src/ZF/Constructible/Formula.thy
author paulson
Fri Jul 19 18:06:31 2002 +0200 (2002-07-19)
changeset 13398 1cadd412da48
parent 13385 31df66ca0780
child 13505 52a16cb7fefb
permissions -rw-r--r--
Towards relativization and absoluteness of formula_rec
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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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            | Nand ("p: formula", "q: formula")
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            | Forall ("p: formula")
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declare formula.intros [TC]
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constdefs Neg :: "i=>i"
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    "Neg(p) == Nand(p,p)"
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constdefs And :: "[i,i]=>i"
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    "And(p,q) == Neg(Nand(p,q))"
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constdefs Or :: "[i,i]=>i"
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    "Or(p,q) == Nand(Neg(p),Neg(q))"
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constdefs Implies :: "[i,i]=>i"
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    "Implies(p,q) == Nand(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 Neg_type [TC]: "p \<in> formula ==> Neg(p) \<in> formula"
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by (simp add: Neg_def) 
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lemma And_type [TC]: "[| p \<in> formula; q \<in> formula |] ==> And(p,q) \<in> formula"
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by (simp add: And_def) 
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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,Nand(p,q)) =
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      (\<lambda>env \<in> list(A). not ((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_Nand_iff [simp]:
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  "env \<in> list(A) 
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   ==> (sats(A, Nand(p,q), env)) <-> ~ (sats(A,p,env) & sats(A,q,env))" 
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by (simp add: Bool.and_def Bool.not_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_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: Neg_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: And_def) 
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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 not_iff_sats:
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      "[| P <-> sats(A,p,env); env \<in> list(A)|]
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       ==> (~P) <-> sats(A, Neg(p), env)"
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by simp
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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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lemmas FOL_iff_sats = 
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        mem_iff_sats equal_iff_sats not_iff_sats conj_iff_sats
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        disj_iff_sats imp_iff_sats iff_iff_sats imp_iff_sats ball_iff_sats
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        bex_iff_sats
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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(Nand(p,q)) =
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      (\<lambda>lev \<in> nat. Nand (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(Nand(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_Neg [simp]: "arity(Neg(p)) = arity(p)"
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by (simp add: Neg_def) 
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lemma arity_And [simp]: "arity(And(p,q)) = arity(p) \<union> arity(q)"
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by (simp add: And_def) 
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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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paulson@13223
   307
lemma arity_sats1_iff:
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   308
  "[| arity(p) \<le> succ(length(env)); p \<in> formula; x \<in> A; env \<in> list(A); 
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   309
    extra \<in> list(A) |]
paulson@13223
   310
   ==> sats(A, p, Cons(x, env @ extra)) <-> sats(A, p, Cons(x, env))"
paulson@13223
   311
apply (insert arity_sats_iff [of p extra A "Cons(x,env)"])
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   312
apply simp 
paulson@13223
   313
done
paulson@13223
   314
paulson@13223
   315
(*the following two lemmas prevent huge case splits in arity_incr_bv_lemma*)
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   316
lemma incr_var_lemma:
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   317
     "[| x \<in> nat; y \<in> nat; lev \<le> x |]
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   318
      ==> succ(x) \<union> incr_var(y,lev) = succ(x \<union> y)"
paulson@13223
   319
apply (simp add: incr_var_def Ord_Un_if, auto)
paulson@13223
   320
  apply (blast intro: leI)
paulson@13223
   321
 apply (simp add: not_lt_iff_le)  
paulson@13223
   322
 apply (blast intro: le_anti_sym) 
paulson@13223
   323
apply (blast dest: lt_trans2) 
paulson@13223
   324
done
paulson@13223
   325
paulson@13223
   326
lemma incr_And_lemma:
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   327
     "y < x ==> y \<union> succ(x) = succ(x \<union> y)"
paulson@13223
   328
apply (simp add: Ord_Un_if lt_Ord lt_Ord2 succ_lt_iff) 
paulson@13223
   329
apply (blast dest: lt_asym) 
paulson@13223
   330
done
paulson@13223
   331
paulson@13223
   332
lemma arity_incr_bv_lemma [rule_format]:
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   333
  "p \<in> formula 
paulson@13223
   334
   ==> \<forall>n \<in> nat. arity (incr_bv(p) ` n) = 
paulson@13223
   335
                 (if n < arity(p) then succ(arity(p)) else arity(p))"
paulson@13223
   336
apply (induct_tac p) 
paulson@13223
   337
apply (simp_all add: imp_disj not_lt_iff_le Un_least_lt_iff lt_Un_iff le_Un_iff
paulson@13223
   338
                     succ_Un_distrib [symmetric] incr_var_lt incr_var_le
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   339
                     Un_commute incr_var_lemma arity_type nat_imp_quasinat
paulson@13269
   340
            split: split_nat_case) 
paulson@13269
   341
 txt{*the Forall case reduces to linear arithmetic*}
paulson@13269
   342
 prefer 2
paulson@13269
   343
 apply clarify 
paulson@13269
   344
 apply (blast dest: lt_trans1) 
paulson@13269
   345
txt{*left with the And case*}
paulson@13223
   346
apply safe
paulson@13223
   347
 apply (blast intro: incr_And_lemma lt_trans1) 
paulson@13223
   348
apply (subst incr_And_lemma)
paulson@13269
   349
 apply (blast intro: lt_trans1) 
paulson@13269
   350
apply (simp add: Un_commute)
paulson@13223
   351
done
paulson@13223
   352
paulson@13223
   353
lemma arity_incr_boundvars_eq:
paulson@13223
   354
  "p \<in> formula
paulson@13223
   355
   ==> arity(incr_boundvars(p)) =
paulson@13223
   356
        (if 0 < arity(p) then succ(arity(p)) else arity(p))"
paulson@13223
   357
apply (insert arity_incr_bv_lemma [of p 0])
paulson@13223
   358
apply (simp add: incr_boundvars_def) 
paulson@13223
   359
done
paulson@13223
   360
paulson@13223
   361
lemma arity_iterates_incr_boundvars_eq:
paulson@13223
   362
  "[| p \<in> formula; n \<in> nat |]
paulson@13223
   363
   ==> arity(incr_boundvars^n(p)) =
paulson@13223
   364
         (if 0 < arity(p) then n #+ arity(p) else arity(p))"
paulson@13223
   365
apply (induct_tac n) 
paulson@13223
   366
apply (simp_all add: arity_incr_boundvars_eq not_lt_iff_le) 
paulson@13223
   367
done
paulson@13223
   368
paulson@13223
   369
paulson@13298
   370
subsection{*Renaming all but the first bound variable*}
paulson@13223
   371
paulson@13223
   372
constdefs incr_bv1 :: "i => i"
paulson@13223
   373
    "incr_bv1(p) == incr_bv(p)`1"
paulson@13223
   374
paulson@13223
   375
paulson@13223
   376
lemma incr_bv1_type [TC]: "p \<in> formula ==> incr_bv1(p) \<in> formula"
paulson@13223
   377
by (simp add: incr_bv1_def) 
paulson@13223
   378
paulson@13223
   379
(*For renaming all but the bound variable at level 0*)
paulson@13223
   380
lemma sats_incr_bv1_iff [rule_format]:
paulson@13223
   381
  "[| p \<in> formula; env \<in> list(A); x \<in> A; y \<in> A |]
paulson@13223
   382
   ==> sats(A, incr_bv1(p), Cons(x, Cons(y, env))) <-> 
paulson@13223
   383
       sats(A, p, Cons(x,env))"
paulson@13223
   384
apply (insert sats_incr_bv_iff [of p env A y "Cons(x,Nil)"])
paulson@13223
   385
apply (simp add: incr_bv1_def) 
paulson@13223
   386
done
paulson@13223
   387
paulson@13223
   388
lemma formula_add_params1 [rule_format]:
paulson@13223
   389
  "[| p \<in> formula; n \<in> nat; x \<in> A |]
paulson@13223
   390
   ==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A). 
