src/HOL/Integ/Equiv.ML

converse -> ^-1

(*  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 (fast_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 (fast_tac (!claset addIs [compI]) 1);
qed "refl_comp_subset";

goalw Equiv.thy [equiv_def]
    "!!A r. equiv A r ==> r^-1 O r = r";
by (rtac equalityI 1);
by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1
     ORELSE etac conjE 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 (Step_tac 1);
by (fast_tac (!claset addIs [compI]) 3);
by (ALLGOALS (fast_tac (!claset addIs [compI])));
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 (Step_tac 1);
by (rtac ImageI 1);
by (Fast_tac 2);
by (Fast_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 (Fast_tac 1);
qed "equiv_class_eq";

goalw Equiv.thy [equiv_def,refl_def]
    "!!A r. [| equiv A r;  a: A |] ==> a: r^^{a}";
by (Fast_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 (Fast_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 (Fast_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 (Step_tac 1);
by ((rtac equiv_class_eq 1) THEN (assume_tac 1) THEN (assume_tac 1));
by ((rtac eq_equiv_class 3) THEN 
    (assume_tac 4) THEN (assume_tac 4) THEN (assume_tac 3));
by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
    (assume_tac 1) THEN (dtac SigmaD1 1) THEN (assume_tac 1));
by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
    (assume_tac 1) THEN (dtac SigmaD2 1) THEN (assume_tac 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 (Step_tac 1);
by ((rtac eq_equiv_class 1) THEN 
    (assume_tac 1) THEN (assume_tac 1) THEN (assume_tac 1));
by ((rtac equiv_class_eq 1) THEN 
    (assume_tac 1) THEN (assume_tac 1));
qed "eq_equiv_class_iff";

(*** Quotients ***)

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

goalw Equiv.thy [quotient_def] "!!A. x:A ==> r^^{x}: A/r";
by (Fast_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);
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 (Fast_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 (Step_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 (Step_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 (Fast_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 (Step_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 (Fast_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 (Step_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 (Step_tac 1);
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 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 (rewrite_goals_tac [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);
ba 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";