author | blanchet |
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parent 61798 | 27f3c10b0b50 |
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permissions | -rw-r--r-- |
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(* Title: ZF/Constructible/Formula.thy |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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*) |
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section \<open>First-Order Formulas and the Definition of the Class L\<close> |
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theory Formula imports Main begin |
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subsection\<open>Internalized formulas of FOL\<close> |
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text\<open>De Bruijn representation. |
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Unbound variables get their denotations from an environment.\<close> |
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consts formula :: i |
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datatype |
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"formula" = Member ("x \<in> nat", "y \<in> nat") |
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| Equal ("x \<in> nat", "y \<in> nat") |
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| Nand ("p \<in> formula", "q \<in> formula") |
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| Forall ("p \<in> formula") |
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declare formula.intros [TC] |
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definition |
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Neg :: "i=>i" where |
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"Neg(p) == Nand(p,p)" |
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definition |
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And :: "[i,i]=>i" where |
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"And(p,q) == Neg(Nand(p,q))" |
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definition |
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Or :: "[i,i]=>i" where |
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"Or(p,q) == Nand(Neg(p),Neg(q))" |
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definition |
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Implies :: "[i,i]=>i" where |
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"Implies(p,q) == Nand(p,Neg(q))" |
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definition |
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Iff :: "[i,i]=>i" where |
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"Iff(p,q) == And(Implies(p,q), Implies(q,p))" |
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definition |
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Exists :: "i=>i" where |
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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" where |
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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) \<longleftrightarrow> 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) \<longleftrightarrow> 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)) \<longleftrightarrow> ~ (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) \<longleftrightarrow> (\<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\<open>Dividing line between primitive and derived connectives\<close> |
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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) \<longleftrightarrow> ~ 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)) \<longleftrightarrow> 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)) \<longleftrightarrow> 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)) \<longleftrightarrow> (sats(A,p,env) \<longrightarrow> 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)) \<longleftrightarrow> (sats(A,p,env) \<longleftrightarrow> 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) \<longleftrightarrow> (\<exists>x\<in>A. sats(A, p, Cons(x,env)))" |
