src/ZF/Constructible/Formula.thy
author paulson
Thu Jul 04 16:59:54 2002 +0200 (2002-07-04)
changeset 13298 b4f370679c65
parent 13291 a73ab154f75c
child 13306 6eebcddee32b
permissions -rw-r--r--
Constructible: some separation axioms
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header {* First-Order Formulas and the Definition of the Class L *}
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theory Formula = Main:
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subsection{*Internalized formulas of FOL*}
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text{*De Bruijn representation.
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  Unbound variables get their denotations from an environment.*}
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consts   formula :: i
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datatype
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  "formula" = Member ("x: nat", "y: nat")
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            | Equal  ("x: nat", "y: nat")
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            | Neg ("p: formula")
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            | And ("p: formula", "q: formula")
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            | Forall ("p: formula")
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declare formula.intros [TC]
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constdefs Or :: "[i,i]=>i"
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    "Or(p,q) == Neg(And(Neg(p),Neg(q)))"
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constdefs Implies :: "[i,i]=>i"
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    "Implies(p,q) == Neg(And(p,Neg(q)))"
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constdefs Iff :: "[i,i]=>i"
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    "Iff(p,q) == And(Implies(p,q), Implies(q,p))"
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constdefs Exists :: "i=>i"
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    "Exists(p) == Neg(Forall(Neg(p)))";
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lemma Or_type [TC]: "[| p \<in> formula; q \<in> formula |] ==> Or(p,q) \<in> formula"
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by (simp add: Or_def) 
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lemma Implies_type [TC]:
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     "[| p \<in> formula; q \<in> formula |] ==> Implies(p,q) \<in> formula"
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by (simp add: Implies_def) 
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lemma Iff_type [TC]:
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     "[| p \<in> formula; q \<in> formula |] ==> Iff(p,q) \<in> formula"
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by (simp add: Iff_def) 
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lemma Exists_type [TC]: "p \<in> formula ==> Exists(p) \<in> formula"
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by (simp add: Exists_def) 
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consts   satisfies :: "[i,i]=>i"
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primrec (*explicit lambda is required because the environment varies*)
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  "satisfies(A,Member(x,y)) = 
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      (\<lambda>env \<in> list(A). bool_of_o (nth(x,env) \<in> nth(y,env)))"
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  "satisfies(A,Equal(x,y)) = 
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      (\<lambda>env \<in> list(A). bool_of_o (nth(x,env) = nth(y,env)))"
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  "satisfies(A,Neg(p)) = 
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      (\<lambda>env \<in> list(A). not(satisfies(A,p)`env))"
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  "satisfies(A,And(p,q)) =
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      (\<lambda>env \<in> list(A). (satisfies(A,p)`env) and (satisfies(A,q)`env))"
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  "satisfies(A,Forall(p)) = 
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      (\<lambda>env \<in> list(A). bool_of_o (\<forall>x\<in>A. satisfies(A,p) ` (Cons(x,env)) = 1))"
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lemma "p \<in> formula ==> satisfies(A,p) \<in> list(A) -> bool"
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by (induct_tac p, simp_all) 
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syntax sats :: "[i,i,i] => o"
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translations "sats(A,p,env)" == "satisfies(A,p)`env = 1"
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lemma [simp]:
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  "env \<in> list(A) 
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   ==> sats(A, Member(x,y), env) <-> nth(x,env) \<in> nth(y,env)"
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by simp
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lemma [simp]:
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  "env \<in> list(A) 
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   ==> sats(A, Equal(x,y), env) <-> nth(x,env) = nth(y,env)"
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by simp
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lemma sats_Neg_iff [simp]:
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  "env \<in> list(A) 
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   ==> sats(A, Neg(p), env) <-> ~ sats(A,p,env)"
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by (simp add: Bool.not_def cond_def) 
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lemma sats_And_iff [simp]:
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  "env \<in> list(A) 
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   ==> (sats(A, And(p,q), env)) <-> sats(A,p,env) & sats(A,q,env)"
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by (simp add: Bool.and_def cond_def) 
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lemma sats_Forall_iff [simp]:
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  "env \<in> list(A) 
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   ==> sats(A, Forall(p), env) <-> (\<forall>x\<in>A. sats(A, p, Cons(x,env)))"
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by simp
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declare satisfies.simps [simp del]; 
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subsection{*Dividing line between primitive and derived connectives*}
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lemma sats_Or_iff [simp]:
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  "env \<in> list(A) 
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   ==> (sats(A, Or(p,q), env)) <-> sats(A,p,env) | sats(A,q,env)"
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by (simp add: Or_def)
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lemma sats_Implies_iff [simp]:
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  "env \<in> list(A) 
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   ==> (sats(A, Implies(p,q), env)) <-> (sats(A,p,env) --> sats(A,q,env))"
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by (simp add: Implies_def, blast) 
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lemma sats_Iff_iff [simp]:
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  "env \<in> list(A) 
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   ==> (sats(A, Iff(p,q), env)) <-> (sats(A,p,env) <-> sats(A,q,env))"
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by (simp add: Iff_def, blast) 
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lemma sats_Exists_iff [simp]:
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  "env \<in> list(A) 
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   ==> sats(A, Exists(p), env) <-> (\<exists>x\<in>A. sats(A, p, Cons(x,env)))"
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by (simp add: Exists_def)
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subsubsection{*Derived rules to help build up formulas*}
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lemma mem_iff_sats:
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      "[| nth(i,env) = x; nth(j,env) = y; env \<in> list(A)|]
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       ==> (x\<in>y) <-> sats(A, Member(i,j), env)" 
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by (simp add: satisfies.simps)
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lemma equal_iff_sats:
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      "[| nth(i,env) = x; nth(j,env) = y; env \<in> list(A)|]
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       ==> (x=y) <-> sats(A, Equal(i,j), env)" 
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by (simp add: satisfies.simps)
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lemma conj_iff_sats:
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      "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env \<in> list(A)|]
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       ==> (P & Q) <-> sats(A, And(p,q), env)"
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by (simp add: sats_And_iff)
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lemma disj_iff_sats:
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      "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env \<in> list(A)|]
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       ==> (P | Q) <-> sats(A, Or(p,q), env)"
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by (simp add: sats_Or_iff)
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lemma imp_iff_sats:
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      "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env \<in> list(A)|]
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       ==> (P --> Q) <-> sats(A, Implies(p,q), env)"
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by (simp add: sats_Forall_iff) 
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lemma iff_iff_sats:
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      "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env \<in> list(A)|]
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       ==> (P <-> Q) <-> sats(A, Iff(p,q), env)"
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by (simp add: sats_Forall_iff) 
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lemma imp_iff_sats:
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      "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env \<in> list(A)|]
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       ==> (P --> Q) <-> sats(A, Implies(p,q), env)"
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by (simp add: sats_Forall_iff) 
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lemma ball_iff_sats:
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      "[| !!x. x\<in>A ==> P(x) <-> sats(A, p, Cons(x, env)); env \<in> list(A)|]
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       ==> (\<forall>x\<in>A. P(x)) <-> sats(A, Forall(p), env)"
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by (simp add: sats_Forall_iff) 
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lemma bex_iff_sats:
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      "[| !!x. x\<in>A ==> P(x) <-> sats(A, p, Cons(x, env)); env \<in> list(A)|]
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       ==> (\<exists>x\<in>A. P(x)) <-> sats(A, Exists(p), env)"
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by (simp add: sats_Exists_iff) 
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constdefs incr_var :: "[i,i]=>i"
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    "incr_var(x,lev) == if x<lev then x else succ(x)"
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lemma incr_var_lt: "x<lev ==> incr_var(x,lev) = x"
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by (simp add: incr_var_def)
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lemma incr_var_le: "lev\<le>x ==> incr_var(x,lev) = succ(x)"
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apply (simp add: incr_var_def) 
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apply (blast dest: lt_trans1) 
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done
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consts   incr_bv :: "i=>i"
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primrec
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  "incr_bv(Member(x,y)) = 
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      (\<lambda>lev \<in> nat. Member (incr_var(x,lev), incr_var(y,lev)))"
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  "incr_bv(Equal(x,y)) = 
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      (\<lambda>lev \<in> nat. Equal (incr_var(x,lev), incr_var(y,lev)))"
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  "incr_bv(Neg(p)) = 
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      (\<lambda>lev \<in> nat. Neg(incr_bv(p)`lev))"
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  "incr_bv(And(p,q)) =
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      (\<lambda>lev \<in> nat. And (incr_bv(p)`lev, incr_bv(q)`lev))"
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  "incr_bv(Forall(p)) = 
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      (\<lambda>lev \<in> nat. Forall (incr_bv(p) ` succ(lev)))"
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constdefs incr_boundvars :: "i => i"
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    "incr_boundvars(p) == incr_bv(p)`0"
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lemma [TC]: "x \<in> nat ==> incr_var(x,lev) \<in> nat"
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by (simp add: incr_var_def) 
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lemma incr_bv_type [TC]: "p \<in> formula ==> incr_bv(p) \<in> nat -> formula"
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by (induct_tac p, simp_all) 
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lemma incr_boundvars_type [TC]: "p \<in> formula ==> incr_boundvars(p) \<in> formula"
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by (simp add: incr_boundvars_def) 
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(*Obviously DPow is closed under complements and finite intersections and
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unions.  Needs an inductive lemma to allow two lists of parameters to 
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be combined.*)
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lemma sats_incr_bv_iff [rule_format]:
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  "[| p \<in> formula; env \<in> list(A); x \<in> A |]
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   ==> \<forall>bvs \<in> list(A). 
