src/ZF/EquivClass.ML
author paulson
Fri, 16 Feb 1996 18:00:47 +0100
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child 2469 b50b8c0eec01
permissions -rw-r--r--
Elimination of fully-functorial style. Type tactic changed to a type abbrevation (from a datatype). Constructor tactic and function apply deleted.
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(*  Title:      ZF/EquivClass.ML
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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For EquivClass.thy.  Equivalence relations in Zermelo-Fraenkel Set Theory 
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*)
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val RSLIST = curry (op MRS);
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open EquivClass;
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(*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***)
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(** first half: equiv(A,r) ==> converse(r) O r = r **)
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goalw EquivClass.thy [trans_def,sym_def]
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    "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
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by (fast_tac (ZF_cs addSEs [converseD,compE]) 1);
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qed "sym_trans_comp_subset";
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goalw EquivClass.thy [refl_def]
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    "!!A r. [| refl(A,r); r <= A*A |] ==> r <= converse(r) O r";
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by (fast_tac (ZF_cs addSIs [converseI] addIs [compI]) 1);
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qed "refl_comp_subset";
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goalw EquivClass.thy [equiv_def]
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    "!!A r. equiv(A,r) ==> converse(r) O r = r";
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by (fast_tac (subset_cs addSIs [equalityI, sym_trans_comp_subset, 
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                                refl_comp_subset]) 1);
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qed "equiv_comp_eq";
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(*second half*)
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goalw EquivClass.thy [equiv_def,refl_def,sym_def,trans_def]
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    "!!A r. [| converse(r) O r = r;  domain(r) = A |] ==> equiv(A,r)";
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by (etac equalityE 1);
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by (subgoal_tac "ALL x y. <x,y> : r --> <y,x> : r" 1);
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by (safe_tac ZF_cs);
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by (fast_tac (ZF_cs addSIs [converseI] addIs [compI]) 3);
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by (ALLGOALS (fast_tac 
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              (ZF_cs addSIs [converseI] addIs [compI] addSEs [compE])));
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qed "comp_equivI";
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(** Equivalence classes **)
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(*Lemma for the next result*)
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goalw EquivClass.thy [trans_def,sym_def]
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    "!!A r. [| sym(r);  trans(r);  <a,b>: r |] ==> r``{a} <= r``{b}";
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by (fast_tac ZF_cs 1);
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qed "equiv_class_subset";
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goalw EquivClass.thy [equiv_def]
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    "!!A r. [| equiv(A,r);  <a,b>: r |] ==> r``{a} = r``{b}";
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by (safe_tac (subset_cs addSIs [equalityI, equiv_class_subset]));
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by (rewtac sym_def);
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by (fast_tac ZF_cs 1);
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qed "equiv_class_eq";
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goalw EquivClass.thy [equiv_def,refl_def]
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    "!!A r. [| equiv(A,r);  a: A |] ==> a: r``{a}";
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by (fast_tac ZF_cs 1);
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qed "equiv_class_self";
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(*Lemma for the next result*)
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goalw EquivClass.thy [equiv_def,refl_def]
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    "!!A r. [| equiv(A,r);  r``{b} <= r``{a};  b: A |] ==> <a,b>: r";
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by (fast_tac ZF_cs 1);
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qed "subset_equiv_class";
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val prems = goal EquivClass.thy
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    "[| r``{a} = r``{b};  equiv(A,r);  b: A |] ==> <a,b>: r";
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by (REPEAT (resolve_tac (prems @ [equalityD2, subset_equiv_class]) 1));
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qed "eq_equiv_class";
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(*thus r``{a} = r``{b} as well*)
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goalw EquivClass.thy [equiv_def,trans_def,sym_def]
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    "!!A r. [| equiv(A,r);  x: (r``{a} Int r``{b}) |] ==> <a,b>: r";
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by (fast_tac ZF_cs 1);
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qed "equiv_class_nondisjoint";
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goalw EquivClass.thy [equiv_def] "!!A r. equiv(A,r) ==> r <= A*A";
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by (safe_tac subset_cs);
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qed "equiv_type";
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goal EquivClass.thy
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    "!!A r. equiv(A,r) ==> <x,y>: r <-> r``{x} = r``{y} & x:A & y:A";
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by (fast_tac (ZF_cs addIs [eq_equiv_class, equiv_class_eq]
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                    addDs [equiv_type]) 1);
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qed "equiv_class_eq_iff";
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goal EquivClass.thy
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    "!!A r. [| equiv(A,r);  x: A;  y: A |] ==> r``{x} = r``{y} <-> <x,y>: r";
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by (fast_tac (ZF_cs addIs [eq_equiv_class, equiv_class_eq]
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                    addDs [equiv_type]) 1);
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qed "eq_equiv_class_iff";
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(*** Quotients ***)
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(** Introduction/elimination rules -- needed? **)
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val prems = goalw EquivClass.thy [quotient_def] "x:A ==> r``{x}: A/r";
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by (rtac RepFunI 1);
