     \
1.211 -\       !! y z w. [| w: A;  <y,z> : r |] ==> b(y,w) = b(z,w);      \
1.212 -\       !! y z w. [| w: A;  <y,z> : r |] ==> b(w,y) = b(w,z)       \
1.213 -\    |] ==> congruent2(r,b)";
1.214 -by (cut_facts_tac prems 1);
1.215 -by Safe_tac;
1.216 -by (rtac trans 1);
1.217 -by (REPEAT (ares_tac prems 1
1.218 -     ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));
1.219 -qed "congruent2I";
1.220 -
1.221 -val [equivA,commute,congt] = Goal
1.222 -    "[| equiv(A,r);     \
1.223 -\       !! y z. [| y: A;  z: A |] ==> b(y,z) = b(z,y);        \
1.224 -\       !! y z w. [| w: A;  <y,z>: r |] ==> b(w,y) = b(w,z)     \
1.225 -\    |] ==> congruent2(r,b)";
1.226 -by (resolve_tac [equivA RS congruent2I] 1);
1.227 -by (rtac (commute RS trans) 1);
1.228 -by (rtac (commute RS trans RS sym) 3);
1.229 -by (rtac sym 5);
1.230 -by (REPEAT (ares_tac [congt] 1
1.231 -     ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
1.232 -qed "congruent2_commuteI";
1.233 -
1.234 -(*Obsolete?*)
1.235 -val [equivA,ZinA,congt,commute] = Goalw [quotient_def]
1.236 -    "[| equiv(A,r);  Z: A//r;  \
1.237 -\       !!w. [| w: A |] ==> congruent(r, %z. b(w,z));    \
1.238 -\       !!x y. [| x: A;  y: A |] ==> b(y,x) = b(x,y)    \
1.239 -\    |] ==> congruent(r, %w. UN z: Z. b(w,z))";
1.240 -val congt' = rewrite_rule [congruent_def] congt;
1.241 -by (cut_facts_tac [ZinA] 1);
1.242 -by (rewtac congruent_def);
1.243 -by Safe_tac;
1.244 -by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
1.245 -by (assume_tac 1);
1.246 -by (asm_simp_tac (simpset() addsimps [commute,
1.247 -                                     [equivA, congt] MRS UN_equiv_class]) 1);
1.248 -by (REPEAT (ares_tac [congt' RS spec RS spec RS mp] 1));
1.249 -qed "congruent_commuteI";
```
```     2.1 --- a/src/ZF/Integ/EquivClass.thy	Fri Sep 20 11:49:38 2002 +0200
2.2 +++ b/src/ZF/Integ/EquivClass.thy	Sat Sep 21 21:10:34 2002 +0200
2.3 @@ -3,21 +3,260 @@
2.4      Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
2.5      Copyright   1994  University of Cambridge
2.6
2.7 -Equivalence relations in Zermelo-Fraenkel Set Theory
2.8  *)
2.9
2.10 -EquivClass = Trancl + Perm +
2.12 +
2.13 +theory EquivClass = Trancl + Perm:
2.14
2.15  constdefs
2.16
2.17 -  quotient    :: [i,i]=>i    (infixl "'/'/" 90)  (*set of equiv classes*)
2.18 +  quotient   :: "[i,i]=>i"    (infixl "'/'/" 90)  (*set of equiv classes*)
2.19        "A//r == {r``{x} . x:A}"
2.20
2.21 -  congruent   :: [i,i=>i]=>o
2.22 +  congruent  :: "[i,i=>i]=>o"
2.23        "congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)"
2.24
2.25 -  congruent2  :: [i,[i,i]=>i]=>o
2.26 -      "congruent2(r,b) == ALL y1 z1 y2 z2.
2.27 +  congruent2 :: "[i,[i,i]=>i]=>o"
2.28 +      "congruent2(r,b) == ALL y1 z1 y2 z2.