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           sats(A, incr_bv(p) ` length(bvs), bvs @ Cons(x,env)) <-> 
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           sats(A, p, bvs@env)"
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apply (induct_tac p)
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apply (simp_all add: incr_var_def nth_append succ_lt_iff length_type)
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apply (auto simp add: diff_succ not_lt_iff_le)
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done
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(*UNUSED*)
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lemma sats_incr_boundvars_iff:
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  "[| p \<in> formula; env \<in> list(A); x \<in> A |]
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   ==> sats(A, incr_boundvars(p), Cons(x,env)) <-> sats(A, p, env)"
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apply (insert sats_incr_bv_iff [of p env A x Nil])
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apply (simp add: incr_boundvars_def) 
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done
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(*UNUSED
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lemma formula_add_params [rule_format]:
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  "[| p \<in> formula; n \<in> nat |]
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   ==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A). 
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         length(bvs) = n --> 
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         sats(A, iterates(incr_boundvars,n,p), bvs@env) <-> sats(A, p, env)"
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apply (induct_tac n, simp, clarify) 
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apply (erule list.cases)
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apply (auto simp add: sats_incr_boundvars_iff)  
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done
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*)
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consts   arity :: "i=>i"
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primrec
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  "arity(Member(x,y)) = succ(x) \<union> succ(y)"
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  "arity(Equal(x,y)) = succ(x) \<union> succ(y)"
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  "arity(Neg(p)) = arity(p)"
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  "arity(And(p,q)) = arity(p) \<union> arity(q)"
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  "arity(Forall(p)) = nat_case(0, %x. x, arity(p))"
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lemma arity_type [TC]: "p \<in> formula ==> arity(p) \<in> nat"
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by (induct_tac p, simp_all) 
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lemma arity_Or [simp]: "arity(Or(p,q)) = arity(p) \<union> arity(q)"
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by (simp add: Or_def) 
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lemma arity_Implies [simp]: "arity(Implies(p,q)) = arity(p) \<union> arity(q)"
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by (simp add: Implies_def) 
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lemma arity_Iff [simp]: "arity(Iff(p,q)) = arity(p) \<union> arity(q)"
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by (simp add: Iff_def, blast)
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lemma arity_Exists [simp]: "arity(Exists(p)) = nat_case(0, %x. x, arity(p))"
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by (simp add: Exists_def) 
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lemma arity_sats_iff [rule_format]:
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  "[| p \<in> formula; extra \<in> list(A) |]
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   ==> \<forall>env \<in> list(A). 
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           arity(p) \<le> length(env) --> 
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           sats(A, p, env @ extra) <-> sats(A, p, env)"
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apply (induct_tac p)
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apply (simp_all add: nth_append Un_least_lt_iff arity_type nat_imp_quasinat
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                split: split_nat_case, auto) 
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done
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lemma arity_sats1_iff:
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  "[| arity(p) \<le> succ(length(env)); p \<in> formula; x \<in> A; env \<in> list(A); 
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    extra \<in> list(A) |]
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   ==> sats(A, p, Cons(x, env @ extra)) <-> sats(A, p, Cons(x, env))"
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apply (insert arity_sats_iff [of p extra A "Cons(x,env)"])
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apply simp 
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done
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(*the following two lemmas prevent huge case splits in arity_incr_bv_lemma*)
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lemma incr_var_lemma:
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     "[| x \<in> nat; y \<in> nat; lev \<le> x |]
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      ==> succ(x) \<union> incr_var(y,lev) = succ(x \<union> y)"
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apply (simp add: incr_var_def Ord_Un_if, auto)
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  apply (blast intro: leI)
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 apply (simp add: not_lt_iff_le)  
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 apply (blast intro: le_anti_sym) 
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apply (blast dest: lt_trans2) 
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done
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lemma incr_And_lemma:
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     "y < x ==> y \<union> succ(x) = succ(x \<union> y)"
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apply (simp add: Ord_Un_if lt_Ord lt_Ord2 succ_lt_iff) 
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apply (blast dest: lt_asym) 
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   307
done
paulson@13223
   308
paulson@13223
   309
lemma arity_incr_bv_lemma [rule_format]:
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   310
  "p \<in> formula 
paulson@13223
   311
   ==> \<forall>n \<in> nat. arity (incr_bv(p) ` n) = 
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   312
                 (if n < arity(p) then succ(arity(p)) else arity(p))"
paulson@13223
   313
apply (induct_tac p) 
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   314
apply (simp_all add: imp_disj not_lt_iff_le Un_least_lt_iff lt_Un_iff le_Un_iff
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   315
                     succ_Un_distrib [symmetric] incr_var_lt incr_var_le
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   316
                     Un_commute incr_var_lemma arity_type nat_imp_quasinat
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   317
            split: split_nat_case) 
paulson@13269
   318
 txt{*the Forall case reduces to linear arithmetic*}
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   319
 prefer 2
paulson@13269
   320
 apply clarify 
paulson@13269
   321
 apply (blast dest: lt_trans1) 
paulson@13269
   322
txt{*left with the And case*}
paulson@13223
   323
apply safe
paulson@13223
   324
 apply (blast intro: incr_And_lemma lt_trans1) 
paulson@13223
   325
apply (subst incr_And_lemma)
paulson@13269
   326
 apply (blast intro: lt_trans1) 
paulson@13269
   327
apply (simp add: Un_commute)
paulson@13223
   328
done
paulson@13223
   329
paulson@13223
   330
lemma arity_incr_boundvars_eq:
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   331
  "p \<in> formula
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   332
   ==> arity(incr_boundvars(p)) =
paulson@13223
   333
        (if 0 < arity(p) then succ(arity(p)) else arity(p))"
paulson@13223
   334
apply (insert arity_incr_bv_lemma [of p 0])
paulson@13223
   335
apply (simp add: incr_boundvars_def) 
paulson@13223
   336
done
paulson@13223
   337
paulson@13223
   338
lemma arity_iterates_incr_boundvars_eq:
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   339
  "[| p \<in> formula; n \<in> nat |]
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   340
   ==> arity(incr_boundvars^n(p)) =
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   341
         (if 0 < arity(p) then n #+ arity(p) else arity(p))"
paulson@13223
   342
apply (induct_tac n) 
paulson@13223
   343
apply (simp_all add: arity_incr_boundvars_eq not_lt_iff_le) 
paulson@13223
   344
done
paulson@13223
   345
paulson@13223
   346
paulson@13298
   347
subsection{*Renaming all but the first bound variable*}
paulson@13223
   348
paulson@13223
   349
constdefs incr_bv1 :: "i => i"
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   350
    "incr_bv1(p) == incr_bv(p)`1"
paulson@13223
   351
paulson@13223
   352
paulson@13223
   353
lemma incr_bv1_type [TC]: "p \<in> formula ==> incr_bv1(p) \<in> formula"
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   354
by (simp add: incr_bv1_def) 
paulson@13223
   355
paulson@13223
   356
(*For renaming all but the bound variable at level 0*)
paulson@13223
   357
lemma sats_incr_bv1_iff [rule_format]:
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   358
  "[| p \<in> formula; env \<in> list(A); x \<in> A; y \<in> A |]
paulson@13223
   359
   ==> sats(A, incr_bv1(p), Cons(x, Cons(y, env))) <-> 
paulson@13223
   360
       sats(A, p, Cons(x,env))"
paulson@13223
   361
apply (insert sats_incr_bv_iff [of p env A y "Cons(x,Nil)"])
paulson@13223
   362
apply (simp add: incr_bv1_def) 
paulson@13223
   363
done
paulson@13223
   364
paulson@13223
   365
lemma formula_add_params1 [rule_format]:
paulson@13223
   366
  "[| p \<in> formula; n \<in> nat; x \<in> A |]
paulson@13223
   367
   ==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A). 