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by (simp add: Exists_def) |
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subsubsection\<open>Derived rules to help build up formulas\<close> |
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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) \<longleftrightarrow> 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) \<longleftrightarrow> 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 \<longleftrightarrow> sats(A,p,env); env \<in> list(A)|] |
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==> (~P) \<longleftrightarrow> sats(A, Neg(p), env)" |
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by simp |
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lemma conj_iff_sats: |
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"[| P \<longleftrightarrow> sats(A,p,env); Q \<longleftrightarrow> sats(A,q,env); env \<in> list(A)|] |
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==> (P & Q) \<longleftrightarrow> 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 \<longleftrightarrow> sats(A,p,env); Q \<longleftrightarrow> sats(A,q,env); env \<in> list(A)|] |
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==> (P | Q) \<longleftrightarrow> 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 \<longleftrightarrow> sats(A,p,env); Q \<longleftrightarrow> sats(A,q,env); env \<in> list(A)|] |
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==> (P \<longleftrightarrow> Q) \<longleftrightarrow> 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 \<longleftrightarrow> sats(A,p,env); Q \<longleftrightarrow> sats(A,q,env); env \<in> list(A)|] |
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==> (P \<longrightarrow> Q) \<longleftrightarrow> 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) \<longleftrightarrow> sats(A, p, Cons(x, env)); env \<in> list(A)|] |
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==> (\<forall>x\<in>A. P(x)) \<longleftrightarrow> 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) \<longleftrightarrow> sats(A, p, Cons(x, env)); env \<in> list(A)|] |
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==> (\<exists>x\<in>A. P(x)) \<longleftrightarrow> 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\<open>Arity of a Formula: Maximum Free de Bruijn Index\<close> |
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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) \<longrightarrow> |
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sats(A, p, env @ extra) \<longleftrightarrow> 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)) \<longleftrightarrow> 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\<open>Renaming Some de Bruijn Variables\<close> |
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definition |
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incr_var :: "[i,i]=>i" where |
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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\<open>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.\<close> |
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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)) \<longleftrightarrow> |
|
13223 | 295 |
sats(A, p, bvs@env)" |
296 |
apply (induct_tac p) |
|
297 |
apply (simp_all add: incr_var_def nth_append succ_lt_iff length_type) |
|
298 |
apply (auto simp add: diff_succ not_lt_iff_le) |
|
299 |
done |
|
300 |
||
301 |
||
302 |
(*the following two lemmas prevent huge case splits in arity_incr_bv_lemma*) |
|
303 |
lemma incr_var_lemma: |
|
13687 | 304 |
"[| x \<in> nat; y \<in> nat; nq \<le> x |] |
305 |