paulson@13223
   391
          length(bvs) = n --> 
paulson@13223
   392
          sats(A, iterates(incr_bv1, n, p), Cons(x, bvs@env)) <-> 
paulson@13223
   393
          sats(A, p, Cons(x,env))"
paulson@13223
   394
apply (induct_tac n, simp, clarify) 
paulson@13223
   395
apply (erule list.cases)
paulson@13223
   396
apply (simp_all add: sats_incr_bv1_iff) 
paulson@13223
   397
done
paulson@13223
   398
paulson@13223
   399
paulson@13223
   400
lemma arity_incr_bv1_eq:
paulson@13223
   401
  "p \<in> formula
paulson@13223
   402
   ==> arity(incr_bv1(p)) =
paulson@13223
   403
        (if 1 < arity(p) then succ(arity(p)) else arity(p))"
paulson@13223
   404
apply (insert arity_incr_bv_lemma [of p 1])
paulson@13223
   405
apply (simp add: incr_bv1_def) 
paulson@13223
   406
done
paulson@13223
   407
paulson@13223
   408
lemma arity_iterates_incr_bv1_eq:
paulson@13223
   409
  "[| p \<in> formula; n \<in> nat |]
paulson@13223
   410
   ==> arity(incr_bv1^n(p)) =
paulson@13223
   411
         (if 1 < arity(p) then n #+ arity(p) else arity(p))"
paulson@13223
   412
apply (induct_tac n) 
paulson@13298
   413
apply (simp_all add: arity_incr_bv1_eq)
paulson@13223
   414
apply (simp add: not_lt_iff_le)
paulson@13223
   415
apply (blast intro: le_trans add_le_self2 arity_type) 
paulson@13223
   416
done
paulson@13223
   417
paulson@13223
   418
paulson@13223
   419
(*Definable powerset operation: Kunen's definition 1.1, page 165.*)
paulson@13223
   420
constdefs DPow :: "i => i"
paulson@13223
   421
  "DPow(A) == {X \<in> Pow(A). 
paulson@13223
   422
               \<exists>env \<in> list(A). \<exists>p \<in> formula. 
paulson@13223
   423
                 arity(p) \<le> succ(length(env)) & 
paulson@13223
   424
                 X = {x\<in>A. sats(A, p, Cons(x,env))}}"
paulson@13223
   425
paulson@13223
   426
lemma DPowI:
paulson@13291
   427
  "[|env \<in> list(A);  p \<in> formula;  arity(p) \<le> succ(length(env))|]
paulson@13223
   428
   ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
paulson@13223
   429
by (simp add: DPow_def, blast) 
paulson@13223
   430
paulson@13291
   431
text{*With this rule we can specify @{term p} later.*}
paulson@13291
   432
lemma DPowI2 [rule_format]:
paulson@13291
   433
  "[|\<forall>x\<in>A. P(x) <-> sats(A, p, Cons(x,env));
paulson@13291
   434
     env \<in> list(A);  p \<in> formula;  arity(p) \<le> succ(length(env))|]
paulson@13291
   435
   ==> {x\<in>A. P(x)} \<in> DPow(A)"
paulson@13291
   436
by (simp add: DPow_def, blast) 
paulson@13291
   437
paulson@13223
   438
lemma DPowD:
paulson@13223
   439
  "X \<in> DPow(A) 
paulson@13223
   440
   ==> X <= A &
paulson@13223
   441
       (\<exists>env \<in> list(A). 
paulson@13223
   442
        \<exists>p \<in> formula. arity(p) \<le> succ(length(env)) & 
paulson@13223
   443
                      X = {x\<in>A. sats(A, p, Cons(x,env))})"
paulson@13223
   444
by (simp add: DPow_def) 
paulson@13223
   445
paulson@13223
   446
lemmas DPow_imp_subset = DPowD [THEN conjunct1]
paulson@13223
   447
paulson@13223
   448
(*Lemma 1.2*)
paulson@13223
   449
lemma "[| p \<in> formula; env \<in> list(A); arity(p) \<le> succ(length(env)) |] 
paulson@13223
   450
       ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
paulson@13223
   451
by (blast intro: DPowI)
paulson@13223
   452
paulson@13223
   453
lemma DPow_subset_Pow: "DPow(A) <= Pow(A)"
paulson@13223
   454
by (simp add: DPow_def, blast)
paulson@13223
   455
paulson@13223
   456
lemma empty_in_DPow: "0 \<in> DPow(A)"
paulson@13223
   457
apply (simp add: DPow_def)
paulson@13339
   458
apply (rule_tac x=Nil in bexI) 
paulson@13223
   459
 apply (rule_tac x="Neg(Equal(0,0))" in bexI) 
paulson@13223
   460
  apply (auto simp add: Un_least_lt_iff) 
paulson@13223
   461
done
paulson@13223
   462
paulson@13223
   463
lemma Compl_in_DPow: "X \<in> DPow(A) ==> (A-X) \<in> DPow(A)"
paulson@13223
   464
apply (simp add: DPow_def, clarify, auto) 
paulson@13223
   465
apply (rule bexI) 
paulson@13223
   466
 apply (rule_tac x="Neg(p)" in bexI) 
paulson@13223
   467
  apply auto 
paulson@13223
   468
done
paulson@13223
   469
paulson@13223
   470
lemma Int_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Int Y \<in> DPow(A)"
paulson@13223
   471
apply (simp add: DPow_def, auto) 
paulson@13223
   472
apply (rename_tac envp p envq q) 
paulson@13223
   473
apply (rule_tac x="envp@envq" in bexI) 
paulson@13223
   474
 apply (rule_tac x="And(p, iterates(incr_bv1,length(envp),q))" in bexI)
paulson@13223
   475
  apply typecheck
paulson@13223
   476
apply (rule conjI) 
paulson@13223
   477
(*finally check the arity!*)
paulson@13223
   478
 apply (simp add: arity_iterates_incr_bv1_eq length_app Un_least_lt_iff)
paulson@13223
   479
 apply (force intro: add_le_self le_trans) 
paulson@13223
   480
apply (simp add: arity_sats1_iff formula_add_params1, blast) 
paulson@13223
   481
done
paulson@13223
   482
paulson@13223
   483
lemma Un_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Un Y \<in> DPow(A)"
paulson@13223
   484
apply (subgoal_tac "X Un Y = A - ((A-X) Int (A-Y))") 
paulson@13223
   485
apply (simp add: Int_in_DPow Compl_in_DPow) 
paulson@13223
   486
apply (simp add: DPow_def, blast) 
paulson@13223
   487
done
paulson@13223
   488
paulson@13223
   489
lemma singleton_in_DPow: "x \<in> A ==> {x} \<in> DPow(A)"
paulson@13223
   490
apply (simp add: DPow_def)
paulson@13223
   491
apply (rule_tac x="Cons(x,Nil)" in bexI) 
