diff -r 37b22343c35a -r f9796358e66f src/ZF/Integ/EquivClass.thy --- a/src/ZF/Integ/EquivClass.thy Fri Sep 20 11:49:38 2002 +0200 +++ b/src/ZF/Integ/EquivClass.thy Sat Sep 21 21:10:34 2002 +0200 @@ -3,21 +3,260 @@ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge -Equivalence relations in Zermelo-Fraenkel Set Theory *) -EquivClass = Trancl + Perm + +header{*Equivalence Relations*} + +theory EquivClass = Trancl + Perm: constdefs - quotient :: [i,i]=>i (infixl "'/'/" 90) (*set of equiv classes*) + quotient :: "[i,i]=>i" (infixl "'/'/" 90) (*set of equiv classes*) "A//r == {r``{x} . x:A}" - congruent :: [i,i=>i]=>o + congruent :: "[i,i=>i]=>o" "congruent(r,b) == ALL y z. :r --> b(y)=b(z)" - congruent2 :: [i,[i,i]=>i]=>o - "congruent2(r,b) == ALL y1 z1 y2 z2. + congruent2 :: "[i,[i,i]=>i]=>o" + "congruent2(r,b) == ALL y1 z1 y2 z2. :r --> :r --> b(y1,y2) = b(z1,z2)" +subsection{*Suppes, Theorem 70: + @{term r} is an equiv relation iff @{term "converse(r) O r = r"}*} + +(** first half: equiv(A,r) ==> converse(r) O r = r **) + +lemma sym_trans_comp_subset: + "[| sym(r); trans(r) |] ==> converse(r) O r <= r" +apply (unfold trans_def sym_def, blast) +done + +lemma refl_comp_subset: + "[| refl(A,r); r <= A*A |] ==> r <= converse(r) O r" +apply (unfold refl_def, blast) +done + +lemma equiv_comp_eq: + "equiv(A,r) ==> converse(r) O r = r" +apply (unfold equiv_def) +apply (blast del: subsetI + intro!: sym_trans_comp_subset refl_comp_subset) +done + +(*second half*) +lemma comp_equivI: + "[| converse(r) O r = r; domain(r) = A |] ==> equiv(A,r)" +apply (unfold equiv_def refl_def sym_def trans_def) +apply (erule equalityE) +apply (subgoal_tac "ALL x y. : r --> : r", blast+) +done + +(** Equivalence classes **) + +(*Lemma for the next result*) +lemma equiv_class_subset: + "[| sym(r); trans(r); : r |] ==> r``{a} <= r``{b}" +by (unfold trans_def sym_def, blast) + +lemma equiv_class_eq: + "[| equiv(A,r); : r |] ==> r``{a} = r``{b}" +apply (unfold equiv_def) +apply (safe del: subsetI intro!: equalityI equiv_class_subset) +apply (unfold sym_def, blast) +done + +lemma equiv_class_self: + "[| equiv(A,r); a: A |] ==> a: r``{a}" +by (unfold equiv_def refl_def, blast) + +(*Lemma for the next result*) +lemma subset_equiv_class: + "[| equiv(A,r); r``{b} <= r``{a}; b: A |] ==> : r" +by (unfold equiv_def refl_def, blast) + +lemma eq_equiv_class: "[| r``{a} = r``{b}; equiv(A,r); b: A |] ==> : r" +by (assumption | rule equalityD2 subset_equiv_class)+ + +(*thus r``{a} = r``{b} as well*) +lemma equiv_class_nondisjoint: + "[| equiv(A,r); x: (r``{a} Int r``{b}) |] ==> : r" +by (unfold equiv_def trans_def sym_def, blast) + +lemma equiv_type: "equiv(A,r) ==> r <= A*A" +by (unfold equiv_def, blast) + +lemma equiv_class_eq_iff: + "equiv(A,r) ==> : r <-> r``{x} = r``{y} & x:A & y:A" +by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type) + +lemma eq_equiv_class_iff: + "[| equiv(A,r); x: A; y: A |] ==> r``{x} = r``{y} <-> : r" +by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type) + +(*** Quotients ***) + +(** Introduction/elimination rules -- needed? **) + +lemma quotientI [TC]: "x:A ==> r``{x}: A//r" +apply (unfold quotient_def) +apply (erule RepFunI) +done + +lemma quotientE: + "[| X: A//r; !!x. [| X = r``{x}; x:A |] ==> P |] ==> P" +by (unfold quotient_def, blast) + +lemma Union_quotient: + "equiv(A,r) ==> Union(A//r) = A" +by (unfold equiv_def refl_def quotient_def, blast) + +lemma quotient_disj: + "[| equiv(A,r); X: A//r; Y: A//r |] ==> X=Y | (X Int Y <= 0)" +apply (unfold quotient_def) +apply (safe intro!: equiv_class_eq, assumption) +apply (unfold equiv_def trans_def sym_def, blast) +done + +subsection{*Defining Unary Operations upon Equivalence Classes*} + +(** Could have a locale with the premises equiv(A,r) and congruent(r,b) +**) + +(*Conversion rule*) +lemma UN_equiv_class: + "[| equiv(A,r); congruent(r,b); a: A |] ==> (UN x:r``{a}. b(x)) = b(a)" +apply (rule trans [OF refl [THEN UN_cong] UN_constant]) +apply (erule_tac [2] equiv_class_self) +prefer 2 apply assumption +apply (unfold equiv_def sym_def congruent_def, blast) +done + +(*type checking