(* Title: ZF/equalities.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)
section\<open>Basic Equalities and Inclusions\<close>
theory equalities imports pair begin
text\<open>These cover union, intersection, converse, domain, range, etc. Philippe
de Groote proved many of the inclusions.\<close>
lemma in_mono: "A\<subseteq>B ==> x\<in>A \<longrightarrow> x\<in>B"
by blast
lemma the_eq_0 [simp]: "(THE x. False) = 0"
by (blast intro: the_0)
subsection\<open>Bounded Quantifiers\<close>
text \<open>\medskip
The following are not added to the default simpset because
(a) they duplicate the body and (b) there are no similar rules for \<open>Int\<close>.\<close>
lemma ball_Un: "(\<forall>x \<in> A\<union>B. P(x)) \<longleftrightarrow> (\<forall>x \<in> A. P(x)) & (\<forall>x \<in> B. P(x))"
by blast
lemma bex_Un: "(\<exists>x \<in> A\<union>B. P(x)) \<longleftrightarrow> (\<exists>x \<in> A. P(x)) | (\<exists>x \<in> B. P(x))"
by blast
lemma ball_UN: "(\<forall>z \<in> (\<Union>x\<in>A. B(x)). P(z)) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B(x). P(z))"
by blast
lemma bex_UN: "(\<exists>z \<in> (\<Union>x\<in>A. B(x)). P(z)) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B(x). P(z))"
by blast
subsection\<open>Converse of a Relation\<close>
lemma converse_iff [simp]: "<a,b>\<in> converse(r) \<longleftrightarrow> <b,a>\<in>r"
by (unfold converse_def, blast)
lemma converseI [intro!]: "<a,b>\<in>r ==> <b,a>\<in>converse(r)"
by (unfold converse_def, blast)
lemma converseD: "<a,b> \<in> converse(r) ==> <b,a> \<in> r"
by (unfold converse_def, blast)
lemma converseE [elim!]:
"[| yx \<in> converse(r);
!!x y. [| yx=<y,x>; <x,y>\<in>r |] ==> P |]
==> P"
by (unfold converse_def, blast)
lemma converse_converse: "r\<subseteq>Sigma(A,B) ==> converse(converse(r)) = r"
by blast
lemma converse_type: "r\<subseteq>A*B ==> converse(r)\<subseteq>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 \<subseteq> Sigma(X,Y) ==> converse(A) \<subseteq> converse(B) \<longleftrightarrow> A \<subseteq> B"
by blast
subsection\<open>Finite Set Constructions Using @{term cons}\<close>
lemma cons_subsetI: "[| a\<in>C; B\<subseteq>C |] ==> cons(a,B) \<subseteq> C"
by blast
lemma subset_consI: "B \<subseteq> cons(a,B)"
by blast
lemma cons_subset_iff [iff]: "cons(a,B)\<subseteq>C \<longleftrightarrow> a\<in>C & B\<subseteq>C"
by blast
(*A safe special case of subset elimination, adding no new variables
[| cons(a,B) \<subseteq> C; [| a \<in> C; B \<subseteq> C |] ==> R |] ==> R *)
lemmas cons_subsetE = cons_subset_iff [THEN iffD1, THEN conjE]
lemma subset_empty_iff: "A\<subseteq>0 \<longleftrightarrow> A=0"
by blast
lemma subset_cons_iff: "C\<subseteq>cons(a,B) \<longleftrightarrow> C\<subseteq>B | (a\<in>C & C-{a} \<subseteq> B)"
by blast
(* cons_def refers to Upair; reversing the equality LOOPS in rewriting!*)
lemma cons_eq: "{a} \<union> 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\<in>C then B-C else cons(a,B-C))"
by auto
lemma equal_singleton [rule_format]: "[| a: C; \<forall>y\<in>C. y=b |] ==> C = {b}"
by blast
lemma [simp]: "cons(a,cons(a,B)) = cons(a,B)"
by blast
(** singletons **)
lemma singleton_subsetI: "a\<in>C ==> {a} \<subseteq> C"
