Quelle  equalities.thy   Sprache: Isabelle

(*  Title:      ZF/equalities.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge
*)

section‹Basic Equalities and Inclusions›

theory equalities imports pair begin

text‹These cover union, intersection, converse, domain, range, etc.  Philippe
de Groote proved many of the inclusions.›

lemma in_mono: "A\B \ x\A \ x\B"
by blast

lemma the_eq_0 [simp]: "(THE x. False) = 0"
by (blast intro: the_0)

subsection‹Bounded Quantifiers›
text ‹\medskip

  The following are not added to the default simpset because
  (a) they duplicate the body and (b) there are no similar rules for ‹Int›.›

lemma ball_Un: "(\x \ A\B. P(x)) \ (\x \ A. P(x)) \ (\x \ B. P(x))"
  by blast

lemma bex_Un: "(\x \ A\B. P(x)) \ (\x \ A. P(x)) | (\x \ B. P(x))"
  by blast

lemma ball_UN: "(\z \ (\x\A. B(x)). P(z)) \ (\x\A. \z \ B(x). P(z))"
  by blast

lemma bex_UN: "(\z \ (\x\A. B(x)). P(z)) \ (\x\A. \z\B(x). P(z))"
  by blast

subsection‹Converse of a Relation›

lemma converse_iff [simp]: "\a,b\\ converse(r) \ \b,a\\r"
by (unfold converse_def, blast)

lemma converseI [intro!]: "\a,b\\r \ \b,a\\converse(r)"
by (unfold converse_def, blast)

lemma converseD: "\a,b\ \ converse(r) \ \b,a\ \ r"
by (unfold converse_def, blast)

lemma converseE [elim!]:
    "\yx \ converse(r);
        ∧x y. [yx=⟨y,x⟩;  ⟨x,y⟩∈r] ==> P]
     ==> P"
by (unfold converse_def, blast)

lemma converse_converse: "r\Sigma(A,B) \ converse(converse(r)) = r"
by blast

lemma converse_type: "r\A*B \ converse(r)\B*A"
by blast

lemma converse_prod [simp]: "converse(A*B) = B*A"
by blast

lemma converse_empty [simp]: "converse(0) = 0"
by blast

lemma converse_subset_iff:
     "A \ Sigma(X,Y) \ converse(A) \ converse(B) \ A \ B"
by blast


subsection‹Finite Set Constructions Using 🍋‹cons››

lemma cons_subsetI: "\a\C; B\C\ \ cons(a,B) \ C"
by blast

lemma subset_consI: "B \ cons(a,B)"
by blast

lemma cons_subset_iff [iff]: "cons(a,B)\C \ a\C \ B\C"
by blast

(*A safe special case of subset elimination, adding no new variables
  \<lbrakk>cons(a,B) \<subseteq> C; \<lbrakk>a \<in> C; B \<subseteq> C\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R *)
lemmas cons_subsetE = cons_subset_iff [THEN iffD1, THEN conjE]

lemma subset_empty_iff: "A\0 \ A=0"
by blast

lemma subset_cons_iff: "C\cons(a,B) \ C\B | (a\C \ C-{a} \ B)"
by blast

(* cons_def refers to Upair; reversing the equality LOOPS in rewriting!*)
lemma cons_eq: "{a} \ B = cons(a,B)"
by blast

lemma cons_commute: "cons(a, cons(b, C)) = cons(b, cons(a, C))"
by blast

lemma cons_absorb: "a: B \ cons(a,B) = B"
by blast

lemma cons_Diff: "a: B \ cons(a, B-{a}) = B"
by blast

lemma Diff_cons_eq: "cons(a,B) - C = (if a\C then B-C else cons(a,B-C))"
by auto

lemma equal_singleton: "\a: C; \y. y \C \ y=b\ \ C = {b}"
by blast

lemma [simp]: "cons(a,cons(a,B)) = cons(a,B)"
by blast

(** singletons **)

lemma singleton_subsetI: "a\C \ {a} \ C"
by blast

lemma singleton_subsetD: "{a} \ C \ a\C"
by blast


(** succ **)

