src/ZF/Constructible/Formula.thy
author wenzelm
Tue Nov 07 19:40:13 2006 +0100 (2006-11-07)
changeset 21233 5a5c8ea5f66a
parent 16417 9bc16273c2d4
child 21404 eb85850d3eb7
permissions -rw-r--r--
tuned specifications;
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(*  Title:      ZF/Constructible/Formula.thy
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    ID: $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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*)
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header {* First-Order Formulas and the Definition of the Class L *}
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theory Formula imports Main begin
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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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definition Neg :: "i=>i"
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    "Neg(p) == Nand(p,p)"
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definition And :: "[i,i]=>i"
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    "And(p,q) == Neg(Nand(p,q))"
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definition Or :: "[i,i]=>i"
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    "Or(p,q) == Nand(Neg(p),Neg(q))"
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definition Implies :: "[i,i]=>i"
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    "Implies(p,q) == Nand(p,Neg(q))"
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definition Iff :: "[i,i]=>i"
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    "Iff(p,q) == And(Implies(p,q), Implies(q,p))"
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definition 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 set: formula) simp_all
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abbreviation
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  sats :: "[i,i,i] => o"
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  "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 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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subsection{*Arity of a Formula: Maximum Free de Bruijn Index*}
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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)) = Arith.pred(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)) = Arith.pred(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: Arith.pred_def nth_append Un_least_lt_iff 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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subsection{*Renaming Some de Bruijn Variables*}
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definition incr_var :: "[i,i]=>i"
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    "incr_var(x,nq) == if x<nq then x else succ(x)"
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lemma incr_var_lt: "x<nq ==> incr_var(x,nq) = x"
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by (simp add: incr_var_def)
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lemma incr_var_le: "nq\<le>x ==> incr_var(x,nq) = 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>nq \<in> nat. Member (incr_var(x,nq), incr_var(y,nq)))"
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  "incr_bv(Equal(x,y)) = 
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      (\<lambda>nq \<in> nat. Equal (incr_var(x,nq), incr_var(y,nq)))"
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  "incr_bv(Nand(p,q)) =
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      (\<lambda>nq \<in> nat. Nand (incr_bv(p)`nq, incr_bv(q)`nq))"
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  "incr_bv(Forall(p)) = 
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      (\<lambda>nq \<in> nat. Forall (incr_bv(p) ` succ(nq)))"
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lemma [TC]: "x \<in> nat ==> incr_var(x,nq) \<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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text{*Obviously, @{term DPow} is closed under complements and finite
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intersections and unions.  Needs an inductive lemma to allow two lists of
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parameters to 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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(*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; nq \<le> x |]
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      ==> succ(x) \<union> incr_var(y,nq) = 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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   310
apply (blast dest: lt_asym) 
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   311
done
paulson@13223
   312
paulson@13223
   313
lemma arity_incr_bv_lemma [rule_format]:
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   314
  "p \<in> formula 
paulson@13223
   315
   ==> \<forall>n \<in> nat. arity (incr_bv(p) ` n) = 
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   316
                 (if n < arity(p) then succ(arity(p)) else arity(p))"
paulson@13223
   317
apply (induct_tac p) 
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   318
apply (simp_all add: imp_disj not_lt_iff_le Un_least_lt_iff lt_Un_iff le_Un_iff
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   319
                     succ_Un_distrib [symmetric] incr_var_lt incr_var_le
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   320
                     Un_commute incr_var_lemma Arith.pred_def nat_imp_quasinat
paulson@13269
   321
            split: split_nat_case) 
paulson@13269
   322
 txt{*the Forall case reduces to linear arithmetic*}
paulson@13269
   323
 prefer 2
paulson@13269
   324
 apply clarify 
paulson@13269
   325
 apply (blast dest: lt_trans1) 
paulson@13269
   326
txt{*left with the And case*}
paulson@13223
   327
apply safe
paulson@13223
   328
 apply (blast intro: incr_And_lemma lt_trans1) 
paulson@13223
   329
apply (subst incr_And_lemma)
paulson@13269
   330
 apply (blast intro: lt_trans1) 
paulson@13269
   331
apply (simp add: Un_commute)
paulson@13223
   332
done
paulson@13223
   333
paulson@13223
   334
paulson@13647
   335
subsection{*Renaming all but the First de Bruijn Variable*}
paulson@13223
   336
wenzelm@21233
   337
definition incr_bv1 :: "i => i"
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   338
    "incr_bv1(p) == incr_bv(p)`1"
paulson@13223
   339
paulson@13223
   340
paulson@13223
   341
lemma incr_bv1_type [TC]: "p \<in> formula ==> incr_bv1(p) \<in> formula"
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   342
by (simp add: incr_bv1_def) 
paulson@13223
   343
paulson@13223
   344
(*For renaming all but the bound variable at level 0*)
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   345
lemma sats_incr_bv1_iff:
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   346
  "[| p \<in> formula; env \<in> list(A); x \<in> A; y \<in> A |]
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   347
   ==> sats(A, incr_bv1(p), Cons(x, Cons(y, env))) <-> 
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   348
       sats(A, p, Cons(x,env))"
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   349
apply (insert sats_incr_bv_iff [of p env A y "Cons(x,Nil)"])
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   350
apply (simp add: incr_bv1_def) 
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   351
done
paulson@13223
   352
paulson@13223
   353
lemma formula_add_params1 [rule_format]:
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   354
  "[| p \<in> formula; n \<in> nat; x \<in> A |]
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   355
   ==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A). 
paulson@13223
   356
          length(bvs) = n --> 
paulson@13223
   357
          sats(A, iterates(incr_bv1, n, p), Cons(x, bvs@env)) <-> 
paulson@13223
   358
          sats(A, p, Cons(x,env))"
paulson@13223
   359
apply (induct_tac n, simp, clarify) 
paulson@13223
   360
apply (erule list.cases)
paulson@13223
   361
apply (simp_all add: sats_incr_bv1_iff) 
paulson@13223
   362
done
paulson@13223
   363
paulson@13223
   364
paulson@13223
   365
lemma arity_incr_bv1_eq:
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   366
  "p \<in> formula
paulson@13223
   367
   ==> arity(incr_bv1(p)) =
paulson@13223
   368
        (if 1 < arity(p) then succ(arity(p)) else arity(p))"
paulson@13223
   369
apply (insert arity_incr_bv_lemma [of p 1])
paulson@13223
   370
apply (simp add: incr_bv1_def) 
paulson@13223
   371
done
paulson@13223
   372
paulson@13223
   373
lemma arity_iterates_incr_bv1_eq:
paulson@13223
   374
  "[| p \<in> formula; n \<in> nat |]
paulson@13223
   375
   ==> arity(incr_bv1^n(p)) =
paulson@13223
   376
         (if 1 < arity(p) then n #+ arity(p) else arity(p))"
paulson@13223
   377
apply (induct_tac n) 
paulson@13298
   378
apply (simp_all add: arity_incr_bv1_eq)
paulson@13223
   379
apply (simp add: not_lt_iff_le)
paulson@13223
   380
apply (blast intro: le_trans add_le_self2 arity_type) 
paulson@13223
   381
done
paulson@13223
   382
paulson@13223
   383
paulson@13647
   384
paulson@13647
   385
subsection{*Definable Powerset*}
paulson@13647
   386
paulson@13647
   387
text{*The definable powerset operation: Kunen's definition VI 1.1, page 165.*}
wenzelm@21233
   388
definition DPow :: "i => i"
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   389
  "DPow(A) == {X \<in> Pow(A). 
