src/ZF/ex/equiv.ML
author lcp
Tue Aug 16 18:58:42 1994 +0200 (1994-08-16)
changeset 532 851df239ac8b
parent 7 268f93ab3bc4
permissions -rw-r--r--
ZF/Makefile,ROOT.ML, ZF/ex/Integ.thy: updated for EquivClass
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(*  Title: 	ZF/ex/equiv.ML
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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For equiv.thy.  Equivalence relations in Zermelo-Fraenkel 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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(*** 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]
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    "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
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by (fast_tac (ZF_cs addSEs [converseD,compE]) 1);
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val sym_trans_comp_subset = result();
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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 (ZF_cs addSIs [converseI] addIs [compI]) 1);
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val refl_comp_subset = result();
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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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val equiv_comp_eq = result();
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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 ZF_cs);
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by (fast_tac (ZF_cs addSIs [converseI] addIs [compI]) 3);
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by (ALLGOALS (fast_tac 
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	      (ZF_cs addSIs [converseI] addIs [compI] addSEs [compE])));
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by flexflex_tac;
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val comp_equivI = result();
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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 (fast_tac ZF_cs 1);
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val equiv_class_subset = result();
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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 ZF_cs 1);
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val equiv_class_eq = result();
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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 ZF_cs 1);
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val equiv_class_self = result();
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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 ZF_cs 1);
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val subset_equiv_class = result();
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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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val eq_equiv_class = result();
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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 ZF_cs 1);
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val equiv_class_nondisjoint = result();
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val [major] = goalw Equiv.thy [equiv_def,refl_def]
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    "equiv(A,r) ==> r <= A*A";
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by (rtac (major RS conjunct1 RS conjunct1) 1);
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val equiv_type = result();
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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 (fast_tac (ZF_cs addIs [eq_equiv_class, equiv_class_eq]
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		    addDs [equiv_type]) 1);
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val equiv_class_eq_iff = result();
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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 (fast_tac (ZF_cs addIs [eq_equiv_class, equiv_class_eq]
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		    addDs [equiv_type]) 1);
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val eq_equiv_class_iff = result();
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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 RepFunI 1);
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by (resolve_tac prems 1);
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val quotientI = result();
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val major::prems = 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 (rtac (major RS RepFunE) 1);
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by (eresolve_tac prems 1);
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by (assume_tac 1);
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val quotientE = result();
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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 eq_cs 1);
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val Union_quotient = result();
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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 (ZF_cs 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 ZF_cs 1);
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val quotient_disj = result();
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(**** Defining unary operations upon equivalence classes ****)
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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 (etac equiv_class_self 1);
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by (assume_tac 1);
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by (rewrite_goals_tac [equiv_def,sym_def,congruent_def]);
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by (fast_tac ZF_cs 1);
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val UN_equiv_class = result();
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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 ZF_cs);
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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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val UN_equiv_class_type = result();
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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 ZF_cs);
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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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val UN_equiv_class_inject = result();
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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 ZF_cs 1);
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val congruent2_implies_congruent = result();
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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 ZF_cs);
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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 (ZF_ss 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 ZF_cs 1);
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val congruent2_implies_congruent_UN = result();
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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 (ZF_ss 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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val UN_equiv_class2 = result();
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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 ZF_cs);
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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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val UN_equiv_class_type2 = result();
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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 ZF_cs);
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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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val congruent2I = result();
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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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val congruent2_commuteI = result();
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(***OBSOLETE VERSION
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(*Rules congruentI and congruentD would simplify use of rewriting below*)
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val [equivA,ZinA,congt,commute] = goalw Equiv.thy [quotient_def]
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    "[| equiv(A,r);  Z: A/r;  \
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\       !!w. [| w: A |] ==> congruent(r, %z.b(w,z));	\
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\       !!x y. [| x: A;  y: A |] ==> b(y,x) = b(x,y)	\
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\    |] ==> congruent(r, %w. UN z: Z. b(w,z))";
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val congt' = rewrite_rule [congruent_def] congt;
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by (cut_facts_tac [ZinA,congt] 1);
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by (rewtac congruent_def);
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by (safe_tac ZF_cs);
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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 (ZF_ss addsimps [congt RS (equivA RS UN_equiv_class)]) 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' RS spec RS spec RS mp] 1));
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val congruent_commuteI = result();
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***)