Theory EquivClass
section‹Equivalence Relations›
theory EquivClass imports Trancl Perm begin
definition
quotient :: "[i,i]⇒i" (infixl ‹'/'/› 90) 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)"
subsection‹Suppes, Theorem 70:
\<^term>‹r› is an equiv relation iff \<^term>‹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
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
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 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)+
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)
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›
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
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)
apply (simp (no_asm_simp) add: UN_equiv_class)
done
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 (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⟧
⟹ (⋃x1 ∈ r1``{a1}. ⋃x2 ∈ r2``{a2}. b(x1,x2)) = b(a1,a2)"
by (simp add: UN_equiv_class congruent2_implies_congruent
congruent2_implies_congruent_UN)
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
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
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))"
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