==> succ(x) \<union> incr_var(y,nq) = succ(x \<union> y)" |
|
13223 | 306 |
apply (simp add: incr_var_def Ord_Un_if, auto) |
307 |
apply (blast intro: leI) |
|
46823 | 308 |
apply (simp add: not_lt_iff_le) |
309 |
apply (blast intro: le_anti_sym) |
|
310 |
apply (blast dest: lt_trans2) |
|
13223 | 311 |
done |
312 |
||
313 |
lemma incr_And_lemma: |
|
314 |
"y < x ==> y \<union> succ(x) = succ(x \<union> y)" |
|
46823 | 315 |
apply (simp add: Ord_Un_if lt_Ord lt_Ord2 succ_lt_iff) |
316 |
apply (blast dest: lt_asym) |
|
13223 | 317 |
done |
318 |
||
319 |
lemma arity_incr_bv_lemma [rule_format]: |
|
46823 | 320 |
"p \<in> formula |
321 |
==> \<forall>n \<in> nat. arity (incr_bv(p) ` n) = |
|
13223 | 322 |
(if n < arity(p) then succ(arity(p)) else arity(p))" |
46823 | 323 |
apply (induct_tac p) |
13223 | 324 |
apply (simp_all add: imp_disj not_lt_iff_le Un_least_lt_iff lt_Un_iff le_Un_iff |
325 |
succ_Un_distrib [symmetric] incr_var_lt incr_var_le |
|
13647 | 326 |
Un_commute incr_var_lemma Arith.pred_def nat_imp_quasinat |
46823 | 327 |
split: split_nat_case) |
60770 | 328 |
txt\<open>the Forall case reduces to linear arithmetic\<close> |
13269 | 329 |
prefer 2 |
46823 | 330 |
apply clarify |
331 |
apply (blast dest: lt_trans1) |
|
60770 | 332 |
txt\<open>left with the And case\<close> |
13223 | 333 |
apply safe |
46823 | 334 |
apply (blast intro: incr_And_lemma lt_trans1) |
13223 | 335 |
apply (subst incr_And_lemma) |
46823 | 336 |
apply (blast intro: lt_trans1) |
13269 | 337 |
apply (simp add: Un_commute) |
13223 | 338 |
done |
339 |
||
340 |
||
60770 | 341 |
subsection\<open>Renaming all but the First de Bruijn Variable\<close> |
13223 | 342 |
|
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343 |
definition |
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344 |
incr_bv1 :: "i => i" where |
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|
345 |
"incr_bv1(p) == incr_bv(p)`1" |
13223 | 346 |
|
347 |
||
348 |
lemma incr_bv1_type [TC]: "p \<in> formula ==> incr_bv1(p) \<in> formula" |
|
46823 | 349 |
by (simp add: incr_bv1_def) |
13223 | 350 |
|
351 |
(*For renaming all but the bound variable at level 0*) |
|
13647 | 352 |
lemma sats_incr_bv1_iff: |
13223 | 353 |
"[| p \<in> formula; env \<in> list(A); x \<in> A; y \<in> A |] |
46823 | 354 |
==> sats(A, incr_bv1(p), Cons(x, Cons(y, env))) \<longleftrightarrow> |
13223 | 355 |
sats(A, p, Cons(x,env))" |
356 |
apply (insert sats_incr_bv_iff [of p env A y "Cons(x,Nil)"]) |
|
46823 | 357 |
apply (simp add: incr_bv1_def) |
13223 | 358 |
done |
359 |
||
360 |
lemma formula_add_params1 [rule_format]: |
|
361 |
"[| p \<in> formula; n \<in> nat; x \<in> A |] |
|
46823 | 362 |
==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A). |
363 |
length(bvs) = n \<longrightarrow> |
|
364 |
sats(A, iterates(incr_bv1, n, p), Cons(x, bvs@env)) \<longleftrightarrow> |
|
13223 | 365 |
sats(A, p, Cons(x,env))" |
46823 | 366 |
apply (induct_tac n, simp, clarify) |
13223 | 367 |
apply (erule list.cases) |
46823 | 368 |
apply (simp_all add: sats_incr_bv1_iff) |
13223 | 369 |
done |
370 |
||
371 |
||
372 |
lemma arity_incr_bv1_eq: |
|
373 |
"p \<in> formula |
|
374 |
==> arity(incr_bv1(p)) = |
|
375 |
(if 1 < arity(p) then succ(arity(p)) else arity(p))" |
|
376 |
apply (insert arity_incr_bv_lemma [of p 1]) |
|
46823 | 377 |
apply (simp add: incr_bv1_def) |
13223 | 378 |
done |
379 |
||
380 |
lemma arity_iterates_incr_bv1_eq: |
|
381 |
"[| p \<in> formula; n \<in> nat |] |
|
382 |
==> arity(incr_bv1^n(p)) = |
|
383 |
(if 1 < arity(p) then n #+ arity(p) else arity(p))" |
|
46823 | 384 |
apply (induct_tac n) |
13298 | 385 |
apply (simp_all add: arity_incr_bv1_eq) |
13223 | 386 |
apply (simp add: not_lt_iff_le) |
46823 | 387 |
apply (blast intro: le_trans add_le_self2 arity_type) |
13223 | 388 |
done |
389 |
||
390 |
||
13647 | 391 |
|
60770 | 392 |
subsection\<open>Definable Powerset\<close> |
