# Theory Set

section ‹Extending FOL by a modified version of HOL set theory›

theory Set
imports FOL
begin

declare [[eta_contract]]

typedecl 'a set
instance set :: ("term") "term" ..

subsection ‹Set comprehension and membership›

axiomatization Collect :: "['a  o]  'a set"
and mem :: "['a, 'a set]  o"  (infixl ":" 50)
where mem_Collect_iff: "(a : Collect(P))  P(a)"
and set_extension: "A = B  (ALL x. x:A  x:B)"

syntax
"_Coll" :: "[idt, o]  'a set"  ("(1{_./ _})")
translations
"{x. P}" == "CONST Collect(λx. P)"

lemma CollectI: "P(a)  a : {x. P(x)}"
apply (rule mem_Collect_iff [THEN iffD2])
apply assumption
done

lemma CollectD: "a : {x. P(x)}  P(a)"
apply (erule mem_Collect_iff [THEN iffD1])
done

lemmas CollectE = CollectD [elim_format]

lemma set_ext: "(x. x:A  x:B)  A = B"
apply (rule set_extension [THEN iffD2])
apply simp
done

subsection ‹Bounded quantifiers›

definition Ball :: "['a set, 'a  o]  o"
where "Ball(A, P) == ALL x. x:A  P(x)"

definition Bex :: "['a set, 'a  o]  o"
where "Bex(A, P) == EX x. x:A  P(x)"

syntax
"_Ball" :: "[idt, 'a set, o]  o"  ("(ALL _:_./ _)" [0, 0, 0] 10)
"_Bex" :: "[idt, 'a set, o]  o"  ("(EX _:_./ _)" [0, 0, 0] 10)
translations
"ALL x:A. P"  == "CONST Ball(A, λx. P)"
"EX x:A. P"   == "CONST Bex(A, λx. P)"

lemma ballI: "(x. x:A  P(x))  ALL x:A. P(x)"

lemma bspec: "ALL x:A. P(x); x:A  P(x)"

lemma ballE: "ALL x:A. P(x); P(x)  Q; ¬ x:A  Q  Q"
unfolding Ball_def by blast

lemma bexI: "P(x); x:A  EX x:A. P(x)"
unfolding Bex_def by blast

lemma bexCI: "EX x:A. ¬P(x)  P(a); a:A  EX x:A. P(x)"
unfolding Bex_def by blast

lemma bexE: "EX x:A. P(x); x. x:A; P(x)  Q  Q"
unfolding Bex_def by blast

(*Trival rewrite rule;   (! x:A.P)=P holds only if A is nonempty!*)
lemma ball_rew: "(ALL x:A. True)  True"
by (blast intro: ballI)

subsubsection ‹Congruence rules›

lemma ball_cong:
"A = A'; x. x:A'  P(x)  P'(x)
(ALL x:A. P(x))  (ALL x:A'. P'(x))"
by (blast intro: ballI elim: ballE)

lemma bex_cong:
"A = A'; x. x:A'  P(x)  P'(x)
(EX x:A. P(x))  (EX x:A'. P'(x))"
by (blast intro: bexI elim: bexE)

subsection ‹Further operations›

definition subset :: "['a set, 'a set]  o"  (infixl "<=" 50)
where "A <= B == ALL x:A. x:B"

definition mono :: "['a set  'b set]  o"
where "mono(f) == (ALL A B. A <= B  f(A) <= f(B))"

definition singleton :: "'a  'a set"  ("{_}")
where "{a} == {x. x=a}"

definition empty :: "'a set"  ("{}")
where "{} == {x. False}"

definition Un :: "['a set, 'a set]  'a set"  (infixl "Un" 65)
where "A Un B == {x. x:A | x:B}"

definition Int :: "['a set, 'a set]  'a set"  (infixl "Int" 70)
where "A Int B == {x. x:A  x:B}"

definition Compl :: "('a set)  'a set"
where "Compl(A) == {x. ¬x:A}"

subsection ‹Big Intersection / Union›

definition INTER :: "['a set, 'a  'b set]  'b set"
where "INTER(A, B) == {y. ALL x:A. y: B(x)}"

definition UNION :: "['a set, 'a  'b set]  'b set"
where "UNION(A, B) == {y. EX x:A. y: B(x)}"

syntax
"_INTER" :: "[idt, 'a set, 'b set]  'b set"  ("(INT _:_./ _)" [0, 0, 0] 10)
"_UNION" :: "[idt, 'a set, 'b set]  'b set"  ("(UN _:_./ _)" [0, 0, 0] 10)
translations
"INT x:A. B" == "CONST INTER(A, λx. B)"
"UN x:A. B" == "CONST UNION(A, λx. B)"

definition Inter :: "(('a set)set)  'a set"
where "Inter(S) == (INT x:S. x)"

definition Union :: "(('a set)set)  'a set"
where "Union(S) == (UN x:S. x)"

subsection ‹Rules for subsets›

lemma subsetI: "(x. x:A  x:B)  A <= B"
unfolding subset_def by (blast intro: ballI)