paulson@13223
   390
               \<exists>env \<in> list(A). \<exists>p \<in> formula. 
paulson@13223
   391
                 arity(p) \<le> succ(length(env)) & 
paulson@13223
   392
                 X = {x\<in>A. sats(A, p, Cons(x,env))}}"
paulson@13223
   393
paulson@13223
   394
lemma DPowI:
paulson@13291
   395
  "[|env \<in> list(A);  p \<in> formula;  arity(p) \<le> succ(length(env))|]
paulson@13223
   396
   ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
paulson@13223
   397
by (simp add: DPow_def, blast) 
paulson@13223
   398
paulson@13291
   399
text{*With this rule we can specify @{term p} later.*}
paulson@13291
   400
lemma DPowI2 [rule_format]:
paulson@13291
   401
  "[|\<forall>x\<in>A. P(x) <-> sats(A, p, Cons(x,env));
paulson@13291
   402
     env \<in> list(A);  p \<in> formula;  arity(p) \<le> succ(length(env))|]
paulson@13291
   403
   ==> {x\<in>A. P(x)} \<in> DPow(A)"
paulson@13291
   404
by (simp add: DPow_def, blast) 
paulson@13291
   405
paulson@13223
   406
lemma DPowD:
paulson@13223
   407
  "X \<in> DPow(A) 
paulson@13223
   408
   ==> X <= A &
paulson@13223
   409
       (\<exists>env \<in> list(A). 
paulson@13223
   410
        \<exists>p \<in> formula. arity(p) \<le> succ(length(env)) & 
paulson@13223
   411
                      X = {x\<in>A. sats(A, p, Cons(x,env))})"
paulson@13223
   412
by (simp add: DPow_def) 
paulson@13223
   413
paulson@13223
   414
lemmas DPow_imp_subset = DPowD [THEN conjunct1]
paulson@13223
   415
paulson@13647
   416
(*Kunen's Lemma VI 1.2*)
paulson@13223
   417
lemma "[| p \<in> formula; env \<in> list(A); arity(p) \<le> succ(length(env)) |] 
paulson@13223
   418
       ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
paulson@13223
   419
by (blast intro: DPowI)
paulson@13223
   420
paulson@13223
   421
lemma DPow_subset_Pow: "DPow(A) <= Pow(A)"
paulson@13223
   422
by (simp add: DPow_def, blast)
paulson@13223
   423
paulson@13223
   424
lemma empty_in_DPow: "0 \<in> DPow(A)"
paulson@13223
   425
apply (simp add: DPow_def)
paulson@13339
   426
apply (rule_tac x=Nil in bexI) 
paulson@13223
   427
 apply (rule_tac x="Neg(Equal(0,0))" in bexI) 
paulson@13223
   428
  apply (auto simp add: Un_least_lt_iff) 
paulson@13223
   429
done
paulson@13223
   430
paulson@13223
   431
lemma Compl_in_DPow: "X \<in> DPow(A) ==> (A-X) \<in> DPow(A)"
paulson@13223
   432
apply (simp add: DPow_def, clarify, auto) 
paulson@13223
   433
apply (rule bexI) 
paulson@13223
   434
 apply (rule_tac x="Neg(p)" in bexI) 
paulson@13223
   435
  apply auto 
paulson@13223
   436
done
paulson@13223
   437
paulson@13223
   438
lemma Int_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Int Y \<in> DPow(A)"
paulson@13223
   439
apply (simp add: DPow_def, auto) 
paulson@13223
   440
apply (rename_tac envp p envq q) 
paulson@13223
   441
apply (rule_tac x="envp@envq" in bexI) 
paulson@13223
   442
 apply (rule_tac x="And(p, iterates(incr_bv1,length(envp),q))" in bexI)
paulson@13223
   443
  apply typecheck
paulson@13223
   444
apply (rule conjI) 
paulson@13223
   445
(*finally check the arity!*)
paulson@13223
   446
 apply (simp add: arity_iterates_incr_bv1_eq length_app Un_least_lt_iff)
paulson@13223
   447
 apply (force intro: add_le_self le_trans) 
paulson@13223
   448
apply (simp add: arity_sats1_iff formula_add_params1, blast) 
paulson@13223
   449
done
paulson@13223
   450
paulson@13223
   451
lemma Un_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Un Y \<in> DPow(A)"
paulson@13223
   452
apply (subgoal_tac "X Un Y = A - ((A-X) Int (A-Y))") 
paulson@13223
   453
apply (simp add: Int_in_DPow Compl_in_DPow) 
paulson@13223
   454
apply (simp add: DPow_def, blast) 
paulson@13223
   455
done
paulson@13223
   456
paulson@13651
   457
lemma singleton_in_DPow: "a \<in> A ==> {a} \<in> DPow(A)"
paulson@13223
   458
apply (simp add: DPow_def)
paulson@13651
   459
apply (rule_tac x="Cons(a,Nil)" in bexI) 
paulson@13223
   460
 apply (rule_tac x="Equal(0,1)" in bexI) 
paulson@13223
   461
  apply typecheck
paulson@13223
   462
apply (force simp add: succ_Un_distrib [symmetric])  
paulson@13223
   463
done
paulson@13223
   464
paulson@13223
   465
lemma cons_in_DPow: "[| a \<in> A; X \<in> DPow(A) |] ==> cons(a,X) \<in> DPow(A)"
paulson@13223
   466
apply (rule cons_eq [THEN subst]) 
paulson@13223
   467
apply (blast intro: singleton_in_DPow Un_in_DPow) 
paulson@13223
   468
done
paulson@13223
   469
paulson@13223
   470
(*Part of Lemma 1.3*)
paulson@13223
   471
lemma Fin_into_DPow: "X \<in> Fin(A) ==> X \<in> DPow(A)"
paulson@13223
   472
apply (erule Fin.induct) 
paulson@13223
   473
 apply (rule empty_in_DPow) 
paulson@13223
   474
apply (blast intro: cons_in_DPow) 
paulson@13223
   475
done
paulson@13223
   476
paulson@13651
   477
text{*@{term DPow} is not monotonic.  For example, let @{term A} be some
paulson@13651
   478
non-constructible set of natural numbers, and let @{term B} be @{term nat}.