paulson@13223
   492
 apply (rule_tac x="Equal(0,1)" in bexI) 
paulson@13223
   493
  apply typecheck
paulson@13223
   494
apply (force simp add: succ_Un_distrib [symmetric])  
paulson@13223
   495
done
paulson@13223
   496
paulson@13223
   497
lemma cons_in_DPow: "[| a \<in> A; X \<in> DPow(A) |] ==> cons(a,X) \<in> DPow(A)"
paulson@13223
   498
apply (rule cons_eq [THEN subst]) 
paulson@13223
   499
apply (blast intro: singleton_in_DPow Un_in_DPow) 
paulson@13223
   500
done
paulson@13223
   501
paulson@13223
   502
(*Part of Lemma 1.3*)
paulson@13223
   503
lemma Fin_into_DPow: "X \<in> Fin(A) ==> X \<in> DPow(A)"
paulson@13223
   504
apply (erule Fin.induct) 
paulson@13223
   505
 apply (rule empty_in_DPow) 
paulson@13223
   506
apply (blast intro: cons_in_DPow) 
paulson@13223
   507
done
paulson@13223
   508
paulson@13223
   509
(*DPow is not monotonic.  For example, let A be some non-constructible set
paulson@13223
   510
  of natural numbers, and let B be nat.  Then A<=B and obviously A : DPow(A)
paulson@13223
   511
  but A ~: DPow(B).*)
paulson@13223
   512
lemma DPow_mono: "A : DPow(B) ==> DPow(A) <= DPow(B)"
paulson@13223
   513
apply (simp add: DPow_def, auto) 
paulson@13223
   514
(*must use the formula defining A in B to relativize the new formula...*)
paulson@13223
   515
oops
paulson@13223
   516
paulson@13223
   517
lemma DPow_0: "DPow(0) = {0}" 
paulson@13223
   518
by (blast intro: empty_in_DPow dest: DPow_imp_subset)
paulson@13223
   519
paulson@13223
   520
lemma Finite_Pow_subset_Pow: "Finite(A) ==> Pow(A) <= DPow(A)" 
paulson@13223
   521
by (blast intro: Fin_into_DPow Finite_into_Fin Fin_subset)
paulson@13223
   522
paulson@13223
   523
lemma Finite_DPow_eq_Pow: "Finite(A) ==> DPow(A) = Pow(A)"
paulson@13223
   524
apply (rule equalityI) 
paulson@13223
   525
apply (rule DPow_subset_Pow) 
paulson@13223
   526
apply (erule Finite_Pow_subset_Pow) 
paulson@13223
   527
done
paulson@13223
   528
paulson@13223
   529
(*This may be true but the proof looks difficult, requiring relativization 
paulson@13223
   530
lemma DPow_insert: "DPow (cons(a,A)) = DPow(A) Un {cons(a,X) . X: DPow(A)}"
paulson@13223
   531
apply (rule equalityI, safe)
paulson@13223
   532
oops
paulson@13223
   533
*)
paulson@13223
   534
paulson@13298
   535
paulson@13298
   536
subsection{*Internalized formulas for basic concepts*}
paulson@13298
   537
paulson@13298
   538
subsubsection{*The subset relation*}
paulson@13298
   539
paulson@13298
   540
constdefs subset_fm :: "[i,i]=>i"
paulson@13298
   541
    "subset_fm(x,y) == Forall(Implies(Member(0,succ(x)), Member(0,succ(y))))"
paulson@13298
   542
paulson@13298
   543
lemma subset_type [TC]: "[| x \<in> nat; y \<in> nat |] ==> subset_fm(x,y) \<in> formula"
paulson@13298
   544
by (simp add: subset_fm_def) 
paulson@13298
   545
paulson@13298
   546
lemma arity_subset_fm [simp]:
paulson@13298
   547
     "[| x \<in> nat; y \<in> nat |] ==> arity(subset_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13298
   548
by (simp add: subset_fm_def succ_Un_distrib [symmetric]) 
paulson@13298
   549
paulson@13298
   550
lemma sats_subset_fm [simp]:
paulson@13298
   551
   "[|x < length(env); y \<in> nat; env \<in> list(A); Transset(A)|]
paulson@13298
   552
    ==> sats(A, subset_fm(x,y), env) <-> nth(x,env) \<subseteq> nth(y,env)"
paulson@13298
   553
apply (frule lt_length_in_nat, assumption)  
paulson@13298
   554
apply (simp add: subset_fm_def Transset_def) 
paulson@13298
   555
apply (blast intro: nth_type) 
paulson@13298
   556
done
paulson@13298
   557
paulson@13298
   558
subsubsection{*Transitive sets*}
paulson@13298
   559
paulson@13298
   560
constdefs transset_fm :: "i=>i"
paulson@13298
   561
   "transset_fm(x) == Forall(Implies(Member(0,succ(x)), subset_fm(0,succ(x))))"
paulson@13298
   562
paulson@13298
   563
lemma transset_type [TC]: "x \<in> nat ==> transset_fm(x) \<in> formula"
paulson@13298
   564
by (simp add: transset_fm_def) 
paulson@13298
   565
paulson@13298
   566
lemma arity_transset_fm [simp]:
paulson@13298
   567
     "x \<in> nat ==> arity(transset_fm(x)) = succ(x)"
paulson@13298
   568
by (simp add: transset_fm_def succ_Un_distrib [symmetric]) 
paulson@13298
   569
paulson@13298
   570
lemma sats_transset_fm [simp]:
paulson@13298
   571
   "[|x < length(env); env \<in> list(A); Transset(A)|]
paulson@13298
   572
    ==> sats(A, transset_fm(x), env) <-> Transset(nth(x,env))"
paulson@13298
   573
apply (frule lt_nat_in_nat, erule length_type) 
paulson@13298
   574
apply (simp add: transset_fm_def Transset_def) 
paulson@13298
   575
apply (blast intro: nth_type) 
paulson@13298
   576
done
paulson@13298
   577
paulson@13298
   578
subsubsection{*Ordinals*}
paulson@13298
   579
paulson@13298
   580
constdefs ordinal_fm :: "i=>i"
paulson@13298
   581
   "ordinal_fm(x) == 
paulson@13298
   582
      And(transset_fm(x), Forall(Implies(Member(0,succ(x)), transset_fm(0))))"
paulson@13298
   583
paulson@13298
   584
lemma ordinal_type [TC]: "x \<in> nat ==> ordinal_fm(x) \<in> formula"
paulson@13298
   585
by (simp add: ordinal_fm_def) 
paulson@13298
   586
paulson@13298
   587
lemma arity_ordinal_fm [simp]:
paulson@13298
   588
     "x \<in> nat ==> arity(ordinal_fm(x)) = succ(x)"
paulson@13298
   589
by (simp add: ordinal_fm_def succ_Un_distrib [symmetric]) 
paulson@13298
   590
paulson@13306
   591
lemma sats_ordinal_fm:
paulson@13298
   592
   "[|x < length(env); env \<in> list(A); Transset(A)|]