by blast
lemma singleton_subsetD: "{a} \<subseteq> C ==> a\<in>C"
by blast
(** succ **)
lemma subset_succI: "i \<subseteq> 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\<in>j; i\<subseteq>j |] ==> succ(i)\<subseteq>j"
by (unfold succ_def, blast)
lemma succ_subsetE:
"[| succ(i) \<subseteq> j; [| i\<in>j; i\<subseteq>j |] ==> P |] ==> P"
by (unfold succ_def, blast)
lemma succ_subset_iff: "succ(a) \<subseteq> B \<longleftrightarrow> (a \<subseteq> B & a \<in> B)"
by (unfold succ_def, blast)
subsection\<open>Binary Intersection\<close>
(** Intersection is the greatest lower bound of two sets **)
lemma Int_subset_iff: "C \<subseteq> A \<inter> B \<longleftrightarrow> C \<subseteq> A & C \<subseteq> B"
by blast
lemma Int_lower1: "A \<inter> B \<subseteq> A"
by blast
lemma Int_lower2: "A \<inter> B \<subseteq> B"
by blast
lemma Int_greatest: "[| C\<subseteq>A; C\<subseteq>B |] ==> C \<subseteq> A \<inter> B"
by blast
lemma Int_cons: "cons(a,B) \<inter> C \<subseteq> cons(a, B \<inter> C)"
by blast
lemma Int_absorb [simp]: "A \<inter> A = A"
by blast
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
by blast
lemma Int_commute: "A \<inter> B = B \<inter> A"
by blast
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
by blast
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> 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 \<subseteq> A ==> A \<inter> B = B"
by blast
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
by blast
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
by blast
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
by blast
lemma subset_Int_iff: "A\<subseteq>B \<longleftrightarrow> A \<inter> B = A"
by (blast elim!: equalityE)
lemma subset_Int_iff2: "A\<subseteq>B \<longleftrightarrow> B \<inter> A = A"
by (blast elim!: equalityE)
lemma Int_Diff_eq: "C\<subseteq>A ==> (A-B) \<inter> C = C-B"
by blast
lemma Int_cons_left:
"cons(a,A) \<inter> B = (if a \<in> B then cons(a, A \<inter> B) else A \<inter> B)"
by auto
lemma Int_cons_right:
"A \<inter> cons(a, B) = (if a \<in> A then cons(a, A \<inter> B) else A \<inter> B)"
by auto
lemma cons_Int_distrib: "cons(x, A \<inter> B) = cons(x, A) \<inter> cons(x, B)"
by auto
subsection\<open>Binary Union\<close>
(** Union is the least upper bound of two sets *)
lemma Un_subset_iff: "A \<union> B \<subseteq> C \<longleftrightarrow> A \<subseteq> C & B \<subseteq> C"
by blast
lemma Un_upper1: "A \<subseteq> A \<union> B"
by blast
lemma Un_upper2: "B \<subseteq> A \<union> B"
by blast
lemma Un_least: "[| A\<subseteq>C; B\<subseteq>C |] ==> A \<union> B \<subseteq> C"
by blast
lemma Un_cons: "cons(a,B) \<union> C = cons(a, B \<union> C)"
by blast
lemma Un_absorb [simp]: "A \<union> A = A"
by blast
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
by blast
lemma Un_commute: "A \<union> B = B \<union> A"
by blast
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
by blast
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> 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 \<subseteq> B ==> A \<union> B = B"
by blast
lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
by blast