paulson@13721
   479
Then @{term "A<=B"} and obviously @{term "A \<in> DPow(A)"} but @{term "A ~:
paulson@13651
   480
DPow(B)"}.*}
paulson@13651
   481
paulson@13651
   482
(*This may be true but the proof looks difficult, requiring relativization 
paulson@13651
   483
lemma DPow_insert: "DPow (cons(a,A)) = DPow(A) Un {cons(a,X) . X: DPow(A)}"
paulson@13651
   484
apply (rule equalityI, safe)
paulson@13223
   485
oops
paulson@13651
   486
*)
paulson@13223
   487
paulson@13223
   488
lemma Finite_Pow_subset_Pow: "Finite(A) ==> Pow(A) <= DPow(A)" 
paulson@13223
   489
by (blast intro: Fin_into_DPow Finite_into_Fin Fin_subset)
paulson@13223
   490
paulson@13223
   491
lemma Finite_DPow_eq_Pow: "Finite(A) ==> DPow(A) = Pow(A)"
paulson@13223
   492
apply (rule equalityI) 
paulson@13223
   493
apply (rule DPow_subset_Pow) 
paulson@13223
   494
apply (erule Finite_Pow_subset_Pow) 
paulson@13223
   495
done
paulson@13223
   496
paulson@13651
   497
paulson@13651
   498
subsection{*Internalized Formulas for the Ordinals*}
paulson@13223
   499
paulson@13651
   500
text{*The @{text sats} theorems below differ from the usual form in that they
paulson@13651
   501
include an element of absoluteness.  That is, they relate internalized
paulson@13651
   502
formulas to real concepts such as the subset relation, rather than to the
paulson@13651
   503
relativized concepts defined in theory @{text Relative}.  This lets us prove
paulson@13651
   504
the theorem as @{text Ords_in_DPow} without first having to instantiate the
paulson@13651
   505
locale @{text M_trivial}.  Note that the present theory does not even take
paulson@13651
   506
@{text Relative} as a parent.*}
paulson@13298
   507
paulson@13298
   508
subsubsection{*The subset relation*}
paulson@13298
   509
wenzelm@21233
   510
definition subset_fm :: "[i,i]=>i"
paulson@13298
   511
    "subset_fm(x,y) == Forall(Implies(Member(0,succ(x)), Member(0,succ(y))))"
paulson@13298
   512
paulson@13298
   513
lemma subset_type [TC]: "[| x \<in> nat; y \<in> nat |] ==> subset_fm(x,y) \<in> formula"
paulson@13298
   514
by (simp add: subset_fm_def) 
paulson@13298
   515
paulson@13298
   516
lemma arity_subset_fm [simp]:
paulson@13298
   517
     "[| x \<in> nat; y \<in> nat |] ==> arity(subset_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13298
   518
by (simp add: subset_fm_def succ_Un_distrib [symmetric]) 
paulson@13298
   519
paulson@13298
   520
lemma sats_subset_fm [simp]:
paulson@13298
   521
   "[|x < length(env); y \<in> nat; env \<in> list(A); Transset(A)|]
paulson@13298
   522
    ==> sats(A, subset_fm(x,y), env) <-> nth(x,env) \<subseteq> nth(y,env)"
paulson@13298
   523
apply (frule lt_length_in_nat, assumption)  
paulson@13298
   524
apply (simp add: subset_fm_def Transset_def) 
paulson@13298
   525
apply (blast intro: nth_type) 
paulson@13298
   526
done
paulson@13298
   527
paulson@13298
   528
subsubsection{*Transitive sets*}
paulson@13298
   529
wenzelm@21233
   530
definition transset_fm :: "i=>i"
paulson@13298
   531
   "transset_fm(x) == Forall(Implies(Member(0,succ(x)), subset_fm(0,succ(x))))"
paulson@13298
   532
paulson@13298
   533
lemma transset_type [TC]: "x \<in> nat ==> transset_fm(x) \<in> formula"
paulson@13298
   534
by (simp add: transset_fm_def) 
paulson@13298
   535
paulson@13298
   536
lemma arity_transset_fm [simp]:
paulson@13298
   537
     "x \<in> nat ==> arity(transset_fm(x)) = succ(x)"
paulson@13298
   538
by (simp add: transset_fm_def succ_Un_distrib [symmetric]) 
paulson@13298
   539
paulson@13298
   540
lemma sats_transset_fm [simp]:
paulson@13298
   541
   "[|x < length(env); env \<in> list(A); Transset(A)|]
paulson@13298
   542
    ==> sats(A, transset_fm(x), env) <-> Transset(nth(x,env))"
paulson@13298
   543
apply (frule lt_nat_in_nat, erule length_type) 
paulson@13298
   544
apply (simp add: transset_fm_def Transset_def) 
paulson@13298
   545
apply (blast intro: nth_type) 
paulson@13298
   546
done
paulson@13298
   547
paulson@13298
   548
subsubsection{*Ordinals*}
paulson@13298
   549
wenzelm@21233
   550
definition ordinal_fm :: "i=>i"
paulson@13298
   551
   "ordinal_fm(x) == 
paulson@13298
   552
      And(transset_fm(x), Forall(Implies(Member(0,succ(x)), transset_fm(0))))"
paulson@13298
   553
paulson@13298
   554
lemma ordinal_type [TC]: "x \<in> nat ==> ordinal_fm(x) \<in> formula"
paulson@13298
   555
by (simp add: ordinal_fm_def) 
paulson@13298
   556
paulson@13298
   557
lemma arity_ordinal_fm [simp]:
paulson@13298
   558
     "x \<in> nat ==> arity(ordinal_fm(x)) = succ(x)"
paulson@13298
   559
by (simp add: ordinal_fm_def succ_Un_distrib [symmetric]) 
paulson@13298
   560
paulson@13306
   561
lemma sats_ordinal_fm:
paulson@13298
   562
   "[|x < length(env); env \<in> list(A); Transset(A)|]
paulson@13298
   563
    ==> sats(A, ordinal_fm(x), env) <-> Ord(nth(x,env))"
paulson@13298
   564
apply (frule lt_nat_in_nat, erule length_type) 
paulson@13298
   565