13647 | 393 |
|
60770 | 394 |
text\<open>The definable powerset operation: Kunen's definition VI 1.1, page 165.\<close> |
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|
395 |
definition |
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|
396 |
DPow :: "i => i" where |
46823 | 397 |
"DPow(A) == {X \<in> Pow(A). |
398 |
\<exists>env \<in> list(A). \<exists>p \<in> formula. |
|
399 |
arity(p) \<le> succ(length(env)) & |
|
13223 | 400 |
X = {x\<in>A. sats(A, p, Cons(x,env))}}" |
401 |
||
402 |
lemma DPowI: |
|
13291 | 403 |
"[|env \<in> list(A); p \<in> formula; arity(p) \<le> succ(length(env))|] |
13223 | 404 |
==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)" |
46823 | 405 |
by (simp add: DPow_def, blast) |
13223 | 406 |
|
60770 | 407 |
text\<open>With this rule we can specify @{term p} later.\<close> |
13291 | 408 |
lemma DPowI2 [rule_format]: |
46823 | 409 |
"[|\<forall>x\<in>A. P(x) \<longleftrightarrow> sats(A, p, Cons(x,env)); |
13291 | 410 |
env \<in> list(A); p \<in> formula; arity(p) \<le> succ(length(env))|] |
411 |
==> {x\<in>A. P(x)} \<in> DPow(A)" |
|
46823 | 412 |
by (simp add: DPow_def, blast) |
13291 | 413 |
|
13223 | 414 |
lemma DPowD: |
46823 | 415 |
"X \<in> DPow(A) |
416 |
==> X \<subseteq> A & |
|
417 |
(\<exists>env \<in> list(A). |
|
418 |
\<exists>p \<in> formula. arity(p) \<le> succ(length(env)) & |
|
13223 | 419 |
X = {x\<in>A. sats(A, p, Cons(x,env))})" |
46823 | 420 |
by (simp add: DPow_def) |
13223 | 421 |
|
422 |
lemmas DPow_imp_subset = DPowD [THEN conjunct1] |
|
423 |
||
13647 | 424 |
(*Kunen's Lemma VI 1.2*) |
46823 | 425 |
lemma "[| p \<in> formula; env \<in> list(A); arity(p) \<le> succ(length(env)) |] |
13223 | 426 |
==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)" |
427 |
by (blast intro: DPowI) |
|
428 |
||
46823 | 429 |
lemma DPow_subset_Pow: "DPow(A) \<subseteq> Pow(A)" |
13223 | 430 |
by (simp add: DPow_def, blast) |
431 |
||
432 |
lemma empty_in_DPow: "0 \<in> DPow(A)" |
|
433 |
apply (simp add: DPow_def) |
|
46823 | 434 |
apply (rule_tac x=Nil in bexI) |
435 |
apply (rule_tac x="Neg(Equal(0,0))" in bexI) |
|
436 |
apply (auto simp add: Un_least_lt_iff) |
|
13223 | 437 |
done |
438 |
||
439 |
lemma Compl_in_DPow: "X \<in> DPow(A) ==> (A-X) \<in> DPow(A)" |
|
46823 | 440 |
apply (simp add: DPow_def, clarify, auto) |
441 |
apply (rule bexI) |
|
442 |
apply (rule_tac x="Neg(p)" in bexI) |
|
443 |
apply auto |
|
13223 | 444 |
done |
445 |
||
46823 | 446 |
lemma Int_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X \<inter> Y \<in> DPow(A)" |
447 |
apply (simp add: DPow_def, auto) |
|
448 |
apply (rename_tac envp p envq q) |
|
449 |
apply (rule_tac x="envp@envq" in bexI) |
|
13223 | 450 |
apply (rule_tac x="And(p, iterates(incr_bv1,length(envp),q))" in bexI) |
451 |
apply typecheck |
|
46823 | 452 |
apply (rule conjI) |
13223 | 453 |
(*finally check the arity!*) |
454 |
apply (simp add: arity_iterates_incr_bv1_eq length_app Un_least_lt_iff) |
|
46823 | 455 |
apply (force intro: add_le_self le_trans) |
456 |
apply (simp add: arity_sats1_iff formula_add_params1, blast) |
|
13223 | 457 |
done |
458 |
||
46823 | 459 |
lemma Un_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X \<union> Y \<in> DPow(A)" |
460 |
apply (subgoal_tac "X \<union> Y = A - ((A-X) \<inter> (A-Y))") |
|
461 |
apply (simp add: Int_in_DPow Compl_in_DPow) |
|
462 |
apply (simp add: DPow_def, blast) |
|
13223 | 463 |
done |
464 |
||
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|
465 |
lemma singleton_in_DPow: "a \<in> A ==> {a} \<in> DPow(A)" |
13223 | 466 |
apply (simp add: DPow_def) |
46823 | 467 |
apply (rule_tac x="Cons(a,Nil)" in bexI) |
468 |
apply (rule_tac x="Equal(0,1)" in bexI) |
|
13223 | 469 |
apply typecheck |
46823 | 470 |
apply (force simp add: succ_Un_distrib [symmetric]) |
13223 | 471 |
done |
472 |
||
473 |