paulson@13298
   593
    ==> sats(A, ordinal_fm(x), env) <-> Ord(nth(x,env))"
paulson@13298
   594
apply (frule lt_nat_in_nat, erule length_type) 
paulson@13298
   595
apply (simp add: ordinal_fm_def Ord_def Transset_def)
paulson@13298
   596
apply (blast intro: nth_type) 
paulson@13298
   597
done
paulson@13298
   598
paulson@13298
   599
paulson@13223
   600
subsection{* Constant Lset: Levels of the Constructible Universe *}
paulson@13223
   601
paulson@13223
   602
constdefs Lset :: "i=>i"
paulson@13223
   603
    "Lset(i) == transrec(i, %x f. \<Union>y\<in>x. DPow(f`y))"
paulson@13223
   604
paulson@13223
   605
text{*NOT SUITABLE FOR REWRITING -- RECURSIVE!*}
paulson@13223
   606
lemma Lset: "Lset(i) = (UN j:i. DPow(Lset(j)))"
paulson@13223
   607
by (subst Lset_def [THEN def_transrec], simp)
paulson@13223
   608
paulson@13223
   609
lemma LsetI: "[|y\<in>x; A \<in> DPow(Lset(y))|] ==> A \<in> Lset(x)";
paulson@13223
   610
by (subst Lset, blast)
paulson@13223
   611
paulson@13223
   612
lemma LsetD: "A \<in> Lset(x) ==> \<exists>y\<in>x. A \<in> DPow(Lset(y))";
paulson@13223
   613
apply (insert Lset [of x]) 
paulson@13223
   614
apply (blast intro: elim: equalityE) 
paulson@13223
   615
done
paulson@13223
   616
paulson@13223
   617
subsubsection{* Transitivity *}
paulson@13223
   618
paulson@13223
   619
lemma elem_subset_in_DPow: "[|X \<in> A; X \<subseteq> A|] ==> X \<in> DPow(A)"
paulson@13223
   620
apply (simp add: Transset_def DPow_def)
paulson@13223
   621
apply (rule_tac x="[X]" in bexI) 
paulson@13223
   622
 apply (rule_tac x="Member(0,1)" in bexI) 
paulson@13223
   623
  apply (auto simp add: Un_least_lt_iff) 
paulson@13223
   624
done
paulson@13223
   625
paulson@13223
   626
lemma Transset_subset_DPow: "Transset(A) ==> A <= DPow(A)"
paulson@13223
   627
apply clarify  
paulson@13223
   628
apply (simp add: Transset_def)
paulson@13223
   629
apply (blast intro: elem_subset_in_DPow) 
paulson@13223
   630
done
paulson@13223
   631
paulson@13223
   632
lemma Transset_DPow: "Transset(A) ==> Transset(DPow(A))"
paulson@13223
   633
apply (simp add: Transset_def) 
paulson@13223
   634
apply (blast intro: elem_subset_in_DPow dest: DPowD) 
paulson@13223
   635
done
paulson@13223
   636
paulson@13223
   637
text{*Kunen's VI, 1.6 (a)*}
paulson@13223
   638
lemma Transset_Lset: "Transset(Lset(i))"
paulson@13223
   639
apply (rule_tac a=i in eps_induct)
paulson@13223
   640
apply (subst Lset)
paulson@13223
   641
apply (blast intro!: Transset_Union_family Transset_Un Transset_DPow)
paulson@13223
   642
done
paulson@13223
   643
paulson@13291
   644
lemma mem_Lset_imp_subset_Lset: "a \<in> Lset(i) ==> a \<subseteq> Lset(i)"
paulson@13291
   645
apply (insert Transset_Lset) 
paulson@13291
   646
apply (simp add: Transset_def) 
paulson@13291
   647
done
paulson@13291
   648
paulson@13223
   649
subsubsection{* Monotonicity *}
paulson@13223
   650
paulson@13223
   651
text{*Kunen's VI, 1.6 (b)*}
paulson@13223
   652
lemma Lset_mono [rule_format]:
paulson@13223
   653
     "ALL j. i<=j --> Lset(i) <= Lset(j)"
paulson@13223
   654
apply (rule_tac a=i in eps_induct)
paulson@13223
   655
apply (rule impI [THEN allI])
paulson@13223
   656
apply (subst Lset)
paulson@13223
   657
apply (subst Lset, blast) 
paulson@13223
   658
done
paulson@13223
   659
paulson@13223
   660
text{*This version lets us remove the premise @{term "Ord(i)"} sometimes.*}
paulson@13223
   661
lemma Lset_mono_mem [rule_format]:
paulson@13223
   662
     "ALL j. i:j --> Lset(i) <= Lset(j)"
paulson@13223
   663
apply (rule_tac a=i in eps_induct)
paulson@13223
   664
apply (rule impI [THEN allI])
paulson@13223
   665
apply (subst Lset, auto) 
paulson@13223
   666
apply (rule rev_bexI, assumption)
paulson@13223
   667
apply (blast intro: elem_subset_in_DPow dest: LsetD DPowD) 
paulson@13223
   668
done
paulson@13223
   669
paulson@13291
   670
text{*Useful with Reflection to bump up the ordinal*}
paulson@13291
   671
lemma subset_Lset_ltD: "[|A \<subseteq> Lset(i); i < j|] ==> A \<subseteq> Lset(j)"
paulson@13291
   672
by (blast dest: ltD [THEN Lset_mono_mem]) 
paulson@13291
   673
paulson@13223
   674
subsubsection{* 0, successor and limit equations fof Lset *}
paulson@13223
   675
paulson@13223
   676
lemma Lset_0 [simp]: "Lset(0) = 0"
paulson@13223
   677
by (subst Lset, blast)
paulson@13223
   678
paulson@13223
   679
lemma Lset_succ_subset1: "DPow(Lset(i)) <= Lset(succ(i))"
paulson@13223
   680
by (subst Lset, rule succI1 [THEN RepFunI, THEN Union_upper])
paulson@13223
   681
paulson@13223
   682
lemma Lset_succ_subset2: "Lset(succ(i)) <= DPow(Lset(i))"
paulson@13223
   683
apply (subst Lset, rule UN_least)
paulson@13223
   684
apply (erule succE) 
paulson@13223
   685
 apply blast 
paulson@13223
   686
apply clarify
paulson@13223
   687
apply (rule elem_subset_in_DPow)
paulson@13223
   688
 apply (subst Lset)
paulson@13223
   689
 apply blast 
paulson@13223
   690
apply (blast intro: dest: DPowD Lset_mono_mem) 
paulson@13223
   691
done
paulson@13223
   692
paulson@13223
   693
lemma Lset_succ: "Lset(succ(i)) = DPow(Lset(i))"
paulson@13223
   694
by (intro equalityI Lset_succ_subset1 Lset_succ_subset2) 
paulson@13223
   695
paulson@13223
   696
lemma Lset_Union [simp]: "Lset(\<Union>(X)) = (\<Union>y\<in>X. Lset(y))"
paulson@13223
   697
apply (subst Lset)
paulson@13223
   698
apply (rule equalityI)
paulson@13223
   699
 txt{*first inclusion*}
