(* Title: ZF/EquivClass.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
For EquivClass.thy. Equivalence relations in Zermelo-Fraenkel Set Theory
*)
val RSLIST = curry (op MRS);
open EquivClass;
(*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***)
(** first half: equiv(A,r) ==> converse(r) O r = r **)
goalw EquivClass.thy [trans_def,sym_def]
"!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
by (fast_tac (ZF_cs addSEs [converseD,compE]) 1);
qed "sym_trans_comp_subset";
goalw EquivClass.thy [refl_def]
"!!A r. [| refl(A,r); r <= A*A |] ==> r <= converse(r) O r";
by (fast_tac (ZF_cs addSIs [converseI] addIs [compI]) 1);
qed "refl_comp_subset";
goalw EquivClass.thy [equiv_def]
"!!A r. equiv(A,r) ==> converse(r) O r = r";
by (fast_tac (subset_cs addSIs [equalityI, sym_trans_comp_subset,
refl_comp_subset]) 1);
qed "equiv_comp_eq";
(*second half*)
goalw EquivClass.thy [equiv_def,refl_def,sym_def,trans_def]
"!!A r. [| converse(r) 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 (safe_tac ZF_cs);
by (fast_tac (ZF_cs addSIs [converseI] addIs [compI]) 3);
by (ALLGOALS (fast_tac
(ZF_cs addSIs [converseI] addIs [compI] addSEs [compE])));
by flexflex_tac;
qed "comp_equivI";
(** Equivalence classes **)
(*Lemma for the next result*)
goalw EquivClass.thy [trans_def,sym_def]
"!!A r. [| sym(r); trans(r); <a,b>: r |] ==> r``{a} <= r``{b}";
by (fast_tac ZF_cs 1);
qed "equiv_class_subset";
goalw EquivClass.thy [equiv_def]
"!!A r. [| equiv(A,r); <a,b>: r |] ==> r``{a} = r``{b}";
by (safe_tac (subset_cs addSIs [equalityI, equiv_class_subset]));
by (rewrite_goals_tac [sym_def]);
by (fast_tac ZF_cs 1);
qed "equiv_class_eq";
goalw EquivClass.thy [equiv_def,refl_def]
"!!A r. [| equiv(A,r); a: A |] ==> a: r``{a}";
by (fast_tac ZF_cs 1);
qed "equiv_class_self";
(*Lemma for the next result*)
goalw EquivClass.thy [equiv_def,refl_def]
"!!A r. [| equiv(A,r); r``{b} <= r``{a}; b: A |] ==> <a,b>: r";
by (fast_tac ZF_cs 1);
qed "subset_equiv_class";
val prems = goal EquivClass.thy
"[| r``{a} = r``{b}; equiv(A,r); b: A |] ==> <a,b>: r";
by (REPEAT (resolve_tac (prems @ [equalityD2, subset_equiv_class]) 1));
qed "eq_equiv_class";
(*thus r``{a} = r``{b} as well*)
goalw EquivClass.thy [equiv_def,trans_def,sym_def]
"!!A r. [| equiv(A,r); x: (r``{a} Int r``{b}) |] ==> <a,b>: r";
by (fast_tac ZF_cs 1);
qed "equiv_class_nondisjoint";
goalw EquivClass.thy [equiv_def] "!!A r. equiv(A,r) ==> r <= A*A";
by (safe_tac ZF_cs);
qed "equiv_type";
goal EquivClass.thy
"!!A r. equiv(A,r) ==> <x,y>: r <-> r``{x} = r``{y} & x:A & y:A";
by (fast_tac (ZF_cs addIs [eq_equiv_class, equiv_class_eq]
addDs [equiv_type]) 1);
qed "equiv_class_eq_iff";
goal EquivClass.thy
"!!A r. [| equiv(A,r); x: A; y: A |] ==> r``{x} = r``{y} <-> <x,y>: r";
by (fast_tac (ZF_cs addIs [eq_equiv_class, equiv_class_eq]
addDs [equiv_type]) 1);
qed "eq_equiv_class_iff";
(*** Quotients ***)
(** Introduction/elimination rules -- needed? **)
val prems = goalw EquivClass.thy [quotient_def] "x:A ==> r``{x}: A/r";
by (rtac RepFunI 1);
by (resolve_tac prems 1);
qed "quotientI";
val major::prems = goalw EquivClass.thy [quotient_def]
"[| X: A/r; !!x. [| X = r``{x}; x:A |] ==> P |] \
\ ==> P";
by (rtac (major RS RepFunE) 1);
by (eresolve_tac prems 1);
by (assume_tac 1);
qed "quotientE";
goalw EquivClass.thy [equiv_def,refl_def,quotient_def]
"!!A r. equiv(A,r) ==> Union(A/r) = A";
by (fast_tac eq_cs 1);
qed "Union_quotient";
goalw EquivClass.thy [quotient_def]