lemma subset_succI: "i \ succ(i)"
by blast

(*But if j is an ordinal or is transitive, then @{term"i\<in>j"} implies @{term"i\<subseteq>j"}!
  See @{text"Ord_succ_subsetI}*)
lemma succ_subsetI: "\i\j; i\j\ \ succ(i)\j"
by (unfold succ_def, blast)

lemma succ_subsetE:
    "\succ(i) \ j; \i\j; i\j\ \ P\ \ P"
by (unfold succ_def, blast)

lemma succ_subset_iff: "succ(a) \ B \ (a \ B \ a \ B)"
by (unfold succ_def, blast)


subsection‹Binary Intersection›

(** Intersection is the greatest lower bound of two sets **)

lemma Int_subset_iff: "C \ A \ B \ C \ A \ C \ B"
by blast

lemma Int_lower1: "A \ B \ A"
by blast

lemma Int_lower2: "A \ B \ B"
by blast

lemma Int_greatest: "\C\A; C\B\ \ C \ A \ B"
by blast

lemma Int_cons: "cons(a,B) \ C \ cons(a, B \ C)"
by blast

lemma Int_absorb [simp]: "A \ A = A"
by blast

lemma Int_left_absorb: "A \ (A \ B) = A \ B"
by blast

lemma Int_commute: "A \ B = B \ A"
by blast

lemma Int_left_commute: "A \ (B \ C) = B \ (A \ C)"
by blast

lemma Int_assoc: "(A \ B) \ C = A \ (B \ C)"
by blast

(*Intersection is an AC-operator*)
lemmas Int_ac= Int_assoc Int_left_absorb Int_commute Int_left_commute

lemma Int_absorb1: "B \ A \ A \ B = B"
  by blast

lemma Int_absorb2: "A \ B \ A \ B = A"
  by blast

lemma Int_Un_distrib: "A \ (B \ C) = (A \ B) \ (A \ C)"
by blast

lemma Int_Un_distrib2: "(B \ C) \ A = (B \ A) \ (C \ A)"
by blast

lemma subset_Int_iff: "A\B \ A \ B = A"
by (blast elim!: equalityE)

lemma subset_Int_iff2: "A\B \ B \ A = A"
by (blast elim!: equalityE)

lemma Int_Diff_eq: "C\A \ (A-B) \ C = C-B"
by blast

lemma Int_cons_left:
     "cons(a,A) \ B = (if a \ B then cons(a, A \ B) else A \ B)"
by auto

lemma Int_cons_right:
     "A \ cons(a, B) = (if a \ A then cons(a, A \ B) else A \ B)"
by auto

lemma cons_Int_distrib: "cons(x, A \ B) = cons(x, A) \ cons(x, B)"
by auto

subsection‹Binary Union›

(** Union is the least upper bound of two sets *)

lemma Un_subset_iff: "A \ B \ C \ A \ C \ B \ C"
by blast

lemma Un_upper1: "A \ A \ B"
by blast

lemma Un_upper2: "B \ A \ B"
by blast

lemma Un_least: "\A\C; B\C\ \ A \ B \ C"
by blast

lemma Un_cons: "cons(a,B) \ C = cons(a, B \ C)"
by blast

lemma Un_absorb [simp]: "A \ A = A"
by blast

lemma Un_left_absorb: "A \ (A \ B) = A \ B"
by blast

lemma Un_commute: "A \ B = B \ A"
by blast

lemma Un_left_commute: "A \ (B \ C) = B \ (A \ C)"
by blast

lemma Un_assoc: "(A \ B) \ C = A \ (B \ C)"
by blast

(*Union is an AC-operator*)
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute

lemma Un_absorb1: "A \ B \ A \ B = B"
  by blast

lemma Un_absorb2: "B \ A \ A \ B = A"
  by blast

lemma Un_Int_distrib: "(A \ B) \ C = (A \ C) \ (B \ C)"
by blast

lemma subset_Un_iff: "A\B \ A \ B = B"
by (blast elim!: equalityE)

lemma subset_Un_iff2: "A\B \ B \ A = B"
by (blast elim!: equalityE)