paulson@13223
   700
 apply (rule UN_least)
paulson@13223
   701
 apply (erule UnionE)
paulson@13223
   702
 apply (rule subset_trans)
paulson@13223
   703
  apply (erule_tac [2] UN_upper, subst Lset, erule UN_upper)
paulson@13223
   704
txt{*opposite inclusion*}
paulson@13223
   705
apply (rule UN_least)
paulson@13223
   706
apply (subst Lset, blast)
paulson@13223
   707
done
paulson@13223
   708
paulson@13223
   709
subsubsection{* Lset applied to Limit ordinals *}
paulson@13223
   710
paulson@13223
   711
lemma Limit_Lset_eq:
paulson@13223
   712
    "Limit(i) ==> Lset(i) = (\<Union>y\<in>i. Lset(y))"
paulson@13223
   713
by (simp add: Lset_Union [symmetric] Limit_Union_eq)
paulson@13223
   714
paulson@13223
   715
lemma lt_LsetI: "[| a: Lset(j);  j<i |] ==> a : Lset(i)"
paulson@13223
   716
by (blast dest: Lset_mono [OF le_imp_subset [OF leI]])
paulson@13223
   717
paulson@13223
   718
lemma Limit_LsetE:
paulson@13223
   719
    "[| a: Lset(i);  ~R ==> Limit(i);
paulson@13223
   720
        !!x. [| x<i;  a: Lset(x) |] ==> R
paulson@13223
   721
     |] ==> R"
paulson@13223
   722
apply (rule classical)
paulson@13223
   723
apply (rule Limit_Lset_eq [THEN equalityD1, THEN subsetD, THEN UN_E])
paulson@13223
   724
  prefer 2 apply assumption
paulson@13223
   725
 apply blast 
paulson@13223
   726
apply (blast intro: ltI  Limit_is_Ord)
paulson@13223
   727
done
paulson@13223
   728
paulson@13223
   729
subsubsection{* Basic closure properties *}
paulson@13223
   730
paulson@13223
   731
lemma zero_in_Lset: "y:x ==> 0 : Lset(x)"
paulson@13223
   732
by (subst Lset, blast intro: empty_in_DPow)
paulson@13223
   733
paulson@13223
   734
lemma notin_Lset: "x \<notin> Lset(x)"
paulson@13223
   735
apply (rule_tac a=x in eps_induct)
paulson@13223
   736
apply (subst Lset)
paulson@13223
   737
apply (blast dest: DPowD)  
paulson@13223
   738
done
paulson@13223
   739
paulson@13223
   740
paulson@13298
   741
subsection{*Constructible Ordinals: Kunen's VI, 1.9 (b)*}
paulson@13223
   742
paulson@13223
   743
text{*The subset consisting of the ordinals is definable.*}
paulson@13223
   744
lemma Ords_in_DPow: "Transset(A) ==> {x \<in> A. Ord(x)} \<in> DPow(A)"
paulson@13223
   745
apply (simp add: DPow_def Collect_subset) 
paulson@13339
   746
apply (rule_tac x=Nil in bexI) 
paulson@13223
   747
 apply (rule_tac x="ordinal_fm(0)" in bexI) 
paulson@13223
   748
apply (simp_all add: sats_ordinal_fm)
paulson@13223
   749
done 
paulson@13223
   750
paulson@13223
   751
lemma Ords_of_Lset_eq: "Ord(i) ==> {x\<in>Lset(i). Ord(x)} = i"
paulson@13223
   752
apply (erule trans_induct3)
paulson@13223
   753
  apply (simp_all add: Lset_succ Limit_Lset_eq Limit_Union_eq)
paulson@13223
   754
txt{*The successor case remains.*} 
paulson@13223
   755
apply (rule equalityI)
paulson@13223
   756
txt{*First inclusion*}
paulson@13223
   757
 apply clarify  
paulson@13223
   758
 apply (erule Ord_linear_lt, assumption) 
paulson@13223
   759
   apply (blast dest: DPow_imp_subset ltD notE [OF notin_Lset]) 
paulson@13223
   760
  apply blast 
paulson@13223
   761
 apply (blast dest: ltD)
paulson@13223
   762
txt{*Opposite inclusion, @{term "succ(x) \<subseteq> DPow(Lset(x)) \<inter> ON"}*}
paulson@13223
   763
apply auto
paulson@13223
   764
txt{*Key case: *}
paulson@13223
   765
  apply (erule subst, rule Ords_in_DPow [OF Transset_Lset]) 
paulson@13223
   766
 apply (blast intro: elem_subset_in_DPow dest: OrdmemD elim: equalityE) 
paulson@13223
   767
apply (blast intro: Ord_in_Ord) 
paulson@13223
   768
done
paulson@13223
   769
paulson@13223
   770
paulson@13223
   771
lemma Ord_subset_Lset: "Ord(i) ==> i \<subseteq> Lset(i)"
paulson@13223
   772
by (subst Ords_of_Lset_eq [symmetric], assumption, fast)
paulson@13223
   773
paulson@13223
   774
lemma Ord_in_Lset: "Ord(i) ==> i \<in> Lset(succ(i))"
paulson@13223
   775
apply (simp add: Lset_succ)
paulson@13223
   776
apply (subst Ords_of_Lset_eq [symmetric], assumption, 
paulson@13223
   777
       rule Ords_in_DPow [OF Transset_Lset]) 
paulson@13223
   778
done
paulson@13223
   779
paulson@13223
   780
subsubsection{* Unions *}
paulson@13223
   781
paulson@13223
   782
lemma Union_in_Lset:
paulson@13223
   783
     "X \<in> Lset(j) ==> Union(X) \<in> Lset(succ(j))"
paulson@13223
   784
apply (insert Transset_Lset)
paulson@13223
   785
apply (rule LsetI [OF succI1])
paulson@13223
   786
apply (simp add: Transset_def DPow_def) 
paulson@13223
   787
apply (intro conjI, blast)
paulson@13223
   788
txt{*Now to create the formula @{term "\<exists>y. y \<in> X \<and> x \<in> y"} *}
paulson@13223
   789
apply (rule_tac x="Cons(X,Nil)" in bexI) 
paulson@13223
   790
 apply (rule_tac x="Exists(And(Member(0,2), Member(1,0)))" in bexI) 
paulson@13223
   791
  apply typecheck
paulson@13223
   792
apply (simp add: succ_Un_distrib [symmetric], blast) 
paulson@13223
   793
done
paulson@13223
   794
paulson@13223
   795
lemma Union_in_LLimit:
paulson@13223
   796
     "[| X: Lset(i);  Limit(i) |] ==> Union(X) : Lset(i)"
paulson@13223
   797
apply (rule Limit_LsetE, assumption+)
paulson@13223
   798
apply (blast intro: Limit_has_succ lt_LsetI Union_in_Lset)
paulson@13223
   799
done
paulson@13223
   800
paulson@13223
   801
subsubsection{* Finite sets and ordered pairs *}
paulson@13223
   802
paulson@13223
   803
lemma singleton_in_Lset: "a: Lset(i) ==> {a} : Lset(succ(i))"
paulson@13223
   804
by (simp add: Lset_succ singleton_in_DPow) 