(*Rule in Modus Ponens style*)
lemma subsetD: "A <= B; c:A  c:B"
unfolding subset_def by (blast elim: ballE)

(*Classical elimination rule*)
lemma subsetCE: "A <= B; ¬(c:A)  P; c:B  P  P"
by (blast dest: subsetD)

lemma subset_refl: "A <= A"
by (blast intro: subsetI)

lemma subset_trans: "A <= B; B <= C  A <= C"
by (blast intro: subsetI dest: subsetD)

subsection ‹Rules for equality›

(*Anti-symmetry of the subset relation*)
lemma subset_antisym: "A <= B; B <= A  A = B"
by (blast intro: set_ext dest: subsetD)

lemmas equalityI = subset_antisym

(* Equality rules from ZF set theory -- are they appropriate here? *)
lemma equalityD1: "A = B  A<=B"
and equalityD2: "A = B  B<=A"

lemma equalityE: "A = B; A <= B; B <= A  P  P"

lemma equalityCE: "A = B; c:A; c:B  P; ¬ c:A; ¬ c:B  P  P"
by (blast elim: equalityE subsetCE)

lemma trivial_set: "{x. x:A} = A"
by (blast intro: equalityI subsetI CollectI dest: CollectD)

subsection ‹Rules for binary union›

lemma UnI1: "c:A  c : A Un B"
and UnI2: "c:B  c : A Un B"
unfolding Un_def by (blast intro: CollectI)+

(*Classical introduction rule: no commitment to A vs B*)
lemma UnCI: "(¬c:B  c:A)  c : A Un B"
by (blast intro: UnI1 UnI2)

lemma UnE: "c : A Un B; c:A  P; c:B  P  P"
unfolding Un_def by (blast dest: CollectD)

subsection ‹Rules for small intersection›

lemma IntI: "c:A; c:B  c : A Int B"
unfolding Int_def by (blast intro: CollectI)

lemma IntD1: "c : A Int B  c:A"
and IntD2: "c : A Int B  c:B"
unfolding Int_def by (blast dest: CollectD)+

lemma IntE: "c : A Int B; c:A; c:B  P  P"
by (blast dest: IntD1 IntD2)

subsection ‹Rules for set complement›

lemma ComplI: "(c:A  False)  c : Compl(A)"
unfolding Compl_def by (blast intro: CollectI)

(*This form, with negated conclusion, works well with the Classical prover.
Negated assumptions behave like formulae on the right side of the notional
turnstile...*)
lemma ComplD: "c : Compl(A)  ¬c:A"
unfolding Compl_def by (blast dest: CollectD)

lemmas ComplE = ComplD [elim_format]

subsection ‹Empty sets›

lemma empty_eq: "{x. False} = {}"

lemma emptyD: "a : {}  P"
unfolding empty_def by (blast dest: CollectD)

lemmas emptyE = emptyD [elim_format]

lemma not_emptyD:
assumes "¬ A={}"
shows "EX x. x:A"
proof -
have "¬ (EX x. x:A)  A = {}"
by (rule equalityI) (blast intro!: subsetI elim!: emptyD)+
with assms show ?thesis by blast
qed

subsection ‹Singleton sets›

lemma singletonI: "a : {a}"
unfolding singleton_def by (blast intro: CollectI)

lemma singletonD: "b : {a}  b=a"
unfolding singleton_def by (blast dest: CollectD)

lemmas singletonE = singletonD [elim_format]

subsection ‹Unions of families›

(*The order of the premises presupposes that A is rigid; b may be flexible*)
lemma UN_I: "a:A; b: B(a)  b: (UN x:A. B(x))"
unfolding UNION_def by (blast intro: bexI CollectI)

lemma UN_E: "b : (UN x:A. B(x)); x. x:A; b: B(x)  R  R"
unfolding UNION_def by (blast dest: CollectD elim: bexE)

lemma UN_cong: "A = B; x. x:B  C(x) = D(x)  (UN x:A. C(x)) = (UN x:B. D(x))"
by (simp add: UNION_def cong: bex_cong)

subsection ‹Intersections of families›

lemma INT_I: "(x. x:A  b: B(x))  b : (INT x:A. B(x))"
unfolding INTER_def by (blast intro: CollectI ballI)

lemma INT_D: "b : (INT x:A. B(x)); a:A  b: B(a)"
unfolding INTER_def by (blast dest: CollectD bspec)