apply (simp add: ordinal_fm_def Ord_def Transset_def)
paulson@13298
   566
apply (blast intro: nth_type) 
paulson@13298
   567
done
paulson@13298
   568
paulson@13651
   569
text{*The subset consisting of the ordinals is definable.  Essential lemma for
paulson@13651
   570
@{text Ord_in_Lset}.  This result is the objective of the present subsection.*}
paulson@13651
   571
theorem Ords_in_DPow: "Transset(A) ==> {x \<in> A. Ord(x)} \<in> DPow(A)"
paulson@13651
   572
apply (simp add: DPow_def Collect_subset) 
paulson@13651
   573
apply (rule_tac x=Nil in bexI) 
paulson@13651
   574
 apply (rule_tac x="ordinal_fm(0)" in bexI) 
paulson@13651
   575
apply (simp_all add: sats_ordinal_fm)
paulson@13651
   576
done 
paulson@13651
   577
paulson@13298
   578
paulson@13223
   579
subsection{* Constant Lset: Levels of the Constructible Universe *}
paulson@13223
   580
wenzelm@21233
   581
definition
paulson@13651
   582
  Lset :: "i=>i"
paulson@13223
   583
    "Lset(i) == transrec(i, %x f. \<Union>y\<in>x. DPow(f`y))"
paulson@13223
   584
paulson@13651
   585
  L :: "i=>o" --{*Kunen's definition VI 1.5, page 167*}
paulson@13651
   586
    "L(x) == \<exists>i. Ord(i) & x \<in> Lset(i)"
paulson@13651
   587
  
paulson@13223
   588
text{*NOT SUITABLE FOR REWRITING -- RECURSIVE!*}
paulson@13223
   589
lemma Lset: "Lset(i) = (UN j:i. DPow(Lset(j)))"
paulson@13223
   590
by (subst Lset_def [THEN def_transrec], simp)
paulson@13223
   591
paulson@13223
   592
lemma LsetI: "[|y\<in>x; A \<in> DPow(Lset(y))|] ==> A \<in> Lset(x)";
paulson@13223
   593
by (subst Lset, blast)
paulson@13223
   594
paulson@13223
   595
lemma LsetD: "A \<in> Lset(x) ==> \<exists>y\<in>x. A \<in> DPow(Lset(y))";
paulson@13223
   596
apply (insert Lset [of x]) 
paulson@13223
   597
apply (blast intro: elim: equalityE) 
paulson@13223
   598
done
paulson@13223
   599
paulson@13223
   600
subsubsection{* Transitivity *}
paulson@13223
   601
paulson@13223
   602
lemma elem_subset_in_DPow: "[|X \<in> A; X \<subseteq> A|] ==> X \<in> DPow(A)"
paulson@13223
   603
apply (simp add: Transset_def DPow_def)
paulson@13223
   604
apply (rule_tac x="[X]" in bexI) 
paulson@13223
   605
 apply (rule_tac x="Member(0,1)" in bexI) 
paulson@13223
   606
  apply (auto simp add: Un_least_lt_iff) 
paulson@13223
   607
done
paulson@13223
   608
paulson@13223
   609
lemma Transset_subset_DPow: "Transset(A) ==> A <= DPow(A)"
paulson@13223
   610
apply clarify  
paulson@13223
   611
apply (simp add: Transset_def)
paulson@13223
   612
apply (blast intro: elem_subset_in_DPow) 
paulson@13223
   613
done
paulson@13223
   614
paulson@13223
   615
lemma Transset_DPow: "Transset(A) ==> Transset(DPow(A))"
paulson@13223
   616
apply (simp add: Transset_def) 
paulson@13223
   617
apply (blast intro: elem_subset_in_DPow dest: DPowD) 
paulson@13223
   618
done
paulson@13223
   619
paulson@13651
   620
text{*Kunen's VI 1.6 (a)*}
paulson@13223
   621
lemma Transset_Lset: "Transset(Lset(i))"
paulson@13223
   622
apply (rule_tac a=i in eps_induct)
paulson@13223
   623
apply (subst Lset)
paulson@13223
   624
apply (blast intro!: Transset_Union_family Transset_Un Transset_DPow)
paulson@13223
   625
done
paulson@13223
   626
paulson@13291
   627
lemma mem_Lset_imp_subset_Lset: "a \<in> Lset(i) ==> a \<subseteq> Lset(i)"
paulson@13291
   628
apply (insert Transset_Lset) 
paulson@13291
   629
apply (simp add: Transset_def) 
paulson@13291
   630
done
paulson@13291
   631
paulson@13223
   632
subsubsection{* Monotonicity *}
paulson@13223
   633
paulson@13651
   634
text{*Kunen's VI 1.6 (b)*}
paulson@13223
   635
lemma Lset_mono [rule_format]:
paulson@13223
   636
     "ALL j. i<=j --> Lset(i) <= Lset(j)"
paulson@15481
   637
proof (induct i rule: eps_induct, intro allI impI)
paulson@15481
   638
  fix x j
paulson@15481
   639
  assume "\<forall>y\<in>x. \<forall>j. y \<subseteq> j \<longrightarrow> Lset(y) \<subseteq> Lset(j)"
paulson@15481
   640
     and "x \<subseteq> j"
paulson@15481
   641
  thus "Lset(x) \<subseteq> Lset(j)"
paulson@15481
   642
    by (force simp add: Lset [of x] Lset [of j]) 
paulson@15481
   643
qed
paulson@13223
   644
paulson@13223
   645
text{*This version lets us remove the premise @{term "Ord(i)"} sometimes.*}
paulson@13223
   646
lemma Lset_mono_mem [rule_format]:
paulson@13223
   647
     "ALL j. i:j --> Lset(i) <= Lset(j)"
paulson@15481
   648
proof (induct i rule: eps_induct, intro allI impI)
paulson@15481
   649
  fix x j
paulson@15481
   650
  assume "\<forall>y\<in>x. \<forall>j. y \<in> j \<longrightarrow> Lset(y) \<subseteq> Lset(j)"
paulson@15481
   651
     and "x \<in> j"
paulson@15481
   652
  thus "Lset(x) \<subseteq> Lset(j)"
paulson@15481
   653
    by (force simp add: Lset [of j] 
paulson@15481
   654
              intro!: bexI intro: elem_subset_in_DPow dest: LsetD DPowD) 
paulson@15481
   655
qed
paulson@15481
   656
paulson@13223
   657
paulson@13291
   658
text{*Useful with Reflection to bump up the ordinal*}