lemma Un_Int_distrib: "(A \<inter> B) \<union> C = (A \<union> C) \<inter> (B \<union> C)"
by blast
lemma subset_Un_iff: "A\<subseteq>B \<longleftrightarrow> A \<union> B = B"
by (blast elim!: equalityE)
lemma subset_Un_iff2: "A\<subseteq>B \<longleftrightarrow> B \<union> A = B"
by (blast elim!: equalityE)
lemma Un_empty [iff]: "(A \<union> B = 0) \<longleftrightarrow> (A = 0 & B = 0)"
by blast
lemma Un_eq_Union: "A \<union> B = \<Union>({A, B})"
by blast
subsection\<open>Set Difference\<close>
lemma Diff_subset: "A-B \<subseteq> A"
by blast
lemma Diff_contains: "[| C\<subseteq>A; C \<inter> B = 0 |] ==> C \<subseteq> A-B"
by blast
lemma subset_Diff_cons_iff: "B \<subseteq> A - cons(c,C) \<longleftrightarrow> B\<subseteq>A-C & c \<notin> B"
by blast
lemma Diff_cancel: "A - A = 0"
by blast
lemma Diff_triv: "A \<inter> 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 \<longleftrightarrow> A \<subseteq> B"
by (blast elim: equalityE)
(*NOT SUITABLE FOR REWRITING since {a} == cons(a,0)*)
lemma Diff_cons: "A - cons(a,B) = A - B - {a}"
by blast
(*NOT SUITABLE FOR REWRITING since {a} == cons(a,0)*)
lemma Diff_cons2: "A - cons(a,B) = A - {a} - B"
by blast
lemma Diff_disjoint: "A \<inter> (B-A) = 0"
by blast
lemma Diff_partition: "A\<subseteq>B ==> A \<union> (B-A) = B"
by blast
lemma subset_Un_Diff: "A \<subseteq> B \<union> (A - B)"
by blast
lemma double_complement: "[| A\<subseteq>B; B\<subseteq>C |] ==> B-(C-A) = A"
by blast
lemma double_complement_Un: "(A \<union> B) - (B-A) = A"
by blast
lemma Un_Int_crazy:
"(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
apply blast
done
lemma Diff_Un: "A - (B \<union> C) = (A-B) \<inter> (A-C)"
by blast
lemma Diff_Int: "A - (B \<inter> C) = (A-B) \<union> (A-C)"
by blast
lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
by blast
lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
by blast
lemma Diff_Int_distrib: "C \<inter> (A-B) = (C \<inter> A) - (C \<inter> B)"
by blast
lemma Diff_Int_distrib2: "(A-B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
by blast
(*Halmos, Naive Set Theory, page 16.*)
lemma Un_Int_assoc_iff: "(A \<inter> B) \<union> C = A \<inter> (B \<union> C) \<longleftrightarrow> C\<subseteq>A"
by (blast elim!: equalityE)
subsection\<open>Big Union and Intersection\<close>
(** Big Union is the least upper bound of a set **)
lemma Union_subset_iff: "\<Union>(A) \<subseteq> C \<longleftrightarrow> (\<forall>x\<in>A. x \<subseteq> C)"
by blast
lemma Union_upper: "B\<in>A ==> B \<subseteq> \<Union>(A)"
by blast
lemma Union_least: "[| !!x. x\<in>A ==> x\<subseteq>C |] ==> \<Union>(A) \<subseteq> C"
by blast
lemma Union_cons [simp]: "\<Union>(cons(a,B)) = a \<union> \<Union>(B)"
by blast
lemma Union_Un_distrib: "\<Union>(A \<union> B) = \<Union>(A) \<union> \<Union>(B)"
by blast
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>(A) \<inter> \<Union>(B)"
by blast
lemma Union_disjoint: "\<Union>(C) \<inter> A = 0 \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = 0)"
by (blast elim!: equalityE)
lemma Union_empty_iff: "\<Union>(A) = 0 \<longleftrightarrow> (\<forall>B\<in>A. B=0)"
by blast
lemma Int_Union2: "\<Union>(B) \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
