Theory EquivClass

(*  Title:      ZF/EquivClass.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

section‹Equivalence Relations›

theory EquivClass imports Trancl Perm begin

  quotient   :: "[i,i]i"    (infixl '/'/ 90)  (*set of equiv classes*)  where
      "A//r  {r``{x} . x  A}"

  congruent  :: "[i,ii]o"  where
      "congruent(r,b)  y z. y,z:r  b(y)=b(z)"

  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)"

  RESPECTS ::"[ii, i]  o"  (infixr respects 80) where
  "f respects r  congruent(r,f)"

  RESPECTS2 ::"[iii, 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:
    termr is an equiv relation iff termconverse(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)

(*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+)

(** 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)

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)

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)

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  (xr``{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)

(*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
      (xX. b(x))  B"
apply (unfold quotient_def, safe)
apply (simp (no_asm_simp) add: UN_equiv_class)

(*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;
        (xX. b(x))=(yY. b(y));  X  A//r;  Y  A//r;
        x y. x  A; y  A; b(x)=b(y)  x,y:r
apply (unfold quotient_def, safe)
apply (rule equiv_class_eq, assumption)
apply (simp add: UN_equiv_class [of A r b])

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)

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

(*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
  (x1X1. x2X2. b(x1,x2))  B"
apply (unfold quotient_def, safe)
apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
                    congruent2_implies_congruent quotientI)

(*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)
apply (unfold congruent2_def equiv_def refl_def, safe)
apply (blast intro: trans)

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)+

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. zZ. 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)