lemma cons_in_DPow: "[| a \<in> A; X \<in> DPow(A) |] ==> cons(a,X) \<in> DPow(A)" |
|
46823 | 474 |
apply (rule cons_eq [THEN subst]) |
475 |
apply (blast intro: singleton_in_DPow Un_in_DPow) |
|
13223 | 476 |
done |
477 |
||
478 |
(*Part of Lemma 1.3*) |
|
479 |
lemma Fin_into_DPow: "X \<in> Fin(A) ==> X \<in> DPow(A)" |
|
46823 | 480 |
apply (erule Fin.induct) |
481 |
apply (rule empty_in_DPow) |
|
482 |
apply (blast intro: cons_in_DPow) |
|
13223 | 483 |
done |
484 |
||
60770 | 485 |
text\<open>@{term DPow} is not monotonic. For example, let @{term A} be some |
13651
ac80e101306a
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|
486 |
non-constructible set of natural numbers, and let @{term B} be @{term nat}. |
46823 | 487 |
Then @{term "A<=B"} and obviously @{term "A \<in> DPow(A)"} but @{term "A \<notin> |
60770 | 488 |
DPow(B)"}.\<close> |
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|
489 |
|
46823 | 490 |
(*This may be true but the proof looks difficult, requiring relativization |
46953 | 491 |
lemma DPow_insert: "DPow (cons(a,A)) = DPow(A) \<union> {cons(a,X) . X \<in> DPow(A)}" |
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|
492 |
apply (rule equalityI, safe) |
13223 | 493 |
oops |
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|
494 |
*) |
13223 | 495 |
|
46823 | 496 |
lemma Finite_Pow_subset_Pow: "Finite(A) ==> Pow(A) \<subseteq> DPow(A)" |
13223 | 497 |
by (blast intro: Fin_into_DPow Finite_into_Fin Fin_subset) |
498 |
||
499 |
lemma Finite_DPow_eq_Pow: "Finite(A) ==> DPow(A) = Pow(A)" |
|
46823 | 500 |
apply (rule equalityI) |
501 |
apply (rule DPow_subset_Pow) |
|
502 |
apply (erule Finite_Pow_subset_Pow) |
|
13223 | 503 |
done |
504 |
||
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|
505 |
|
60770 | 506 |
subsection\<open>Internalized Formulas for the Ordinals\<close> |
13223 | 507 |
|
61798 | 508 |
text\<open>The \<open>sats\<close> theorems below differ from the usual form in that they |
13651
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|
509 |
include an element of absoluteness. That is, they relate internalized |
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|
510 |
formulas to real concepts such as the subset relation, rather than to the |
61798 | 511 |
relativized concepts defined in theory \<open>Relative\<close>. This lets us prove |
512 |
the theorem as \<open>Ords_in_DPow\<close> without first having to instantiate the |
|
513 |
locale \<open>M_trivial\<close>. Note that the present theory does not even take |
|
514 |
\<open>Relative\<close> as a parent.\<close> |
|
13298 | 515 |
|
60770 | 516 |
subsubsection\<open>The subset relation\<close> |
13298 | 517 |
|
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|
518 |
definition |
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|
519 |
subset_fm :: "[i,i]=>i" where |
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|
520 |
"subset_fm(x,y) == Forall(Implies(Member(0,succ(x)), Member(0,succ(y))))" |
13298 | 521 |
|
522 |
lemma subset_type [TC]: "[| x \<in> nat; y \<in> nat |] ==> subset_fm(x,y) \<in> formula" |
|
46823 | 523 |
by (simp add: subset_fm_def) |
13298 | 524 |
|
525 |
lemma arity_subset_fm [simp]: |
|
526 |
"[| x \<in> nat; y \<in> nat |] ==> arity(subset_fm(x,y)) = succ(x) \<union> succ(y)" |
|
46823 | 527 |
by (simp add: subset_fm_def succ_Un_distrib [symmetric]) |
13298 | 528 |
|
529 |
lemma sats_subset_fm [simp]: |
|
530 |
"[|x < length(env); y \<in> nat; env \<in> list(A); Transset(A)|] |
|
46823 | 531 |
==> sats(A, subset_fm(x,y), env) \<longleftrightarrow> nth(x,env) \<subseteq> nth(y,env)" |
532 |
apply (frule lt_length_in_nat, assumption) |
|
533 |
apply (simp add: subset_fm_def Transset_def) |
|
534 |
apply (blast intro: nth_type) |
|
13298 | 535 |
done |
536 |
||
60770 | 537 |
subsubsection\<open>Transitive sets\<close> |
13298 | 538 |
|
21404
