src/HOL/Integ/Equiv.ML
author paulson
Wed, 11 Sep 1996 18:46:07 +0200
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(*  Title:      Equiv.ML
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    ID:         $Id$
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    Authors:    Riccardo Mattolini, Dip. Sistemi e Informatica
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                Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994 Universita' di Firenze
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    Copyright   1993  University of Cambridge
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Equivalence relations in HOL Set Theory 
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*)
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val RSLIST = curry (op MRS);
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open Equiv;
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Delrules [equalityI];
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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 Equiv.thy [trans_def,sym_def,converse_def]
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    "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
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by (fast_tac (!claset addSEs [converseD]) 1);
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qed "sym_trans_comp_subset";
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goalw Equiv.thy [refl_def]
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    "!!A r. refl A r ==> r <= converse(r) O r";
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by (fast_tac (!claset addIs [compI]) 1);
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qed "refl_comp_subset";
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goalw Equiv.thy [equiv_def]
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    "!!A r. equiv A r ==> converse(r) O r = r";
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by (rtac equalityI 1);
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by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1
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     ORELSE etac conjE 1));
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qed "equiv_comp_eq";
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(*second half*)
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goalw Equiv.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 (!claset));
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by (fast_tac (!claset addSIs [converseI] addIs [compI]) 3);
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by (ALLGOALS (fast_tac (!claset 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 Equiv.thy [equiv_def,trans_def,sym_def]
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    "!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} <= r^^{b}";
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by (safe_tac (!claset));
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by (rtac ImageI 1);
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by (Fast_tac 2);
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by (Fast_tac 1);
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qed "equiv_class_subset";
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goal Equiv.thy "!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} = r^^{b}";
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by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1));
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by (rewrite_goals_tac [equiv_def,sym_def]);
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by (Fast_tac 1);
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qed "equiv_class_eq";
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val prems = goalw Equiv.thy [equiv_def,refl_def]
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    "[| equiv A r;  a: A |] ==> a: r^^{a}";
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by (cut_facts_tac prems 1);
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by (Fast_tac 1);
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qed "equiv_class_self";
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(*Lemma for the next result*)
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goalw Equiv.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 1);
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qed "subset_equiv_class";
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val prems = goal Equiv.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 Equiv.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 1);
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qed "equiv_class_nondisjoint";
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val [major] = goalw Equiv.thy [equiv_def,refl_def]
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    "equiv A r ==> r <= A Times A";
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by (rtac (major RS conjunct1 RS conjunct1) 1);
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qed "equiv_type";
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goal Equiv.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 (safe_tac (!claset));
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by ((rtac equiv_class_eq 1) THEN (assume_tac 1) THEN (assume_tac 1));
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by ((rtac eq_equiv_class 3) THEN 
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    (assume_tac 4) THEN (assume_tac 4) THEN (assume_tac 3));
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by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
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    (assume_tac 1) THEN (dtac SigmaD1 1) THEN (assume_tac 1));
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by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
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    (assume_tac 1) THEN (dtac SigmaD2 1) THEN (assume_tac 1));
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qed "equiv_class_eq_iff";
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goal Equiv.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 (safe_tac (!claset));
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by ((rtac eq_equiv_class 1) THEN 
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    (assume_tac 1) THEN (assume_tac 1) THEN (assume_tac 1));
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by ((rtac equiv_class_eq 1) THEN 
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    (assume_tac 1) THEN (assume_tac 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 Equiv.thy [quotient_def] "x:A ==> r^^{x}: A/r";
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by (rtac UN_I 1);
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by (resolve_tac prems 1);
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by (rtac singletonI 1);
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qed "quotientI";
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val [major,minor] = goalw Equiv.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 (resolve_tac [major RS UN_E] 1);
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by (rtac minor 1);
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by (assume_tac 2);
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by (Fast_tac 1);
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qed "quotientE";
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(** Not needed by Theory Integ --> bypassed **)
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(**goalw Equiv.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 1);
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qed "Union_quotient";
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**)
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(** Not needed by Theory Integ --> bypassed **)
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(*goalw Equiv.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 (!claset 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 1);
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qed "quotient_disj";
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**)
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(**** Defining unary operations upon equivalence classes ****)
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(* theorem needed to prove UN_equiv_class *)
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goal Set.thy "!!A. [| a:A; ! y:A. b(y)=b(a) |] ==> (UN y:A. b(y))=b(a)";
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by (fast_tac (!claset addSEs [equalityE] addSIs [equalityI]) 1);
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qed "UN_singleton_lemma";
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val UN_singleton = ballI RSN (2,UN_singleton_lemma);
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(** These proofs really require as local premises
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     equiv A r;  congruent r b
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**)
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(*Conversion rule*)
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val prems as [equivA,bcong,_] = goal Equiv.thy
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    "[| equiv A r;  congruent r b;  a: A |] ==> (UN x:r^^{a}. b(x)) = b(a)";
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by (cut_facts_tac prems 1);
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by (rtac UN_singleton 1);
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by (rtac equiv_class_self 1);
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by (assume_tac 1);
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by (assume_tac 1);
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by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
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by (Fast_tac 1);
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qed "UN_equiv_class";
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(*Resolve th against the "local" premises*)
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val localize = RSLIST [equivA,bcong];
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(*type checking of  UN x:r``{a}. b(x) *)
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val _::_::prems = goalw Equiv.thy [quotient_def]
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    "[| equiv A r;  congruent r b;  X: A/r;     \
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\       !!x.  x : A ==> b(x) : B |]     \
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\    ==> (UN x:X. b(x)) : B";
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by (cut_facts_tac prems 1);
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by (safe_tac (!claset));
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by (rtac (localize UN_equiv_class RS ssubst) 1);
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by (REPEAT (ares_tac prems 1));
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qed "UN_equiv_class_type";
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(*Sufficient conditions for injectiveness.  Could weaken premises!