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by (resolve_tac prems 1);
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qed "quotientI";
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val major::prems = goalw EquivClass.thy [quotient_def]
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    "[| X: A/r;  !!x. [| X = r``{x};  x:A |] ==> P |]   \
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\    ==> P";
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by (rtac (major RS RepFunE) 1);
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by (eresolve_tac prems 1);
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by (assume_tac 1);
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qed "quotientE";
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goalw EquivClass.thy [equiv_def,refl_def,quotient_def]
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    "!!A r. equiv(A,r) ==> Union(A/r) = A";
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by (fast_tac eq_cs 1);
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qed "Union_quotient";
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goalw EquivClass.thy [quotient_def]
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    "!!A r. [| equiv(A,r);  X: A/r;  Y: A/r |] ==> X=Y | (X Int Y <= 0)";
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by (safe_tac (ZF_cs addSIs [equiv_class_eq]));
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by (assume_tac 1);
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by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
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by (fast_tac ZF_cs 1);
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diff changeset
   125
qed "quotient_disj";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   126
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   127
(**** Defining unary operations upon equivalence classes ****)
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   128
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   129
(** These proofs really require as local premises
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   130
     equiv(A,r);  congruent(r,b)
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   131
**)
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   132
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   133
(*Conversion rule*)
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   134
val prems as [equivA,bcong,_] = goal EquivClass.thy
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   135
    "[| equiv(A,r);  congruent(r,b);  a: A |] ==> (UN x:r``{a}. b(x)) = b(a)";
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   136
by (cut_facts_tac prems 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   137
by (rtac ([refl RS UN_cong, UN_constant] MRS trans) 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   138
by (etac equiv_class_self 2);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   139
by (assume_tac 2);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   140
by (rewrite_goals_tac [equiv_def,sym_def,congruent_def]);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   141
by (fast_tac ZF_cs 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   142
qed "UN_equiv_class";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   143
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   144
(*type checking of  UN x:r``{a}. b(x) *)
946
c0f4ae3fda92 Removed "localize"; instead, proofs refer to their own
lcp
parents: 760
diff changeset
   145
val prems = goalw EquivClass.thy [quotient_def]
1461
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clasohm
parents: 1076
diff changeset
   146
    "[| equiv(A,r);  congruent(r,b);  X: A/r;   \
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1076
diff changeset
   147
\       !!x.  x : A ==> b(x) : B |]     \
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   148
\    ==> (UN x:X. b(x)) : B";
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   149
by (cut_facts_tac prems 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   150
by (safe_tac ZF_cs);
946
c0f4ae3fda92 Removed "localize"; instead, proofs refer to their own
lcp
parents: 760
diff changeset
   151
by (asm_simp_tac (ZF_ss addsimps (UN_equiv_class::prems)) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   152
qed "UN_equiv_class_type";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   153
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   154
(*Sufficient conditions for injectiveness.  Could weaken premises!
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   155
  major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   156
*)
946
c0f4ae3fda92 Removed "localize"; instead, proofs refer to their own
lcp
parents: 760
diff changeset
   157
val prems = goalw EquivClass.thy [quotient_def]
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   158
    "[| equiv(A,r);   congruent(r,b);  \
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   159
\       (UN x:X. b(x))=(UN y:Y. b(y));  X: A/r;  Y: A/r;  \
1461
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clasohm
parents: 1076
diff changeset
   160
\       !!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |]         \
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   161
\    ==> X=Y";
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   162
by (cut_facts_tac prems 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   163
by (safe_tac ZF_cs);
946
c0f4ae3fda92 Removed "localize"; instead, proofs refer to their own
lcp
parents: 760
diff changeset
   164
by (rtac equiv_class_eq 1);
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   165
by (REPEAT (ares_tac prems 1));
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   166
by (etac box_equals 1);
946
c0f4ae3fda92 Removed "localize"; instead, proofs refer to their own
lcp
parents: 760
diff changeset
   167
by (REPEAT (ares_tac [UN_equiv_class] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   168
qed "UN_equiv_class_inject";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   169
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   170
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   171
(**** Defining binary operations upon equivalence classes ****)
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   172
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   173
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   174
goalw EquivClass.thy [congruent_def,congruent2_def,equiv_def,refl_def]
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   175
    "!!A r. [| equiv(A,r);  congruent2(r,b);  a: A |] ==> congruent(r,b(a))";
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   176
by (fast_tac ZF_cs 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   177
qed "congruent2_implies_congruent";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   178
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   179
val equivA::prems = goalw EquivClass.thy [congruent_def]
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   180
    "[| equiv(A,r);  congruent2(r,b);  a: A |] ==> \
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   181
\    congruent(r, %x1. UN x2:r``{a}. b(x1,x2))";
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   182
by (cut_facts_tac (equivA::prems) 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   183
by (safe_tac ZF_cs);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   184
by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   185