by blast
(** Big Intersection is the greatest lower bound of a nonempty set **)
lemma Inter_subset_iff: "A\<noteq>0 ==> C \<subseteq> \<Inter>(A) \<longleftrightarrow> (\<forall>x\<in>A. C \<subseteq> x)"
by blast
lemma Inter_lower: "B\<in>A ==> \<Inter>(A) \<subseteq> B"
by blast
lemma Inter_greatest: "[| A\<noteq>0; !!x. x\<in>A ==> C\<subseteq>x |] ==> C \<subseteq> \<Inter>(A)"
by blast
(** Intersection of a family of sets **)
lemma INT_lower: "x\<in>A ==> (\<Inter>x\<in>A. B(x)) \<subseteq> B(x)"
by blast
lemma INT_greatest: "[| A\<noteq>0; !!x. x\<in>A ==> C\<subseteq>B(x) |] ==> C \<subseteq> (\<Inter>x\<in>A. B(x))"
by force
lemma Inter_0 [simp]: "\<Inter>(0) = 0"
by (unfold Inter_def, blast)
lemma Inter_Un_subset:
"[| z\<in>A; z\<in>B |] ==> \<Inter>(A) \<union> \<Inter>(B) \<subseteq> \<Inter>(A \<inter> B)"
by blast
(* A good challenge: Inter is ill-behaved on the empty set *)
lemma Inter_Un_distrib:
"[| A\<noteq>0; B\<noteq>0 |] ==> \<Inter>(A \<union> B) = \<Inter>(A) \<inter> \<Inter>(B)"
by blast
lemma Union_singleton: "\<Union>({b}) = b"
by blast
lemma Inter_singleton: "\<Inter>({b}) = b"
by blast
lemma Inter_cons [simp]:
"\<Inter>(cons(a,B)) = (if B=0 then a else a \<inter> \<Inter>(B))"
by force
subsection\<open>Unions and Intersections of Families\<close>
lemma subset_UN_iff_eq: "A \<subseteq> (\<Union>i\<in>I. B(i)) \<longleftrightarrow> A = (\<Union>i\<in>I. A \<inter> B(i))"
by (blast elim!: equalityE)
lemma UN_subset_iff: "(\<Union>x\<in>A. B(x)) \<subseteq> C \<longleftrightarrow> (\<forall>x\<in>A. B(x) \<subseteq> C)"
by blast
lemma UN_upper: "x\<in>A ==> B(x) \<subseteq> (\<Union>x\<in>A. B(x))"
by (erule RepFunI [THEN Union_upper])
lemma UN_least: "[| !!x. x\<in>A ==> B(x)\<subseteq>C |] ==> (\<Union>x\<in>A. B(x)) \<subseteq> C"
by blast
lemma Union_eq_UN: "\<Union>(A) = (\<Union>x\<in>A. x)"
by blast
lemma Inter_eq_INT: "\<Inter>(A) = (\<Inter>x\<in>A. x)"
by (unfold Inter_def, blast)
lemma UN_0 [simp]: "(\<Union>i\<in>0. A(i)) = 0"
by blast
lemma UN_singleton: "(\<Union>x\<in>A. {x}) = A"
by blast
lemma UN_Un: "(\<Union>i\<in> A \<union> B. C(i)) = (\<Union>i\<in> A. C(i)) \<union> (\<Union>i\<in>B. C(i))"
by blast
lemma INT_Un: "(\<Inter>i\<in>I \<union> J. A(i)) =
(if I=0 then \<Inter>j\<in>J. A(j)
else if J=0 then \<Inter>i\<in>I. A(i)
else ((\<Inter>i\<in>I. A(i)) \<inter> (\<Inter>j\<in>J. A(j))))"
by (simp, blast intro!: equalityI)
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B(y)). C(x)) = (\<Union>y\<in>A. \<Union>x\<in> B(y). C(x))"
by blast
(*Halmos, Naive Set Theory, page 35.*)
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A(i)) = (\<Union>i\<in>I. B \<inter> A(i))"
by blast
lemma Un_INT_distrib: "I\<noteq>0 ==> B \<union> (\<Inter>i\<in>I. A(i)) = (\<Inter>i\<in>I. B \<union> A(i))"
by auto
lemma Int_UN_distrib2:
"(\<Union>i\<in>I. A(i)) \<inter> (\<Union>j\<in>J. B(j)) = (\<Union>i\<in>I. \<Union>j\<in>J. A(i) \<inter> B(j))"
by blast
lemma Un_INT_distrib2: "[| I\<noteq>0; J\<noteq>0 |] ==>
(\<Inter>i\<in>I. A(i)) \<union> (\<Inter>j\<in>J. B(j)) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A(i) \<union> B(j))"