"!!A r. [| equiv(A,r); X: A/r; Y: A/r |] ==> X=Y | (X Int Y <= 0)";
by (safe_tac (ZF_cs addSIs [equiv_class_eq]));
by (assume_tac 1);
by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
by (fast_tac ZF_cs 1);
qed "quotient_disj";
(**** Defining unary operations upon equivalence classes ****)
(** These proofs really require as local premises
equiv(A,r); congruent(r,b)
**)
(*Conversion rule*)
val prems as [equivA,bcong,_] = goal EquivClass.thy
"[| equiv(A,r); congruent(r,b); a: A |] ==> (UN x:r``{a}. b(x)) = b(a)";
by (cut_facts_tac prems 1);
by (rtac ([refl RS UN_cong, UN_constant] MRS trans) 1);
by (etac equiv_class_self 2);
by (assume_tac 2);
by (rewrite_goals_tac [equiv_def,sym_def,congruent_def]);
by (fast_tac ZF_cs 1);
qed "UN_equiv_class";
(*Resolve th against the "local" premises*)
val localize = RSLIST [equivA,bcong];
(*type checking of UN x:r``{a}. b(x) *)
val _::_::prems = goalw EquivClass.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 (safe_tac ZF_cs);
by (rtac (localize UN_equiv_class RS ssubst) 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 EquivClass.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 (safe_tac ZF_cs);
by (rtac (equivA RS equiv_class_eq) 1);
by (REPEAT (ares_tac prems 1));
by (etac box_equals 1);
by (REPEAT (ares_tac [localize UN_equiv_class] 1));
qed "UN_equiv_class_inject";
(**** Defining binary operations upon equivalence classes ****)
goalw EquivClass.thy [congruent_def,congruent2_def,equiv_def,refl_def]
"!!A r. [| equiv(A,r); congruent2(r,b); a: A |] ==> congruent(r,b(a))";
by (fast_tac ZF_cs 1);
qed "congruent2_implies_congruent";
val equivA::prems = goalw EquivClass.thy [congruent_def]
"[| equiv(A,r); congruent2(r,b); a: A |] ==> \
\ congruent(r, %x1. UN x2:r``{a}. b(x1,x2))";
by (cut_facts_tac (equivA::prems) 1);
by (safe_tac ZF_cs);
by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
by (assume_tac 1);
by (asm_simp_tac (ZF_ss addsimps [equivA RS UN_equiv_class,
congruent2_implies_congruent]) 1);
by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
by (fast_tac ZF_cs 1);
qed "congruent2_implies_congruent_UN";
val prems as equivA::_ = goal EquivClass.thy
"[| 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 (cut_facts_tac prems 1);
by (asm_simp_tac (ZF_ss addsimps [equivA RS UN_equiv_class,
congruent2_implies_congruent,
congruent2_implies_congruent_UN]) 1);
qed "UN_equiv_class2";
(*type checking*)
val prems = goalw EquivClass.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 (safe_tac ZF_cs);
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 EquivClass.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 (safe_tac ZF_cs);
by (rtac trans 1);
by (REPEAT (ares_tac prems 1
ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));
qed "congruent2I";
val [equivA,commute,congt] = goal EquivClass.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";
(*Obsolete?*)
val [equivA,ZinA,congt,commute] = goalw EquivClass.thy [quotient_def]
"[| equiv(A,r); Z: A/r; \
\ !!w. [| w: A |] ==> congruent(r, %z.b(w,z)); \
\ !!x y. [| x: A; y: A |] ==> b(y,x) = b(x,y) \
\ |] ==> congruent(r, %w. UN z: Z. b(w,z))";
val congt' = rewrite_rule [congruent_def] congt;
by (cut_facts_tac [ZinA] 1);
by (rewtac congruent_def);
by (safe_tac ZF_cs);
by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
by (assume_tac 1);
by (asm_simp_tac (ZF_ss addsimps [commute,
[equivA, congt] MRS UN_equiv_class]) 1);
by (REPEAT (ares_tac [congt' RS spec RS spec RS mp] 1));
qed "congruent_commuteI";