lemma Un_empty [iff]: "(A \ B = 0) \ (A = 0 \ B = 0)"
by blast

lemma Un_eq_Union: "A \ B = \({A, B})"
by blast

subsection‹Set Difference›

lemma Diff_subset: "A-B \ A"
by blast

lemma Diff_contains: "\C\A; C \ B = 0\ \ C \ A-B"
by blast

lemma subset_Diff_cons_iff: "B \ A - cons(c,C) \ B\A-C \ c \ B"
by blast

lemma Diff_cancel: "A - A = 0"
by blast

lemma Diff_triv: "A \ B = 0 \ A - B = A"
by blast

lemma empty_Diff [simp]: "0 - A = 0"
by blast

lemma Diff_0 [simp]: "A - 0 = A"
by blast

lemma Diff_eq_0_iff: "A - B = 0 \ A \ B"
by (blast elim: equalityE)

(*NOT SUITABLE FOR REWRITING since {a} \<equiv> cons(a,0)*)
lemma Diff_cons: "A - cons(a,B) = A - B - {a}"
by blast

(*NOT SUITABLE FOR REWRITING since {a} \<equiv> cons(a,0)*)
lemma Diff_cons2: "A - cons(a,B) = A - {a} - B"
by blast

lemma Diff_disjoint: "A \ (B-A) = 0"
by blast

lemma Diff_partition: "A\B \ A \ (B-A) = B"
by blast

lemma subset_Un_Diff: "A \ B \ (A - B)"
by blast

lemma double_complement: "\A\B; B\C\ \ B-(C-A) = A"
by blast

lemma double_complement_Un: "(A \ B) - (B-A) = A"
by blast

lemma Un_Int_crazy:
 "(A \ B) \ (B \ C) \ (C \ A) = (A \ B) \ (B \ C) \ (C \ A)"
apply blast
done

lemma Diff_Un: "A - (B \ C) = (A-B) \ (A-C)"
by blast

lemma Diff_Int: "A - (B \ C) = (A-B) \ (A-C)"
by blast

lemma Un_Diff: "(A \ B) - C = (A - C) \ (B - C)"
by blast

lemma Int_Diff: "(A \ B) - C = A \ (B - C)"
by blast

lemma Diff_Int_distrib: "C \ (A-B) = (C \ A) - (C \ B)"
by blast

lemma Diff_Int_distrib2: "(A-B) \ C = (A \ C) - (B \ C)"
by blast

(*Halmos, Naive Set Theory, page 16.*)
lemma Un_Int_assoc_iff: "(A \ B) \ C = A \ (B \ C) \ C\A"
by (blast elim!: equalityE)


subsection‹Big Union and Intersection›

(** Big Union is the least upper bound of a set  **)

lemma Union_subset_iff: "\(A) \ C \ (\x\A. x \ C)"
by blast

lemma Union_upper: "B\A \ B \ \(A)"
by blast

lemma Union_least: "\\x. x\A \ x\C\ \ \(A) \ C"
by blast

lemma Union_cons [simp]: "\(cons(a,B)) = a \ \(B)"
by blast

lemma Union_Un_distrib: "\(A \ B) = \(A) \ \(B)"
by blast

lemma Union_Int_subset: "\(A \ B) \ \(A) \ \(B)"
by blast

lemma Union_disjoint: "\(C) \ A = 0 \ (\B\C. B \ A = 0)"
by (blast elim!: equalityE)

lemma Union_empty_iff: "\(A) = 0 \ (\B\A. B=0)"
by blast

lemma Int_Union2: "\(B) \ A = (\C\B. C \ A)"
by blast

(** Big Intersection is the greatest lower bound of a nonempty set **)

lemma Inter_subset_iff: "A\0 \ C \ \(A) \ (\x\A. C \ x)"
by blast

lemma Inter_lower: "B\A \ \(A) \ B"
by blast

lemma Inter_greatest: "\A\0; \x. x\A \ C\x\ \ C \ \(A)"
by blast

(** Intersection of a family of sets  **)

lemma INT_lower: "x\A \ (\x\A. B(x)) \ B(x)"
by blast

lemma INT_greatest: "\A\0; \x. x\A \ C\B(x)\ \ C \ (\x\A. B(x))"
by force

lemma Inter_0 [simp]: "\(0) = 0"
by (unfold Inter_def, blast)