paulson@13223
   368
          length(bvs) = n --> 
paulson@13223
   369
          sats(A, iterates(incr_bv1, n, p), Cons(x, bvs@env)) <-> 
paulson@13223
   370
          sats(A, p, Cons(x,env))"
paulson@13223
   371
apply (induct_tac n, simp, clarify) 
paulson@13223
   372
apply (erule list.cases)
paulson@13223
   373
apply (simp_all add: sats_incr_bv1_iff) 
paulson@13223
   374
done
paulson@13223
   375
paulson@13223
   376
paulson@13223
   377
lemma arity_incr_bv1_eq:
paulson@13223
   378
  "p \<in> formula
paulson@13223
   379
   ==> arity(incr_bv1(p)) =
paulson@13223
   380
        (if 1 < arity(p) then succ(arity(p)) else arity(p))"
paulson@13223
   381
apply (insert arity_incr_bv_lemma [of p 1])
paulson@13223
   382
apply (simp add: incr_bv1_def) 
paulson@13223
   383
done
paulson@13223
   384
paulson@13223
   385
lemma arity_iterates_incr_bv1_eq:
paulson@13223
   386
  "[| p \<in> formula; n \<in> nat |]
paulson@13223
   387
   ==> arity(incr_bv1^n(p)) =
paulson@13223
   388
         (if 1 < arity(p) then n #+ arity(p) else arity(p))"
paulson@13223
   389
apply (induct_tac n) 
paulson@13298
   390
apply (simp_all add: arity_incr_bv1_eq)
paulson@13223
   391
apply (simp add: not_lt_iff_le)
paulson@13223
   392
apply (blast intro: le_trans add_le_self2 arity_type) 
paulson@13223
   393
done
paulson@13223
   394
paulson@13223
   395
paulson@13223
   396
(*Definable powerset operation: Kunen's definition 1.1, page 165.*)
paulson@13223
   397
constdefs DPow :: "i => i"
paulson@13223
   398
  "DPow(A) == {X \<in> Pow(A). 
paulson@13223
   399
               \<exists>env \<in> list(A). \<exists>p \<in> formula. 
paulson@13223
   400
                 arity(p) \<le> succ(length(env)) & 
paulson@13223
   401
                 X = {x\<in>A. sats(A, p, Cons(x,env))}}"
paulson@13223
   402
paulson@13223
   403
lemma DPowI:
paulson@13291
   404
  "[|env \<in> list(A);  p \<in> formula;  arity(p) \<le> succ(length(env))|]
paulson@13223
   405
   ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
paulson@13223
   406
by (simp add: DPow_def, blast) 
paulson@13223
   407
paulson@13291
   408
text{*With this rule we can specify @{term p} later.*}
paulson@13291
   409
lemma DPowI2 [rule_format]:
paulson@13291
   410
  "[|\<forall>x\<in>A. P(x) <-> sats(A, p, Cons(x,env));
paulson@13291
   411
     env \<in> list(A);  p \<in> formula;  arity(p) \<le> succ(length(env))|]
paulson@13291
   412
   ==> {x\<in>A. P(x)} \<in> DPow(A)"
paulson@13291
   413
by (simp add: DPow_def, blast) 
paulson@13291
   414
paulson@13223
   415
lemma DPowD:
paulson@13223
   416
  "X \<in> DPow(A) 
paulson@13223
   417
   ==> X <= A &
paulson@13223
   418
       (\<exists>env \<in> list(A). 
