src/HOL/Integ/Equiv.ML

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(*  Title:      Equiv.ML
    ID:         $Id$
    Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge

Equivalence relations in HOL Set Theory 
*)

(*** Suppes, Theorem 70: r is an equiv relation iff r^-1 O r = r ***)

(** first half: equiv A r ==> r^-1 O r = r **)

Goalw [trans_def,sym_def,converse_def]
     "[| sym(r); trans(r) |] ==> r^-1 O r <= r";
by (Blast_tac 1);
qed "sym_trans_comp_subset";

Goalw [refl_def] "refl A r ==> r <= r^-1 O r";
by (Blast_tac 1);
qed "refl_comp_subset";

Goalw [equiv_def] "equiv A r ==> r^-1 O r = r";
by (Clarify_tac 1);
by (rtac equalityI 1);
by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1));
qed "equiv_comp_eq";

(*second half*)
Goalw [equiv_def,refl_def,sym_def,trans_def]
     "[| r^-1 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 (ALLGOALS Fast_tac);
qed "comp_equivI";

(** Equivalence classes **)

(*Lemma for the next result*)
Goalw [equiv_def,trans_def,sym_def]
     "[| equiv A r;  (a,b): r |] ==> r``{a} <= r``{b}";
by (Blast_tac 1);
qed "equiv_class_subset";

Goal "[| 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 (Blast_tac 1);
qed "equiv_class_eq";

Goalw [equiv_def,refl_def] "[| equiv A r;  a: A |] ==> a: r``{a}";
by (Blast_tac 1);
qed "equiv_class_self";

(*Lemma for the next result*)
Goalw [equiv_def,refl_def]
    "[| equiv A r;  r``{b} <= r``{a};  b: A |] ==> (a,b): r";
by (Blast_tac 1);
qed "subset_equiv_class";

Goal "[| r``{a} = r``{b};  equiv A r;  b: A |] ==> (a,b): r";
by (REPEAT (ares_tac [equalityD2, subset_equiv_class] 1));
qed "eq_equiv_class";

(*thus r``{a} = r``{b} as well*)
Goalw [equiv_def,trans_def,sym_def]
     "[| equiv A r;  x: (r``{a} Int r``{b}) |] ==> (a,b): r";
by (Blast_tac 1);
qed "equiv_class_nondisjoint";

Goalw [equiv_def,refl_def] "equiv A r ==> r <= A <*> A";
by (Blast_tac 1);
qed "equiv_type";

Goal "equiv A r ==> ((x,y): r) = (r``{x} = r``{y} & x:A & y:A)";
by (blast_tac (claset() addSIs [equiv_class_eq]
	                addDs [eq_equiv_class, equiv_type]) 1);
qed "equiv_class_eq_iff";

Goal "[| equiv A r;  x: A;  y: A |] ==> (r``{x} = r``{y}) = ((x,y): r)";
by (blast_tac (claset() addSIs [equiv_class_eq]
	                addDs [eq_equiv_class, equiv_type]) 1);
qed "eq_equiv_class_iff";

(*** Quotients ***)

(** Introduction/elimination rules -- needed? **)

Goalw [quotient_def] "x:A ==> r``{x}: A//r";
by (Blast_tac 1);
qed "quotientI";

val [major,minor] = Goalw [quotient_def]
    "[| X:(A//r);  !!x. [| X = r``{x};  x:A |] ==> P |]  \
\    ==> P";
by (resolve_tac [major RS UN_E] 1);
by (rtac minor 1);
by (assume_tac 2);
by (Fast_tac 1);   (*Blast_tac FAILS to prove it*)
qed "quotientE";

Goalw [equiv_def,refl_def,quotient_def]
     "equiv A r ==> Union(A//r) = A";
by (Blast_tac 1);
qed "Union_quotient";

Goalw [quotient_def]
     "[| equiv A r;  X: A//r;  Y: A//r |] ==> X=Y | (X Int Y = {})";
by (Clarify_tac 1);
by (rtac equiv_class_eq 1);
by (assume_tac 1);
by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
by (Blast_tac 1);
qed "quotient_disj";


(**** Defining unary operations upon equivalence classes ****)

(* theorem needed to prove UN_equiv_class *)
Goal "[| a:A;  ALL y:A. b(y)=c |] ==> (UN y:A. b(y))=c";
by Auto_tac;
qed "UN_constant_eq";


(** Could introduce a LOCALE with the assumptions
     equiv A r;  congruent r b
**)

(*Conversion rule*)
Goal "[| equiv A r;  congruent r b;  a: A |] \
\     ==> (UN x:r``{a}. b(x)) = b(a)";
by (rtac (equiv_class_self RS UN_constant_eq) 1 THEN REPEAT (assume_tac 1));
by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
by (blast_tac (claset() delrules [equalityI]) 1);
qed "UN_equiv_class";

(*type checking of  UN x:r`{a}. b(x) *)
val prems = Goalw [quotient_def]
    "[| equiv A r;  congruent r b;  X: A//r;     \
\       !!x.  x : A ==> b(x) : B |]             \
\    ==> (UN x:X. b(x)) : B";
by (cut_facts_tac prems 1);
by (Clarify_tac 1);
by (stac UN_equiv_class 1);
by (REPEAT (ares_tac prems 1));
qed "UN_equiv_class_type";