lemma Inter_Un_subset:
     "\z\A; z\B\ \ \(A) \ \(B) \ \(A \ B)"
by blast

(* A good challenge: Inter is ill-behaved on the empty set *)
lemma Inter_Un_distrib:
     "\A\0; B\0\ \ \(A \ B) = \(A) \ \(B)"
by blast

lemma Union_singleton: "\({b}) = b"
by blast

lemma Inter_singleton: "\({b}) = b"
by blast

lemma Inter_cons [simp]:
     "\(cons(a,B)) = (if B=0 then a else a \ \(B))"
by force

subsection‹Unions and Intersections of Families›

lemma subset_UN_iff_eq: "A \ (\i\I. B(i)) \ A = (\i\I. A \ B(i))"
by (blast elim!: equalityE)

lemma UN_subset_iff: "(\x\A. B(x)) \ C \ (\x\A. B(x) \ C)"
by blast

lemma UN_upper: "x\A \ B(x) \ (\x\A. B(x))"
by (erule RepFunI [THEN Union_upper])

lemma UN_least: "\\x. x\A \ B(x)\C\ \ (\x\A. B(x)) \ C"
by blast

lemma Union_eq_UN: "\(A) = (\x\A. x)"
by blast

lemma Inter_eq_INT: "\(A) = (\x\A. x)"
by (unfold Inter_def, blast)

lemma UN_0 [simp]: "(\i\0. A(i)) = 0"
by blast

lemma UN_singleton: "(\x\A. {x}) = A"
by blast

lemma UN_Un: "(\i\ A \ B. C(i)) = (\i\ A. C(i)) \ (\i\B. C(i))"
by blast

lemma INT_Un: "(\i\I \ J. A(i)) =
               (if I=0 then ∩j∈J. A(j)
                       else if J=0 then ∩i∈I. A(i)
                       else ((∩i∈I. A(i)) ∩  (∩j∈J. A(j))))"
by (simp, blast intro!: equalityI)

lemma UN_UN_flatten: "(\x \ (\y\A. B(y)). C(x)) = (\y\A. \x\ B(y). C(x))"
by blast

(*Halmos, Naive Set Theory, page 35.*)
lemma Int_UN_distrib: "B \ (\i\I. A(i)) = (\i\I. B \ A(i))"
by blast

lemma Un_INT_distrib: "I\0 \ B \ (\i\I. A(i)) = (\i\I. B \ A(i))"
by auto

lemma Int_UN_distrib2:
     "(\i\I. A(i)) \ (\j\J. B(j)) = (\i\I. \j\J. A(i) \ B(j))"
by blast

lemma Un_INT_distrib2: "\I\0; J\0\ \
      (∩i∈I. A(i)) ∪ (∩j∈J. B(j)) = (∩i∈I. ∩j∈J. A(i) ∪ B(j))"
by auto

lemma UN_constant [simp]: "(\y\A. c) = (if A=0 then 0 else c)"
by force

lemma INT_constant [simp]: "(\y\A. c) = (if A=0 then 0 else c)"
by force

lemma UN_RepFun [simp]: "(\y\ RepFun(A,f). B(y)) = (\x\A. B(f(x)))"
by blast

lemma INT_RepFun [simp]: "(\x\RepFun(A,f). B(x)) = (\a\A. B(f(a)))"
by (auto simp add: Inter_def)

lemma INT_Union_eq:
     "0 \ A \ (\x\ \(A). B(x)) = (\y\A. \x\y. B(x))"
apply (subgoal_tac "\x\A. x\0")
   prefer 2 apply blast
apply (force simp add: Inter_def ball_conj_distrib)
done

lemma INT_UN_eq:
     "(\x\A. B(x) \ 0)
      ==> (∩z∈ (∪x∈A. B(x)). C(z)) = (∩x∈A. ∩z∈ B(x). C(z))"
apply (subst INT_Union_eq, blast)
apply (simp add: Inter_def)
done


(** Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:
    Union of a family of unions **)

lemma UN_Un_distrib:
     "(\i\I. A(i) \ B(i)) = (\i\I. A(i)) \ (\i\I. B(i))"
by blast

lemma INT_Int_distrib:
     "I\0 \ (\i\I. A(i) \ B(i)) = (\i\I. A(i)) \ (\i\I. B(i))"
by (blast elim!: not_emptyE)

lemma UN_Int_subset:
     "(\z\I \ J. A(z)) \ (\z\I. A(z)) \ (\z\J. A(z))"
by blast

(** Devlin, page 12, exercise 5: Complements **)

lemma Diff_UN: "I\0 \ B - (\i\I. A(i)) = (\i\I. B - A(i))"
by (blast elim!: not_emptyE)

lemma Diff_INT: "I\0 \ B - (\i\I. A(i)) = (\i\I. B - A(i))"
by (blast elim!: not_emptyE)


(** Unions and Intersections with General Sum **)

(*Not suitable for rewriting: LOOPS!*)
lemma Sigma_cons1: "Sigma(cons(a,B), C) = ({a}*C(a)) \ Sigma(B,C)"
by blast

(*Not suitable for rewriting: LOOPS!*)
lemma Sigma_cons2: "A * cons(b,B) = A*{b} \ A*B"
by blast

lemma Sigma_succ1: "Sigma(succ(A), B) = ({A}*B(A)) \ Sigma(A,B)"
by blast

lemma Sigma_succ2: "A * succ(B) = A*{B} \ A*B"
by blast

lemma SUM_UN_distrib1:
     "(\x \ (\y\A. C(y)). B(x)) = (\y\A. \x\C(y). B(x))"
by blast

lemma SUM_UN_distrib2:
     "(\i\I. \j\J. C(i,j)) = (\j\J. \i\I. C(i,j))"
by blast

lemma SUM_Un_distrib1:
     "(\i\I \ J. C(i)) = (\i\I. C(i)) \ (\j\J. C(j))"
by blast

lemma SUM_Un_distrib2:
     "(\i\I. A(i) \ B(i)) = (\i\I. A(i)) \ (\i\I. B(i))"
by blast

(*First-order version of the above, for rewriting*)
lemma prod_Un_distrib2: "I * (A \ B) = I*A \ I*B"
by (rule SUM_Un_distrib2)

lemma SUM_Int_distrib1:
     "(\i\I \ J. C(i)) = (\i\I. C(i)) \ (\j\J. C(j))"
by blast

lemma SUM_Int_distrib2:
     "(\i\I. A(i) \ B(i)) = (\i\I. A(i)) \ (\i\I. B(i))"
by blast

(*First-order version of the above, for rewriting*)
lemma prod_Int_distrib2: "I * (A \ B) = I*A \ I*B"
by (rule SUM_Int_distrib2)

(*Cf Aczel, Non-Well-Founded Sets, page 115*)
lemma SUM_eq_UN: "(\i\I. A(i)) = (\i\I. {i} * A(i))"
by blast

lemma times_subset_iff:
     "(A'*B' \ A*B) \ (A' = 0 | B' = 0 | (A'\A) \ (B'\B))"
by blast

lemma Int_Sigma_eq:
     "(\x \ A'. B'(x)) \ (\x \ A. B(x)) = (\x \ A' \ A. B'(x) \ B(x))"
by blast

(** Domain **)

lemma domain_iff: "a: domain(r) \ (\y. \a,y\\ r)"
by (unfold domain_def, blast)

lemma domainI [intro]: "\a,b\\ r \ a: domain(r)"
by (unfold domain_def, blast)

lemma domainE [elim!]:
    "\a \ domain(r); \y. \a,y\\ r \ P\ \ P"
by (unfold domain_def, blast)

lemma domain_subset: "domain(Sigma(A,B)) \ A"
by blast

lemma domain_of_prod: "b\B \ domain(A*B) = A"
by blast

lemma domain_0 [simp]: "domain(0) = 0"
by blast

lemma domain_cons [simp]: "domain(cons(\a,b\,r)) = cons(a, domain(r))"
by blast

lemma domain_Un_eq [simp]: "domain(A \ B) = domain(A) \ domain(B)"
by blast

lemma domain_Int_subset: "domain(A \ B) \ domain(A) \ domain(B)"
by blast

lemma domain_Diff_subset: "domain(A) - domain(B) \ domain(A - B)"
by blast

lemma domain_UN: "domain(\x\A. B(x)) = (\x\A. domain(B(x)))"
by blast

lemma domain_Union: "domain(\(A)) = (\x\A. domain(x))"
by blast


(** Range **)