paulson@13223
   419
        \<exists>p \<in> formula. arity(p) \<le> succ(length(env)) & 
paulson@13223
   420
                      X = {x\<in>A. sats(A, p, Cons(x,env))})"
paulson@13223
   421
by (simp add: DPow_def) 
paulson@13223
   422
paulson@13223
   423
lemmas DPow_imp_subset = DPowD [THEN conjunct1]
paulson@13223
   424
paulson@13223
   425
(*Lemma 1.2*)
paulson@13223
   426
lemma "[| p \<in> formula; env \<in> list(A); arity(p) \<le> succ(length(env)) |] 
paulson@13223
   427
       ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
paulson@13223
   428
by (blast intro: DPowI)
paulson@13223
   429
paulson@13223
   430
lemma DPow_subset_Pow: "DPow(A) <= Pow(A)"
paulson@13223
   431
by (simp add: DPow_def, blast)
paulson@13223
   432
paulson@13223
   433
lemma empty_in_DPow: "0 \<in> DPow(A)"
paulson@13223
   434
apply (simp add: DPow_def)
paulson@13223
   435
apply (rule_tac x="Nil" in bexI) 
paulson@13223
   436
 apply (rule_tac x="Neg(Equal(0,0))" in bexI) 
paulson@13223
   437
  apply (auto simp add: Un_least_lt_iff) 
paulson@13223
   438
done
paulson@13223
   439
paulson@13223
   440
lemma Compl_in_DPow: "X \<in> DPow(A) ==> (A-X) \<in> DPow(A)"
paulson@13223
   441
apply (simp add: DPow_def, clarify, auto) 
paulson@13223
   442
apply (rule bexI) 
paulson@13223
   443
 apply (rule_tac x="Neg(p)" in bexI) 
paulson@13223
   444
  apply auto 
paulson@13223
   445
done
paulson@13223
   446
paulson@13223
   447
lemma Int_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Int Y \<in> DPow(A)"
paulson@13223
   448
apply (simp add: DPow_def, auto) 
paulson@13223
   449
apply (rename_tac envp p envq q) 
paulson@13223
   450
apply (rule_tac x="envp@envq" in bexI) 
paulson@13223
   451
 apply (rule_tac x="And(p, iterates(incr_bv1,length(envp),q))" in bexI)
paulson@13223
   452
  apply typecheck
paulson@13223
   453
apply (rule conjI) 
paulson@13223
   454
(*finally check the arity!*)
paulson@13223
   455
 apply (simp add: arity_iterates_incr_bv1_eq length_app Un_least_lt_iff)
paulson@13223
   456
 apply (force intro: add_le_self le_trans) 
paulson@13223
   457
apply (simp add: arity_sats1_iff formula_add_params1, blast) 
paulson@13223
   458
done
paulson@13223
   459
paulson@13223
   460
lemma Un_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Un Y \<in> DPow(A)"
paulson@13223
   461
apply (subgoal_tac "X Un Y = A - ((A-X) Int (A-Y))") 
paulson@13223
   462
apply (simp add: Int_in_DPow Compl_in_DPow) 
paulson@13223
   463
apply (simp add: DPow_def, blast) 
paulson@13223
   464
done
paulson@13223
   465
paulson@13223
   466
lemma singleton_in_DPow: "x \<in> A ==> {x} \<in> DPow(A)"
paulson@13223
   467
apply (simp add: DPow_def)
paulson@13223
   468
apply (rule_tac x="Cons(x,Nil)" in bexI) 
paulson@13223
   469
 apply (rule_tac x="Equal(0,1)" in bexI) 
paulson@13223
   470
  apply typecheck
paulson@13223
   471
apply (force simp add: succ_Un_distrib [symmetric])  
paulson@13223
   472
done
paulson@13223
   473
paulson@13223
   474
lemma cons_in_DPow: "[| a \<in> A; X \<in> DPow(A) |] ==> cons(a,X) \<in> DPow(A)"
paulson@13223
   475
apply (rule cons_eq [THEN subst]) 
paulson@13223
   476
apply (blast intro: singleton_in_DPow Un_in_DPow) 
paulson@13223
   477
done
paulson@13223
   478
paulson@13223
   479
(*Part of Lemma 1.3*)
paulson@13223
   480
lemma Fin_into_DPow: "X \<in> Fin(A) ==> X \<in> DPow(A)"
paulson@13223
   481
apply (erule Fin.induct) 
paulson@13223
   482
 apply (rule empty_in_DPow) 
paulson@13223
   483
apply (blast intro: cons_in_DPow) 
paulson@13223
   484
done
paulson@13223
   485
paulson@13223
   486
(*DPow is not monotonic.  For example, let A be some non-constructible set
paulson@13223
   487
  of natural numbers, and let B be nat.  Then A<=B and obviously A : DPow(A)
paulson@13223
   488
  but A ~: DPow(B).*)
paulson@13223
   489
lemma DPow_mono: "A : DPow(B) ==> DPow(A) <= DPow(B)"
paulson@13223
   490
apply (simp add: DPow_def, auto) 
paulson@13223
   491
(*must use the formula defining A in B to relativize the new formula...*)
paulson@13223
   492
oops
paulson@13223
   493
paulson@13223
   494
lemma DPow_0: "DPow(0) = {0}" 
paulson@13223
   495
by (blast intro: empty_in_DPow dest: DPow_imp_subset)
paulson@13223
   496
paulson@13223
   497
lemma Finite_Pow_subset_Pow: "Finite(A) ==> Pow(A) <= DPow(A)" 
paulson@13223
   498
by (blast intro: Fin_into_DPow Finite_into_Fin Fin_subset)
paulson@13223
   499
paulson@13223
   500
lemma Finite_DPow_eq_Pow: "Finite(A) ==> DPow(A) = Pow(A)"
paulson@13223
   501
apply (rule equalityI) 
paulson@13223
   502
apply (rule DPow_subset_Pow) 
paulson@13223
   503
apply (erule Finite_Pow_subset_Pow) 
paulson@13223
   504
done
paulson@13223
   505
paulson@13223
   506
(*This may be true but the proof looks difficult, requiring relativization 
paulson@13223
   507
lemma DPow_insert: "DPow (cons(a,A)) = DPow(A) Un {cons(a,X) . X: DPow(A)}"
paulson@13223
   508
apply (rule equalityI, safe)
paulson@13223
   509
oops
paulson@13223
   510
*)
paulson@13223
   511
paulson@13298
   512
paulson@13298
   513
subsection{*Internalized formulas for basic concepts*}
paulson@13298
   514
paulson@13298
   515
subsubsection{*The subset relation*}
paulson@13298
   516
paulson@13298
   517
lemma lt_length_in_nat:
paulson@13298
   518
   "[|x < length(xs); xs \<in> list(A)|] ==> x \<in> nat"
paulson@13298
   519
apply (frule lt_nat_in_nat, typecheck) 
paulson@13298
   520
done
paulson@13298
   521
paulson@13298
   522
constdefs subset_fm :: "[i,i]=>i"
paulson@13298
   523
    "subset_fm(x,y) == Forall(Implies(Member(0,succ(x)), Member(0,succ(y))))"
paulson@13298
   524
paulson@13298
   525
lemma subset_type [TC]: "[| x \<in> nat; y \<in> nat |] ==> subset_fm(x,y) \<in> formula"
paulson@13298
   526
by (simp add: subset_fm_def) 
paulson@13298
   527
paulson@13298
   528
lemma arity_subset_fm [simp]:
paulson@13298
   529
     "[| x \<in> nat; y \<in> nat |] ==> arity(subset_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13298
   530
by (simp add: subset_fm_def succ_Un_distrib [symmetric]) 
paulson@13298
   531
paulson@13298
   532
lemma sats_subset_fm [simp]:
paulson@13298
   533
   "[|x < length(env); y \<in> nat; env \<in> list(A); Transset(A)|]
paulson@13298
   534
    ==> sats(A, subset_fm(x,y), env) <-> nth(x,env) \<subseteq> nth(y,env)"