by auto
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A=0 then 0 else c)"
by force
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A=0 then 0 else c)"
by force
lemma UN_RepFun [simp]: "(\<Union>y\<in> RepFun(A,f). B(y)) = (\<Union>x\<in>A. B(f(x)))"
by blast
lemma INT_RepFun [simp]: "(\<Inter>x\<in>RepFun(A,f). B(x)) = (\<Inter>a\<in>A. B(f(a)))"
by (auto simp add: Inter_def)
lemma INT_Union_eq:
"0 \<notin> A ==> (\<Inter>x\<in> \<Union>(A). B(x)) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B(x))"
apply (subgoal_tac "\<forall>x\<in>A. x\<noteq>0")
prefer 2 apply blast
apply (force simp add: Inter_def ball_conj_distrib)
done
lemma INT_UN_eq:
"(\<forall>x\<in>A. B(x) \<noteq> 0)
==> (\<Inter>z\<in> (\<Union>x\<in>A. B(x)). C(z)) = (\<Inter>x\<in>A. \<Inter>z\<in> 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:
"(\<Union>i\<in>I. A(i) \<union> B(i)) = (\<Union>i\<in>I. A(i)) \<union> (\<Union>i\<in>I. B(i))"
by blast
lemma INT_Int_distrib:
"I\<noteq>0 ==> (\<Inter>i\<in>I. A(i) \<inter> B(i)) = (\<Inter>i\<in>I. A(i)) \<inter> (\<Inter>i\<in>I. B(i))"
by (blast elim!: not_emptyE)
lemma UN_Int_subset:
"(\<Union>z\<in>I \<inter> J. A(z)) \<subseteq> (\<Union>z\<in>I. A(z)) \<inter> (\<Union>z\<in>J. A(z))"
by blast
(** Devlin, page 12, exercise 5: Complements **)
lemma Diff_UN: "I\<noteq>0 ==> B - (\<Union>i\<in>I. A(i)) = (\<Inter>i\<in>I. B - A(i))"
by (blast elim!: not_emptyE)
lemma Diff_INT: "I\<noteq>0 ==> B - (\<Inter>i\<in>I. A(i)) = (\<Union>i\<in>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)) \<union> Sigma(B,C)"
by blast
(*Not suitable for rewriting: LOOPS!*)
lemma Sigma_cons2: "A * cons(b,B) = A*{b} \<union> A*B"
by blast
lemma Sigma_succ1: "Sigma(succ(A), B) = ({A}*B(A)) \<union> Sigma(A,B)"
by blast
lemma Sigma_succ2: "A * succ(B) = A*{B} \<union> A*B"
by blast
lemma SUM_UN_distrib1:
"(\<Sum>x \<in> (\<Union>y\<in>A. C(y)). B(x)) = (\<Union>y\<in>A. \<Sum>x\<in>C(y). B(x))"
by blast
lemma SUM_UN_distrib2:
"(\<Sum>i\<in>I. \<Union>j\<in>J. C(i,j)) = (\<Union>j\<in>J. \<Sum>i\<in>I. C(i,j))"
by blast
lemma SUM_Un_distrib1:
"(\<Sum>i\<in>I \<union> J. C(i)) = (\<Sum>i\<in>I. C(i)) \<union> (\<Sum>j\<in>J. C(j))"
by blast
lemma SUM_Un_distrib2:
"(\<Sum>i\<in>I. A(i) \<union> B(i)) = (\<Sum>i\<in>I. A(i)) \<union> (\<Sum>i\<in>I. B(i))"
by blast
(*First-order version of the above, for rewriting*)
lemma prod_Un_distrib2: "I * (A \<union> B) = I*A \<union> I*B"
by (rule SUM_Un_distrib2)
lemma SUM_Int_distrib1:
"(\<Sum>i\<in>I \<inter> J. C(i)) = (\<Sum>i\<in>I. C(i)) \<inter> (\<Sum>j\<in>J. C(j))"
by blast
lemma SUM_Int_distrib2:
"(\<Sum>i\<in>I. A(i) \<inter> B(i)) = (\<Sum>i\<in>I. A(i)) \<inter> (\<Sum>i\<in>I. B(i))"
by blast
(*First-order version of the above, for rewriting*)
lemma prod_Int_distrib2: "I * (A \<inter> B) = I*A \<inter> I*B"
by (rule SUM_Int_distrib2)
(*Cf Aczel, Non-Well-Founded Sets, page 115*)
lemma SUM_eq_UN: "(\<Sum>i\<in>I. A(i)) = (\<Union>i\<in>I. {i} * A(i))"
by blast
lemma times_subset_iff:
"(A'*B' \<subseteq> A*B) \<longleftrightarrow> (A' = 0 | B' = 0 | (A'\<subseteq>A) & (B'\<subseteq>B))"
by blast
lemma Int_Sigma_eq:
"(\<Sum>x \<in> A'. B'(x)) \<inter> (\<Sum>x \<in> A. B(x)) = (\<Sum>x \<in> A' \<inter> A. B'(x) \<inter> B(x))"
by blast
(** Domain **)
lemma domain_iff: "a: domain(r) \<longleftrightarrow> (\<exists>y. <a,y>\<in> r)"
by (unfold domain_def, blast)
lemma domainI [intro]: "<a,b>\<in> r ==> a: domain(r)"
by (unfold domain_def, blast)
lemma domainE [elim!]:
"[| a \<in> domain(r); !!y. <a,y>\<in> r ==> P |] ==> P"
by (unfold domain_def, blast)
lemma domain_subset: "domain(Sigma(A,B)) \<subseteq> A"
by blast
lemma domain_of_prod: "b\<in>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 \<union> B) = domain(A) \<union> domain(B)"
by blast
lemma domain_Int_subset: "domain(A \<inter> B) \<subseteq> domain(A) \<inter> domain(B)"
by blast
lemma domain_Diff_subset: "domain(A) - domain(B) \<subseteq> domain(A - B)"
by blast
lemma domain_UN: "domain(\<Union>x\<in>A. B(x)) = (\<Union>x\<in>A. domain(B(x)))"
by blast
lemma domain_Union: "domain(\<Union>(A)) = (\<Union>x\<in>A. domain(x))"
by blast
(** Range **)
lemma rangeI [intro]: "<a,b>\<in> r ==> b \<in> range(r)"
apply (unfold range_def)
apply (erule converseI [THEN domainI])
done
lemma rangeE [elim!]: "[| b \<in> range(r); !!x. <x,b>\<in> r ==> P |] ==> P"
by (unfold range_def, blast)
lemma range_subset: "range(A*B) \<subseteq> B"
apply (unfold range_def)
apply (subst converse_prod)
apply (rule domain_subset)
done
lemma range_of_prod: "a\<in>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 \<union> B) = range(A) \<union> range(B)"
by blast
lemma range_Int_subset: "range(A \<inter> B) \<subseteq> range(A) \<inter> range(B)"
by blast
lemma range_Diff_subset: "range(A) - range(B) \<subseteq> 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>\<in> r ==> a \<in> field(r)"
by (unfold field_def, blast)
lemma fieldI2: "<a,b>\<in> r ==> b \<in> field(r)"
by (unfold field_def, blast)
lemma fieldCI [intro]:
"(~ <c,a>\<in>r ==> <a,b>\<in> r) ==> a \<in> field(r)"
apply (unfold field_def, blast)
done
lemma fieldE [elim!]:
"[| a \<in> field(r);
!!x. <a,x>\<in> r ==> P;
!!x. <x,a>\<in> r ==> P |] ==> P"
by (unfold field_def, blast)
lemma field_subset: "field(A*B) \<subseteq> A \<union> B"
by blast
lemma domain_subset_field: "domain(r) \<subseteq> field(r)"
apply (unfold field_def)
apply (rule Un_upper1)
done
lemma range_subset_field: "range(r) \<subseteq> field(r)"
apply (unfold field_def)
apply (rule Un_upper2)
done
lemma domain_times_range: "r \<subseteq> Sigma(A,B) ==> r \<subseteq> domain(r)*range(r)"
by blast
lemma field_times_field: "r \<subseteq> Sigma(A,B) ==> r \<subseteq> field(r)*field(r)"
by blast
lemma relation_field_times_field: "relation(r) ==> r \<subseteq> 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 \<union> B) = field(A) \<union> field(B)"
by blast
lemma field_Int_subset: "field(A \<inter> B) \<subseteq> field(A) \<inter> field(B)"
by blast
lemma field_Diff_subset: "field(A) - field(B) \<subseteq> 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: "(\<forall>x\<in>S. \<exists>A B. x \<subseteq> A*B) ==>
\<Union>(S) \<subseteq> domain(\<Union>(S)) * range(\<Union>(S))"
by blast
(** The Union of 2 relations is a relation (Lemma for fun_Un) **)