lemma rangeI [intro]: "\a,b\\ r \ b \ range(r)"
  unfolding range_def
apply (erule converseI [THEN domainI])
done

lemma rangeE [elim!]: "\b \ range(r); \x. \x,b\\ r \ P\ \ P"
by (unfold range_def, blast)

lemma range_subset: "range(A*B) \ B"
  unfolding range_def
apply (subst converse_prod)
apply (rule domain_subset)
done

lemma range_of_prod: "a\A \ range(A*B) = B"
by blast

lemma range_0 [simp]: "range(0) = 0"
by blast

lemma range_cons [simp]: "range(cons(\a,b\,r)) = cons(b, range(r))"
by blast

lemma range_Un_eq [simp]: "range(A \ B) = range(A) \ range(B)"
by blast

lemma range_Int_subset: "range(A \ B) \ range(A) \ range(B)"
by blast

lemma range_Diff_subset: "range(A) - range(B) \ range(A - B)"
by blast

lemma domain_converse [simp]: "domain(converse(r)) = range(r)"
by blast

lemma range_converse [simp]: "range(converse(r)) = domain(r)"
by blast


(** Field **)

lemma fieldI1: "\a,b\\ r \ a \ field(r)"
by (unfold field_def, blast)

lemma fieldI2: "\a,b\\ r \ b \ field(r)"
by (unfold field_def, blast)

lemma fieldCI [intro]:
    "(\ \c,a\\r \ \a,b\\ r) \ a \ field(r)"
apply (unfold field_def, blast)
done

lemma fieldE [elim!]:
     "\a \ field(r);
         ∧x. ⟨a,x⟩∈ r ==> P;
         ∧x. ⟨x,a⟩∈ r ==> P] ==> P"
by (unfold field_def, blast)

lemma field_subset: "field(A*B) \ A \ B"
by blast

lemma domain_subset_field: "domain(r) \ field(r)"
  unfolding field_def
apply (rule Un_upper1)
done

lemma range_subset_field: "range(r) \ field(r)"
  unfolding field_def
apply (rule Un_upper2)
done

lemma domain_times_range: "r \ Sigma(A,B) \ r \ domain(r)*range(r)"
by blast

lemma field_times_field: "r \ Sigma(A,B) \ r \ field(r)*field(r)"
by blast

lemma relation_field_times_field: "relation(r) \ r \ field(r)*field(r)"
by (simp add: relation_def, blast)

lemma field_of_prod: "field(A*A) = A"
by blast

lemma field_0 [simp]: "field(0) = 0"
by blast

lemma field_cons [simp]: "field(cons(\a,b\,r)) = cons(a, cons(b, field(r)))"
by blast

lemma field_Un_eq [simp]: "field(A \ B) = field(A) \ field(B)"
by blast

lemma field_Int_subset: "field(A \ B) \ field(A) \ field(B)"
by blast

lemma field_Diff_subset: "field(A) - field(B) \ field(A - B)"
by blast

lemma field_converse [simp]: "field(converse(r)) = field(r)"
by blast

(** The Union of a set of relations is a relation -- Lemma for fun_Union **)
lemma rel_Union: "(\x\S. \A B. x \ A*B) \
                  ∪(S) ⊆ domain(∪(S)) * range(∪(S))"
by blast

(** The Union of 2 relations is a relation (Lemma for fun_Un)  **)
lemma rel_Un: "\r \ A*B; s \ C*D\ \ (r \ s) \ (A \ C) * (B \ D)"
by blast

lemma domain_Diff_eq: "\\a,c\ \ r; c\b\ \ domain(r-{\a,b\}) = domain(r)"
by blast

lemma range_Diff_eq: "\\c,b\ \ r; c\a\ \ range(r-{\a,b\}) = range(r)"
by blast


subsection‹Image of a Set under a Function or Relation›

lemma image_iff: "b \ r``A \ (\x\A. \x,b\\r)"
by (unfold image_def, blast)