by (assume_tac 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   186
by (asm_simp_tac (ZF_ss addsimps [equivA RS UN_equiv_class,
1461
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clasohm
parents: 1076
diff changeset
   187
                                 congruent2_implies_congruent]) 1);
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   188
by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   189
by (fast_tac ZF_cs 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   190
qed "congruent2_implies_congruent_UN";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   191
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   192
val prems as equivA::_ = goal EquivClass.thy
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   193
    "[| equiv(A,r);  congruent2(r,b);  a1: A;  a2: A |]  \
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   194
\    ==> (UN x1:r``{a1}. UN x2:r``{a2}. b(x1,x2)) = b(a1,a2)";
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   195
by (cut_facts_tac prems 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   196
by (asm_simp_tac (ZF_ss addsimps [equivA RS UN_equiv_class,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1076
diff changeset
   197
                                 congruent2_implies_congruent,
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1076
diff changeset
   198
                                 congruent2_implies_congruent_UN]) 1);
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   199
qed "UN_equiv_class2";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   200
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   201
(*type checking*)
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   202
val prems = goalw EquivClass.thy [quotient_def]
1461
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clasohm
parents: 1076
diff changeset
   203
    "[| equiv(A,r);  congruent2(r,b);                   \
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1076
diff changeset
   204
\       X1: A/r;  X2: A/r;                              \
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1076
diff changeset
   205
\       !!x1 x2.  [| x1: A; x2: A |] ==> b(x1,x2) : B   \
749
4b605159b5a5 equiv_comp_eq: simplified proof
lcp
parents: 723
diff changeset
   206
\    |] ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   207
by (cut_facts_tac prems 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   208
by (safe_tac ZF_cs);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   209
by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
1461
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clasohm
parents: 1076
diff changeset
   210
                             congruent2_implies_congruent_UN,
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1076
diff changeset
   211
                             congruent2_implies_congruent, quotientI]) 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   212
qed "UN_equiv_class_type2";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   213
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   214
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   215
(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   216
  than the direct proof*)
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   217
val prems = goalw EquivClass.thy [congruent2_def,equiv_def,refl_def]
1461
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clasohm
parents: 1076
diff changeset
   218
    "[| equiv(A,r);     \
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   219
\       !! y z w. [| w: A;  <y,z> : r |] ==> b(y,w) = b(z,w);      \
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   220
\       !! y z w. [| w: A;  <y,z> : r |] ==> b(w,y) = b(w,z)       \
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   221
\    |] ==> congruent2(r,b)";
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   222
by (cut_facts_tac prems 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   223
by (safe_tac ZF_cs);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   224
by (rtac trans 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   225
by (REPEAT (ares_tac prems 1
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   226
     ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   227
qed "congruent2I";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   228
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   229
val [equivA,commute,congt] = goal EquivClass.thy
1461
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clasohm
parents: 1076
diff changeset
   230
    "[| equiv(A,r);     \
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   231
\       !! y z. [| y: A;  z: A |] ==> b(y,z) = b(z,y);        \
1461
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clasohm
parents: 1076
diff changeset
   232
\       !! y z w. [| w: A;  <y,z>: r |] ==> b(w,y) = b(w,z)     \
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   233
\    |] ==> congruent2(r,b)";
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   234
by (resolve_tac [equivA RS congruent2I] 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   235
by (rtac (commute RS trans) 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   236
by (rtac (commute RS trans RS sym) 3);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   237
by (rtac sym 5);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   238
by (REPEAT (ares_tac [congt] 1
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   239
     ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   240
qed "congruent2_commuteI";
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   241
723
82caba9e130f ZF/EquivClass/congruent_commuteI: uncommented and simplified proof
lcp
parents: 535
diff changeset
   242
(*Obsolete?*)
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   243
val [equivA,ZinA,congt,commute] = goalw EquivClass.thy [quotient_def]
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   244
    "[| equiv(A,r);  Z: A/r;  \
1461
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clasohm
parents: 1076
diff changeset
   245
\       !!w. [| w: A |] ==> congruent(r, %z.b(w,z));    \
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1076
diff changeset
   246
\       !!x y. [| x: A;  y: A |] ==> b(y,x) = b(x,y)    \
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   247
\    |] ==> congruent(r, %w. UN z: Z. b(w,z))";
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   248
val congt' = rewrite_rule [congruent_def] congt;
723
82caba9e130f ZF/EquivClass/congruent_commuteI: uncommented and simplified proof
lcp
parents: 535
diff changeset
   249
by (cut_facts_tac [ZinA] 1);
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   250
by (rewtac congruent_def);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   251
by (safe_tac ZF_cs);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   252
by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   253
by (assume_tac 1);
723
82caba9e130f ZF/EquivClass/congruent_commuteI: uncommented and simplified proof
lcp
parents: 535
diff changeset
   254
by (asm_simp_tac (ZF_ss addsimps [commute,
1461
6bcb44e4d6e5 expanded tabs
clasohm
parents: 1076
diff changeset
   255
                                  [equivA, congt] MRS UN_equiv_class]) 1);
535
9d62c7e08699 ZF/ex/Equiv.thy,.ML: moved to main directory as EquivClass.thy,.ML
lcp
parents:
diff changeset
   256
by (REPEAT (ares_tac [congt' RS spec RS spec RS mp] 1));
760
f0200e91b272 added qed and qed_goal[w]
clasohm
parents: 749
diff changeset
   257
qed "congruent_commuteI";