(* Title: ZF/ZF.thy ID: $Id: ZF.thy,v 1.56 2007/10/07 19:19:33 wenzelm Exp $ Author: Lawrence C Paulson and Martin D Coen, CU Computer Laboratory Copyright 1993 University of Cambridge *) header{*Zermelo-Fraenkel Set Theory*} theory ZF imports FOL begin ML {* reset eta_contract *} global typedecl i arities i :: "term" consts "0" :: "i" ("0") --{*the empty set*} Pow :: "i => i" --{*power sets*} Inf :: "i" --{*infinite set*} text {*Bounded Quantifiers *} consts Ball :: "[i, i => o] => o" Bex :: "[i, i => o] => o" text {*General Union and Intersection *} consts Union :: "i => i" Inter :: "i => i" text {*Variations on Replacement *} consts PrimReplace :: "[i, [i, i] => o] => i" Replace :: "[i, [i, i] => o] => i" RepFun :: "[i, i => i] => i" Collect :: "[i, i => o] => i" text{*Definite descriptions -- via Replace over the set "1"*} consts The :: "(i => o) => i" (binder "THE " 10) If :: "[o, i, i] => i" ("(if (_)/ then (_)/ else (_))" [10] 10) abbreviation (input) old_if :: "[o, i, i] => i" ("if '(_,_,_')") where "if(P,a,b) == If(P,a,b)" text {*Finite Sets *} consts Upair :: "[i, i] => i" cons :: "[i, i] => i" succ :: "i => i" text {*Ordered Pairing *} consts Pair :: "[i, i] => i" fst :: "i => i" snd :: "i => i" split :: "[[i, i] => 'a, i] => 'a::{}" --{*for pattern-matching*} text {*Sigma and Pi Operators *} consts Sigma :: "[i, i => i] => i" Pi :: "[i, i => i] => i" text {*Relations and Functions *} consts "domain" :: "i => i" range :: "i => i" field :: "i => i" converse :: "i => i" relation :: "i => o" --{*recognizes sets of pairs*} "function" :: "i => o" --{*recognizes functions; can have non-pairs*} Lambda :: "[i, i => i] => i" restrict :: "[i, i] => i" text {*Infixes in order of decreasing precedence *} consts Image :: "[i, i] => i" (infixl "``" 90) --{*image*} vimage :: "[i, i] => i" (infixl "-``" 90) --{*inverse image*} "apply" :: "[i, i] => i" (infixl "`" 90) --{*function application*} "Int" :: "[i, i] => i" (infixl "Int" 70) --{*binary intersection*} "Un" :: "[i, i] => i" (infixl "Un" 65) --{*binary union*} Diff :: "[i, i] => i" (infixl "-" 65) --{*set difference*} Subset :: "[i, i] => o" (infixl "<=" 50) --{*subset relation*} mem :: "[i, i] => o" (infixl ":" 50) --{*membership relation*} abbreviation not_mem :: "[i, i] => o" (infixl "~:" 50) --{*negated membership relation*} where "x ~: y == ~ (x : y)" abbreviation cart_prod :: "[i, i] => i" (infixr "*" 80) --{*Cartesian product*} where "A * B == Sigma(A, %_. B)" abbreviation function_space :: "[i, i] => i" (infixr "->" 60) --{*function space*} where "A -> B == Pi(A, %_. B)" nonterminals "is" patterns syntax "" :: "i => is" ("_") "@Enum" :: "[i, is] => is" ("_,/ _") "@Finset" :: "is => i" ("{(_)}") "@Tuple" :: "[i, is] => i" ("<(_,/ _)>") "@Collect" :: "[pttrn, i, o] => i" ("(1{_: _ ./ _})") "@Replace" :: "[pttrn, pttrn, i, o] => i" ("(1{_ ./ _: _, _})") "@RepFun" :: "[i, pttrn, i] => i" ("(1{_ ./ _: _})" [51,0,51]) "@INTER" :: "[pttrn, i, i] => i" ("(3INT _:_./ _)" 10) "@UNION" :: "[pttrn, i, i] => i" ("(3UN _:_./ _)" 10) "@PROD" :: "[pttrn, i, i] => i" ("(3PROD _:_./ _)" 10) "@SUM" :: "[pttrn, i, i] => i" ("(3SUM _:_./ _)" 10) "@lam" :: "[pttrn, i, i] => i" ("(3lam _:_./ _)" 10) "@Ball" :: "[pttrn, i, o] => o" ("(3ALL _:_./ _)" 10) "@Bex" :: "[pttrn, i, o] => o" ("(3EX _:_./ _)" 10) (** Patterns -- extends pre-defined type "pttrn" used in abstractions **) "@pattern" :: "patterns => pttrn" ("<_>") "" :: "pttrn => patterns" ("_") "@patterns" :: "[pttrn, patterns] => patterns" ("_,/_") translations "{x, xs}" == "cons(x, {xs})" "{x}" == "cons(x, 0)" "{x:A. P}" == "Collect(A, %x. P)" "{y. x:A, Q}" == "Replace(A, %x y. Q)" "{b. x:A}" == "RepFun(A, %x. b)" "INT x:A. B" == "Inter({B. x:A})" "UN x:A. B" == "Union({B. x:A})" "PROD x:A. B" == "Pi(A, %x. B)" "SUM x:A. B" == "Sigma(A, %x. B)" "lam x:A. f" == "Lambda(A, %x. f)" "ALL x:A. P" == "Ball(A, %x. P)" "EX x:A. P" == "Bex(A, %x. P)" "<x, y, z>" == "<x, <y, z>>" "<x, y>" == "Pair(x, y)" "%<x,y,zs>.b" == "split(%x <y,zs>.b)" "%<x,y>.b" == "split(%x y. b)" notation (xsymbols) cart_prod (infixr "×" 80) and Int (infixl "∩" 70) and Un (infixl "∪" 65) and function_space (infixr "->" 60) and Subset (infixl "⊆" 50) and mem (infixl "∈" 50) and not_mem (infixl "∉" 50) and Union ("\<Union>_" [90] 90) and Inter ("\<Inter>_" [90] 90) syntax (xsymbols) "@Collect" :: "[pttrn, i, o] => i" ("(1{_ ∈ _ ./ _})") "@Replace" :: "[pttrn, pttrn, i, o] => i" ("(1{_ ./ _ ∈ _, _})") "@RepFun" :: "[i, pttrn, i] => i" ("(1{_ ./ _ ∈ _})" [51,0,51]) "@UNION" :: "[pttrn, i, i] => i" ("(3\<Union>_∈_./ _)" 10) "@INTER" :: "[pttrn, i, i] => i" ("(3\<Inter>_∈_./ _)" 10) "@PROD" :: "[pttrn, i, i] => i" ("(3Π_∈_./ _)" 10) "@SUM" :: "[pttrn, i, i] => i" ("(3Σ_∈_./ _)" 10) "@lam" :: "[pttrn, i, i] => i" ("(3λ_∈_./ _)" 10) "@Ball" :: "[pttrn, i, o] => o" ("(3∀_∈_./ _)" 10) "@Bex" :: "[pttrn, i, o] => o" ("(3∃_∈_./ _)" 10) "@Tuple" :: "[i, is] => i" ("〈(_,/ _)〉") "@pattern" :: "patterns => pttrn" ("〈_〉") notation (HTML output) cart_prod (infixr "×" 80) and Int (infixl "∩" 70) and Un (infixl "∪" 65) and Subset (infixl "⊆" 50) and mem (infixl "∈" 50) and not_mem (infixl "∉" 50) and Union ("\<Union>_" [90] 90) and Inter ("\<Inter>_" [90] 90) syntax (HTML output) "@Collect" :: "[pttrn, i, o] => i" ("(1{_ ∈ _ ./ _})") "@Replace" :: "[pttrn, pttrn, i, o] => i" ("(1{_ ./ _ ∈ _, _})") "@RepFun" :: "[i, pttrn, i] => i" ("(1{_ ./ _ ∈ _})" [51,0,51]) "@UNION" :: "[pttrn, i, i] => i" ("(3\<Union>_∈_./ _)" 10) "@INTER" :: "[pttrn, i, i] => i" ("(3\<Inter>_∈_./ _)" 10) "@PROD" :: "[pttrn, i, i] => i" ("(3Π_∈_./ _)" 10) "@SUM" :: "[pttrn, i, i] => i" ("(3Σ_∈_./ _)" 10) "@lam" :: "[pttrn, i, i] => i" ("(3λ_∈_./ _)" 10) "@Ball" :: "[pttrn, i, o] => o" ("(3∀_∈_./ _)" 10) "@Bex" :: "[pttrn, i, o] => o" ("(3∃_∈_./ _)" 10) "@Tuple" :: "[i, is] => i" ("〈(_,/ _)〉") "@pattern" :: "patterns => pttrn" ("〈_〉") finalconsts 0 Pow Inf Union PrimReplace mem defs (*don't try to use constdefs: the declaration order is tightly constrained*) (* Bounded Quantifiers *) Ball_def: "Ball(A, P) == ∀x. x∈A --> P(x)" Bex_def: "Bex(A, P) == ∃x. x∈A & P(x)" subset_def: "A <= B == ∀x∈A. x∈B" local axioms (* ZF axioms -- see Suppes p.238 Axioms for Union, Pow and Replace state existence only, uniqueness is derivable using extensionality. *) extension: "A = B <-> A <= B & B <= A" Union_iff: "A ∈ Union(C) <-> (∃B∈C. A∈B)" Pow_iff: "A ∈ Pow(B) <-> A <= B" (*We may name this set, though it is not uniquely defined.*) infinity: "0∈Inf & (∀y∈Inf. succ(y): Inf)" (*This formulation facilitates case analysis on A.*) foundation: "A=0 | (∃x∈A. ∀y∈x. y~:A)" (*Schema axiom since predicate P is a higher-order variable*) replacement: "(∀x∈A. ∀y z. P(x,y) & P(x,z) --> y=z) ==> b ∈ PrimReplace(A,P) <-> (∃x∈A. P(x,b))" defs (* Derived form of replacement, restricting P to its functional part. The resulting set (for functional P) is the same as with PrimReplace, but the rules are simpler. *) Replace_def: "Replace(A,P) == PrimReplace(A, %x y. (EX!z. P(x,z)) & P(x,y))" (* Functional form of replacement -- analgous to ML's map functional *) RepFun_def: "RepFun(A,f) == {y . x∈A, y=f(x)}" (* Separation and Pairing can be derived from the Replacement and Powerset Axioms using the following definitions. *) Collect_def: "Collect(A,P) == {y . x∈A, x=y & P(x)}" (*Unordered pairs (Upair) express binary union/intersection and cons; set enumerations translate as {a,...,z} = cons(a,...,cons(z,0)...)*) Upair_def: "Upair(a,b) == {y. x∈Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)}" cons_def: "cons(a,A) == Upair(a,a) Un A" succ_def: "succ(i) == cons(i, i)" (* Difference, general intersection, binary union and small intersection *) Diff_def: "A - B == { x∈A . ~(x∈B) }" Inter_def: "Inter(A) == { x∈Union(A) . ∀y∈A. x∈y}" Un_def: "A Un B == Union(Upair(A,B))" Int_def: "A Int B == Inter(Upair(A,B))" (* definite descriptions *) the_def: "The(P) == Union({y . x ∈ {0}, P(y)})" if_def: "if(P,a,b) == THE z. P & z=a | ~P & z=b" (* this "symmetric" definition works better than {{a}, {a,b}} *) Pair_def: "<a,b> == {{a,a}, {a,b}}" fst_def: "fst(p) == THE a. ∃b. p=<a,b>" snd_def: "snd(p) == THE b. ∃a. p=<a,b>" split_def: "split(c) == %p. c(fst(p), snd(p))" Sigma_def: "Sigma(A,B) == \<Union>x∈A. \<Union>y∈B(x). {<x,y>}" (* Operations on relations *) (*converse of relation r, inverse of function*) converse_def: "converse(r) == {z. w∈r, ∃x y. w=<x,y> & z=<y,x>}" domain_def: "domain(r) == {x. w∈r, ∃y. w=<x,y>}" range_def: "range(r) == domain(converse(r))" field_def: "field(r) == domain(r) Un range(r)" relation_def: "relation(r) == ∀z∈r. ∃x y. z = <x,y>" function_def: "function(r) == ∀x y. <x,y>:r --> (∀y'. <x,y'>:r --> y=y')" image_def: "r `` A == {y : range(r) . ∃x∈A. <x,y> : r}" vimage_def: "r -`` A == converse(r)``A" (* Abstraction, application and Cartesian product of a family of sets *) lam_def: "Lambda(A,b) == {<x,b(x)> . x∈A}" apply_def: "f`a == Union(f``{a})" Pi_def: "Pi(A,B) == {f∈Pow(Sigma(A,B)). A<=domain(f) & function(f)}" (* Restrict the relation r to the domain A *) restrict_def: "restrict(r,A) == {z : r. ∃x∈A. ∃y. z = <x,y>}" subsection {* Substitution*} (*Useful examples: singletonI RS subst_elem, subst_elem RSN (2,IntI) *) lemma subst_elem: "[| b∈A; a=b |] ==> a∈A" by (erule ssubst, assumption) subsection{*Bounded universal quantifier*} lemma ballI [intro!]: "[| !!x. x∈A ==> P(x) |] ==> ∀x∈A. P(x)" by (simp add: Ball_def) lemmas strip = impI allI ballI lemma bspec [dest?]: "[| ∀x∈A. P(x); x: A |] ==> P(x)" by (simp add: Ball_def) (*Instantiates x first: better for automatic theorem proving?*) lemma rev_ballE [elim]: "[| ∀x∈A. P(x); x~:A ==> Q; P(x) ==> Q |] ==> Q" by (simp add: Ball_def, blast) lemma ballE: "[| ∀x∈A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q" by blast (*Used in the datatype package*) lemma rev_bspec: "[| x: A; ∀x∈A. P(x) |] ==> P(x)" by (simp add: Ball_def) (*Trival rewrite rule; (∀x∈A.P)<->P holds only if A is nonempty!*) lemma ball_triv [simp]: "(∀x∈A. P) <-> ((∃x. x∈A) --> P)" by (simp add: Ball_def) (*Congruence rule for rewriting*) lemma ball_cong [cong]: "[| A=A'; !!x. x∈A' ==> P(x) <-> P'(x) |] ==> (∀x∈A. P(x)) <-> (∀x∈A'. P'(x))" by (simp add: Ball_def) lemma atomize_ball: "(!!x. x ∈ A ==> P(x)) == Trueprop (∀x∈A. P(x))" by (simp only: Ball_def atomize_all atomize_imp) lemmas [symmetric, rulify] = atomize_ball and [symmetric, defn] = atomize_ball subsection{*Bounded existential quantifier*} lemma bexI [intro]: "[| P(x); x: A |] ==> ∃x∈A. P(x)" by (simp add: Bex_def, blast) (*The best argument order when there is only one x∈A*) lemma rev_bexI: "[| x∈A; P(x) |] ==> ∃x∈A. P(x)" by blast (*Not of the general form for such rules; ~∃has become ALL~ *) lemma bexCI: "[| ∀x∈A. ~P(x) ==> P(a); a: A |] ==> ∃x∈A. P(x)" by blast lemma bexE [elim!]: "[| ∃x∈A. P(x); !!x. [| x∈A; P(x) |] ==> Q |] ==> Q" by (simp add: Bex_def, blast) (*We do not even have (∃x∈A. True) <-> True unless A is nonempty!!*) lemma bex_triv [simp]: "(∃x∈A. P) <-> ((∃x. x∈A) & P)" by (simp add: Bex_def) lemma bex_cong [cong]: "[| A=A'; !!x. x∈A' ==> P(x) <-> P'(x) |] ==> (∃x∈A. P(x)) <-> (∃x∈A'. P'(x))" by (simp add: Bex_def cong: conj_cong) subsection{*Rules for subsets*} lemma subsetI [intro!]: "(!!x. x∈A ==> x∈B) ==> A <= B" by (simp add: subset_def) (*Rule in Modus Ponens style [was called subsetE] *) lemma subsetD [elim]: "[| A <= B; c∈A |] ==> c∈B" apply (unfold subset_def) apply (erule bspec, assumption) done (*Classical elimination rule*) lemma subsetCE [elim]: "[| A <= B; c~:A ==> P; c∈B ==> P |] ==> P" by (simp add: subset_def, blast) (*Sometimes useful with premises