paulson@13291
   659
lemma subset_Lset_ltD: "[|A \<subseteq> Lset(i); i < j|] ==> A \<subseteq> Lset(j)"
paulson@13291
   660
by (blast dest: ltD [THEN Lset_mono_mem]) 
paulson@13291
   661
paulson@13651
   662
subsubsection{* 0, successor and limit equations for Lset *}
paulson@13223
   663
paulson@13223
   664
lemma Lset_0 [simp]: "Lset(0) = 0"
paulson@13223
   665
by (subst Lset, blast)
paulson@13223
   666
paulson@13223
   667
lemma Lset_succ_subset1: "DPow(Lset(i)) <= Lset(succ(i))"
paulson@13223
   668
by (subst Lset, rule succI1 [THEN RepFunI, THEN Union_upper])
paulson@13223
   669
paulson@13223
   670
lemma Lset_succ_subset2: "Lset(succ(i)) <= DPow(Lset(i))"
paulson@13223
   671
apply (subst Lset, rule UN_least)
paulson@13223
   672
apply (erule succE) 
paulson@13223
   673
 apply blast 
paulson@13223
   674
apply clarify
paulson@13223
   675
apply (rule elem_subset_in_DPow)
paulson@13223
   676
 apply (subst Lset)
paulson@13223
   677
 apply blast 
paulson@13223
   678
apply (blast intro: dest: DPowD Lset_mono_mem) 
paulson@13223
   679
done
paulson@13223
   680
paulson@13223
   681
lemma Lset_succ: "Lset(succ(i)) = DPow(Lset(i))"
paulson@13223
   682
by (intro equalityI Lset_succ_subset1 Lset_succ_subset2) 
paulson@13223
   683
paulson@13223
   684
lemma Lset_Union [simp]: "Lset(\<Union>(X)) = (\<Union>y\<in>X. Lset(y))"
paulson@13223
   685
apply (subst Lset)
paulson@13223
   686
apply (rule equalityI)
paulson@13223
   687
 txt{*first inclusion*}
paulson@13223
   688
 apply (rule UN_least)
paulson@13223
   689
 apply (erule UnionE)
paulson@13223
   690
 apply (rule subset_trans)
paulson@13223
   691
  apply (erule_tac [2] UN_upper, subst Lset, erule UN_upper)
paulson@13223
   692
txt{*opposite inclusion*}
paulson@13223
   693
apply (rule UN_least)
paulson@13223
   694
apply (subst Lset, blast)
paulson@13223
   695
done
paulson@13223
   696
paulson@13223
   697
subsubsection{* Lset applied to Limit ordinals *}
paulson@13223
   698
paulson@13223
   699
lemma Limit_Lset_eq:
paulson@13223
   700
    "Limit(i) ==> Lset(i) = (\<Union>y\<in>i. Lset(y))"
paulson@13223
   701
by (simp add: Lset_Union [symmetric] Limit_Union_eq)
paulson@13223
   702
paulson@13721
   703
lemma lt_LsetI: "[| a: Lset(j);  j<i |] ==> a \<in> Lset(i)"
paulson@13223
   704
by (blast dest: Lset_mono [OF le_imp_subset [OF leI]])
paulson@13223
   705
paulson@13223
   706
lemma Limit_LsetE:
paulson@13223
   707
    "[| a: Lset(i);  ~R ==> Limit(i);
paulson@13223
   708
        !!x. [| x<i;  a: Lset(x) |] ==> R
paulson@13223
   709
     |] ==> R"
paulson@13223
   710
apply (rule classical)
paulson@13223
   711
apply (rule Limit_Lset_eq [THEN equalityD1, THEN subsetD, THEN UN_E])
paulson@13223
   712
  prefer 2 apply assumption
paulson@13223
   713
 apply blast 
paulson@13223
   714
apply (blast intro: ltI  Limit_is_Ord)
paulson@13223
   715
done
paulson@13223
   716
paulson@13223
   717
subsubsection{* Basic closure properties *}
paulson@13223
   718
paulson@13721
   719
lemma zero_in_Lset: "y:x ==> 0 \<in> Lset(x)"
paulson@13223
   720
by (subst Lset, blast intro: empty_in_DPow)
paulson@13223
   721
paulson@13223
   722
lemma notin_Lset: "x \<notin> Lset(x)"
paulson@13223
   723
apply (rule_tac a=x in eps_induct)
paulson@13223
   724
apply (subst Lset)
paulson@13223
   725
apply (blast dest: DPowD)  
paulson@13223
   726
done
paulson@13223
   727
paulson@13223
   728
paulson@13651
   729
subsection{*Constructible Ordinals: Kunen's VI 1.9 (b)*}
paulson@13223
   730
paulson@13223
   731
lemma Ords_of_Lset_eq: "Ord(i) ==> {x\<in>Lset(i). Ord(x)} = i"
paulson@13223
   732
apply (erule trans_induct3)
paulson@13223
   733
  apply (simp_all add: Lset_succ Limit_Lset_eq Limit_Union_eq)
paulson@13223
   734
txt{*The successor case remains.*} 
paulson@13223
   735
apply (rule equalityI)
paulson@13223
   736
txt{*First inclusion*}
paulson@13223
   737
 apply clarify  
paulson@13223
   738
 apply (erule Ord_linear_lt, assumption) 
paulson@13223
   739
   apply (blast dest: DPow_imp_subset ltD notE [OF notin_Lset]) 
paulson@13223
   740
  apply blast 
paulson@13223
   741
 apply (blast dest: ltD)
paulson@13223
   742
txt{*Opposite inclusion, @{term "succ(x) \<subseteq> DPow(Lset(x)) \<inter> ON"}*}
paulson@13223
   743
apply auto
paulson@13223
   744
txt{*Key case: *}
paulson@13223
   745
  apply (erule subst, rule Ords_in_DPow [OF Transset_Lset]) 
paulson@13223
   746
 apply (blast intro: elem_subset_in_DPow dest: OrdmemD elim: equalityE) 
paulson@13223
   747
apply (blast intro: Ord_in_Ord) 
paulson@13223
   748
done
paulson@13223
   749
paulson@13223
   750
paulson@13223
   751
lemma Ord_subset_Lset: "Ord(i) ==> i \<subseteq> Lset(i)"
paulson@13223
   752
by (subst Ords_of_Lset_eq [symmetric], assumption, fast)
paulson@13223
   753
paulson@13223
   754
lemma Ord_in_Lset: "Ord(i) ==> i \<in> Lset(succ(i))"
paulson@13223
   755
apply (simp add: Lset_succ)
paulson@13223
   756
apply (subst Ords_of_Lset_eq [symmetric], assumption, 