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|
539 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
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|
540 |
transset_fm :: "i=>i" where |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
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changeset
|
541 |
"transset_fm(x) == Forall(Implies(Member(0,succ(x)), subset_fm(0,succ(x))))" |
13298 | 542 |
|
543 |
lemma transset_type [TC]: "x \<in> nat ==> transset_fm(x) \<in> formula" |
|
46823 | 544 |
by (simp add: transset_fm_def) |
13298 | 545 |
|
546 |
lemma arity_transset_fm [simp]: |
|
547 |
"x \<in> nat ==> arity(transset_fm(x)) = succ(x)" |
|
46823 | 548 |
by (simp add: transset_fm_def succ_Un_distrib [symmetric]) |
13298 | 549 |
|
550 |
lemma sats_transset_fm [simp]: |
|
551 |
"[|x < length(env); env \<in> list(A); Transset(A)|] |
|
46823 | 552 |
==> sats(A, transset_fm(x), env) \<longleftrightarrow> Transset(nth(x,env))" |
553 |
apply (frule lt_nat_in_nat, erule length_type) |
|
554 |
apply (simp add: transset_fm_def Transset_def) |
|
555 |
apply (blast intro: nth_type) |
|
13298 | 556 |
done |
557 |
||
60770 | 558 |
subsubsection\<open>Ordinals\<close> |
13298 | 559 |
|
21404
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|
560 |
definition |
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|
561 |
ordinal_fm :: "i=>i" where |
46823 | 562 |
"ordinal_fm(x) == |
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|
563 |
And(transset_fm(x), Forall(Implies(Member(0,succ(x)), transset_fm(0))))" |
13298 | 564 |
|
565 |
lemma ordinal_type [TC]: "x \<in> nat ==> ordinal_fm(x) \<in> formula" |
|
46823 | 566 |
by (simp add: ordinal_fm_def) |
13298 | 567 |
|
568 |
lemma arity_ordinal_fm [simp]: |
|
569 |
"x \<in> nat ==> arity(ordinal_fm(x)) = succ(x)" |
|
46823 | 570 |
by (simp add: ordinal_fm_def succ_Un_distrib [symmetric]) |
13298 | 571 |
|
13306 | 572 |
lemma sats_ordinal_fm: |
13298 | 573 |
"[|x < length(env); env \<in> list(A); Transset(A)|] |
46823 | 574 |
==> sats(A, ordinal_fm(x), env) \<longleftrightarrow> Ord(nth(x,env))" |
575 |
apply (frule lt_nat_in_nat, erule length_type) |
|
13298 | 576 |
apply (simp add: ordinal_fm_def Ord_def Transset_def) |
46823 | 577 |
apply (blast intro: nth_type) |
13298 | 578 |
done |
579 |
||
60770 | 580 |
text\<open>The subset consisting of the ordinals is definable. Essential lemma for |
61798 | 581 |
\<open>Ord_in_Lset\<close>. This result is the objective of the present subsection.\<close> |
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|
582 |
theorem Ords_in_DPow: "Transset(A) ==> {x \<in> A. Ord(x)} \<in> DPow(A)" |
46823 | 583 |
apply (simp add: DPow_def Collect_subset) |
584 |
apply (rule_tac x=Nil in bexI) |
|
585 |
apply (rule_tac x="ordinal_fm(0)" in bexI) |
|
13651
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Cosmetic changes suggested by writing the paper. Deleted some
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changeset
|
586 |
apply (simp_all add: sats_ordinal_fm) |
46823 | 587 |
done |
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Cosmetic changes suggested by writing the paper. Deleted some
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changeset
|
588 |
|
13298 | 589 |
|
60770 | 590 |
subsection\<open>Constant Lset: Levels of the Constructible Universe\<close> |
13223 | 591 |
|
21233 | 592 |
definition |
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more robust syntax for definition/abbreviation/notation;
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|
593 |
Lset :: "i=>i" where |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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changeset
|
594 |
"Lset(i) == transrec(i, %x f. \<Union>y\<in>x. DPow(f`y))" |
13223 | 595 |
|
21404
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more robust syntax for definition/abbreviation/notation;