paulson@13298
   535
apply (frule lt_length_in_nat, assumption)  
paulson@13298
   536
apply (simp add: subset_fm_def Transset_def) 
paulson@13298
   537
apply (blast intro: nth_type) 
paulson@13298
   538
done
paulson@13298
   539
paulson@13298
   540
subsubsection{*Transitive sets*}
paulson@13298
   541
paulson@13298
   542
constdefs transset_fm :: "i=>i"
paulson@13298
   543
   "transset_fm(x) == Forall(Implies(Member(0,succ(x)), subset_fm(0,succ(x))))"
paulson@13298
   544
paulson@13298
   545
lemma transset_type [TC]: "x \<in> nat ==> transset_fm(x) \<in> formula"
paulson@13298
   546
by (simp add: transset_fm_def) 
paulson@13298
   547
paulson@13298
   548
lemma arity_transset_fm [simp]:
paulson@13298
   549
     "x \<in> nat ==> arity(transset_fm(x)) = succ(x)"
paulson@13298
   550
by (simp add: transset_fm_def succ_Un_distrib [symmetric]) 
paulson@13298
   551
paulson@13298
   552
lemma sats_transset_fm [simp]:
paulson@13298
   553
   "[|x < length(env); env \<in> list(A); Transset(A)|]
paulson@13298
   554
    ==> sats(A, transset_fm(x), env) <-> Transset(nth(x,env))"
paulson@13298
   555
apply (frule lt_nat_in_nat, erule length_type) 
paulson@13298
   556
apply (simp add: transset_fm_def Transset_def) 
paulson@13298
   557
apply (blast intro: nth_type) 
paulson@13298
   558
done
paulson@13298
   559
paulson@13298
   560
subsubsection{*Ordinals*}
paulson@13298
   561
paulson@13298
   562
constdefs ordinal_fm :: "i=>i"
paulson@13298
   563
   "ordinal_fm(x) == 
paulson@13298
   564
      And(transset_fm(x), Forall(Implies(Member(0,succ(x)), transset_fm(0))))"
paulson@13298
   565
paulson@13298
   566
lemma ordinal_type [TC]: "x \<in> nat ==> ordinal_fm(x) \<in> formula"
paulson@13298
   567
by (simp add: ordinal_fm_def) 
paulson@13298
   568
paulson@13298
   569
lemma arity_ordinal_fm [simp]:
paulson@13298
   570
     "x \<in> nat ==> arity(ordinal_fm(x)) = succ(x)"
paulson@13298
   571
by (simp add: ordinal_fm_def succ_Un_distrib [symmetric]) 
paulson@13298
   572
paulson@13298
   573
lemma sats_ordinal_fm [simp]:
paulson@13298
   574
   "[|x < length(env); env \<in> list(A); Transset(A)|]
paulson@13298
   575
    ==> sats(A, ordinal_fm(x), env) <-> Ord(nth(x,env))"
paulson@13298
   576
apply (frule lt_nat_in_nat, erule length_type) 
paulson@13298
   577
apply (simp add: ordinal_fm_def Ord_def Transset_def)
paulson@13298
   578
apply (blast intro: nth_type) 
paulson@13298
   579
done
paulson@13298
   580
paulson@13298
   581
paulson@13223
   582
subsection{* Constant Lset: Levels of the Constructible Universe *}
paulson@13223
   583
paulson@13223
   584
constdefs Lset :: "i=>i"
paulson@13223
   585
    "Lset(i) == transrec(i, %x f. \<Union>y\<in>x. DPow(f`y))"
paulson@13223
   586
paulson@13223
   587
text{*NOT SUITABLE FOR REWRITING -- RECURSIVE!*}
paulson@13223
   588
lemma Lset: "Lset(i) = (UN j:i. DPow(Lset(j)))"
paulson@13223
   589
by (subst Lset_def [THEN def_transrec], simp)
paulson@13223
   590
paulson@13223
   591
lemma LsetI: "[|y\<in>x; A \<in> DPow(Lset(y))|] ==> A \<in> Lset(x)";
paulson@13223
   592
by (subst Lset, blast)
paulson@13223
   593
paulson@13223
   594
lemma LsetD: "A \<in> Lset(x) ==> \<exists>y\<in>x. A \<in> DPow(Lset(y))";
paulson@13223
   595
apply (insert Lset [of x]) 
paulson@13223
   596
apply (blast intro: elim: equalityE) 
paulson@13223
   597
done
paulson@13223
   598
paulson@13223
   599
subsubsection{* Transitivity *}
paulson@13223
   600
paulson@13223
   601
lemma elem_subset_in_DPow: "[|X \<in> A; X \<subseteq> A|] ==> X \<in> DPow(A)"
paulson@13223
   602
apply (simp add: Transset_def DPow_def)
paulson@13223
   603
apply (rule_tac x="[X]" in bexI) 
paulson@13223
   604
 apply (rule_tac x="Member(0,1)" in bexI) 
paulson@13223
   605
  apply (auto simp add: Un_least_lt_iff) 
paulson@13223
   606
done
paulson@13223
   607
paulson@13223
   608
lemma Transset_subset_DPow: "Transset(A) ==> A <= DPow(A)"
paulson@13223
   609
apply clarify  
paulson@13223
   610
apply (simp add: Transset_def)
paulson@13223
   611
apply (blast intro: elem_subset_in_DPow) 
paulson@13223
   612
done
paulson@13223
   613
paulson@13223
   614
lemma Transset_DPow: "Transset(A) ==> Transset(DPow(A))"
paulson@13223
   615
apply (simp add: Transset_def) 
paulson@13223
   616
apply (blast intro: elem_subset_in_DPow dest: DPowD) 
paulson@13223
   617
done
paulson@13223
   618
paulson@13223
   619
text{*Kunen's VI, 1.6 (a)*}
paulson@13223
   620
lemma Transset_Lset: "Transset(Lset(i))"
paulson@13223
   621
apply (rule_tac a=i in eps_induct)
paulson@13223
   622
apply (subst Lset)
paulson@13223
   623
apply (blast intro!: Transset_Union_family Transset_Un Transset_DPow)
paulson@13223
   624
done
paulson@13223
   625
paulson@13291
   626
lemma mem_Lset_imp_subset_Lset: "a \<in> Lset(i) ==> a \<subseteq> Lset(i)"
paulson@13291
   627
apply (insert Transset_Lset) 
paulson@13291
   628
apply (simp add: Transset_def) 
paulson@13291
   629
done
paulson@13291
   630
paulson@13223
   631
subsubsection{* Monotonicity *}
paulson@13223
   632
paulson@13223
   633
text{*Kunen's VI, 1.6 (b)*}
paulson@13223
   634
lemma Lset_mono [rule_format]:
paulson@13223
   635
     "ALL j. i<=j --> Lset(i) <= Lset(j)"
paulson@13223
   636
apply (rule_tac a=i in eps_induct)
paulson@13223
   637
apply (rule impI [THEN allI])
paulson@13223
   638
apply (subst Lset)
paulson@13223
   639
apply (subst Lset, blast) 
paulson@13223
   640
done
paulson@13223
   641
paulson@13223
   642
text{*This version lets us remove the premise @{term "Ord(i)"} sometimes.*}
paulson@13223
   643
lemma Lset_mono_mem [rule_format]:
paulson@13223
   644
     "ALL j. i:j --> Lset(i) <= Lset(j)"
paulson@13223
   645
apply (rule_tac a=i in eps_induct)
paulson@13223
   646
apply (rule impI [THEN allI])
paulson@13223
   647
apply (subst Lset, auto) 
paulson@13223
   648
apply (rule rev_bexI, assumption)
paulson@13223
   649
apply (blast intro: elem_subset_in_DPow dest: LsetD DPowD) 
paulson@13223
   650
done
paulson@13223
   651
paulson@13291
   652
text{*Useful with Reflection to bump up the ordinal*}
paulson@13291
   653
lemma subset_Lset_ltD: "[|A \<subseteq> Lset(i); i < j|] ==> A \<subseteq> Lset(j)"
paulson@13291
   654
by (blast dest: ltD [THEN Lset_mono_mem]) 
paulson@13291
