(* Title: Equiv.ML
ID: $Id$
Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Equivalence relations in HOL Set Theory
*)
val RSLIST = curry (op MRS);
open Equiv;
Delrules [equalityI];
(*** 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 Equiv.thy [trans_def,sym_def,inverse_def]
"!!r. [| sym(r); trans(r) |] ==> r^-1 O r <= r";
by (blast_tac (claset() addSEs [inverseD]) 1);
qed "sym_trans_comp_subset";
goalw Equiv.thy [refl_def]
"!!A r. refl A r ==> r <= r^-1 O r";
by (Blast_tac 1);
qed "refl_comp_subset";
goalw Equiv.thy [equiv_def]
"!!A r. 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.thy [equiv_def,refl_def,sym_def,trans_def]
"!!A r. [| 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.thy [equiv_def,trans_def,sym_def]
"!!A r. [| equiv A r; (a,b): r |] ==> r^^{a} <= r^^{b}";
by (Blast_tac 1);
qed "equiv_class_subset";
goal Equiv.thy "!!A r. [| 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.thy [equiv_def,refl_def]
"!!A r. [| equiv A r; a: A |] ==> a: r^^{a}";
by (Blast_tac 1);
qed "equiv_class_self";
(*Lemma for the next result*)
goalw Equiv.thy [equiv_def,refl_def]
"!!A r. [| equiv A r; r^^{b} <= r^^{a}; b: A |] ==> (a,b): r";
by (Blast_tac 1);
qed "subset_equiv_class";
goal Equiv.thy
"!!A r. [| 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.thy [equiv_def,trans_def,sym_def]
"!!A r. [| equiv A r; x: (r^^{a} Int r^^{b}) |] ==> (a,b): r";
by (Blast_tac 1);
qed "equiv_class_nondisjoint";
val [major] = goalw Equiv.thy [equiv_def,refl_def]
"equiv A r ==> r <= A Times A";
by (rtac (major RS conjunct1 RS conjunct1) 1);
qed "equiv_type";
goal Equiv.thy
"!!A r. 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.thy
"!!A r. [| 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 Equiv.thy [quotient_def] "!!A. x:A ==> r^^{x}: A/r";
by (Blast_tac 1);
qed "quotientI";
val [major,minor] = goalw Equiv.thy [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.thy [equiv_def,refl_def,quotient_def]
"!!A r. equiv A r ==> Union(A/r) = A";
by (blast_tac (claset() addSIs [equalityI]) 1);
qed "Union_quotient";
goalw Equiv.thy [quotient_def]
"!!A r. [| equiv A r; X: A/r; Y: A/r |] ==> X=Y | (X Int Y = {})";
by (safe_tac (claset() addSIs [equiv_class_eq]));
by (assume_tac 1);
by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
by (blast_tac (claset() addSIs [equalityI]) 1);
qed "quotient_disj";
(**** Defining unary operations upon equivalence classes ****)
(* theorem needed to prove UN_equiv_class *)
goal Set.thy "!!A. [| a:A; ! y:A. b(y)=b(a) |] ==> (UN y:A. b(y))=b(a)";
by (fast_tac (claset() addSEs [equalityE] addSIs [equalityI]) 1);
qed "UN_singleton_lemma";
val UN_singleton = ballI RSN (2,UN_singleton_lemma);
(** These proofs really require the local premises
equiv A r; congruent r b
**)
(*Conversion rule*)
goal Equiv.thy "!!A r. [| equiv A r; congruent r b; a: A |] \
\ ==> (UN x:r^^{a}. b(x)) = b(a)";
by (rtac (equiv_class_self RS UN_singleton) 1 THEN REPEAT (assume_tac 1));
by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
by (Blast_tac 1);
qed "UN_equiv_class";
(*type checking of UN x:r``{a}. b(x) *)
val prems = goalw Equiv.thy [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 Equiv.thy [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 Equiv.thy [congruent_def,congruent2_def,equiv_def,refl_def]
"!!A r. [| equiv A r; congruent2 r b; a: A |] ==> congruent r (b a)";
by (Blast_tac 1);
qed "congruent2_implies_congruent";
goalw Equiv.thy [congruent_def]
"!!A r. [| 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 1);
qed "congruent2_implies_congruent_UN";
goal Equiv.thy
"!!A r. [| 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 Equiv.thy [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";
(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
than the direct proof*)
val prems = goalw Equiv.thy [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.thy
"[| 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 Kammüller ***)
(*Recall that equiv A r implies r <= A Times A (equiv_type) *)
goal thy "!!A. [| finite A; r <= A Times 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 thy [quotient_def]
"!!A. [| finite A; r <= A Times A; X : A/r |] ==> finite X";
by (rtac finite_subset 1);
by (assume_tac 2);
by (Blast_tac 1);
qed "finite_equiv_class";
goal thy "!!A. [| finite A; equiv A r; ! 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() delrules [equalityI] addEs [quotient_disj RS disjE]) 4);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps [Union_quotient, equiv_type,
finite_quotient])));
qed "equiv_imp_dvd_card";