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|
596 |
definition |
61798 | 597 |
L :: "i=>o" where \<comment>\<open>Kunen's definition VI 1.5, page 167\<close> |
21404
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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
598 |
"L(x) == \<exists>i. Ord(i) & x \<in> Lset(i)" |
46823 | 599 |
|
60770 | 600 |
text\<open>NOT SUITABLE FOR REWRITING -- RECURSIVE!\<close> |
46823 | 601 |
lemma Lset: "Lset(i) = (\<Union>j\<in>i. DPow(Lset(j)))" |
13223 | 602 |
by (subst Lset_def [THEN def_transrec], simp) |
603 |
||
58860 | 604 |
lemma LsetI: "[|y\<in>x; A \<in> DPow(Lset(y))|] ==> A \<in> Lset(x)" |
13223 | 605 |
by (subst Lset, blast) |
606 |
||
58860 | 607 |
lemma LsetD: "A \<in> Lset(x) ==> \<exists>y\<in>x. A \<in> DPow(Lset(y))" |
46823 | 608 |
apply (insert Lset [of x]) |
609 |
apply (blast intro: elim: equalityE) |
|
13223 | 610 |
done |
611 |
||
60770 | 612 |
subsubsection\<open>Transitivity\<close> |
13223 | 613 |
|
614 |
lemma elem_subset_in_DPow: "[|X \<in> A; X \<subseteq> A|] ==> X \<in> DPow(A)" |
|
615 |
apply (simp add: Transset_def DPow_def) |
|
46823 | 616 |
apply (rule_tac x="[X]" in bexI) |
617 |
apply (rule_tac x="Member(0,1)" in bexI) |
|
618 |
apply (auto simp add: Un_least_lt_iff) |
|
13223 | 619 |
done |
620 |
||
46823 | 621 |
lemma Transset_subset_DPow: "Transset(A) ==> A \<subseteq> DPow(A)" |
622 |
apply clarify |
|
13223 | 623 |
apply (simp add: Transset_def) |
46823 | 624 |
apply (blast intro: elem_subset_in_DPow) |
13223 | 625 |
done |
626 |
||
627 |
lemma Transset_DPow: "Transset(A) ==> Transset(DPow(A))" |
|
46823 | 628 |
apply (simp add: Transset_def) |
629 |
apply (blast intro: elem_subset_in_DPow dest: DPowD) |
|
13223 | 630 |
done |
631 |
||
60770 | 632 |
text\<open>Kunen's VI 1.6 (a)\<close> |
13223 | 633 |
lemma Transset_Lset: "Transset(Lset(i))" |
634 |
apply (rule_tac a=i in eps_induct) |
|
635 |
apply (subst Lset) |
|
636 |
apply (blast intro!: Transset_Union_family Transset_Un Transset_DPow) |
|
637 |
done |
|
638 |
||
13291 | 639 |
lemma mem_Lset_imp_subset_Lset: "a \<in> Lset(i) ==> a \<subseteq> Lset(i)" |
46823 | 640 |
apply (insert Transset_Lset) |
641 |
apply (simp add: Transset_def) |
|
13291 | 642 |
done |
643 |
||
60770 | 644 |
subsubsection\<open>Monotonicity\<close> |
13223 | 645 |
|
60770 | 646 |
text\<open>Kunen's VI 1.6 (b)\<close> |
13223 | 647 |
lemma Lset_mono [rule_format]: |
46823 | 648 |
"\<forall>j. i<=j \<longrightarrow> Lset(i) \<subseteq> Lset(j)" |
15481 | 649 |
proof (induct i rule: eps_induct, intro allI impI) |
650 |
fix x j |
|
651 |
assume "\<forall>y\<in>x. \<forall>j. y \<subseteq> j \<longrightarrow> Lset(y) \<subseteq> Lset(j)" |
|
652 |
and "x \<subseteq> j" |
|
653 |
thus "Lset(x) \<subseteq> Lset(j)" |
|
46823 | 654 |
by (force simp add: Lset [of x] Lset [of j]) |
15481 | 655 |
qed |
13223 | 656 |
|
60770 | 657 |
text\<open>This version lets us remove the premise @{term "Ord(i)"} sometimes.\<close> |
13223 | 658 |
lemma Lset_mono_mem [rule_format]: |
46953 | 659 |
"\<forall>j. i \<in> j \<longrightarrow> Lset(i) \<subseteq> Lset(j)" |
15481 | 660 |
proof (induct i rule: eps_induct, intro allI impI) |
661 |
fix x j |
|
662 |
assume "\<forall>y\<in>x. \<forall>j. y \<in> j \<longrightarrow> Lset(y) \<subseteq> Lset(j)" |
|
663 |
and "x \<in> j" |
|
664 |
thus "Lset(x) \<subseteq> Lset(j)" |
|
46823 | 665 |
by (force simp add: Lset [of j] |
666 |
intro!: bexI intro: elem_subset_in_DPow dest: LsetD DPowD) |
|
15481 | 667 |
qed |
668 |
||
13223 | 669 |
|
60770 | 670 |
text\<open>Useful with Reflection to bump up the ordinal\<close> |
13291 | 671 |
lemma subset_Lset_ltD: "[|A \<subseteq> Lset(i); i < j|] ==> A \<subseteq> Lset(j)" |
46823 | 672 |
by (blast dest: ltD [THEN Lset_mono_mem]) |
13291 | 673 |
|
60770 | 674 |
subsubsection\<open>0, successor and limit equations for Lset\<close> |