   655
paulson@13223
   656
subsubsection{* 0, successor and limit equations fof Lset *}
paulson@13223
   657
paulson@13223
   658
lemma Lset_0 [simp]: "Lset(0) = 0"
paulson@13223
   659
by (subst Lset, blast)
paulson@13223
   660
paulson@13223
   661
lemma Lset_succ_subset1: "DPow(Lset(i)) <= Lset(succ(i))"
paulson@13223
   662
by (subst Lset, rule succI1 [THEN RepFunI, THEN Union_upper])
paulson@13223
   663
paulson@13223
   664
lemma Lset_succ_subset2: "Lset(succ(i)) <= DPow(Lset(i))"
paulson@13223
   665
apply (subst Lset, rule UN_least)
paulson@13223
   666
apply (erule succE) 
paulson@13223
   667
 apply blast 
paulson@13223
   668
apply clarify
paulson@13223
   669
apply (rule elem_subset_in_DPow)
paulson@13223
   670
 apply (subst Lset)
paulson@13223
   671
 apply blast 
paulson@13223
   672
apply (blast intro: dest: DPowD Lset_mono_mem) 
paulson@13223
   673
done
paulson@13223
   674
paulson@13223
   675
lemma Lset_succ: "Lset(succ(i)) = DPow(Lset(i))"
paulson@13223
   676
by (intro equalityI Lset_succ_subset1 Lset_succ_subset2) 
paulson@13223
   677
paulson@13223
   678
lemma Lset_Union [simp]: "Lset(\<Union>(X)) = (\<Union>y\<in>X. Lset(y))"
paulson@13223
   679
apply (subst Lset)
paulson@13223
   680
apply (rule equalityI)
paulson@13223
   681
 txt{*first inclusion*}
paulson@13223
   682
 apply (rule UN_least)
paulson@13223
   683
 apply (erule UnionE)
paulson@13223
   684
 apply (rule subset_trans)
paulson@13223
   685
  apply (erule_tac [2] UN_upper, subst Lset, erule UN_upper)
paulson@13223
   686
txt{*opposite inclusion*}
paulson@13223
   687
apply (rule UN_least)
paulson@13223
   688
apply (subst Lset, blast)
paulson@13223
   689
done
paulson@13223
   690
paulson@13223
   691
subsubsection{* Lset applied to Limit ordinals *}
paulson@13223
   692
paulson@13223
   693
lemma Limit_Lset_eq:
paulson@13223
   694
    "Limit(i) ==> Lset(i) = (\<Union>y\<in>i. Lset(y))"
paulson@13223
   695
by (simp add: Lset_Union [symmetric] Limit_Union_eq)
paulson@13223
   696
paulson@13223
   697
lemma lt_LsetI: "[| a: Lset(j);  j<i |] ==> a : Lset(i)"
paulson@13223
   698
by (blast dest: Lset_mono [OF le_imp_subset [OF leI]])
paulson@13223
   699
paulson@13223
   700
lemma Limit_LsetE:
paulson@13223
   701
    "[| a: Lset(i);  ~R ==> Limit(i);
paulson@13223
   702
        !!x. [| x<i;  a: Lset(x) |] ==> R
paulson@13223
   703
     |] ==> R"
paulson@13223
   704
apply (rule classical)
paulson@13223
   705
apply (rule Limit_Lset_eq [THEN equalityD1, THEN subsetD, THEN UN_E])
paulson@13223
   706
  prefer 2 apply assumption
paulson@13223
   707
 apply blast 
paulson@13223
   708
apply (blast intro: ltI  Limit_is_Ord)
paulson@13223
   709
done
paulson@13223
   710
paulson@13223
   711
subsubsection{* Basic closure properties *}
paulson@13223
   712
paulson@13223
   713
lemma zero_in_Lset: "y:x ==> 0 : Lset(x)"
paulson@13223
   714
by (subst Lset, blast intro: empty_in_DPow)
paulson@13223
   715
paulson@13223
   716
lemma notin_Lset: "x \<notin> Lset(x)"
paulson@13223
   717
apply (rule_tac a=x in eps_induct)
paulson@13223
   718
apply (subst Lset)
paulson@13223
   719
apply (blast dest: DPowD)  
paulson@13223
   720
done
paulson@13223
   721
paulson@13223
   722
paulson@13298
   723
subsection{*Constructible Ordinals: Kunen's VI, 1.9 (b)*}
paulson@13223
   724
paulson@13223
   725
text{*The subset consisting of the ordinals is definable.*}
paulson@13223
   726
lemma Ords_in_DPow: "Transset(A) ==> {x \<in> A. Ord(x)} \<in> DPow(A)"
paulson@13223
   727
apply (simp add: DPow_def Collect_subset) 
paulson@13223
   728
apply (rule_tac x="Nil" in bexI) 
paulson@13223
   729
 apply (rule_tac x="ordinal_fm(0)" in bexI) 
paulson@13223
   730
apply (simp_all add: sats_ordinal_fm)
paulson@13223
   731
done 
paulson@13223
   732
paulson@13223
   733
lemma Ords_of_Lset_eq: "Ord(i) ==> {x\<in>Lset(i). Ord(x)} = i"
paulson@13223
   734
apply (erule trans_induct3)
paulson@13223
   735
  apply (simp_all add: Lset_succ Limit_Lset_eq Limit_Union_eq)
paulson@13223
   736
txt{*The successor case remains.*} 
paulson@13223
   737
apply (rule equalityI)
paulson@13223
   738
txt{*First inclusion*}
paulson@13223
   739
 apply clarify  
paulson@13223
   740
 apply (erule Ord_linear_lt, assumption) 
paulson@13223
   741
   apply (blast dest: DPow_imp_subset ltD notE [OF notin_Lset]) 
paulson@13223
   742
  apply blast 
paulson@13223
   743
 apply (blast dest: ltD)
paulson@13223
   744
txt{*Opposite inclusion, @{term "succ(x) \<subseteq> DPow(Lset(x)) \<inter> ON"}*}
paulson@13223
   745
apply auto
paulson@13223
   746
txt{*Key case: *}
paulson@13223
   747
  apply (erule subst, rule Ords_in_DPow [OF Transset_Lset]) 
paulson@13223
   748
 apply (blast intro: elem_subset_in_DPow dest: OrdmemD elim: equalityE) 
paulson@13223
   749
apply (blast intro: Ord_in_Ord) 
paulson@13223
   750
done
paulson@13223
   751
paulson@13223
   752
paulson@13223
   753
lemma Ord_subset_Lset: "Ord(i) ==> i \<subseteq> Lset(i)"
paulson@13223
   754
by (subst Ords_of_Lset_eq [symmetric], assumption, fast)
paulson@13223
   755
paulson@13223
   756
lemma Ord_in_Lset: "Ord(i) ==> i \<in> Lset(succ(i))"
paulson@13223
   757
apply (simp add: Lset_succ)
paulson@13223
   758
apply (subst Ords_of_Lset_eq [symmetric], assumption, 
paulson@13223
   759
       rule Ords_in_DPow [OF Transset_Lset]) 
paulson@13223
   760
done
paulson@13223
   761
paulson@13223
   762
subsubsection{* Unions *}
paulson@13223
   763
paulson@13223
   764
lemma Union_in_Lset:
paulson@13223
   765
     "X \<in> Lset(j) ==> Union(X) \<in> Lset(succ(j))"
paulson@13223
   766
apply (insert Transset_Lset)
paulson@13223
   767
apply (rule LsetI [OF succI1])
paulson@13223
   768
apply (simp add: Transset_def DPow_def) 
paulson@13223
   769
apply (intro conjI, blast)
paulson@13223
   770
txt{*Now to create the formula @{term "\<exists>y. y \<in> X \<and> x \<in> y"} *}
paulson@13223
   771
apply (rule_tac x="Cons(X,Nil)" in bexI) 
paulson@13223
   772
 apply (rule_tac x="Exists(And(Member(0,2), Member(1,0)))" in bexI) 
paulson@13223
   773
  apply typecheck
paulson@13223
   774
apply (simp add: succ_Un_distrib [symmetric], blast) 
paulson@13223
   775
done
paulson@13223
   776
paulson@13223
   777
lemma Union_in_LLimit:
paulson@13223
   778