paulson@13223
   805
paulson@13223
   806
lemma doubleton_in_Lset:
paulson@13223
   807
     "[| a: Lset(i);  b: Lset(i) |] ==> {a,b} : Lset(succ(i))"
paulson@13223
   808
by (simp add: Lset_succ empty_in_DPow cons_in_DPow) 
paulson@13223
   809
paulson@13223
   810
lemma Pair_in_Lset:
paulson@13223
   811
    "[| a: Lset(i);  b: Lset(i); Ord(i) |] ==> <a,b> : Lset(succ(succ(i)))"
paulson@13223
   812
apply (unfold Pair_def)
paulson@13223
   813
apply (blast intro: doubleton_in_Lset) 
paulson@13223
   814
done
paulson@13223
   815
paulson@13223
   816
lemmas zero_in_LLimit = Limit_has_0 [THEN ltD, THEN zero_in_Lset, standard]
paulson@13223
   817
paulson@13223
   818
lemma singleton_in_LLimit:
paulson@13223
   819
    "[| a: Lset(i);  Limit(i) |] ==> {a} : Lset(i)"
paulson@13223
   820
apply (erule Limit_LsetE, assumption)
paulson@13223
   821
apply (erule singleton_in_Lset [THEN lt_LsetI])
paulson@13223
   822
apply (blast intro: Limit_has_succ) 
paulson@13223
   823
done
paulson@13223
   824
paulson@13223
   825
lemmas Lset_UnI1 = Un_upper1 [THEN Lset_mono [THEN subsetD], standard]
paulson@13223
   826
lemmas Lset_UnI2 = Un_upper2 [THEN Lset_mono [THEN subsetD], standard]
paulson@13223
   827
paulson@13223
   828
text{*Hard work is finding a single j:i such that {a,b}<=Lset(j)*}
paulson@13223
   829
lemma doubleton_in_LLimit:
paulson@13223
   830
    "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> {a,b} : Lset(i)"
paulson@13223
   831
apply (erule Limit_LsetE, assumption)
paulson@13223
   832
apply (erule Limit_LsetE, assumption)
paulson@13269
   833
apply (blast intro: lt_LsetI [OF doubleton_in_Lset]
paulson@13269
   834
                    Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
paulson@13223
   835
done
paulson@13223
   836
paulson@13223
   837
lemma Pair_in_LLimit:
paulson@13223
   838
    "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> <a,b> : Lset(i)"
paulson@13223
   839
txt{*Infer that a, b occur at ordinals x,xa < i.*}
paulson@13223
   840
apply (erule Limit_LsetE, assumption)
paulson@13223
   841
apply (erule Limit_LsetE, assumption)
paulson@13223
   842
txt{*Infer that succ(succ(x Un xa)) < i *}
paulson@13223
   843
apply (blast intro: lt_Ord lt_LsetI [OF Pair_in_Lset]
paulson@13223
   844
                    Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
paulson@13223
   845
done
paulson@13223
   846
paulson@13223
   847
lemma product_LLimit: "Limit(i) ==> Lset(i) * Lset(i) <= Lset(i)"
paulson@13223
   848
by (blast intro: Pair_in_LLimit)
paulson@13223
   849
paulson@13223
   850
lemmas Sigma_subset_LLimit = subset_trans [OF Sigma_mono product_LLimit]
paulson@13223
   851
paulson@13223
   852
lemma nat_subset_LLimit: "Limit(i) ==> nat \<subseteq> Lset(i)"
paulson@13223
   853
by (blast dest: Ord_subset_Lset nat_le_Limit le_imp_subset Limit_is_Ord)
paulson@13223
   854
paulson@13223
   855
lemma nat_into_LLimit: "[| n: nat;  Limit(i) |] ==> n : Lset(i)"
paulson@13223
   856
by (blast intro: nat_subset_LLimit [THEN subsetD])
paulson@13223
   857
paulson@13223
   858
paulson@13223
   859
subsubsection{* Closure under disjoint union *}
paulson@13223
   860
paulson@13223
   861
lemmas zero_in_LLimit = Limit_has_0 [THEN ltD, THEN zero_in_Lset, standard]
paulson@13223
   862
paulson@13223
   863
lemma one_in_LLimit: "Limit(i) ==> 1 : Lset(i)"
paulson@13223
   864
by (blast intro: nat_into_LLimit)
paulson@13223
   865
paulson@13223
   866
lemma Inl_in_LLimit:
paulson@13223
   867
    "[| a: Lset(i); Limit(i) |] ==> Inl(a) : Lset(i)"
paulson@13223
   868
apply (unfold Inl_def)
paulson@13223
   869
apply (blast intro: zero_in_LLimit Pair_in_LLimit)
paulson@13223
   870
done
paulson@13223
   871
paulson@13223
   872
lemma Inr_in_LLimit:
paulson@13223
   873
    "[| b: Lset(i); Limit(i) |] ==> Inr(b) : Lset(i)"
paulson@13223
   874
apply (unfold Inr_def)
paulson@13223
   875
apply (blast intro: one_in_LLimit Pair_in_LLimit)
paulson@13223
   876
done
paulson@13223
   877
paulson@13223
   878
lemma sum_LLimit: "Limit(i) ==> Lset(i) + Lset(i) <= Lset(i)"
paulson@13223
   879
by (blast intro!: Inl_in_LLimit Inr_in_LLimit)
paulson@13223
   880
paulson@13223
   881
lemmas sum_subset_LLimit = subset_trans [OF sum_mono sum_LLimit]
paulson@13223
   882
paulson@13223
   883
paulson@13223
   884
text{*The constructible universe and its rank function*}
paulson@13223
   885
constdefs
paulson@13223
   886
  L :: "i=>o" --{*Kunen's definition VI, 1.5, page 167*}
paulson@13223
   887
    "L(x) == \<exists>i. Ord(i) & x \<in> Lset(i)"
paulson@13223
   888
  
paulson@13223
   889
  lrank :: "i=>i" --{*Kunen's definition VI, 1.7*}
paulson@13223
   890
    "lrank(x) == \<mu>i. x \<in> Lset(succ(i))"
paulson@13223
   891
paulson@13223
   892
lemma L_I: "[|x \<in> Lset(i); Ord(i)|] ==> L(x)"
paulson@13223
   893
by (simp add: L_def, blast)
paulson@13223
   894
paulson@13223
   895
lemma L_D: "L(x) ==> \<exists>i. Ord(i) & x \<in> Lset(i)"
paulson@13223
   896
by (simp add: L_def)
paulson@13223
   897
paulson@13223
   898
lemma Ord_lrank [simp]: "Ord(lrank(a))"
paulson@13223
   899
by (simp add: lrank_def)
paulson@13223
   900
paulson@13223
   901
lemma Lset_lrank_lt [rule_format]: "Ord(i) ==> x \<in> Lset(i) --> lrank(x) < i"
paulson@13223
   902
apply (erule trans_induct3)
paulson@13223
   903
  apply simp   
paulson@13223
   904
 apply (simp only: lrank_def) 
paulson@13223
   905
 apply (blast intro: Least_le) 
paulson@13223