(*"Classical" elimination rule -- does not require proving X:C *)
lemma INT_E: "b : (INT x:A. B(x)); b: B(a)  R; ¬ a:A  R  R"
unfolding INTER_def by (blast dest: CollectD bspec)

lemma INT_cong: "A = B; x. x:B  C(x) = D(x)  (INT x:A. C(x)) = (INT x:B. D(x))"
by (simp add: INTER_def cong: ball_cong)

subsection ‹Rules for Unions›

(*The order of the premises presupposes that C is rigid; A may be flexible*)
lemma UnionI: "X:C; A:X  A : Union(C)"
unfolding Union_def by (blast intro: UN_I)

lemma UnionE: "A : Union(C); X.  A:X; X:C  R  R"
unfolding Union_def by (blast elim: UN_E)

subsection ‹Rules for Inter›

lemma InterI: "(X. X:C  A:X)  A : Inter(C)"
unfolding Inter_def by (blast intro: INT_I)

(*A "destruct" rule -- every X in C contains A as an element, but
A:X can hold when X:C does not!  This rule is analogous to "spec". *)
lemma InterD: "A : Inter(C);  X:C  A:X"
unfolding Inter_def by (blast dest: INT_D)

(*"Classical" elimination rule -- does not require proving X:C *)
lemma InterE: "A : Inter(C); A:X  R; ¬ X:C  R  R"
unfolding Inter_def by (blast elim: INT_E)

section ‹Derived rules involving subsets; Union and Intersection as lattice operations›

subsection ‹Big Union -- least upper bound of a set›

lemma Union_upper: "B:A  B <= Union(A)"
by (blast intro: subsetI UnionI)

lemma Union_least: "(X. X:A  X<=C)  Union(A) <= C"
by (blast intro: subsetI dest: subsetD elim: UnionE)

subsection ‹Big Intersection -- greatest lower bound of a set›

lemma Inter_lower: "B:A  Inter(A) <= B"
by (blast intro: subsetI dest: InterD)

lemma Inter_greatest: "(X. X:A  C<=X)  C <= Inter(A)"
by (blast intro: subsetI InterI dest: subsetD)

subsection ‹Finite Union -- the least upper bound of 2 sets›

lemma Un_upper1: "A <= A Un B"
by (blast intro: subsetI UnI1)

lemma Un_upper2: "B <= A Un B"
by (blast intro: subsetI UnI2)

lemma Un_least: "A<=C; B<=C  A Un B <= C"
by (blast intro: subsetI elim: UnE dest: subsetD)

subsection ‹Finite Intersection -- the greatest lower bound of 2 sets›

lemma Int_lower1: "A Int B <= A"
by (blast intro: subsetI elim: IntE)

lemma Int_lower2: "A Int B <= B"
by (blast intro: subsetI elim: IntE)

lemma Int_greatest: "C<=A; C<=B  C <= A Int B"
by (blast intro: subsetI IntI dest: subsetD)

subsection ‹Monotonicity›

lemma monoI: "(A B. A <= B  f(A) <= f(B))  mono(f)"
unfolding mono_def by blast

lemma monoD: "mono(f); A <= B  f(A) <= f(B)"
unfolding mono_def by blast

lemma mono_Un: "mono(f)  f(A) Un f(B) <= f(A Un B)"
by (blast intro: Un_least dest: monoD intro: Un_upper1 Un_upper2)

lemma mono_Int: "mono(f)  f(A Int B) <= f(A) Int f(B)"
by (blast intro: Int_greatest dest: monoD intro: Int_lower1 Int_lower2)

subsection ‹Automated reasoning setup›

lemmas [intro!] = ballI subsetI InterI INT_I CollectI ComplI IntI UnCI singletonI
and [intro] = bexI UnionI UN_I
and [elim!] = bexE UnionE UN_E CollectE ComplE IntE UnE emptyE singletonE
and [elim] = ballE InterD InterE INT_D INT_E subsetD subsetCE

lemma mem_rews:
"(a : A Un B)     (a:A | a:B)"
"(a : A Int B)    (a:A  a:B)"
"(a : Compl(B))   (¬a:B)"
"(a : {b})        (a=b)"
"(a : {})          False"
"(a : {x. P(x)})   P(a)"
by blast+