(*Sufficient conditions for injectiveness.  Could weaken premises!
  major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
*)
val prems = Goalw [quotient_def]
    "[| equiv A r;   congruent r b;  \
\       (UN x:X. b(x))=(UN y:Y. b(y));  X: A//r;  Y: A//r;        \
\       !!x y. [| x:A; y:A; b(x)=b(y) |] ==> (x,y):r |]         \
\    ==> X=Y";
by (cut_facts_tac prems 1);
by (Clarify_tac 1);
by (rtac equiv_class_eq 1);
by (REPEAT (ares_tac prems 1));
by (etac box_equals 1);
by (REPEAT (ares_tac [UN_equiv_class] 1));
qed "UN_equiv_class_inject";


(**** Defining binary operations upon equivalence classes ****)


Goalw [congruent_def,congruent2_def,equiv_def,refl_def]
    "[| equiv A r;  congruent2 r b;  a: A |] ==> congruent r (b a)";
by (Blast_tac 1);
qed "congruent2_implies_congruent";

Goalw [congruent_def]
    "[| equiv A r;  congruent2 r b;  a: A |] ==> \
\    congruent r (%x1. UN x2:r``{a}. b x1 x2)";
by (Clarify_tac 1);
by (rtac (equiv_type RS subsetD RS SigmaE2) 1 THEN REPEAT (assume_tac 1));
by (asm_simp_tac (simpset() addsimps [UN_equiv_class,
                                     congruent2_implies_congruent]) 1);
by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
by (blast_tac (claset() delrules [equalityI]) 1);
qed "congruent2_implies_congruent_UN";

Goal "[| equiv A r;  congruent2 r b;  a1: A;  a2: A |]  \
\    ==> (UN x1:r``{a1}. UN x2:r``{a2}. b x1 x2) = b a1 a2";
by (asm_simp_tac (simpset() addsimps [UN_equiv_class,
                                     congruent2_implies_congruent,
                                     congruent2_implies_congruent_UN]) 1);
qed "UN_equiv_class2";

(*type checking*)
val prems = Goalw [quotient_def]
    "[| equiv A r;  congruent2 r b;  \
\       X1: A//r;  X2: A//r;      \
\       !!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 (Clarify_tac 1);
by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
                             congruent2_implies_congruent_UN,
                             congruent2_implies_congruent, quotientI]) 1));
qed "UN_equiv_class_type2";

(*Allows a natural expression of binary operators, without explicit calls
  to "split"*)
Goal "(UN (x1,x2):X. UN (y1,y2):Y. A x1 x2 y1 y2) = \
\     (UN x:X. UN y:Y. (%(x1, x2). (%(y1, y2). A x1 x2 y1 y2) y) x)";
by Auto_tac;
qed "UN_UN_split_split_eq";

(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
  than the direct proof*)
val prems = Goalw [congruent2_def,equiv_def,refl_def]
    "[| equiv A r;      \
\       !! y z w. [| w: A;  (y,z) : r |] ==> b y w = b z w;      \
\       !! y z w. [| w: A;  (y,z) : r |] ==> b w y = b w z       \
\    |] ==> congruent2 r b";
by (cut_facts_tac prems 1);
by (Clarify_tac 1);
by (blast_tac (claset() addIs (trans::prems)) 1);
qed "congruent2I";

val [equivA,commute,congt] = Goal
    "[| equiv A r;      \
\       !! y z. [| y: A;  z: A |] ==> b y z = b z y;        \
\       !! y z w. [| w: A;  (y,z): r |] ==> b w y = b w z       \
\    |] ==> congruent2 r b";
by (resolve_tac [equivA RS congruent2I] 1);
by (rtac (commute RS trans) 1);
by (rtac (commute RS trans RS sym) 3);
by (rtac sym 5);
by (REPEAT (ares_tac [congt] 1
     ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
qed "congruent2_commuteI";


(*** Cardinality results suggested by Florian Kammueller ***)

(*Recall that  equiv A r  implies  r <= A <*> A   (equiv_type) *)
Goal "[| finite A; r <= A <*> A |] ==> finite (A//r)";
by (rtac finite_subset 1);
by (etac (finite_Pow_iff RS iffD2) 2);
by (rewtac quotient_def);
by (Blast_tac 1);
qed "finite_quotient";

Goalw [quotient_def]
    "[| finite A;  r <= A <*> A;  X : A//r |] ==> finite X";
by (rtac finite_subset 1);
by (assume_tac 2);
by (Blast_tac 1);
qed "finite_equiv_class";

Goal "[| finite A;  equiv A r;  ALL X : A//r. k dvd card(X) |] \
\     ==> k dvd card(A)";
by (rtac (Union_quotient RS subst) 1);
by (assume_tac 1);
by (rtac dvd_partition 1);
by (blast_tac (claset() addDs [quotient_disj]) 4);
by (ALLGOALS 
    (asm_simp_tac (simpset() addsimps [Union_quotient, equiv_type, 
				       finite_quotient])));
qed "equiv_imp_dvd_card";