diff -r 8018173a7979 -r b6105462ccd3 src/ZF/ex/equiv.ML --- a/src/ZF/ex/equiv.ML Sat Apr 05 16:18:58 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,268 +0,0 @@ -(* Title: ZF/ex/equiv.ML - ID: $Id$ - Author: Lawrence C Paulson, Cambridge University Computer Laboratory - Copyright 1993 University of Cambridge - -For equiv.thy. Equivalence relations in Zermelo-Fraenkel Set Theory -*) - -val RSLIST = curry (op MRS); - -open Equiv; - -(*** 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 Equiv.thy [trans_def,sym_def] - "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r"; -by (fast_tac (ZF_cs addSEs [converseD,compE]) 1); -val sym_trans_comp_subset = result(); - -goalw Equiv.thy [refl_def] - "!!A r. refl(A,r) ==> r <= converse(r) O r"; -by (fast_tac (ZF_cs addSIs [converseI] addIs [compI]) 1); -val refl_comp_subset = result(); - -goalw Equiv.thy [equiv_def] - "!!A r. equiv(A,r) ==> converse(r) O r = r"; -by (rtac equalityI 1); -by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1 - ORELSE etac conjE 1)); -val equiv_comp_eq = result(); - -(*second half*) -goalw Equiv.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. : r --> : 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; -val comp_equivI = result(); - -(** Equivalence classes **) - -(*Lemma for the next result*) -goalw Equiv.thy [equiv_def,trans_def,sym_def] - "!!A r. [| equiv(A,r); : r |] ==> r``{a} <= r``{b}"; -by (fast_tac ZF_cs 1); -val equiv_class_subset = result(); - -goal Equiv.thy "!!A r. [| equiv(A,r); : 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 ZF_cs 1); -val equiv_class_eq = result(); - -val prems = goalw Equiv.thy [equiv_def,refl_def] - "[| equiv(A,r); a: A |] ==> a: r``{a}"; -by (cut_facts_tac prems 1); -by (fast_tac ZF_cs 1); -val equiv_class_self = result(); - -(*Lemma for the next result*) -goalw Equiv.thy [equiv_def,refl_def] - "!!A r. [| equiv(A,r); r``{b} <= r``{a}; b: A |] ==> : r"; -by (fast_tac ZF_cs 1); -val subset_equiv_class = result(); - -val prems = goal Equiv.thy - "[| r``{a} = r``{b}; equiv(A,r); b: A |] ==> : r"; -by (REPEAT (resolve_tac (prems @ [equalityD2, subset_equiv_class]) 1)); -val eq_equiv_class = result(); - -(*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}) |] ==> : r"; -by (fast_tac ZF_cs 1); -val equiv_class_nondisjoint = result(); - -val [major] = goalw Equiv.thy [equiv_def,refl_def] - "equiv(A,r) ==> r <= A*A"; -by (rtac (major RS conjunct1 RS conjunct1) 1); -val equiv_type = result(); - -goal Equiv.thy - "!!A r. equiv(A,r) ==> : 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); -val equiv_class_eq_iff = result(); - -goal Equiv.thy - "!!A r. [| equiv(A,r); x: A; y: A |] ==> r``{x} = r``{y} <-> : r"; -by (fast_tac (ZF_cs addIs [eq_equiv_class, equiv_class_eq] - addDs [equiv_type]) 1); -val eq_equiv_class_iff = result(); - -(*** Quotients ***) - -(** Introduction/elimination rules -- needed? **) - -val prems = goalw Equiv.thy [quotient_def] "x:A ==> r``{x}: A/r"; -by (rtac RepFunI 1); -by (resolve_tac prems 1); -val quotientI = result(); - -val major::prems = goalw Equiv.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); -val quotientE = result(); - -goalw Equiv.thy [equiv_def,refl_def,quotient_def] - "!!A r. equiv(A,r) ==> Union(A/r) = A"; -by (fast_tac eq_cs 1); -val Union_quotient = result(); - -goalw Equiv.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); -val quotient_disj = result(); - -(**** 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 Equiv.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 UN_singleton 1); -by (etac equiv_class_self 1); -by (assume_tac 1); -by (rewrite_goals_tac [equiv_def,sym_def,congruent_def]); -by (fast_tac ZF_cs 1); -val UN_equiv_class = result(); - -(*Resolve th against the "local" premises*) -val localize = RSLIST [equivA,bcong]; - -(*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 (safe_tac ZF_cs); -by (rtac (localize UN_equiv_class RS ssubst) 1); -by (REPEAT (ares_tac prems 1)); -val UN_equiv_class_type = result(); - -(*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) |] ==> :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)); -val UN_equiv_class_inject = result(); - - -(**** 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 ZF_cs 1); -val congruent2_implies_congruent = result(); - -val equivA::prems = goalw Equiv.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); -val congruent2_implies_congruent_UN = result(); - -val prems as equivA::_ = goal Equiv.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); -val UN_equiv_class2 = result(); - -(*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 (safe_tac ZF_cs); -by (REPEAT (ares_tac (prems@[UN_equiv_class_type, - congruent2_implies_congruent_UN, - congruent2_implies_congruent, quotientI]) 1)); -val UN_equiv_class_type2 = result(); - - -(*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; : r |] ==> b(y,w) = b(z,w); \ -\ !! y z w. [| w: A; : 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)); -val congruent2I = result(); - -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; : 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)); -val congruent2_commuteI = result(); - -(***OBSOLETE VERSION -(*Rules congruentI and congruentD would simplify use of rewriting below*) -val [equivA,ZinA,congt,commute] = goalw Equiv.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,congt] 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 [congt RS (equivA RS UN_equiv_class)]) 1); -by (rtac (commute RS trans) 1); -by (rtac (commute RS trans RS sym) 3); -by (rtac sym 5); -by (REPEAT (ares_tac [congt' RS spec RS spec RS mp] 1)); -val congruent_commuteI = result(); -***)