     "[| X: Lset(i);  Limit(i) |] ==> Union(X) : Lset(i)"
paulson@13223
   779
apply (rule Limit_LsetE, assumption+)
paulson@13223
   780
apply (blast intro: Limit_has_succ lt_LsetI Union_in_Lset)
paulson@13223
   781
done
paulson@13223
   782
paulson@13223
   783
subsubsection{* Finite sets and ordered pairs *}
paulson@13223
   784
paulson@13223
   785
lemma singleton_in_Lset: "a: Lset(i) ==> {a} : Lset(succ(i))"
paulson@13223
   786
by (simp add: Lset_succ singleton_in_DPow) 
paulson@13223
   787
paulson@13223
   788
lemma doubleton_in_Lset:
paulson@13223
   789
     "[| a: Lset(i);  b: Lset(i) |] ==> {a,b} : Lset(succ(i))"
paulson@13223
   790
by (simp add: Lset_succ empty_in_DPow cons_in_DPow) 
paulson@13223
   791
paulson@13223
   792
lemma Pair_in_Lset:
paulson@13223
   793
    "[| a: Lset(i);  b: Lset(i); Ord(i) |] ==> <a,b> : Lset(succ(succ(i)))"
paulson@13223
   794
apply (unfold Pair_def)
paulson@13223
   795
apply (blast intro: doubleton_in_Lset) 
paulson@13223
   796
done
paulson@13223
   797
paulson@13223
   798
lemmas zero_in_LLimit = Limit_has_0 [THEN ltD, THEN zero_in_Lset, standard]
paulson@13223
   799
paulson@13223
   800
lemma singleton_in_LLimit:
paulson@13223
   801
    "[| a: Lset(i);  Limit(i) |] ==> {a} : Lset(i)"
paulson@13223
   802
apply (erule Limit_LsetE, assumption)
paulson@13223
   803
apply (erule singleton_in_Lset [THEN lt_LsetI])
paulson@13223
   804
apply (blast intro: Limit_has_succ) 
paulson@13223
   805
done
paulson@13223
   806
paulson@13223
   807
lemmas Lset_UnI1 = Un_upper1 [THEN Lset_mono [THEN subsetD], standard]
paulson@13223
   808
lemmas Lset_UnI2 = Un_upper2 [THEN Lset_mono [THEN subsetD], standard]
paulson@13223
   809
paulson@13223
   810
text{*Hard work is finding a single j:i such that {a,b}<=Lset(j)*}
paulson@13223
   811
lemma doubleton_in_LLimit:
paulson@13223
   812
    "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> {a,b} : Lset(i)"
paulson@13223
   813
apply (erule Limit_LsetE, assumption)
paulson@13223
   814
apply (erule Limit_LsetE, assumption)
paulson@13269
   815
apply (blast intro: lt_LsetI [OF doubleton_in_Lset]
paulson@13269
   816
                    Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
paulson@13223
   817
done
paulson@13223
   818
paulson@13223
   819
lemma Pair_in_LLimit:
paulson@13223
   820
    "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> <a,b> : Lset(i)"
paulson@13223
   821
txt{*Infer that a, b occur at ordinals x,xa < i.*}
paulson@13223
   822
apply (erule Limit_LsetE, assumption)
paulson@13223
   823
apply (erule Limit_LsetE, assumption)
paulson@13223
   824
txt{*Infer that succ(succ(x Un xa)) < i *}
paulson@13223
   825
apply (blast intro: lt_Ord lt_LsetI [OF Pair_in_Lset]
paulson@13223
   826
                    Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
paulson@13223
   827
done
paulson@13223
   828
paulson@13223
   829
lemma product_LLimit: "Limit(i) ==> Lset(i) * Lset(i) <= Lset(i)"
paulson@13223
   830
by (blast intro: Pair_in_LLimit)
paulson@13223
   831
paulson@13223
   832
lemmas Sigma_subset_LLimit = subset_trans [OF Sigma_mono product_LLimit]
paulson@13223
   833
paulson@13223
   834
lemma nat_subset_LLimit: "Limit(i) ==> nat \<subseteq> Lset(i)"
paulson@13223
   835
by (blast dest: Ord_subset_Lset nat_le_Limit le_imp_subset Limit_is_Ord)
paulson@13223
   836
paulson@13223
   837
lemma nat_into_LLimit: "[| n: nat;  Limit(i) |] ==> n : Lset(i)"
paulson@13223
   838
by (blast intro: nat_subset_LLimit [THEN subsetD])
paulson@13223
   839
paulson@13223
   840
paulson@13223
   841
subsubsection{* Closure under disjoint union *}
paulson@13223
   842
paulson@13223
   843
lemmas zero_in_LLimit = Limit_has_0 [THEN ltD, THEN zero_in_Lset, standard]
paulson@13223
   844
paulson@13223
   845
lemma one_in_LLimit: "Limit(i) ==> 1 : Lset(i)"
paulson@13223
   846
by (blast intro: nat_into_LLimit)
paulson@13223
   847
paulson@13223
   848
lemma Inl_in_LLimit:
paulson@13223
   849
    "[| a: Lset(i); Limit(i) |] ==> Inl(a) : Lset(i)"
paulson@13223
   850
apply (unfold Inl_def)
paulson@13223
   851
apply (blast intro: zero_in_LLimit Pair_in_LLimit)
paulson@13223
   852
done
paulson@13223
   853
paulson@13223
   854
lemma Inr_in_LLimit:
paulson@13223
   855
    "[| b: Lset(i); Limit(i) |] ==> Inr(b) : Lset(i)"
paulson@13223
   856
apply (unfold Inr_def)
paulson@13223
   857
apply (blast intro: one_in_LLimit Pair_in_LLimit)
paulson@13223
   858
done
paulson@13223
   859
paulson@13223
   860
lemma sum_LLimit: "Limit(i) ==> Lset(i) + Lset(i) <= Lset(i)"
paulson@13223
   861
by (blast intro!: Inl_in_LLimit Inr_in_LLimit)
paulson@13223
   862
paulson@13223
   863
lemmas sum_subset_LLimit = subset_trans [OF sum_mono sum_LLimit]
paulson@13223
   864
paulson@13223
   865
paulson@13223
   866
text{*The constructible universe and its rank function*}
paulson@13223
   867
constdefs
paulson@13223
   868
  L :: "i=>o" --{*Kunen's definition VI, 1.5, page 167*}
paulson@13223
   869
    "L(x) == \<exists>i. Ord(i) & x \<in> Lset(i)"
paulson@13223
   870
  
paulson@13223
   871
  lrank :: "i=>i" --{*Kunen's definition VI, 1.7*}
paulson@13223
   872
    "lrank(x) == \<mu>i. x \<in> Lset(succ(i))"
paulson@13223
   873
paulson@13223
   874
lemma L_I: "[|x \<in> Lset(i); Ord(i)|] ==> L(x)"
paulson@13223
   875
by (simp add: L_def, blast)
paulson@13223
   876
paulson@13223
   877
lemma L_D: "L(x) ==> \<exists>i. Ord(i) & x \<in> Lset(i)"
paulson@13223
   878
by (simp add: L_def)
paulson@13223
   879
paulson@13223
   880
lemma Ord_lrank [simp]: "Ord(lrank(a))"
paulson@13223
   881
by (simp add: lrank_def)
paulson@13223
   882
paulson@13223
   883
lemma Lset_lrank_lt [rule_format]: "Ord(i) ==> x \<in> Lset(i) --> lrank(x) < i"
paulson@13223
   884
apply (erule trans_induct3)
paulson@13223
   885
  apply simp   
paulson@13223
   886
 apply (simp only: lrank_def) 
paulson@13223
   887
 apply (blast intro: Least_le) 
paulson@13223
   888
apply (simp_all add: Limit_Lset_eq) 
paulson@13223
   889
apply (blast intro: ltI Limit_is_Ord lt_trans) 
paulson@13223
   890
done
paulson@13223
   891
paulson@13223
   892
text{*Kunen's VI, 1.8, and the proof is much less trivial than the text
paulson@13223
   893
would suggest.  For a start it need the previous lemma, proved by induction.*}
paulson@13223