lemma image_singleton_iff: "b \ r``{a} \ \a,b\\r"
by (rule image_iff [THEN iff_trans], blast)

lemma imageI [intro]: "\\a,b\\ r; a\A\ \ b \ r``A"
by (unfold image_def, blast)

lemma imageE [elim!]:
    "\b: r``A; \x.\\x,b\\ r; x\A\ \ P\ \ P"
by (unfold image_def, blast)

lemma image_subset: "r \ A*B \ r``C \ B"
by blast

lemma image_0 [simp]: "r``0 = 0"
by blast

lemma image_Un [simp]: "r``(A \ B) = (r``A) \ (r``B)"
by blast

lemma image_UN: "r `` (\x\A. B(x)) = (\x\A. r `` B(x))"
by blast

lemma Collect_image_eq:
     "{z \ Sigma(A,B). P(z)} `` C = (\x \ A. {y \ B(x). x \ C \ P(\x,y\)})"
by blast

lemma image_Int_subset: "r``(A \ B) \ (r``A) \ (r``B)"
by blast

lemma image_Int_square_subset: "(r \ A*A)``B \ (r``B) \ A"
by blast

lemma image_Int_square: "B\A \ (r \ A*A)``B = (r``B) \ A"
by blast


(*Image laws for special relations*)
lemma image_0_left [simp]: "0``A = 0"
by blast

lemma image_Un_left: "(r \ s)``A = (r``A) \ (s``A)"
by blast

lemma image_Int_subset_left: "(r \ s)``A \ (r``A) \ (s``A)"
by blast


subsection‹Inverse Image of a Set under a Function or Relation›

lemma vimage_iff:
    "a \ r-``B \ (\y\B. \a,y\\r)"
by (unfold vimage_def image_def converse_def, blast)

lemma vimage_singleton_iff: "a \ r-``{b} \ \a,b\\r"
by (rule vimage_iff [THEN iff_trans], blast)

lemma vimageI [intro]: "\\a,b\\ r; b\B\ \ a \ r-``B"
by (unfold vimage_def, blast)

lemma vimageE [elim!]:
    "\a: r-``B; \x.\\a,x\\ r; x\B\ \ P\ \ P"
apply (unfold vimage_def, blast)
done

lemma vimage_subset: "r \ A*B \ r-``C \ A"
  unfolding vimage_def
apply (erule converse_type [THEN image_subset])
done

lemma vimage_0 [simp]: "r-``0 = 0"
by blast

lemma vimage_Un [simp]: "r-``(A \ B) = (r-``A) \ (r-``B)"
by blast

lemma vimage_Int_subset: "r-``(A \ B) \ (r-``A) \ (r-``B)"
by blast

(*NOT suitable for rewriting*)
lemma vimage_eq_UN: "f -``B = (\y\B. f-``{y})"
by blast

lemma function_vimage_Int:
     "function(f) \ f-``(A \ B) = (f-``A) \ (f-``B)"
by (unfold function_def, blast)

lemma function_vimage_Diff: "function(f) \ f-``(A-B) = (f-``A) - (f-``B)"
by (unfold function_def, blast)

lemma function_image_vimage: "function(f) \ f `` (f-`` A) \ A"
by (unfold function_def, blast)

lemma vimage_Int_square_subset: "(r \ A*A)-``B \ (r-``B) \ A"
by blast

lemma vimage_Int_square: "B\A \ (r \ A*A)-``B = (r-``B) \ A"
by blast



(*Invese image laws for special relations*)
lemma vimage_0_left [simp]: "0-``A = 0"
by blast

lemma vimage_Un_left: "(r \ s)-``A = (r-``A) \ (s-``A)"
by blast

lemma vimage_Int_subset_left: "(r \ s)-``A \ (r-``A) \ (s-``A)"
by blast


(** Converse **)

lemma converse_Un [simp]: "converse(A \ B) = converse(A) \ converse(B)"
by blast

lemma converse_Int [simp]: "converse(A \ B) = converse(A) \ converse(B)"
by blast

lemma converse_Diff [simp]: "converse(A - B) = converse(A) - converse(B)"
by blast

lemma converse_UN [simp]: "converse(\x\A. B(x)) = (\x\A. converse(B(x)))"
by blast