lemma rel_Un: "[| r \<subseteq> A*B; s \<subseteq> C*D |] ==> (r \<union> s) \<subseteq> (A \<union> C) * (B \<union> D)"
by blast
lemma domain_Diff_eq: "[| <a,c> \<in> r; c\<noteq>b |] ==> domain(r-{<a,b>}) = domain(r)"
by blast
lemma range_Diff_eq: "[| <c,b> \<in> r; c\<noteq>a |] ==> range(r-{<a,b>}) = range(r)"
by blast
subsection\<open>Image of a Set under a Function or Relation\<close>
lemma image_iff: "b \<in> r``A \<longleftrightarrow> (\<exists>x\<in>A. <x,b>\<in>r)"
by (unfold image_def, blast)
lemma image_singleton_iff: "b \<in> r``{a} \<longleftrightarrow> <a,b>\<in>r"
by (rule image_iff [THEN iff_trans], blast)
lemma imageI [intro]: "[| <a,b>\<in> r; a\<in>A |] ==> b \<in> r``A"
by (unfold image_def, blast)
lemma imageE [elim!]:
"[| b: r``A; !!x.[| <x,b>\<in> r; x\<in>A |] ==> P |] ==> P"
by (unfold image_def, blast)
lemma image_subset: "r \<subseteq> A*B ==> r``C \<subseteq> B"
by blast
lemma image_0 [simp]: "r``0 = 0"
by blast
lemma image_Un [simp]: "r``(A \<union> B) = (r``A) \<union> (r``B)"
by blast
lemma image_UN: "r `` (\<Union>x\<in>A. B(x)) = (\<Union>x\<in>A. r `` B(x))"
by blast
lemma Collect_image_eq:
"{z \<in> Sigma(A,B). P(z)} `` C = (\<Union>x \<in> A. {y \<in> B(x). x \<in> C & P(<x,y>)})"
by blast
lemma image_Int_subset: "r``(A \<inter> B) \<subseteq> (r``A) \<inter> (r``B)"
by blast
lemma image_Int_square_subset: "(r \<inter> A*A)``B \<subseteq> (r``B) \<inter> A"
by blast
lemma image_Int_square: "B\<subseteq>A ==> (r \<inter> A*A)``B = (r``B) \<inter> A"
by blast
(*Image laws for special relations*)
lemma image_0_left [simp]: "0``A = 0"
by blast
lemma image_Un_left: "(r \<union> s)``A = (r``A) \<union> (s``A)"
by blast
lemma image_Int_subset_left: "(r \<inter> s)``A \<subseteq> (r``A) \<inter> (s``A)"
by blast
subsection\<open>Inverse Image of a Set under a Function or Relation\<close>
lemma vimage_iff:
"a \<in> r-``B \<longleftrightarrow> (\<exists>y\<in>B. <a,y>\<in>r)"
by (unfold vimage_def image_def converse_def, blast)
lemma vimage_singleton_iff: "a \<in> r-``{b} \<longleftrightarrow> <a,b>\<in>r"
by (rule vimage_iff [THEN iff_trans], blast)
lemma vimageI [intro]: "[| <a,b>\<in> r; b\<in>B |] ==> a \<in> r-``B"
by (unfold vimage_def, blast)
lemma vimageE [elim!]:
"[| a: r-``B; !!x.[| <a,x>\<in> r; x\<in>B |] ==> P |] ==> P"
apply (unfold vimage_def, blast)
done
lemma vimage_subset: "r \<subseteq> A*B ==> r-``C \<subseteq> A"
apply (unfold 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 \<union> B) = (r-``A) \<union> (r-``B)"
by blast
lemma vimage_Int_subset: "r-``(A \<inter> B) \<subseteq> (r-``A) \<inter> (r-``B)"
by blast
(*NOT suitable for rewriting*)
lemma vimage_eq_UN: "f -``B = (\<Union>y\<in>B. f-``{y})"
by blast
lemma function_vimage_Int:
"function(f) ==> f-``(A \<inter> B) = (f-``A) \<inter> (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) \<subseteq> A"
by (unfold function_def, blast)
lemma vimage_Int_square_subset: "(r \<inter> A*A)-``B \<subseteq> (r-``B) \<inter> A"
by blast
lemma vimage_Int_square: "B\<subseteq>A ==> (r \<inter> A*A)-``B = (r-``B) \<inter> A"
by blast
(*Invese image laws for special relations*)
lemma vimage_0_left [simp]: "0-``A = 0"
by blast