paulson@13223
   757
       rule Ords_in_DPow [OF Transset_Lset]) 
paulson@13223
   758
done
paulson@13223
   759
paulson@13651
   760
lemma Ord_in_L: "Ord(i) ==> L(i)"
paulson@13651
   761
by (simp add: L_def, blast intro: Ord_in_Lset)
paulson@13651
   762
paulson@13223
   763
subsubsection{* Unions *}
paulson@13223
   764
paulson@13223
   765
lemma Union_in_Lset:
paulson@13687
   766
     "X \<in> Lset(i) ==> Union(X) \<in> Lset(succ(i))"
paulson@13223
   767
apply (insert Transset_Lset)
paulson@13223
   768
apply (rule LsetI [OF succI1])
paulson@13223
   769
apply (simp add: Transset_def DPow_def) 
paulson@13223
   770
apply (intro conjI, blast)
paulson@13223
   771
txt{*Now to create the formula @{term "\<exists>y. y \<in> X \<and> x \<in> y"} *}
paulson@13223
   772
apply (rule_tac x="Cons(X,Nil)" in bexI) 
paulson@13223
   773
 apply (rule_tac x="Exists(And(Member(0,2), Member(1,0)))" in bexI) 
paulson@13223
   774
  apply typecheck
paulson@13223
   775
apply (simp add: succ_Un_distrib [symmetric], blast) 
paulson@13223
   776
done
paulson@13223
   777
paulson@13651
   778
theorem Union_in_L: "L(X) ==> L(Union(X))"
paulson@13687
   779
by (simp add: L_def, blast dest: Union_in_Lset) 
paulson@13651
   780
paulson@13223
   781
subsubsection{* Finite sets and ordered pairs *}
paulson@13223
   782
paulson@13721
   783
lemma singleton_in_Lset: "a: Lset(i) ==> {a} \<in> Lset(succ(i))"
paulson@13223
   784
by (simp add: Lset_succ singleton_in_DPow) 
paulson@13223
   785
paulson@13223
   786
lemma doubleton_in_Lset:
paulson@13721
   787
     "[| a: Lset(i);  b: Lset(i) |] ==> {a,b} \<in> Lset(succ(i))"
paulson@13223
   788
by (simp add: Lset_succ empty_in_DPow cons_in_DPow) 
paulson@13223
   789
paulson@13223
   790
lemma Pair_in_Lset:
paulson@13721
   791
    "[| a: Lset(i);  b: Lset(i); Ord(i) |] ==> <a,b> \<in> Lset(succ(succ(i)))"
paulson@13223
   792
apply (unfold Pair_def)
paulson@13223
   793
apply (blast intro: doubleton_in_Lset) 
paulson@13223
   794
done
paulson@13223
   795
paulson@13223
   796
lemmas Lset_UnI1 = Un_upper1 [THEN Lset_mono [THEN subsetD], standard]
paulson@13223
   797
lemmas Lset_UnI2 = Un_upper2 [THEN Lset_mono [THEN subsetD], standard]
paulson@13223
   798
paulson@13223
   799
text{*Hard work is finding a single j:i such that {a,b}<=Lset(j)*}
paulson@13223
   800
lemma doubleton_in_LLimit:
paulson@13721
   801
    "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> {a,b} \<in> Lset(i)"
paulson@13223
   802
apply (erule Limit_LsetE, assumption)
paulson@13223
   803
apply (erule Limit_LsetE, assumption)
paulson@13269
   804
apply (blast intro: lt_LsetI [OF doubleton_in_Lset]
paulson@13269
   805
                    Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
paulson@13223
   806
done
paulson@13223
   807
paulson@13651
   808
theorem doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})"
paulson@13651
   809
apply (simp add: L_def, clarify) 
paulson@13651
   810
apply (drule Ord2_imp_greater_Limit, assumption) 
paulson@13651
   811
apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord) 
paulson@13651
   812
done
paulson@13651
   813
paulson@13223
   814
lemma Pair_in_LLimit:
paulson@13721
   815
    "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> <a,b> \<in> Lset(i)"
paulson@13223
   816
txt{*Infer that a, b occur at ordinals x,xa < i.*}
paulson@13223
   817
apply (erule Limit_LsetE, assumption)
paulson@13223
   818
apply (erule Limit_LsetE, assumption)
paulson@13223
   819
txt{*Infer that succ(succ(x Un xa)) < i *}
paulson@13223
   820
apply (blast intro: lt_Ord lt_LsetI [OF Pair_in_Lset]
paulson@13223
   821
                    Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
paulson@13223
   822
done
paulson@13223
   823
paulson@13223
   824
paulson@13223
   825
paulson@13651
   826
text{*The rank function for the constructible universe*}
wenzelm@21233
   827
definition
paulson@13651
   828
  lrank :: "i=>i" --{*Kunen's definition VI 1.7*}
skalberg@14171
   829
    "lrank(x) == \<mu> i. x \<in> Lset(succ(i))"
paulson@13223
   830
paulson@13223
   831
lemma L_I: "[|x \<in> Lset(i); Ord(i)|] ==> L(x)"
paulson@13223
   832
by (simp add: L_def, blast)
paulson@13223
   833
paulson@13223
   834
lemma L_D: "L(x) ==> \<exists>i. Ord(i) & x \<in> Lset(i)"
paulson@13223
   835
by (simp add: L_def)
paulson@13223
   836
paulson@13223
   837
lemma Ord_lrank [simp]: "Ord(lrank(a))"
paulson@13223
   838
by (simp add: lrank_def)
paulson@13223
   839
paulson@13223
   840
lemma Lset_lrank_lt [rule_format]: "Ord(i) ==> x \<in> Lset(i) --> lrank(x) < i"
paulson@13223
   841
apply (erule trans_induct3)
paulson@13223
   842
  apply simp   
paulson@13223
   843
 apply (simp only: lrank_def) 
paulson@13223
   844
 apply (blast intro: Least_le) 
paulson@13223
   845
apply (simp_all add: Limit_Lset_eq) 
paulson@13223
   846
apply (blast intro: ltI Limit_is_Ord lt_trans) 
paulson@13223
   847
done
