(* Title: ZF/EquivClass.thy ID: $Id: EquivClass.thy,v 1.9 2007/10/07 19:19:32 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge *) header{*Equivalence Relations*} theory EquivClass imports Trancl Perm begin definition quotient :: "[i,i]=>i" (infixl "'/'/" 90) (*set of equiv classes*) where "A//r == {r``{x} . x:A}" definition congruent :: "[i,i=>i]=>o" where "congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)" definition congruent2 :: "[i,i,[i,i]=>i]=>o" where "congruent2(r1,r2,b) == ALL y1 z1 y2 z2. <y1,z1>:r1 --> <y2,z2>:r2 --> b(y1,y2) = b(z1,z2)" abbreviation RESPECTS ::"[i=>i, i] => o" (infixr "respects" 80) where "f respects r == congruent(r,f)" abbreviation RESPECTS2 ::"[i=>i=>i, i] => o" (infixr "respects2 " 80) where "f respects2 r == congruent2(r,r,f)" --{*Abbreviation for the common case where the relations are identical*} 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" by (unfold trans_def sym_def, blast) lemma refl_comp_subset: "[| refl(A,r); r <= A*A |] ==> r <= converse(r) O r" by (unfold refl_def, blast) 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. <x,y> : r --> <y,x> : r", blast+) done (** Equivalence classes **) (*Lemma for the next result*) lemma equiv_class_subset: "[| sym(r); trans(r); <a,b>: r |] ==> r``{a} <= r``{b}" by (unfold trans_def sym_def, blast) lemma equiv_class_eq: "[| equiv(A,r); <a,b>: 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 |] ==> <a,b>: r" by (unfold equiv_def refl_def, blast) lemma eq_equiv_class: "[| r``{a} = r``{b}; equiv(A,r); b: A |] ==> <a,b>: 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}) |] ==> <a,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) ==> <x,y>: 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} <-> <x,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); b respects r; a: A |] ==> (UN x:r``{a}. b(x)) = b(a)" apply (subgoal_tac "∀x ∈ r``{a}. b(x) = b(a)") apply simp apply (blast intro: equiv_class_self) 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); b respects r; 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); b respects r; (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" apply (unfold quotient_def, safe) apply (rule equiv_class_eq, assumption) apply (simp add: UN_equiv_class [of A r b]) done subsection{*Defining Binary Operations upon Equivalence Classes*} lemma congruent2_implies_congruent: "[| equiv(A,r1); congruent2(r1,r2,b); a: A |] ==> congruent(r2,b(a))" by (unfold congruent_def congruent2_def equiv_def refl_def, blast) lemma congruent2_implies_congruent_UN: "[| equiv(A1,r1); equiv(A2,r2); congruent2(r1,r2,b); a: A2 |] ==> congruent(r1, %x1. \<Union>x2 ∈ r2``{a}. b(x1,x2))" apply (unfold congruent_def, safe) apply (frule equiv_type [THEN subsetD], assumption) apply clarify apply (simp add: UN_equiv_class congruent2_implies_congruent) apply (unfold congruent2_def equiv_def refl_def, blast) done lemma UN_equiv_class2: "[| equiv(A1,r1); equiv(A2,r2); congruent2(r1,r2,b); a1: A1; a2: A2 |] ==> (\<Union>x1 ∈ r1``{a1}. \<Union>x2 ∈ r2``{a2}. b(x1,x2)) = b(a1,a2)" by (simp add: UN_equiv_class congruent2_implies_congruent congruent2_implies_congruent_UN) (*type checking*) lemma UN_equiv_class_type2: "[| equiv(A,r); b respects2 r; 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(A1,r1); equiv(A2,r2); !! y z w. [| w ∈ A2; <y,z> ∈ r1 |] ==> b(y,w) = b(z,w); !! y z w. [| w ∈ A1; <y,z> ∈ r2 |] ==> b(w,y) = b(w,z) |] ==> congruent2(r1,r2,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; <y,z>: r |] ==> b(w,y) = b(w,z)" shows "b respects2 r" apply (insert equivA [THEN equiv_type, THEN subsetD]) apply (rule congruent2I [OF equivA 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 end