lemmas [simp] = trivial_set empty_eq mem_rews
and [cong] = ball_cong bex_cong INT_cong UN_cong

section ‹Equalities involving union, intersection, inclusion, etc.›

subsection ‹Binary Intersection›

lemma Int_absorb: "A Int A = A"
by (blast intro: equalityI)

lemma Int_commute: "A Int B  =  B Int A"
by (blast intro: equalityI)

lemma Int_assoc: "(A Int B) Int C  =  A Int (B Int C)"
by (blast intro: equalityI)

lemma Int_Un_distrib: "(A Un B) Int C  =  (A Int C) Un (B Int C)"
by (blast intro: equalityI)

lemma subset_Int_eq: "(A<=B)  (A Int B = A)"
by (blast intro: equalityI elim: equalityE)

subsection ‹Binary Union›

lemma Un_absorb: "A Un A = A"
by (blast intro: equalityI)

lemma Un_commute: "A Un B  =  B Un A"
by (blast intro: equalityI)

lemma Un_assoc: "(A Un B) Un C  =  A Un (B Un C)"
by (blast intro: equalityI)

lemma Un_Int_distrib: "(A Int B) Un C  =  (A Un C) Int (B Un C)"
by (blast intro: equalityI)

lemma Un_Int_crazy:
"(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)"
by (blast intro: equalityI)

lemma subset_Un_eq: "(A<=B)  (A Un B = B)"
by (blast intro: equalityI elim: equalityE)

subsection ‹Simple properties of Compl› -- complement of a set›

lemma Compl_disjoint: "A Int Compl(A) = {x. False}"
by (blast intro: equalityI)

lemma Compl_partition: "A Un Compl(A) = {x. True}"
by (blast intro: equalityI)

lemma double_complement: "Compl(Compl(A)) = A"
by (blast intro: equalityI)

lemma Compl_Un: "Compl(A Un B) = Compl(A) Int Compl(B)"
by (blast intro: equalityI)

lemma Compl_Int: "Compl(A Int B) = Compl(A) Un Compl(B)"
by (blast intro: equalityI)

lemma Compl_UN: "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))"
by (blast intro: equalityI)

lemma Compl_INT: "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))"
by (blast intro: equalityI)

(*Halmos, Naive Set Theory, page 16.*)
lemma Un_Int_assoc_eq: "((A Int B) Un C = A Int (B Un C))  (C<=A)"
by (blast intro: equalityI elim: equalityE)

subsection ‹Big Union and Intersection›

lemma Union_Un_distrib: "Union(A Un B) = Union(A) Un Union(B)"
by (blast intro: equalityI)

lemma Union_disjoint:
"(Union(C) Int A = {x. False})  (ALL B:C. B Int A = {x. False})"
by (blast intro: equalityI elim: equalityE)

lemma Inter_Un_distrib: "Inter(A Un B) = Inter(A) Int Inter(B)"
by (blast intro: equalityI)

subsection ‹Unions and Intersections of Families›

lemma UN_eq: "(UN x:A. B(x)) = Union({Y. EX x:A. Y=B(x)})"
by (blast intro: equalityI)

(*Look: it has an EXISTENTIAL quantifier*)
lemma INT_eq: "(INT x:A. B(x)) = Inter({Y. EX x:A. Y=B(x)})"
by (blast intro: equalityI)

lemma Int_Union_image: "A Int Union(B) = (UN C:B. A Int C)"
by (blast intro: equalityI)

lemma Un_Inter_image: "A Un Inter(B) = (INT C:B. A Un C)"
by (blast intro: equalityI)

section ‹Monotonicity of various operations›

lemma Union_mono: "A<=B  Union(A) <= Union(B)"
by blast

lemma Inter_anti_mono: "B <= A  Inter(A) <= Inter(B)"
by blast

lemma UN_mono: "A <= B; x. x:A  f(x)<=g(x)  (UN x:A. f(x)) <= (UN x:B. g(x))"
by blast

lemma INT_anti_mono: "B <= A; x. x:A  f(x) <= g(x)  (INT x:A. f(x)) <= (INT x:A. g(x))"
by blast

lemma Un_mono: "A <= C; B <= D  A Un B <= C Un D"
by blast

lemma Int_mono: "A <= C; B <= D  A Int B <= C Int D"
by blast

lemma Compl_anti_mono: "A <= B  Compl(B) <= Compl(A)"
by blast

end