# Theory EquivClass

```(*  Title:      ZF/EquivClass.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section‹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) ≡ ∀y z. ⟨y,z⟩:r ⟶ b(y)=b(z)"

definition
congruent2 :: "[i,i,[i,i]⇒i]⇒o"  where
"congruent2(r1,r2,b) ≡ ∀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"
unfolding 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)"
unfolding equiv_def refl_def sym_def trans_def
apply (erule equalityE)
apply (subgoal_tac "∀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}"
unfolding 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} ∩ 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"
unfolding 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) ⟹ ⋃(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 ∩ Y ⊆ 0)"
unfolding 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⟧ ⟹ (⋃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  @{term"⋃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⟧
⟹ (⋃x∈X. b(x)) ∈ B"
apply (unfold quotient_def, safe)
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;
(⋃x∈X. b(x))=(⋃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, λx1. ⋃x2 ∈ r2``{a}. b(x1,x2))"
apply (unfold congruent_def, safe)
apply (frule equiv_type [THEN subsetD], assumption)
apply clarify
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⟧
⟹ (⋃x1 ∈ r1``{a1}. ⋃x2 ∈ r2``{a2}. b(x1,x2)) = b(a1,a2)"
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
⟧ ⟹ (⋃x1∈X1. ⋃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, λz. b(w,z));
⋀x y. ⟦x ∈ A;  y ∈ A⟧ ⟹ b(y,x) = b(x,y)
⟧ ⟹ congruent(r, λw. ⋃z∈Z. b(w,z))"