2.29             <y1,z1>:r --> <y2,z2>:r --> b(y1,y2) = b(z1,z2)"
2.30
2.31 +subsection{*Suppes, Theorem 70:
2.32 +    @{term r} is an equiv relation iff @{term "converse(r) O r = r"}*}
2.33 +
2.34 +(** first half: equiv(A,r) ==> converse(r) O r = r **)
2.35 +
2.36 +lemma sym_trans_comp_subset:
2.37 +    "[| sym(r); trans(r) |] ==> converse(r) O r <= r"
2.38 +apply (unfold trans_def sym_def, blast)
2.39 +done
2.40 +
2.41 +lemma refl_comp_subset:
2.42 +    "[| refl(A,r); r <= A*A |] ==> r <= converse(r) O r"
2.43 +apply (unfold refl_def, blast)
2.44 +done
2.45 +
2.46 +lemma equiv_comp_eq:
2.47 +    "equiv(A,r) ==> converse(r) O r = r"
2.48 +apply (unfold equiv_def)
2.49 +apply (blast del: subsetI
2.50 +             intro!: sym_trans_comp_subset refl_comp_subset)
2.51 +done
2.52 +
2.53 +(*second half*)
2.54 +lemma comp_equivI:
2.55 +    "[| converse(r) O r = r;  domain(r) = A |] ==> equiv(A,r)"
2.56 +apply (unfold equiv_def refl_def sym_def trans_def)
2.57 +apply (erule equalityE)
2.58 +apply (subgoal_tac "ALL x y. <x,y> : r --> <y,x> : r", blast+)
2.59 +done
2.60 +
2.61 +(** Equivalence classes **)
2.62 +
2.63 +(*Lemma for the next result*)
2.64 +lemma equiv_class_subset:
2.65 +    "[| sym(r);  trans(r);  <a,b>: r |] ==> r``{a} <= r``{b}"
2.66 +by (unfold trans_def sym_def, blast)
2.67 +
2.68 +lemma equiv_class_eq:
2.69 +    "[| equiv(A,r);  <a,b>: r |] ==> r``{a} = r``{b}"
2.70 +apply (unfold equiv_def)
2.71 +apply (safe del: subsetI intro!: equalityI equiv_class_subset)
2.72 +apply (unfold sym_def, blast)
2.73 +done
2.74 +
2.75 +lemma equiv_class_self:
2.76 +    "[| equiv(A,r);  a: A |] ==> a: r``{a}"
2.77 +by (unfold equiv_def refl_def, blast)
2.78 +
2.79 +(*Lemma for the next result*)
2.80 +lemma subset_equiv_class:
2.81 +    "[| equiv(A,r);  r``{b} <= r``{a};  b: A |] ==> <a,b>: r"
2.82 +by (unfold equiv_def refl_def, blast)
2.83 +
2.84 +lemma eq_equiv_class: "[| r``{a} = r``{b};  equiv(A,r);  b: A |] ==> <a,b>: r"
2.85 +by (assumption | rule equalityD2 subset_equiv_class)+
2.86 +
2.87 +(*thus r``{a} = r``{b} as well*)
2.88 +lemma equiv_class_nondisjoint:
2.89 +    "[| equiv(A,r);  x: (r``{a} Int r``{b}) |] ==> <a,b>: r"
2.90 +by (unfold equiv_def trans_def sym_def, blast)
2.91 +
2.92 +lemma equiv_type: "equiv(A,r) ==> r <= A*A"
2.93 +by (unfold equiv_def, blast)
2.94 +
2.95 +lemma equiv_class_eq_iff:
2.96 +     "equiv(A,r) ==> <x,y>: r <-> r``{x} = r``{y} & x:A & y:A"
2.97 +by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type)
2.98 +
2.99 +lemma eq_equiv_class_iff:
2.100 +     "[| equiv(A,r);  x: A;  y: A |] ==> r``{x} = r``{y} <-> <x,y>: r"
2.101 +by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type)
2.102 +
2.103 +(*** Quotients ***)
2.104 +
2.105 +(** Introduction/elimination rules -- needed? **)
2.106 +
2.107 +lemma quotientI [TC]: "x:A ==> r``{x}: A//r"
2.108 +apply (unfold quotient_def)
2.109 +apply (erule RepFunI)
2.110 +done
2.111 +
2.112 +lemma quotientE:
2.113 +    "[| X: A//r;  !!x. [| X = r``{x};  x:A |] ==> P |] ==> P"
2.114 +by (unfold quotient_def, blast)
2.115 +
2.116 +lemma Union_quotient:
2.117 +    "equiv(A,r) ==> Union(A//r) = A"
2.118 +by (unfold equiv_def refl_def quotient_def, blast)
2.119 +
2.120 +lemma quotient_disj:
2.121 +    "[| equiv(A,r);  X: A//r;  Y: A//r |] ==> X=Y | (X Int Y <= 0)"
2.122 +apply (unfold quotient_def)
2.123 +apply (safe intro!: equiv_class_eq, assumption)
2.124 +apply (unfold equiv_def trans_def sym_def, blast)
2.125 +done
2.126 +
2.127 +subsection{*Defining Unary Operations upon Equivalence Classes*}
2.128 +
2.129 +(** Could have a locale with the premises equiv(A,r)  and  congruent(r,b)
2.130 +**)
2.131 +
2.132 +(*Conversion rule*)
2.133 +lemma UN_equiv_class:
2.134 +    "[| equiv(A,r);  congruent(r,b);  a: A |] ==> (UN x:r``{a}. b(x)) = b(a)"
2.135 +apply (rule trans [OF refl [THEN UN_cong] UN_constant])
2.136 +apply (erule_tac [2] equiv_class_self)
2.137 +prefer 2 apply assumption
2.138 +apply (unfold equiv_def sym_def congruent_def, blast)
2.139 +done
2.140 +
2.141 +(*type checking of  UN x:r``{a}. b(x) *)
2.142 +lemma UN_equiv_class_type:
2.143 +    "[| equiv(A,r);  congruent(r,b);  X: A//r;  !!x.  x : A ==> b(x) : B |]
2.144 +     ==> (UN x:X. b(x)) : B"
2.145 +apply (unfold quotient_def, safe)
2.146 +apply (simp (no_asm_simp) add: UN_equiv_class)
2.147 +done
2.148 +
2.149 +(*Sufficient conditions for injectiveness.  Could weaken premises!