paulson@13223
   848
paulson@13651
   849
text{*Kunen's VI 1.8.  The proof is much harder than the text would
paulson@13651
   850
suggest.  For a start, it needs the previous lemma, which is proved by
paulson@13651
   851
induction.*}
paulson@13223
   852
lemma Lset_iff_lrank_lt: "Ord(i) ==> x \<in> Lset(i) <-> L(x) & lrank(x) < i"
paulson@13223
   853
apply (simp add: L_def, auto) 
paulson@13223
   854
 apply (blast intro: Lset_lrank_lt) 
paulson@13223
   855
 apply (unfold lrank_def) 
paulson@13223
   856
apply (drule succI1 [THEN Lset_mono_mem, THEN subsetD]) 
paulson@13223
   857
apply (drule_tac P="\<lambda>i. x \<in> Lset(succ(i))" in LeastI, assumption) 
paulson@13223
   858
apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) 
paulson@13223
   859
done
paulson@13223
   860
paulson@13223
   861
lemma Lset_succ_lrank_iff [simp]: "x \<in> Lset(succ(lrank(x))) <-> L(x)"
paulson@13223
   862
by (simp add: Lset_iff_lrank_lt)
paulson@13223
   863
paulson@13651
   864
text{*Kunen's VI 1.9 (a)*}
paulson@13223
   865
lemma lrank_of_Ord: "Ord(i) ==> lrank(i) = i"
paulson@13223
   866
apply (unfold lrank_def) 
paulson@13223
   867
apply (rule Least_equality) 
paulson@13223
   868
  apply (erule Ord_in_Lset) 
paulson@13223
   869
 apply assumption
paulson@13223
   870
apply (insert notin_Lset [of i]) 
paulson@13223
   871
apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) 
paulson@13223
   872
done
paulson@13223
   873
paulson@13245
   874
paulson@13223
   875
text{*This is lrank(lrank(a)) = lrank(a) *}
paulson@13223
   876
declare Ord_lrank [THEN lrank_of_Ord, simp]
paulson@13223
   877
paulson@13651
   878
text{*Kunen's VI 1.10 *}
paulson@13223
   879
lemma Lset_in_Lset_succ: "Lset(i) \<in> Lset(succ(i))";
paulson@13223
   880
apply (simp add: Lset_succ DPow_def) 
paulson@13339
   881
apply (rule_tac x=Nil in bexI) 
paulson@13223
   882
 apply (rule_tac x="Equal(0,0)" in bexI) 
paulson@13223
   883
apply auto 
paulson@13223
   884
done
paulson@13223
   885
paulson@13223
   886
lemma lrank_Lset: "Ord(i) ==> lrank(Lset(i)) = i"
paulson@13223
   887
apply (unfold lrank_def) 
paulson@13223
   888
apply (rule Least_equality) 
paulson@13223
   889
  apply (rule Lset_in_Lset_succ) 
paulson@13223
   890
 apply assumption
paulson@13223
   891
apply clarify 
paulson@13223
   892
apply (subgoal_tac "Lset(succ(ia)) <= Lset(i)")
paulson@13223
   893
 apply (blast dest: mem_irrefl) 
paulson@13223
   894
apply (blast intro!: le_imp_subset Lset_mono) 
paulson@13223
   895
done
paulson@13223
   896
paulson@13651
   897
text{*Kunen's VI 1.11 *}
paulson@13223
   898
lemma Lset_subset_Vset: "Ord(i) ==> Lset(i) <= Vset(i)";
paulson@13223
   899
apply (erule trans_induct)
paulson@13223
   900
apply (subst Lset) 
paulson@13223
   901
apply (subst Vset) 
paulson@13223
   902
apply (rule UN_mono [OF subset_refl]) 
paulson@13223
   903
apply (rule subset_trans [OF DPow_subset_Pow]) 
paulson@13223
   904
apply (rule Pow_mono, blast) 
paulson@13223
   905
done
paulson@13223
   906
paulson@13651
   907
text{*Kunen's VI 1.12 *}
wenzelm@13535
   908
lemma Lset_subset_Vset': "i \<in> nat ==> Lset(i) = Vset(i)";
paulson@13223
   909
apply (erule nat_induct)
paulson@13223
   910
 apply (simp add: Vfrom_0) 
paulson@13223
   911
apply (simp add: Lset_succ Vset_succ Finite_Vset Finite_DPow_eq_Pow) 
paulson@13223
   912
done
paulson@13223
   913
paulson@13291
   914
text{*Every set of constructible sets is included in some @{term Lset}*} 
paulson@13291
   915
lemma subset_Lset:
paulson@13291
   916
     "(\<forall>x\<in>A. L(x)) ==> \<exists>i. Ord(i) & A \<subseteq> Lset(i)"
paulson@13291
   917
by (rule_tac x = "\<Union>x\<in>A. succ(lrank(x))" in exI, force)
paulson@13291
   918
paulson@13291
   919
lemma subset_LsetE:
paulson@13291
   920
     "[|\<forall>x\<in>A. L(x);
paulson@13291
   921
        !!i. [|Ord(i); A \<subseteq> Lset(i)|] ==> P|]
paulson@13291
   922
      ==> P"
paulson@13291
   923
by (blast dest: subset_Lset) 
paulson@13291
   924
paulson@13651
   925
subsubsection{*For L to satisfy the Powerset axiom *}
paulson@13223
   926
paulson@13223
   927
lemma LPow_env_typing:
paulson@13721
   928
    "[| y \<in> Lset(i); Ord(i); y \<subseteq> X |] 
paulson@13511
   929
     ==> \<exists>z \<in> Pow(X). y \<in> Lset(succ(lrank(z)))"
paulson@13223
   930
by (auto intro: L_I iff: Lset_succ_lrank_iff) 
paulson@13223
   931
paulson@13223
   932
lemma LPow_in_Lset:
paulson@13223
   933
     "[|X \<in> Lset(i); Ord(i)|] ==> \<exists>j. Ord(j) & {y \<in> Pow(X). L(y)} \<in> Lset(j)"
paulson@13223
   934
apply (rule_tac x="succ(\<Union>y \<in> Pow(X). succ(lrank(y)))" in exI)
paulson@13223
   935
apply simp 
paulson@13223
   936
apply (rule LsetI [OF succI1])
paulson@13223
   937
apply (simp add: DPow_def) 
paulson@13223
   938
apply (intro conjI, clarify) 
paulson@13511
   939
 apply (rule_tac a=x in UN_I, simp+)  
paulson@13223
   940
txt{*Now to create the formula @{term "y \<subseteq> X"} *}