   906
apply (simp_all add: Limit_Lset_eq) 
paulson@13223
   907
apply (blast intro: ltI Limit_is_Ord lt_trans) 
paulson@13223
   908
done
paulson@13223
   909
paulson@13223
   910
text{*Kunen's VI, 1.8, and the proof is much less trivial than the text
paulson@13223
   911
would suggest.  For a start it need the previous lemma, proved by induction.*}
paulson@13223
   912
lemma Lset_iff_lrank_lt: "Ord(i) ==> x \<in> Lset(i) <-> L(x) & lrank(x) < i"
paulson@13223
   913
apply (simp add: L_def, auto) 
paulson@13223
   914
 apply (blast intro: Lset_lrank_lt) 
paulson@13223
   915
 apply (unfold lrank_def) 
paulson@13223
   916
apply (drule succI1 [THEN Lset_mono_mem, THEN subsetD]) 
paulson@13223
   917
apply (drule_tac P="\<lambda>i. x \<in> Lset(succ(i))" in LeastI, assumption) 
paulson@13223
   918
apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) 
paulson@13223
   919
done
paulson@13223
   920
paulson@13223
   921
lemma Lset_succ_lrank_iff [simp]: "x \<in> Lset(succ(lrank(x))) <-> L(x)"
paulson@13223
   922
by (simp add: Lset_iff_lrank_lt)
paulson@13223
   923
paulson@13223
   924
text{*Kunen's VI, 1.9 (a)*}
paulson@13223
   925
lemma lrank_of_Ord: "Ord(i) ==> lrank(i) = i"
paulson@13223
   926
apply (unfold lrank_def) 
paulson@13223
   927
apply (rule Least_equality) 
paulson@13223
   928
  apply (erule Ord_in_Lset) 
paulson@13223
   929
 apply assumption
paulson@13223
   930
apply (insert notin_Lset [of i]) 
paulson@13223
   931
apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) 
paulson@13223
   932
done
paulson@13223
   933
paulson@13245
   934
paulson@13245
   935
lemma Ord_in_L: "Ord(i) ==> L(i)"
paulson@13245
   936
by (blast intro: Ord_in_Lset L_I)
paulson@13245
   937
paulson@13223
   938
text{*This is lrank(lrank(a)) = lrank(a) *}
paulson@13223
   939
declare Ord_lrank [THEN lrank_of_Ord, simp]
paulson@13223
   940
paulson@13223
   941
text{*Kunen's VI, 1.10 *}
paulson@13223
   942
lemma Lset_in_Lset_succ: "Lset(i) \<in> Lset(succ(i))";
paulson@13223
   943
apply (simp add: Lset_succ DPow_def) 
paulson@13339
   944
apply (rule_tac x=Nil in bexI) 
paulson@13223
   945
 apply (rule_tac x="Equal(0,0)" in bexI) 
paulson@13223
   946
apply auto 
paulson@13223
   947
done
paulson@13223
   948
paulson@13223
   949
lemma lrank_Lset: "Ord(i) ==> lrank(Lset(i)) = i"
paulson@13223
   950
apply (unfold lrank_def) 
paulson@13223
   951
apply (rule Least_equality) 
paulson@13223
   952
  apply (rule Lset_in_Lset_succ) 
paulson@13223
   953
 apply assumption
paulson@13223
   954
apply clarify 
paulson@13223
   955
apply (subgoal_tac "Lset(succ(ia)) <= Lset(i)")
paulson@13223
   956
 apply (blast dest: mem_irrefl) 
paulson@13223
   957
apply (blast intro!: le_imp_subset Lset_mono) 
paulson@13223
   958
done
paulson@13223
   959
paulson@13223
   960
text{*Kunen's VI, 1.11 *}
paulson@13223
   961
lemma Lset_subset_Vset: "Ord(i) ==> Lset(i) <= Vset(i)";
paulson@13223
   962
apply (erule trans_induct)
paulson@13223
   963
apply (subst Lset) 
paulson@13223
   964
apply (subst Vset) 
paulson@13223
   965
apply (rule UN_mono [OF subset_refl]) 
paulson@13223
   966
apply (rule subset_trans [OF DPow_subset_Pow]) 
paulson@13223
   967
apply (rule Pow_mono, blast) 
paulson@13223
   968
done
paulson@13223
   969
paulson@13223
   970
text{*Kunen's VI, 1.12 *}
paulson@13223
   971
lemma Lset_subset_Vset: "i \<in> nat ==> Lset(i) = Vset(i)";
paulson@13223
   972
apply (erule nat_induct)
paulson@13223
   973
 apply (simp add: Vfrom_0) 
paulson@13223
   974
apply (simp add: Lset_succ Vset_succ Finite_Vset Finite_DPow_eq_Pow) 
paulson@13223
   975
done
paulson@13223
   976
paulson@13291
   977
text{*Every set of constructible sets is included in some @{term Lset}*} 
paulson@13291
   978
lemma subset_Lset:
paulson@13291
   979
     "(\<forall>x\<in>A. L(x)) ==> \<exists>i. Ord(i) & A \<subseteq> Lset(i)"
paulson@13291
   980
by (rule_tac x = "\<Union>x\<in>A. succ(lrank(x))" in exI, force)
paulson@13291
   981
paulson@13291
   982
lemma subset_LsetE:
paulson@13291
   983
     "[|\<forall>x\<in>A. L(x);
paulson@13291
   984
        !!i. [|Ord(i); A \<subseteq> Lset(i)|] ==> P|]
paulson@13291
   985
      ==> P"
paulson@13291
   986
by (blast dest: subset_Lset) 
paulson@13291
   987
paulson@13223
   988
subsection{*For L to satisfy the ZF axioms*}
paulson@13223
   989
paulson@13245
   990
theorem Union_in_L: "L(X) ==> L(Union(X))"
paulson@13223
   991
apply (simp add: L_def, clarify) 
paulson@13223
   992
apply (drule Ord_imp_greater_Limit) 
paulson@13223
   993
apply (blast intro: lt_LsetI Union_in_LLimit Limit_is_Ord) 
paulson@13223
   994
done
paulson@13223
   995
paulson@13245
   996
theorem doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})"
paulson@13223
   997
apply (simp add: L_def, clarify) 
paulson@13223
   998
apply (drule Ord2_imp_greater_Limit, assumption) 
paulson@13223
   999
apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord) 
paulson@13223
  1000
done
paulson@13223
  1001
paulson@13223
  1002
subsubsection{*For L to satisfy Powerset *}
paulson@13223
  1003
paulson@13223
  1004
lemma LPow_env_typing:
paulson@13223
  1005
     "[| y : Lset(i); Ord(i); y \<subseteq> X |] ==> y \<in> (\<Union>y\<in>Pow(X). Lset(succ(lrank(y))))"
paulson@13223
  1006
by (auto intro: L_I iff: Lset_succ_lrank_iff) 
paulson@13223
  1007
paulson@13223
  1008
lemma LPow_in_Lset:
paulson@13223
  1009