2.150 +  major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
2.151 +*)
2.152 +lemma UN_equiv_class_inject:
2.153 +    "[| equiv(A,r);   congruent(r,b);
2.154 +        (UN x:X. b(x))=(UN y:Y. b(y));  X: A//r;  Y: A//r;
2.155 +        !!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |]
2.156 +     ==> X=Y"
2.157 +apply (unfold quotient_def, safe)
2.158 +apply (rule equiv_class_eq, assumption)
2.159 +apply (rotate_tac -2)
2.160 +apply (simp add: UN_equiv_class [of A r b])
2.161 +done
2.162 +
2.163 +
2.164 +subsection{*Defining Binary Operations upon Equivalence Classes*}
2.165 +
2.166 +
2.167 +lemma congruent2_implies_congruent:
2.168 +    "[| equiv(A,r);  congruent2(r,b);  a: A |] ==> congruent(r,b(a))"
2.169 +apply (unfold congruent_def congruent2_def equiv_def refl_def, blast)
2.170 +done
2.171 +
2.172 +lemma congruent2_implies_congruent_UN:
2.173 +    "[| equiv(A,r);  congruent2(r,b);  a: A |] ==>
2.174 +     congruent(r, %x1. UN x2:r``{a}. b(x1,x2))"
2.175 +apply (unfold congruent_def, safe)
2.176 +apply (frule equiv_type [THEN subsetD], assumption)
2.177 +apply clarify
2.178 +apply (simp add: UN_equiv_class [of A r] congruent2_implies_congruent)
2.179 +apply (unfold congruent2_def equiv_def refl_def, blast)
2.180 +done
2.181 +
2.182 +lemma UN_equiv_class2:
2.183 +    "[| equiv(A,r);  congruent2(r,b);  a1: A;  a2: A |]
2.184 +     ==> (UN x1:r``{a1}. UN x2:r``{a2}. b(x1,x2)) = b(a1,a2)"
2.185 +by (simp add: UN_equiv_class [of A r] congruent2_implies_congruent
2.186 +              congruent2_implies_congruent_UN)
2.187 +
2.188 +(*type checking*)
2.189 +lemma UN_equiv_class_type2:
2.190 +    "[| equiv(A,r);  congruent2(r,b);
2.191 +        X1: A//r;  X2: A//r;
2.192 +        !!x1 x2.  [| x1: A; x2: A |] ==> b(x1,x2) : B
2.193 +     |] ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B"
2.194 +apply (unfold quotient_def, safe)
2.195 +apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
2.196 +                    congruent2_implies_congruent quotientI)
2.197 +done
2.198 +
2.199 +
2.200 +(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
2.201 +  than the direct proof*)
2.202 +lemma congruent2I:
2.203 +    "[| equiv(A,r);
2.204 +        !! y z w. [| w: A;  <y,z> : r |] ==> b(y,w) = b(z,w);
2.205 +        !! y z w. [| w: A;  <y,z> : r |] ==> b(w,y) = b(w,z)
2.206 +     |] ==> congruent2(r,b)"
2.207 +apply (unfold congruent2_def equiv_def refl_def, safe)
2.208 +apply (blast intro: trans)
2.209 +done
2.210 +
2.211 +lemma congruent2_commuteI:
2.212 + assumes equivA: "equiv(A,r)"
2.213 +     and commute: "!! y z. [| y: A;  z: A |] ==> b(y,z) = b(z,y)"
2.214 +     and congt:   "!! y z w. [| w: A;  <y,z>: r |] ==> b(w,y) = b(w,z)"
2.215 + shows "congruent2(r,b)"
2.216 +apply (insert equivA [THEN equiv_type, THEN subsetD])
2.217 +apply (rule congruent2I [OF equivA])