of UN x:r``{a}. b(x) *) +lemma UN_equiv_class_type: + "[| equiv(A,r); congruent(r,b); X: A//r; !!x. x : A ==> b(x) : B |] + ==> (UN x:X. b(x)) : B" +apply (unfold quotient_def, safe) +apply (simp (no_asm_simp) add: UN_equiv_class) +done + +(*Sufficient conditions for injectiveness. Could weaken premises! + major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B +*) +lemma UN_equiv_class_inject: + "[| 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" +apply (unfold quotient_def, safe) +apply (rule equiv_class_eq, assumption) +apply (rotate_tac -2) +apply (simp add: UN_equiv_class [of A r b]) +done + + +subsection{*Defining Binary Operations upon Equivalence Classes*} + + +lemma congruent2_implies_congruent: + "[| equiv(A,r); congruent2(r,b); a: A |] ==> congruent(r,b(a))" +apply (unfold congruent_def congruent2_def equiv_def refl_def, blast) +done + +lemma congruent2_implies_congruent_UN: + "[| equiv(A,r); congruent2(r,b); a: A |] ==> + congruent(r, %x1. UN x2:r``{a}. b(x1,x2))" +apply (unfold congruent_def, safe) +apply (frule equiv_type [THEN subsetD], assumption) +apply clarify +apply (simp add: UN_equiv_class [of A r] congruent2_implies_congruent) +apply (unfold congruent2_def equiv_def refl_def, blast) +done + +lemma UN_equiv_class2: + "[| 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 (simp add: UN_equiv_class [of A r] congruent2_implies_congruent + congruent2_implies_congruent_UN) + +(*type checking*) +lemma UN_equiv_class_type2: + "[| 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" +apply (unfold quotient_def, safe) +apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN + congruent2_implies_congruent quotientI) +done + + +(*Suggested by John Harrison -- the two subproofs may be MUCH simpler + than the direct proof*) +lemma congruent2I: + "[| 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)" +apply (unfold congruent2_def equiv_def refl_def, safe) +apply (blast intro: trans) +done + +lemma congruent2_commuteI: + assumes equivA: "equiv(A,r)" + and commute: "!! y z. [| y: A; z: A |] ==> b(y,z) = b(z,y)" + and congt: "!! y z w. [| w: A; : r |] ==> b(w,y) = b(w,z)" + shows "congruent2(r,b)" +apply (insert equivA [THEN equiv_type, THEN subsetD]) +apply (rule congruent2I [OF equivA]) +apply (rule commute [THEN trans]) +apply (rule_tac [3] commute [THEN trans, symmetric]) +apply (rule_tac [5] sym) +apply (blast intro: congt)+ +done + +(*Obsolete?*) +lemma congruent_commuteI: + "[| 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))" +apply (simp (no_asm) add: congruent_def) +apply (safe elim!: quotientE) +apply (frule equiv_type [THEN subsetD], assumption) +apply (simp add: UN_equiv_class [of A r]) +apply (simp add: congruent_def) +done + +ML +{* +val sym_trans_comp_subset = thm "sym_trans_comp_subset"; +val refl_comp_subset = thm "refl_comp_subset"; +val equiv_comp_eq = thm "equiv_comp_eq"; +val comp_equivI = thm "comp_equivI"; +val equiv_class_subset = thm "equiv_class_subset"; +val equiv_class_eq = thm "equiv_class_eq"; +val equiv_class_self = thm "equiv_class_self"; +val subset_equiv_class = thm "subset_equiv_class"; +val eq_equiv_class = thm "eq_equiv_class"; +val equiv_class_nondisjoint = thm "equiv_class_nondisjoint"; +val equiv_type = thm "equiv_type"; +val equiv_class_eq_iff = thm "equiv_class_eq_iff"; +val eq_equiv_class_iff = thm "eq_equiv_class_iff"; +val quotientI = thm "quotientI"; +val quotientE = thm "quotientE"; +val Union_quotient = thm "Union_quotient"; +val quotient_disj = thm "quotient_disj"; +val UN_equiv_class = thm "UN_equiv_class"; +val UN_equiv_class_type = thm "UN_equiv_class_type"; +val UN_equiv_class_inject = thm "UN_equiv_class_inject"; +val congruent2_implies_congruent = thm "congruent2_implies_congruent"; +val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN"; +val congruent_commuteI = thm "congruent_commuteI"; +val UN_equiv_class2 = thm "UN_equiv_class2"; +val UN_equiv_class_type2 = thm "UN_equiv_class_type2"; +val congruent2I = thm "congruent2I"; +val congruent2_commuteI = thm "congruent2_commuteI"; +val congruent_commuteI = thm "congruent_commuteI"; +*} + end