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  major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
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*)
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val _::_::prems = goalw Equiv.thy [quotient_def]
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    "[| equiv A r;   congruent r b;  \
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\       (UN x:X. b(x))=(UN y:Y. b(y));  X: A/r;  Y: A/r;  \
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\       !!x y. [| x:A; y:A; b(x)=b(y) |] ==> (x,y):r |]         \
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\    ==> X=Y";
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by (cut_facts_tac prems 1);
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by (safe_tac ((!claset) delrules [equalityI]));
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by (rtac (equivA RS equiv_class_eq) 1);
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by (REPEAT (ares_tac prems 1));
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by (etac box_equals 1);
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by (REPEAT (ares_tac [localize UN_equiv_class] 1));
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qed "UN_equiv_class_inject";
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(**** Defining binary operations upon equivalence classes ****)
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goalw Equiv.thy [congruent_def,congruent2_def,equiv_def,refl_def]
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    "!!A r. [| equiv A r;  congruent2 r b;  a: A |] ==> congruent r (b a)";
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by (Fast_tac 1);
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qed "congruent2_implies_congruent";
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val equivA::prems = goalw Equiv.thy [congruent_def]
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    "[| equiv A r;  congruent2 r b;  a: A |] ==> \
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\    congruent r (%x1. UN x2:r^^{a}. b x1 x2)";
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by (cut_facts_tac (equivA::prems) 1);
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by (safe_tac (!claset));
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by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
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by (assume_tac 1);
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by (asm_simp_tac (!simpset addsimps [equivA RS UN_equiv_class,
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                                     congruent2_implies_congruent]) 1);
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by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
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by (Fast_tac 1);
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qed "congruent2_implies_congruent_UN";
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val prems as equivA::_ = goal Equiv.thy
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    "[| equiv A r;  congruent2 r b;  a1: A;  a2: A |]  \
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\    ==> (UN x1:r^^{a1}. UN x2:r^^{a2}. b x1 x2) = b a1 a2";
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by (cut_facts_tac prems 1);
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by (asm_simp_tac (!simpset addsimps [equivA RS UN_equiv_class,
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                                     congruent2_implies_congruent,
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                                     congruent2_implies_congruent_UN]) 1);
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qed "UN_equiv_class2";
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(*type checking*)
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val prems = goalw Equiv.thy [quotient_def]
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    "[| equiv A r;  congruent2 r b;  \
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\       X1: A/r;  X2: A/r;      \
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\       !!x1 x2.  [| x1: A; x2: A |] ==> b x1 x2 : B |]    \
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\    ==> (UN x1:X1. UN x2:X2. b x1 x2) : B";
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by (cut_facts_tac prems 1);
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by (safe_tac (!claset));
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by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
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                             congruent2_implies_congruent_UN,
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                             congruent2_implies_congruent, quotientI]) 1));
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qed "UN_equiv_class_type2";
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(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
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  than the direct proof*)
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val prems = goalw Equiv.thy [congruent2_def,equiv_def,refl_def]
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    "[| equiv A r;      \
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\       !! y z w. [| w: A;  (y,z) : r |] ==> b y w = b z w;      \
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\       !! y z w. [| w: A;  (y,z) : r |] ==> b w y = b w z       \
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\    |] ==> congruent2 r b";
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by (cut_facts_tac prems 1);
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by (safe_tac (!claset));
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by (rtac trans 1);
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by (REPEAT (ares_tac prems 1
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     ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));
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qed "congruent2I";
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val [equivA,commute,congt] = goal Equiv.thy
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    "[| equiv A r;      \
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\       !! y z. [| y: A;  z: A |] ==> b y z = b z y;        \
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\       !! y z w. [| w: A;  (y,z): r |] ==> b w y = b w z       \
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\    |] ==> congruent2 r b";
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by (resolve_tac [equivA RS congruent2I] 1);
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by (rtac (commute RS trans) 1);
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by (rtac (commute RS trans RS sym) 3);
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by (rtac sym 5);
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by (REPEAT (ares_tac [congt] 1
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     ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
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qed "congruent2_commuteI";
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