paulson@13223
   941
apply (rule_tac x="Cons(X,Nil)" in bexI) 
paulson@13223
   942
 apply (rule_tac x="subset_fm(0,1)" in bexI) 
paulson@13223
   943
  apply typecheck
paulson@13511
   944
 apply (rule conjI) 
paulson@13223
   945
apply (simp add: succ_Un_distrib [symmetric]) 
paulson@13223
   946
apply (rule equality_iffI) 
paulson@13511
   947
apply (simp add: Transset_UN [OF Transset_Lset] LPow_env_typing)
paulson@13223
   948
apply (auto intro: L_I iff: Lset_succ_lrank_iff) 
paulson@13223
   949
done
paulson@13223
   950
paulson@13245
   951
theorem LPow_in_L: "L(X) ==> L({y \<in> Pow(X). L(y)})"
paulson@13223
   952
by (blast intro: L_I dest: L_D LPow_in_Lset)
paulson@13223
   953
paulson@13385
   954
paulson@13385
   955
subsection{*Eliminating @{term arity} from the Definition of @{term Lset}*}
paulson@13385
   956
paulson@13385
   957
lemma nth_zero_eq_0: "n \<in> nat ==> nth(n,[0]) = 0"
paulson@13385
   958
by (induct_tac n, auto)
paulson@13385
   959
paulson@13385
   960
lemma sats_app_0_iff [rule_format]:
paulson@13385
   961
  "[| p \<in> formula; 0 \<in> A |]
paulson@13385
   962
   ==> \<forall>env \<in> list(A). sats(A,p, env@[0]) <-> sats(A,p,env)"
paulson@13385
   963
apply (induct_tac p)
paulson@13385
   964
apply (simp_all del: app_Cons add: app_Cons [symmetric]
paulson@13385
   965
		add: nth_zero_eq_0 nth_append not_lt_iff_le nth_eq_0)
paulson@13385
   966
done
paulson@13385
   967
paulson@13385
   968
lemma sats_app_zeroes_iff:
paulson@13385
   969
  "[| p \<in> formula; 0 \<in> A; env \<in> list(A); n \<in> nat |]
paulson@13385
   970
   ==> sats(A,p,env @ repeat(0,n)) <-> sats(A,p,env)"
paulson@13385
   971
apply (induct_tac n, simp) 
paulson@13385
   972
apply (simp del: repeat.simps
paulson@13385
   973
            add: repeat_succ_app sats_app_0_iff app_assoc [symmetric]) 
paulson@13385
   974
done
paulson@13385
   975
paulson@13385
   976
lemma exists_bigger_env:
paulson@13385
   977
  "[| p \<in> formula; 0 \<in> A; env \<in> list(A) |]
paulson@13385
   978
   ==> \<exists>env' \<in> list(A). arity(p) \<le> succ(length(env')) & 
paulson@13385
   979
              (\<forall>a\<in>A. sats(A,p,Cons(a,env')) <-> sats(A,p,Cons(a,env)))"
paulson@13385
   980
apply (rule_tac x="env @ repeat(0,arity(p))" in bexI) 
paulson@13385
   981
apply (simp del: app_Cons add: app_Cons [symmetric]
paulson@13385
   982
	    add: length_repeat sats_app_zeroes_iff, typecheck)
paulson@13385
   983
done
paulson@13385
   984
paulson@13385
   985
paulson@13385
   986
text{*A simpler version of @{term DPow}: no arity check!*}
wenzelm@21233
   987
definition DPow' :: "i => i"
paulson@13385
   988
  "DPow'(A) == {X \<in> Pow(A). 
paulson@13385
   989
                \<exists>env \<in> list(A). \<exists>p \<in> formula. 
paulson@13385
   990
                    X = {x\<in>A. sats(A, p, Cons(x,env))}}"
paulson@13385
   991
paulson@13385
   992
lemma DPow_subset_DPow': "DPow(A) <= DPow'(A)";
paulson@13385
   993
by (simp add: DPow_def DPow'_def, blast)
paulson@13385
   994
paulson@13385
   995
lemma DPow'_0: "DPow'(0) = {0}"
paulson@13385
   996
by (auto simp add: DPow'_def)
paulson@13385
   997
paulson@13385
   998
lemma DPow'_subset_DPow: "0 \<in> A ==> DPow'(A) \<subseteq> DPow(A)"
paulson@13385
   999
apply (auto simp add: DPow'_def DPow_def) 
paulson@13385
  1000
apply (frule exists_bigger_env, assumption+, force)  
paulson@13385
  1001
done
paulson@13385
  1002
paulson@13385
  1003
lemma DPow_eq_DPow': "Transset(A) ==> DPow(A) = DPow'(A)"
paulson@13385
  1004
apply (drule Transset_0_disj) 
paulson@13385
  1005
apply (erule disjE) 
paulson@13651
  1006
 apply (simp add: DPow'_0 Finite_DPow_eq_Pow) 
paulson@13385
  1007
apply (rule equalityI)
paulson@13385
  1008
 apply (rule DPow_subset_DPow') 
paulson@13385
  1009
apply (erule DPow'_subset_DPow) 
paulson@13385
  1010
done
paulson@13385
  1011
paulson@13385
  1012
text{*And thus we can relativize @{term Lset} without bothering with
paulson@13385
  1013
      @{term arity} and @{term length}*}
paulson@13385
  1014
lemma Lset_eq_transrec_DPow': "Lset(i) = transrec(i, %x f. \<Union>y\<in>x. DPow'(f`y))"
paulson@13385
  1015
apply (rule_tac a=i in eps_induct)
paulson@13385
  1016
apply (subst Lset)
paulson@13385
  1017
apply (subst transrec)
paulson@13385
  1018
apply (simp only: DPow_eq_DPow' [OF Transset_Lset], simp) 
paulson@13385
  1019
done
paulson@13385
  1020
paulson@13385
  1021
text{*With this rule we can specify @{term p} later and don't worry about
paulson@13385
  1022
      arities at all!*}
paulson@13385
  1023
lemma DPow_LsetI [rule_format]:
paulson@13385
  1024
  "[|\<forall>x\<in>Lset(i). P(x) <-> sats(Lset(i), p, Cons(x,env));
paulson@13385
  1025
     env \<in> list(Lset(i));  p \<in> formula|]
paulson@13385
  1026
   ==> {x\<in>Lset(i). P(x)} \<in> DPow(Lset(i))"
paulson@13385
  1027
by (simp add: DPow_eq_DPow' [OF Transset_Lset] DPow'_def, blast) 
paulson@13385
  1028
paulson@13223
  1029
end