lemma vimage_Un_left: "(r \<union> s)-``A = (r-``A) \<union> (s-``A)"
by blast
lemma vimage_Int_subset_left: "(r \<inter> s)-``A \<subseteq> (r-``A) \<inter> (s-``A)"
by blast
(** Converse **)
lemma converse_Un [simp]: "converse(A \<union> B) = converse(A) \<union> converse(B)"
by blast
lemma converse_Int [simp]: "converse(A \<inter> B) = converse(A) \<inter> converse(B)"
by blast
lemma converse_Diff [simp]: "converse(A - B) = converse(A) - converse(B)"
by blast
lemma converse_UN [simp]: "converse(\<Union>x\<in>A. B(x)) = (\<Union>x\<in>A. converse(B(x)))"
by blast
(*Unfolding Inter avoids using excluded middle on A=0*)
lemma converse_INT [simp]:
"converse(\<Inter>x\<in>A. B(x)) = (\<Inter>x\<in>A. converse(B(x)))"
apply (unfold Inter_def, blast)
done
subsection\<open>Powerset Operator\<close>
lemma Pow_0 [simp]: "Pow(0) = {0}"
by blast
lemma Pow_insert: "Pow (cons(a,A)) = Pow(A) \<union> {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) \<union> Pow(B) \<subseteq> Pow(A \<union> B)"
by blast
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow(B(x))) \<subseteq> Pow(\<Union>x\<in>A. B(x))"
by blast
lemma subset_Pow_Union: "A \<subseteq> Pow(\<Union>(A))"
by blast
lemma Union_Pow_eq [simp]: "\<Union>(Pow(A)) = A"
by blast
lemma Union_Pow_iff: "\<Union>(A) \<in> Pow(B) \<longleftrightarrow> A \<in> Pow(Pow(B))"
by blast
lemma Pow_Int_eq [simp]: "Pow(A \<inter> B) = Pow(A) \<inter> Pow(B)"
by blast
lemma Pow_INT_eq: "A\<noteq>0 ==> Pow(\<Inter>x\<in>A. B(x)) = (\<Inter>x\<in>A. Pow(B(x)))"
by (blast elim!: not_emptyE)
subsection\<open>RepFun\<close>
lemma RepFun_subset: "[| !!x. x\<in>A ==> f(x) \<in> B |] ==> {f(x). x\<in>A} \<subseteq> B"
by blast
lemma RepFun_eq_0_iff [simp]: "{f(x).x\<in>A}=0 \<longleftrightarrow> A=0"
by blast
lemma RepFun_constant [simp]: "{c. x\<in>A} = (if A=0 then 0 else {c})"
by force
subsection\<open>Collect\<close>
lemma Collect_subset: "Collect(A,P) \<subseteq> A"
by blast
lemma Collect_Un: "Collect(A \<union> B, P) = Collect(A,P) \<union> Collect(B,P)"
by blast
lemma Collect_Int: "Collect(A \<inter> B, P) = Collect(A,P) \<inter> Collect(B,P)"
by blast
lemma Collect_Diff: "Collect(A - B, P) = Collect(A,P) - Collect(B,P)"
by blast
lemma Collect_cons: "{x\<in>cons(a,B). P(x)} =
(if P(a) then cons(a, {x\<in>B. P(x)}) else {x\<in>B. P(x)})"
by (simp, blast)
lemma Int_Collect_self_eq: "A \<inter> 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) \<inter> Collect(A,Q) = Collect(A, %x. P(x) & Q(x))"
by blast
lemma Collect_Union_eq [simp]:
"Collect(\<Union>x\<in>A. B(x), P) = (\<Union>x\<in>A. Collect(B(x), P))"
by blast
lemma Collect_Int_left: "{x\<in>A. P(x)} \<inter> B = {x \<in> A \<inter> B. P(x)}"
by blast
lemma Collect_Int_right: "A \<inter> {x\<in>B. P(x)} = {x \<in> A \<inter> B. P(x)}"
by blast
lemma Collect_disj_eq: "{x\<in>A. P(x) | Q(x)} = Collect(A, P) \<union> Collect(A, Q)"
by blast
lemma Collect_conj_eq: "{x\<in>A. P(x) & Q(x)} = Collect(A, P) \<inter> 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 \<open>
val subset_cs =
claset_of (@{context}
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 (@{context} delrules [@{thm equalityI}]);
\<close>
end