in this order*) lemma rev_subsetD: "[| c∈A; A<=B |] ==> c∈B" by blast lemma contra_subsetD: "[| A <= B; c ~: B |] ==> c ~: A" by blast lemma rev_contra_subsetD: "[| c ~: B; A <= B |] ==> c ~: A" by blast lemma subset_refl [simp]: "A <= A" by blast lemma subset_trans: "[| A<=B; B<=C |] ==> A<=C" by blast (*Useful for proving A<=B by rewriting in some cases*) lemma subset_iff: "A<=B <-> (∀x. x∈A --> x∈B)" apply (unfold subset_def Ball_def) apply (rule iff_refl) done subsection{*Rules for equality*} (*Anti-symmetry of the subset relation*) lemma equalityI [intro]: "[| A <= B; B <= A |] ==> A = B" by (rule extension [THEN iffD2], rule conjI) lemma equality_iffI: "(!!x. x∈A <-> x∈B) ==> A = B" by (rule equalityI, blast+) lemmas equalityD1 = extension [THEN iffD1, THEN conjunct1, standard] lemmas equalityD2 = extension [THEN iffD1, THEN conjunct2, standard] lemma equalityE: "[| A = B; [| A<=B; B<=A |] ==> P |] ==> P" by (blast dest: equalityD1 equalityD2) lemma equalityCE: "[| A = B; [| c∈A; c∈B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P" by (erule equalityE, blast) subsection{*Rules for Replace -- the derived form of replacement*} lemma Replace_iff: "b : {y. x∈A, P(x,y)} <-> (∃x∈A. P(x,b) & (∀y. P(x,y) --> y=b))" apply (unfold Replace_def) apply (rule replacement [THEN iff_trans], blast+) done (*Introduction; there must be a unique y such that P(x,y), namely y=b. *) lemma ReplaceI [intro]: "[| P(x,b); x: A; !!y. P(x,y) ==> y=b |] ==> b : {y. x∈A, P(x,y)}" by (rule Replace_iff [THEN iffD2], blast) (*Elimination; may asssume there is a unique y such that P(x,y), namely y=b. *) lemma ReplaceE: "[| b : {y. x∈A, P(x,y)}; !!x. [| x: A; P(x,b); ∀y. P(x,y)-->y=b |] ==> R |] ==> R" by (rule Replace_iff [THEN iffD1, THEN bexE], simp+) (*As above but without the (generally useless) 3rd assumption*) lemma ReplaceE2 [elim!]: "[| b : {y. x∈A, P(x,y)}; !!x. [| x: A; P(x,b) |] ==> R |] ==> R" by (erule ReplaceE, blast) lemma Replace_cong [cong]: "[| A=B; !!x y. x∈B ==> P(x,y) <-> Q(x,y) |] ==> Replace(A,P) = Replace(B,Q)" apply (rule equality_iffI) apply (simp add: Replace_iff) done subsection{*Rules for RepFun*} lemma RepFunI: "a ∈ A ==> f(a) : {f(x). x∈A}" by (simp add: RepFun_def Replace_iff, blast) (*Useful for coinduction proofs*) lemma RepFun_eqI [intro]: "[| b=f(a); a ∈ A |] ==> b : {f(x). x∈A}" apply (erule ssubst) apply (erule RepFunI) done lemma RepFunE [elim!]: "[| b : {f(x). x∈A}; !!x.[| x∈A; b=f(x) |] ==> P |] ==> P" by (simp add: RepFun_def Replace_iff, blast) lemma RepFun_cong [cong]: "[| A=B; !!x. x∈B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)" by (simp add: RepFun_def) lemma RepFun_iff [simp]: "b : {f(x). x∈A} <-> (∃x∈A. b=f(x))" by (unfold Bex_def, blast) lemma triv_RepFun [simp]: "{x. x∈A} = A" by blast subsection{*Rules for Collect -- forming a subset by separation*} (*Separation is derivable from Replacement*) lemma