(*Unfolding Inter avoids using excluded middle on A=0*)
lemma converse_INT [simp]:
     "converse(\x\A. B(x)) = (\x\A. converse(B(x)))"
apply (unfold Inter_def, blast)
done


subsection‹Powerset Operator›

lemma Pow_0 [simp]: "Pow(0) = {0}"
by blast

lemma Pow_insert: "Pow (cons(a,A)) = Pow(A) \ {cons(a,X) . X: Pow(A)}"
apply (rule equalityI, safe)
apply (erule swap)
apply (rule_tac a = "x-{a}" in RepFun_eqI, auto)
done

lemma Un_Pow_subset: "Pow(A) \ Pow(B) \ Pow(A \ B)"
by blast

lemma UN_Pow_subset: "(\x\A. Pow(B(x))) \ Pow(\x\A. B(x))"
by blast

lemma subset_Pow_Union: "A \ Pow(\(A))"
by blast

lemma Union_Pow_eq [simp]: "\(Pow(A)) = A"
by blast

lemma Union_Pow_iff: "\(A) \ Pow(B) \ A \ Pow(Pow(B))"
by blast

lemma Pow_Int_eq [simp]: "Pow(A \ B) = Pow(A) \ Pow(B)"
by blast

lemma Pow_INT_eq: "A\0 \ Pow(\x\A. B(x)) = (\x\A. Pow(B(x)))"
by (blast elim!: not_emptyE)


subsection‹RepFun›

lemma RepFun_subset: "\\x. x\A \ f(x) \ B\ \ {f(x). x\A} \ B"
by blast

lemma RepFun_eq_0_iff [simp]: "{f(x).x\A}=0 \ A=0"
by blast

lemma RepFun_constant [simp]: "{c. x\A} = (if A=0 then 0 else {c})"
by force


subsection‹Collect›

lemma Collect_subset: "Collect(A,P) \ A"
by blast

lemma Collect_Un: "Collect(A \ B, P) = Collect(A,P) \ Collect(B,P)"
by blast

lemma Collect_Int: "Collect(A \ B, P) = Collect(A,P) \ Collect(B,P)"
by blast

lemma Collect_Diff: "Collect(A - B, P) = Collect(A,P) - Collect(B,P)"
by blast

lemma Collect_cons: "{x\cons(a,B). P(x)} =
      (if P(a) then cons(a, {x∈B. P(x)}) else {x∈B. P(x)})"
by (simp, blast)

lemma Int_Collect_self_eq: "A \ Collect(A,P) = Collect(A,P)"
by blast

lemma Collect_Collect_eq [simp]:
     "Collect(Collect(A,P), Q) = Collect(A, \x. P(x) \ Q(x))"
by blast

lemma Collect_Int_Collect_eq:
     "Collect(A,P) \ Collect(A,Q) = Collect(A, \x. P(x) \ Q(x))"
by blast

lemma Collect_Union_eq [simp]:
     "Collect(\x\A. B(x), P) = (\x\A. Collect(B(x), P))"
by blast

lemma Collect_Int_left: "{x\A. P(x)} \ B = {x \ A \ B. P(x)}"
by blast

lemma Collect_Int_right: "A \ {x\B. P(x)} = {x \ A \ B. P(x)}"
by blast

lemma Collect_disj_eq: "{x\A. P(x) | Q(x)} = Collect(A, P) \ Collect(A, Q)"
by blast

lemma Collect_conj_eq: "{x\A. P(x) \ Q(x)} = Collect(A, P) \ Collect(A, Q)"
by blast

lemmas subset_SIs = subset_refl cons_subsetI subset_consI
                    Union_least UN_least Un_least
                    Inter_greatest Int_greatest RepFun_subset
                    Un_upper1 Un_upper2 Int_lower1 Int_lower2

ML ‹
val subset_cs =
  claset_of (🍋
    delrules [@{thm subsetI}, @{thm subsetCE}]
    addSIs @{thms subset_SIs}
    addIs  [@{thm Union_upper}, @{thm Inter_lower}]
    addSEs [@{thm cons_subsetE}]);

val ZF_cs = claset_of (🍋 delrules [@{thm equalityI}]);
›

end