   894
lemma Lset_iff_lrank_lt: "Ord(i) ==> x \<in> Lset(i) <-> L(x) & lrank(x) < i"
paulson@13223
   895
apply (simp add: L_def, auto) 
paulson@13223
   896
 apply (blast intro: Lset_lrank_lt) 
paulson@13223
   897
 apply (unfold lrank_def) 
paulson@13223
   898
apply (drule succI1 [THEN Lset_mono_mem, THEN subsetD]) 
paulson@13223
   899
apply (drule_tac P="\<lambda>i. x \<in> Lset(succ(i))" in LeastI, assumption) 
paulson@13223
   900
apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) 
paulson@13223
   901
done
paulson@13223
   902
paulson@13223
   903
lemma Lset_succ_lrank_iff [simp]: "x \<in> Lset(succ(lrank(x))) <-> L(x)"
paulson@13223
   904
by (simp add: Lset_iff_lrank_lt)
paulson@13223
   905
paulson@13223
   906
text{*Kunen's VI, 1.9 (a)*}
paulson@13223
   907
lemma lrank_of_Ord: "Ord(i) ==> lrank(i) = i"
paulson@13223
   908
apply (unfold lrank_def) 
paulson@13223
   909
apply (rule Least_equality) 
paulson@13223
   910
  apply (erule Ord_in_Lset) 
paulson@13223
   911
 apply assumption
paulson@13223
   912
apply (insert notin_Lset [of i]) 
paulson@13223
   913
apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) 
paulson@13223
   914
done
paulson@13223
   915
paulson@13245
   916
paulson@13245
   917
lemma Ord_in_L: "Ord(i) ==> L(i)"
paulson@13245
   918
by (blast intro: Ord_in_Lset L_I)
paulson@13245
   919
paulson@13223
   920
text{*This is lrank(lrank(a)) = lrank(a) *}
paulson@13223
   921
declare Ord_lrank [THEN lrank_of_Ord, simp]
paulson@13223
   922
paulson@13223
   923
text{*Kunen's VI, 1.10 *}
paulson@13223
   924
lemma Lset_in_Lset_succ: "Lset(i) \<in> Lset(succ(i))";
paulson@13223
   925
apply (simp add: Lset_succ DPow_def) 
paulson@13223
   926
apply (rule_tac x="Nil" in bexI) 
paulson@13223
   927
 apply (rule_tac x="Equal(0,0)" in bexI) 
paulson@13223
   928
apply auto 
paulson@13223
   929
done
paulson@13223
   930
paulson@13223
   931
lemma lrank_Lset: "Ord(i) ==> lrank(Lset(i)) = i"
paulson@13223
   932
apply (unfold lrank_def) 
paulson@13223
   933
apply (rule Least_equality) 
paulson@13223
   934
  apply (rule Lset_in_Lset_succ) 
paulson@13223
   935
 apply assumption
paulson@13223
   936
apply clarify 
paulson@13223
   937
apply (subgoal_tac "Lset(succ(ia)) <= Lset(i)")
paulson@13223
   938
 apply (blast dest: mem_irrefl) 
paulson@13223
   939
apply (blast intro!: le_imp_subset Lset_mono) 
paulson@13223
   940
done
paulson@13223
   941
paulson@13223
   942
text{*Kunen's VI, 1.11 *}
paulson@13223
   943
lemma Lset_subset_Vset: "Ord(i) ==> Lset(i) <= Vset(i)";
paulson@13223
   944
apply (erule trans_induct)
paulson@13223
   945
apply (subst Lset) 
paulson@13223
   946
apply (subst Vset) 
paulson@13223
   947
apply (rule UN_mono [OF subset_refl]) 
paulson@13223
   948
apply (rule subset_trans [OF DPow_subset_Pow]) 
paulson@13223
   949
apply (rule Pow_mono, blast) 
paulson@13223
   950
done
paulson@13223
   951
paulson@13223
   952
text{*Kunen's VI, 1.12 *}
paulson@13223
   953
lemma Lset_subset_Vset: "i \<in> nat ==> Lset(i) = Vset(i)";
paulson@13223
   954
apply (erule nat_induct)
paulson@13223
   955
 apply (simp add: Vfrom_0) 
paulson@13223
   956
apply (simp add: Lset_succ Vset_succ Finite_Vset Finite_DPow_eq_Pow) 
paulson@13223
   957
done
paulson@13223
   958
paulson@13291
   959
text{*Every set of constructible sets is included in some @{term Lset}*} 
paulson@13291
   960
lemma subset_Lset:
paulson@13291
   961
     "(\<forall>x\<in>A. L(x)) ==> \<exists>i. Ord(i) & A \<subseteq> Lset(i)"
paulson@13291
   962
by (rule_tac x = "\<Union>x\<in>A. succ(lrank(x))" in exI, force)
paulson@13291
   963
paulson@13291
   964
lemma subset_LsetE:
paulson@13291
   965
     "[|\<forall>x\<in>A. L(x);
paulson@13291
   966
        !!i. [|Ord(i); A \<subseteq> Lset(i)|] ==> P|]
paulson@13291
   967
      ==> P"
paulson@13291
   968
by (blast dest: subset_Lset) 
paulson@13291
   969
paulson@13223
   970
subsection{*For L to satisfy the ZF axioms*}
paulson@13223
   971
paulson@13245
   972
theorem Union_in_L: "L(X) ==> L(Union(X))"
paulson@13223
   973
apply (simp add: L_def, clarify) 
paulson@13223
   974
apply (drule Ord_imp_greater_Limit) 
paulson@13223
   975
apply (blast intro: lt_LsetI Union_in_LLimit Limit_is_Ord) 
paulson@13223
   976
done
paulson@13223
   977
paulson@13245
   978
theorem doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})"
paulson@13223
   979
apply (simp add: L_def, clarify) 
paulson@13223
   980
apply (drule Ord2_imp_greater_Limit, assumption) 
paulson@13223
   981
apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord) 
paulson@13223
   982
done
paulson@13223
   983
paulson@13223
   984
subsubsection{*For L to satisfy Powerset *}
paulson@13223
   985
paulson@13223
   986
lemma LPow_env_typing:
paulson@13223
   987
     "[| y : Lset(i); Ord(i); y \<subseteq> X |] ==> y \<in> (\<Union>y\<in>Pow(X). Lset(succ(lrank(y))))"
paulson@13223
   988
by (auto intro: L_I iff: Lset_succ_lrank_iff) 
paulson@13223
   989
paulson@13223
   990
lemma LPow_in_Lset:
paulson@13223
   991
     "[|X \<in> Lset(i); Ord(i)|] ==> \<exists>j. Ord(j) & {y \<in> Pow(X). L(y)} \<in> Lset(j)"
paulson@13223
   992
apply (rule_tac x="succ(\<Union>y \<in> Pow(X). succ(lrank(y)))" in exI)
paulson@13223
   993
apply simp 
paulson@13223
   994
apply (rule LsetI [OF succI1])
paulson@13223
   995
apply (simp add: DPow_def) 
paulson@13223
   996
apply (intro conjI, clarify) 
paulson@13223
   997
apply (rule_tac a="x" in UN_I, simp+)  
paulson@13223
   998
txt{*Now to create the formula @{term "y \<subseteq> X"} *}
paulson@13223
   999
apply (rule_tac x="Cons(X,Nil)" in bexI) 
paulson@13223
  1000
 apply (rule_tac x="subset_fm(0,1)" in bexI) 
paulson@13223
  1001
  apply typecheck
paulson@13223
  1002
apply (rule conjI) 
paulson@13223
  1003
apply (simp add: succ_Un_distrib [symmetric]) 
paulson@13223
  1004
apply (rule equality_iffI) 
paulson@13223
  1005
apply (simp add: Transset_UN [OF Transset_Lset] list.Cons [OF LPow_env_typing])
paulson@13223
  1006
apply (auto intro: L_I iff: Lset_succ_lrank_iff) 
paulson@13223
  1007
done
paulson@13223
  1008
paulson@13245
  1009
theorem LPow_in_L: "L(X) ==> L({y \<in> Pow(X). L(y)})"
paulson@13223
  1010
by (blast intro: L_I dest: L_D LPow_in_Lset)
paulson@13223
  1011
paulson@13223
  1012
end