2.218 +apply (rule commute [THEN trans])
2.219 +apply (rule_tac [3] commute [THEN trans, symmetric])
2.220 +apply (rule_tac [5] sym)
2.221 +apply (blast intro: congt)+
2.222 +done
2.223 +
2.224 +(*Obsolete?*)
2.225 +lemma congruent_commuteI:
2.226 +    "[| equiv(A,r);  Z: A//r;
2.227 +        !!w. [| w: A |] ==> congruent(r, %z. b(w,z));
2.228 +        !!x y. [| x: A;  y: A |] ==> b(y,x) = b(x,y)
2.229 +     |] ==> congruent(r, %w. UN z: Z. b(w,z))"
2.230 +apply (simp (no_asm) add: congruent_def)
2.231 +apply (safe elim!: quotientE)
2.232 +apply (frule equiv_type [THEN subsetD], assumption)
2.233 +apply (simp add: UN_equiv_class [of A r])
2.235 +done
2.236 +
2.237 +ML
2.238 +{*
2.239 +val sym_trans_comp_subset = thm "sym_trans_comp_subset";
2.240 +val refl_comp_subset = thm "refl_comp_subset";
2.241 +val equiv_comp_eq = thm "equiv_comp_eq";
2.242 +val comp_equivI = thm "comp_equivI";
2.243 +val equiv_class_subset = thm "equiv_class_subset";
2.244 +val equiv_class_eq = thm "equiv_class_eq";
2.245 +val equiv_class_self = thm "equiv_class_self";
2.246 +val subset_equiv_class = thm "subset_equiv_class";
2.247 +val eq_equiv_class = thm "eq_equiv_class";
2.248 +val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";
2.249 +val equiv_type = thm "equiv_type";
2.250 +val equiv_class_eq_iff = thm "equiv_class_eq_iff";
2.251 +val eq_equiv_class_iff = thm "eq_equiv_class_iff";
2.252 +val quotientI = thm "quotientI";
2.253 +val quotientE = thm "quotientE";
2.254 +val Union_quotient = thm "Union_quotient";
2.255 +val quotient_disj = thm "quotient_disj";
2.256 +val UN_equiv_class = thm "UN_equiv_class";
2.257 +val UN_equiv_class_type = thm "UN_equiv_class_type";
2.258 +val UN_equiv_class_inject = thm "UN_equiv_class_inject";
2.259 +val congruent2_implies_congruent = thm "congruent2_implies_congruent";
2.260 +val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";
2.261 +val congruent_commuteI = thm "congruent_commuteI";
2.262 +val UN_equiv_class2 = thm "UN_equiv_class2";
2.263 +val UN_equiv_class_type2 = thm "UN_equiv_class_type2";
2.264 +val congruent2I = thm "congruent2I";
2.265 +val congruent2_commuteI = thm "congruent2_commuteI";
2.266 +val congruent_commuteI = thm "congruent_commuteI";
2.267 +*}
2.268 +
2.269  end
```
```     3.1 --- a/src/ZF/IsaMakefile	Fri Sep 20 11:49:38 2002 +0200
3.2 +++ b/src/ZF/IsaMakefile	Sat Sep 21 21:10:34 2002 +0200
3.3 @@ -33,7 +33,7 @@
3.4    CardinalArith.thy Cardinal_AC.thy \
3.5    Datatype.ML Datatype.thy Epsilon.thy Finite.thy	\
3.6    Fixedpt.thy Inductive.ML Inductive.thy 	\
3.7 -  InfDatatype.thy Integ/Bin.thy Integ/EquivClass.ML	\
3.8 +  InfDatatype.thy Integ/Bin.thy \
3.9    Integ/EquivClass.thy Integ/Int.thy Integ/IntArith.thy	\
3.10    Integ/IntDiv.thy Integ/int_arith.ML			\
3.11    Let.ML Let.thy List.thy Main.ML Main.thy	\
```