     "[|X \<in> Lset(i); Ord(i)|] ==> \<exists>j. Ord(j) & {y \<in> Pow(X). L(y)} \<in> Lset(j)"
paulson@13223
  1010
apply (rule_tac x="succ(\<Union>y \<in> Pow(X). succ(lrank(y)))" in exI)
paulson@13223
  1011
apply simp 
paulson@13223
  1012
apply (rule LsetI [OF succI1])
paulson@13223
  1013
apply (simp add: DPow_def) 
paulson@13223
  1014
apply (intro conjI, clarify) 
paulson@13339
  1015
apply (rule_tac a=x in UN_I, simp+)  
paulson@13223
  1016
txt{*Now to create the formula @{term "y \<subseteq> X"} *}
paulson@13223
  1017
apply (rule_tac x="Cons(X,Nil)" in bexI) 
paulson@13223
  1018
 apply (rule_tac x="subset_fm(0,1)" in bexI) 
paulson@13223
  1019
  apply typecheck
paulson@13223
  1020
apply (rule conjI) 
paulson@13223
  1021
apply (simp add: succ_Un_distrib [symmetric]) 
paulson@13223
  1022
apply (rule equality_iffI) 
paulson@13223
  1023
apply (simp add: Transset_UN [OF Transset_Lset] list.Cons [OF LPow_env_typing])
paulson@13223
  1024
apply (auto intro: L_I iff: Lset_succ_lrank_iff) 
paulson@13223
  1025
done
paulson@13223
  1026
paulson@13245
  1027
theorem LPow_in_L: "L(X) ==> L({y \<in> Pow(X). L(y)})"
paulson@13223
  1028
by (blast intro: L_I dest: L_D LPow_in_Lset)
paulson@13223
  1029
paulson@13385
  1030
paulson@13385
  1031
subsection{*Eliminating @{term arity} from the Definition of @{term Lset}*}
paulson@13385
  1032
paulson@13385
  1033
paulson@13385
  1034
lemma nth_zero_eq_0: "n \<in> nat ==> nth(n,[0]) = 0"
paulson@13385
  1035
by (induct_tac n, auto)
paulson@13385
  1036
paulson@13385
  1037
lemma sats_app_0_iff [rule_format]:
paulson@13385
  1038
  "[| p \<in> formula; 0 \<in> A |]
paulson@13385
  1039
   ==> \<forall>env \<in> list(A). sats(A,p, env@[0]) <-> sats(A,p,env)"
paulson@13385
  1040
apply (induct_tac p)
paulson@13385
  1041
apply (simp_all del: app_Cons add: app_Cons [symmetric]
paulson@13385
  1042
		add: nth_zero_eq_0 nth_append not_lt_iff_le nth_eq_0)
paulson@13385
  1043
done
paulson@13385
  1044
paulson@13385
  1045
lemma sats_app_zeroes_iff:
paulson@13385
  1046
  "[| p \<in> formula; 0 \<in> A; env \<in> list(A); n \<in> nat |]
paulson@13385
  1047
   ==> sats(A,p,env @ repeat(0,n)) <-> sats(A,p,env)"
paulson@13385
  1048
apply (induct_tac n, simp) 
paulson@13385
  1049
apply (simp del: repeat.simps
paulson@13385
  1050
            add: repeat_succ_app sats_app_0_iff app_assoc [symmetric]) 
paulson@13385
  1051
done
paulson@13385
  1052
paulson@13385
  1053
lemma exists_bigger_env:
paulson@13385
  1054
  "[| p \<in> formula; 0 \<in> A; env \<in> list(A) |]
paulson@13385
  1055
   ==> \<exists>env' \<in> list(A). arity(p) \<le> succ(length(env')) & 
paulson@13385
  1056
              (\<forall>a\<in>A. sats(A,p,Cons(a,env')) <-> sats(A,p,Cons(a,env)))"
paulson@13385
  1057
apply (rule_tac x="env @ repeat(0,arity(p))" in bexI) 
paulson@13385
  1058
apply (simp del: app_Cons add: app_Cons [symmetric]
paulson@13385
  1059
	    add: length_repeat sats_app_zeroes_iff, typecheck)
paulson@13385
  1060
done
paulson@13385
  1061
paulson@13385
  1062
paulson@13385
  1063
text{*A simpler version of @{term DPow}: no arity check!*}
paulson@13385
  1064
constdefs DPow' :: "i => i"
paulson@13385
  1065
  "DPow'(A) == {X \<in> Pow(A). 
paulson@13385
  1066
                \<exists>env \<in> list(A). \<exists>p \<in> formula. 
paulson@13385
  1067
                    X = {x\<in>A. sats(A, p, Cons(x,env))}}"
paulson@13385
  1068
paulson@13385
  1069
lemma DPow_subset_DPow': "DPow(A) <= DPow'(A)";
paulson@13385
  1070
by (simp add: DPow_def DPow'_def, blast)
paulson@13385
  1071
paulson@13385
  1072
lemma DPow'_0: "DPow'(0) = {0}"
paulson@13385
  1073
by (auto simp add: DPow'_def)
paulson@13385
  1074
paulson@13385
  1075
lemma DPow'_subset_DPow: "0 \<in> A ==> DPow'(A) \<subseteq> DPow(A)"
paulson@13385
  1076
apply (auto simp add: DPow'_def DPow_def) 
paulson@13385
  1077
apply (frule exists_bigger_env, assumption+, force)  
paulson@13385
  1078
done
paulson@13385
  1079
paulson@13385
  1080
lemma DPow_eq_DPow': "Transset(A) ==> DPow(A) = DPow'(A)"
paulson@13385
  1081
apply (drule Transset_0_disj) 
paulson@13385
  1082
apply (erule disjE) 
paulson@13385
  1083
 apply (simp add: DPow'_0 DPow_0) 
paulson@13385
  1084
apply (rule equalityI)
paulson@13385
  1085
 apply (rule DPow_subset_DPow') 
paulson@13385
  1086
apply (erule DPow'_subset_DPow) 
paulson@13385
  1087
done
paulson@13385
  1088
paulson@13385
  1089
text{*And thus we can relativize @{term Lset} without bothering with
paulson@13385
  1090
      @{term arity} and @{term length}*}
paulson@13385
  1091
lemma Lset_eq_transrec_DPow': "Lset(i) = transrec(i, %x f. \<Union>y\<in>x. DPow'(f`y))"
paulson@13385
  1092
apply (rule_tac a=i in eps_induct)
paulson@13385
  1093
apply (subst Lset)
paulson@13385
  1094
apply (subst transrec)
paulson@13385
  1095
apply (simp only: DPow_eq_DPow' [OF Transset_Lset], simp) 
paulson@13385
  1096
done
paulson@13385
  1097
paulson@13385
  1098
text{*With this rule we can specify @{term p} later and don't worry about
paulson@13385
  1099
      arities at all!*}
paulson@13385
  1100
lemma DPow_LsetI [rule_format]:
paulson@13385
  1101
  "[|\<forall>x\<in>Lset(i). P(x) <-> sats(Lset(i), p, Cons(x,env));
paulson@13385
  1102
     env \<in> list(Lset(i));  p \<in> formula|]
paulson@13385
  1103
   ==> {x\<in>Lset(i). P(x)} \<in> DPow(Lset(i))"
paulson@13385
  1104
by (simp add: DPow_eq_DPow' [OF Transset_Lset] DPow'_def, blast) 
paulson@13385
  1105
paulson@13223
  1106
end