separation [simp]: "a : {x∈A. P(x)} <-> a∈A & P(a)" by (unfold Collect_def, blast) lemma CollectI [intro!]: "[| a∈A; P(a) |] ==> a : {x∈A. P(x)}" by simp lemma CollectE [elim!]: "[| a : {x∈A. P(x)}; [| a∈A; P(a) |] ==> R |] ==> R" by simp lemma CollectD1: "a : {x∈A. P(x)} ==> a∈A" by (erule CollectE, assumption) lemma CollectD2: "a : {x∈A. P(x)} ==> P(a)" by (erule CollectE, assumption) lemma Collect_cong [cong]: "[| A=B; !!x. x∈B ==> P(x) <-> Q(x) |] ==> Collect(A, %x. P(x)) = Collect(B, %x. Q(x))" by (simp add: Collect_def) subsection{*Rules for Unions*} declare Union_iff [simp] (*The order of the premises presupposes that C is rigid; A may be flexible*) lemma UnionI [intro]: "[| B: C; A: B |] ==> A: Union(C)" by (simp, blast) lemma UnionE [elim!]: "[| A ∈ Union(C); !!B.[| A: B; B: C |] ==> R |] ==> R" by (simp, blast) subsection{*Rules for Unions of families*} (* \<Union>x∈A. B(x) abbreviates Union({B(x). x∈A}) *) lemma UN_iff [simp]: "b : (\<Union>x∈A. B(x)) <-> (∃x∈A. b ∈ B(x))" by (simp add: Bex_def, blast) (*The order of the premises presupposes that A is rigid; b may be flexible*) lemma UN_I: "[| a: A; b: B(a) |] ==> b: (\<Union>x∈A. B(x))" by (simp, blast) lemma UN_E [elim!]: "[| b : (\<Union>x∈A. B(x)); !!x.[| x: A; b: B(x) |] ==> R |] ==> R" by blast lemma UN_cong: "[| A=B; !!x. x∈B ==> C(x)=D(x) |] ==> (\<Union>x∈A. C(x)) = (\<Union>x∈B. D(x))" by simp (*No "Addcongs [UN_cong]" because \<Union>is a combination of constants*) (* UN_E appears before UnionE so that it is tried first, to avoid expensive calls to hyp_subst_tac. Cannot include UN_I as it is unsafe: would enlarge the search space.*) subsection{*Rules for the empty set*} (*The set {x∈0. False} is empty; by foundation it equals 0 See Suppes, page 21.*) lemma not_mem_empty [simp]: "a ~: 0" apply (cut_tac foundation) apply (best dest: equalityD2) done lemmas emptyE [elim!] = not_mem_empty [THEN notE, standard] lemma empty_subsetI [simp]: "0 <= A" by blast lemma equals0I: "[| !!y. y∈A ==> False |] ==> A=0" by blast lemma equals0D [dest]: "A=0 ==> a ~: A" by blast declare sym [THEN equals0D, dest] lemma not_emptyI: "a∈A ==> A ~= 0" by blast lemma not_emptyE: "[| A ~= 0; !!x. x∈A ==> R |] ==> R" by blast subsection{*Rules for Inter*} (*Not obviously useful for proving InterI, InterD, InterE*) lemma Inter_iff: "A ∈ Inter(C) <-> (∀x∈C. A: x) & C≠0" by (simp add: Inter_def Ball_def, blast) (* Intersection is well-behaved only if the family is non-empty! *) lemma InterI [intro!]: "[| !!x. x: C ==> A: x; C≠0 |] ==> A ∈ Inter(C)" by (simp add: Inter_iff) (*A "destruct" rule -- every B in C contains A as an element, but A∈B can hold when B∈C does not! This rule is analogous to "spec". *) lemma InterD [elim]: "[| A ∈ Inter(C); B ∈ C |] ==> A ∈ B" by (unfold Inter_def, blast) (*"Classical" elimination rule -- does not require exhibiting B∈C *) lemma InterE [elim]: "[| A ∈ Inter(C); B~:C ==> R; A∈B ==> R |] ==> R" by (simp add: Inter_def, blast) subsection{*Rules for Intersections of families*} (* \<Inter>x∈A. B(x) abbreviates Inter({B(x). x∈A}) *) lemma INT_iff: "b : (\<Inter>x∈A. B(x)) <-> (∀x∈A. b ∈ B(x)) & A≠0" by (force simp add: Inter_def) lemma INT_I: "[| !!x. x: A ==> b: B(x); A≠0 |] ==> b: (\<Inter>x∈A. B(x))" by blast lemma INT_E: "[| b : (\<Inter>x∈A. B(x)); a: A |] ==> b ∈ B(a)" by blast lemma INT_cong: "[| A=B; !!x. x∈B ==> C(x)=D(x) |] ==> (\<Inter>x∈A. C(x)) = (\<Inter>x∈B. D(x))" by simp (*No "Addcongs [INT_cong]" because \<Inter>is a combination of constants*) subsection{*Rules for Powersets*} lemma PowI: "A <= B ==> A ∈ Pow(B)" by (erule Pow_iff [THEN iffD2]) lemma PowD: "A ∈ Pow(B) ==> A<=B" by (erule Pow_iff [THEN iffD1]) declare Pow_iff [iff] lemmas Pow_bottom = empty_subsetI [THEN PowI] (* 0 ∈ Pow(B) *) lemmas Pow_top = subset_refl [THEN PowI] (* A ∈ Pow(A) *) subsection{*Cantor's Theorem: There is no surjection from a set to its powerset.*} (*The search is undirected. Allowing redundant introduction rules may make it diverge. Variable b represents ANY map, such as (lam x∈A.b(x)): A->Pow(A). *) lemma cantor: "∃S ∈ Pow(A). ∀x∈A. b(x) ~= S" by (best elim!: equalityCE del: ReplaceI RepFun_eqI) (*Functions for ML scripts*) ML {* (*Converts A<=B to x∈A ==> x∈B*) fun impOfSubs th = th RSN (2, @{thm rev_subsetD}); (*Takes assumptions ∀x∈A.P(x) and a∈A; creates assumption P(a)*) val ball_tac = dtac @{thm bspec} THEN' assume_tac *} end
lemma subst_elem:
[| b ∈ A; a = b |] ==> a ∈ A
lemma ballI:
(!!x. x ∈ A ==> P(x)) ==> ∀x∈A. P(x)
lemma strip:
(P ==> Q) ==> P --> Q
(!!x. P(x)) ==> ∀x. P(x)
(!!x. x ∈ A ==> P(x)) ==> ∀x∈A. P(x)
lemma bspec:
[| ∀x∈A. P(x); x ∈ A |] ==> P(x)
lemma rev_ballE:
[| ∀x∈A. P(x); x ∉ A ==> Q; P(x) ==> Q |] ==> Q
lemma ballE:
[| ∀x∈A. P(x); P(x) ==> Q; x ∉ A ==> Q |] ==> Q
lemma rev_bspec:
[| x ∈ A; ∀x∈A. P(x) |] ==> P(x)
lemma ball_triv:
(∀x∈A. P) <-> (∃x. x ∈ A) --> P
lemma ball_cong:
[| A = A'; !!x. x ∈ A' ==> P(x) <-> P'(x) |] ==> (∀x∈A. P(x)) <-> (∀x∈A'. P'(x))
lemma atomize_ball:
(!!x. x ∈ A ==> P(x)) == ∀x∈A. P(x)
lemma
∀x∈A. P(x) == (!!x. x ∈ A ==> P(x))
and
∀x∈A. P(x) == (!!x. x ∈ A ==> P(x))
lemma bexI:
[| P(x); x ∈ A |] ==> ∃x∈A. P(x)
lemma rev_bexI:
[| x ∈ A; P(x) |] ==> ∃x∈A. P(x)
lemma bexCI:
[| ∀x∈A. ¬ P(x) ==> P(a); a ∈ A |] ==> ∃x∈A. P(x)
lemma bexE:
[| ∃x∈A. P(x); !!x. [| x ∈ A; P(x) |] ==> Q |] ==> Q
lemma bex_triv:
(∃x∈A. P) <-> (∃x. x ∈ A) ∧ P
lemma bex_cong:
[| A = A'; !!x. x ∈ A' ==> P(x) <-> P'(x) |] ==> (∃x∈A. P(x)) <-> (∃x∈A'. P'(x))
lemma subsetI:
(!!x. x ∈ A ==> x ∈ B) ==> A ⊆ B
lemma subsetD:
[| A ⊆ B; c ∈ A |] ==> c ∈ B
lemma subsetCE:
[| A ⊆ B; c ∉ A ==> P; c ∈ B ==> P |] ==> P
lemma rev_subsetD:
[| c ∈ A; A ⊆ B |] ==> c ∈ B
lemma contra_subsetD:
[| A ⊆ B; c ∉ B |] ==> c ∉ A
lemma rev_contra_subsetD:
[| c ∉ B; A ⊆ B |] ==> c ∉ A
lemma subset_refl:
A ⊆ A
lemma subset_trans:
[| A ⊆ B; B ⊆ C |] ==> A ⊆ C
lemma subset_iff:
A ⊆ B <-> (∀x. x ∈ A --> x ∈ B)
lemma equalityI:
[| A ⊆ B; B ⊆ A |] ==> A = B
lemma equality_iffI:
(!!x. x ∈ A <-> x ∈ B) ==> A = B
lemma equalityD1:
A = B ==> A ⊆ B
lemma 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
lemma Replace_iff:
b ∈ {y . x ∈ A, P(x, y)} <-> (∃x∈A. P(x, b) ∧ (∀y. P(x, y) --> y = b))
lemma ReplaceI:
[| P(x, b); x ∈ A; !!y. P(x, y) ==> y = b |] ==> b ∈ {y . x ∈ A, P(x, y)}
lemma ReplaceE:
[| b ∈ {y . x ∈ A, P(x, y)};
!!x. [| x ∈ A; P(x, b); ∀y. P(x, y) --> y = b |] ==> R |]
==> R
lemma ReplaceE2:
[| b ∈ {y . x ∈ A, P(x, y)}; !!x. [| x ∈ A; P(x, b) |] ==> R |] ==> R
lemma Replace_cong:
[| A = B; !!x y. x ∈ B ==> P(x, y) <-> Q(x, y) |]
==> Replace(A, P) = Replace(B, Q)
lemma RepFunI:
a ∈ A ==> f(a) ∈ {f(x) . x ∈ A}
lemma RepFun_eqI:
[| b = f(a); a ∈ A |] ==> b ∈ {f(x) . x ∈ A}
lemma RepFunE:
[| b ∈ {f(x) . x ∈ A}; !!x. [| x ∈ A; b = f(x) |] ==> P |] ==> P
lemma RepFun_cong:
[| A = B; !!x. x ∈ B ==> f(x) = g(x) |] ==> RepFun(A, f) = RepFun(B, g)
lemma RepFun_iff:
b ∈ {f(x) . x ∈ A} <-> (∃x∈A. b = f(x))
lemma triv_RepFun:
{x . x ∈ A} = A
lemma separation:
a ∈ {x ∈ A . P(x)} <-> a ∈ A ∧ P(a)
lemma CollectI:
[| a ∈ A; P(a) |] ==> a ∈ {x ∈ A . P(x)}
lemma CollectE:
[| a ∈ {x ∈ A . P(x)}; [| a ∈ A; P(a) |] ==> R |] ==> R
lemma CollectD1:
a ∈ {x ∈ A . P(x)} ==> a ∈ A
lemma CollectD2:
a ∈ {x ∈ A . P(x)} ==> P(a)
lemma Collect_cong:
[| A = B; !!x. x ∈ B ==> P(x) <-> Q(x) |] ==> {x ∈ A . P(x)} = {x ∈ B . Q(x)}
lemma UnionI:
[| B ∈ C; A ∈ B |] ==> A ∈ \<Union>C
lemma UnionE:
[| A ∈ \<Union>C; !!B. [| A ∈ B; B ∈ C |] ==> R |] ==> R
lemma UN_iff:
b ∈ (\<Union>x∈A. B(x)) <-> (∃x∈A. b ∈ B(x))
lemma UN_I:
[| a ∈ A; b ∈ B(a) |] ==> b ∈ (\<Union>x∈A. B(x))
lemma UN_E:
[| b ∈ (\<Union>x∈A. B(x)); !!x. [| x ∈ A; b ∈ B(x) |] ==> R |] ==> R
lemma UN_cong:
[| A = B; !!x. x ∈ B ==> C(x) = D(x) |]
==> (\<Union>x∈A. C(x)) = (\<Union>x∈B. D(x))
lemma not_mem_empty:
a ∉ 0
lemma emptyE:
a ∈ 0 ==> R
lemma empty_subsetI:
0 ⊆ A
lemma equals0I:
(!!y. y ∈ A ==> False) ==> A = 0
lemma equals0D:
A = 0 ==> a ∉ A
lemma not_emptyI:
a ∈ A ==> A ≠ 0
lemma not_emptyE:
[| A ≠ 0; !!x. x ∈ A ==> R |] ==> R
lemma Inter_iff:
A ∈ \<Inter>C <-> (∀x∈C. A ∈ x) ∧ C ≠ 0
lemma InterI:
[| !!x. x ∈ C ==> A ∈ x; C ≠ 0 |] ==> A ∈ \<Inter>C
lemma InterD:
[| A ∈ \<Inter>C; B ∈ C |] ==> A ∈ B
lemma InterE:
[| A ∈ \<Inter>C; B ∉ C ==> R; A ∈ B ==> R |] ==> R
lemma INT_iff:
b ∈ (\<Inter>x∈A. B(x)) <-> (∀x∈A. b ∈ B(x)) ∧ A ≠ 0
lemma INT_I:
[| !!x. x ∈ A ==> b ∈ B(x); A ≠ 0 |] ==> b ∈ (\<Inter>x∈A. B(x))
lemma INT_E:
[| b ∈ (\<Inter>x∈A. B(x)); a ∈ A |] ==> b ∈ B(a)
lemma INT_cong:
[| A = B; !!x. x ∈ B ==> C(x) = D(x) |]
==> (\<Inter>x∈A. C(x)) = (\<Inter>x∈B. D(x))
lemma PowI:
A ⊆ B ==> A ∈ Pow(B)
lemma PowD:
A ∈ Pow(B) ==> A ⊆ B
lemma Pow_bottom:
0 ∈ Pow(B)
lemma Pow_top:
A ∈ Pow(A)
lemma cantor:
∃S∈Pow(A). ∀x∈A. b(x) ≠ S