lemma sym_trans_comp_subset:
[| sym(r); trans(r) |] ==> converse(r) O r ⊆ r
lemma refl_comp_subset:
[| refl(A, r); r ⊆ A × A |] ==> r ⊆ converse(r) O r
lemma equiv_comp_eq:
equiv(A, r) ==> converse(r) O r = r
lemma comp_equivI:
[| converse(r) O r = r; domain(r) = A |] ==> equiv(A, r)
lemma equiv_class_subset:
[| sym(r); trans(r); 〈a, b〉 ∈ r |] ==> r `` {a} ⊆ r `` {b}
lemma equiv_class_eq:
[| equiv(A, r); 〈a, b〉 ∈ r |] ==> r `` {a} = r `` {b}
lemma equiv_class_self:
[| equiv(A, r); a ∈ A |] ==> a ∈ r `` {a}
lemma subset_equiv_class:
[| equiv(A, r); r `` {b} ⊆ r `` {a}; b ∈ A |] ==> 〈a, b〉 ∈ r
lemma eq_equiv_class:
[| r `` {a} = r `` {b}; equiv(A, r); b ∈ A |] ==> 〈a, b〉 ∈ r
lemma equiv_class_nondisjoint:
[| equiv(A, r); x ∈ r `` {a} ∩ r `` {b} |] ==> 〈a, b〉 ∈ r
lemma equiv_type:
equiv(A, r) ==> r ⊆ A × A
lemma equiv_class_eq_iff:
equiv(A, r) ==> 〈x, y〉 ∈ r <-> r `` {x} = r `` {y} ∧ x ∈ A ∧ y ∈ A
lemma eq_equiv_class_iff:
[| equiv(A, r); x ∈ A; y ∈ A |] ==> r `` {x} = r `` {y} <-> 〈x, y〉 ∈ r
lemma quotientI:
x ∈ A ==> r `` {x} ∈ A // r
lemma quotientE:
[| X ∈ A // r; !!x. [| X = r `` {x}; x ∈ A |] ==> P |] ==> P
lemma Union_quotient:
equiv(A, r) ==> \<Union>A // r = A
lemma quotient_disj:
[| equiv(A, r); X ∈ A // r; Y ∈ A // r |] ==> X = Y ∨ X ∩ Y ⊆ 0
lemma UN_equiv_class:
[| equiv(A, r); b respects r; a ∈ A |] ==> (\<Union>x∈r `` {a}. b(x)) = b(a)
lemma UN_equiv_class_type:
[| equiv(A, r); b respects r; X ∈ A // r; !!x. x ∈ A ==> b(x) ∈ B |]
==> (\<Union>x∈X. b(x)) ∈ B
lemma UN_equiv_class_inject:
[| equiv(A, r); b respects r; (\<Union>x∈X. b(x)) = (\<Union>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
lemma congruent2_implies_congruent:
[| equiv(A, r1.0); congruent2(r1.0, r2.0, b); a ∈ A |] ==> b(a) respects r2.0
lemma congruent2_implies_congruent_UN:
[| equiv(A1.0, r1.0); equiv(A2.0, r2.0); congruent2(r1.0, r2.0, b); a ∈ A2.0 |]
==> (λx1. \<Union>x2∈r2.0 `` {a}. b(x1, x2)) respects r1.0
lemma UN_equiv_class2:
[| equiv(A1.0, r1.0); equiv(A2.0, r2.0); congruent2(r1.0, r2.0, b); a1.0 ∈ A1.0;
a2.0 ∈ A2.0 |]
==> (\<Union>x1∈r1.0 `` {a1.0}. \<Union>x2∈r2.0 `` {a2.0}. b(x1, x2)) =
b(a1.0, a2.0)
lemma UN_equiv_class_type2:
[| equiv(A, r); b respects2 r; X1.0 ∈ A // r; X2.0 ∈ A // r;
!!x1 x2. [| x1 ∈ A; x2 ∈ A |] ==> b(x1, x2) ∈ B |]
==> (\<Union>x1∈X1.0. \<Union>x2∈X2.0. b(x1, x2)) ∈ B
lemma congruent2I:
[| equiv(A1.0, r1.0); equiv(A2.0, r2.0);
!!y z w. [| w ∈ A2.0; 〈y, z〉 ∈ r1.0 |] ==> b(y, w) = b(z, w);
!!y z w. [| w ∈ A1.0; 〈y, z〉 ∈ r2.0 |] ==> b(w, y) = b(w, z) |]
==> congruent2(r1.0, r2.0, b)
lemma congruent2_commuteI:
[| 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) |]
==> b respects2 r
lemma congruent_commuteI:
[| equiv(A, r); Z ∈ A // r; !!w. w ∈ A ==> (λz. b(w, z)) respects r;
!!x y. [| x ∈ A; y ∈ A |] ==> b(y, x) = b(x, y) |]
==> (λw. \<Union>z∈Z. b(w, z)) respects r