13223 | 675 |
|
676 |
lemma Lset_0 [simp]: "Lset(0) = 0" |
|
677 |
by (subst Lset, blast) |
|
678 |
||
46823 | 679 |
lemma Lset_succ_subset1: "DPow(Lset(i)) \<subseteq> Lset(succ(i))" |
13223 | 680 |
by (subst Lset, rule succI1 [THEN RepFunI, THEN Union_upper]) |
681 |
||
46823 | 682 |
lemma Lset_succ_subset2: "Lset(succ(i)) \<subseteq> DPow(Lset(i))" |
13223 | 683 |
apply (subst Lset, rule UN_least) |
46823 | 684 |
apply (erule succE) |
685 |
apply blast |
|
13223 | 686 |
apply clarify |
687 |
apply (rule elem_subset_in_DPow) |
|
688 |
apply (subst Lset) |
|
46823 | 689 |
apply blast |
690 |
apply (blast intro: dest: DPowD Lset_mono_mem) |
|
13223 | 691 |
done |
692 |
||
693 |
lemma Lset_succ: "Lset(succ(i)) = DPow(Lset(i))" |
|
46823 | 694 |
by (intro equalityI Lset_succ_subset1 Lset_succ_subset2) |
13223 | 695 |
|
696 |
lemma Lset_Union [simp]: "Lset(\<Union>(X)) = (\<Union>y\<in>X. Lset(y))" |
|
697 |
apply (subst Lset) |
|
698 |
apply (rule equalityI) |
|
60770 | 699 |
txt\<open>first inclusion\<close> |
13223 | 700 |
apply (rule UN_least) |
701 |
apply (erule UnionE) |
|
702 |
apply (rule subset_trans) |
|
703 |
apply (erule_tac [2] UN_upper, subst Lset, erule UN_upper) |
|
60770 | 704 |
txt\<open>opposite inclusion\<close> |
13223 | 705 |
apply (rule UN_least) |
706 |
apply (subst Lset, blast) |
|
707 |
done |
|
708 |
||
60770 | 709 |
subsubsection\<open>Lset applied to Limit ordinals\<close> |
13223 | 710 |
|
711 |
lemma Limit_Lset_eq: |
|
712 |
"Limit(i) ==> Lset(i) = (\<Union>y\<in>i. Lset(y))" |
|
713 |
by (simp add: Lset_Union [symmetric] Limit_Union_eq) |
|
714 |
||
46953 | 715 |
lemma lt_LsetI: "[| a \<in> Lset(j); j<i |] ==> a \<in> Lset(i)" |
13223 | 716 |
by (blast dest: Lset_mono [OF le_imp_subset [OF leI]]) |
717 |
||
718 |
lemma Limit_LsetE: |
|
46953 | 719 |
"[| a \<in> Lset(i); ~R ==> Limit(i); |
720 |
!!x. [| x<i; a \<in> Lset(x) |] ==> R |
|
13223 | 721 |
|] ==> R" |
722 |
apply (rule classical) |
|
723 |
apply (rule Limit_Lset_eq [THEN equalityD1, THEN subsetD, THEN UN_E]) |
|
724 |
prefer 2 apply assumption |
|
46823 | 725 |
apply blast |
13223 | 726 |
apply (blast intro: ltI Limit_is_Ord) |
727 |
done |
|
728 |
||
60770 | 729 |
subsubsection\<open>Basic closure properties\<close> |
13223 | 730 |
|
46953 | 731 |
lemma zero_in_Lset: "y \<in> x ==> 0 \<in> Lset(x)" |
13223 | 732 |
by (subst Lset, blast intro: empty_in_DPow) |
733 |
||
734 |
lemma notin_Lset: "x \<notin> Lset(x)" |
|
735 |
apply (rule_tac a=x in eps_induct) |
|
736 |
apply (subst Lset) |
|
46823 | 737 |
apply (blast dest: DPowD) |
13223 | 738 |
done |
739 |
||
740 |
||
60770 | 741 |
subsection\<open>Constructible Ordinals: Kunen's VI 1.9 (b)\<close> |
13223 | 742 |
|
743 |
lemma Ords_of_Lset_eq: "Ord(i) ==> {x\<in>Lset(i). Ord(x)} = i" |
|
744 |
apply (erule trans_induct3) |
|
745 |
apply (simp_all add: Lset_succ Limit_Lset_eq Limit_Union_eq) |
|
60770 | 746 |
txt\<open>The successor case remains.\<close> |
13223 | 747 |
apply (rule equalityI) |
60770 | 748 |
txt\<open>First inclusion\<close> |
46823 | 749 |
apply clarify |
750 |
apply (erule Ord_linear_lt, assumption) |
|
751 |
apply (blast dest: DPow_imp_subset ltD notE [OF notin_Lset]) |
|
752 |
apply blast |
|
13223 | 753 |
apply (blast dest: ltD) |
60770 | 754 |
txt\<open>Opposite inclusion, @{term "succ(x) \<subseteq> DPow(Lset(x)) \<inter> ON"}\<close> |
13223 | 755 |
apply auto |
60770 | 756 |
txt\<open>Key case:\<close> |
46823 | 757 |
apply (erule subst, rule Ords_in_DPow [OF Transset_Lset]) |
758 |
apply (blast intro: elem_subset_in_DPow dest: OrdmemD elim: equalityE) |
|
759 |
apply (blast intro: Ord_in_Ord) |
|
13223 | 760 |
done |
761 |
||
762 |
||
763 |
lemma Ord_subset_Lset: "Ord(i) ==> i \<subseteq> Lset(i)" |
|
764 |
by (subst Ords_of_Lset_eq [symmetric], assumption, fast) |
|
765 |
||
766 |
lemma Ord_in_Lset: "Ord(i) ==> i \<in> Lset(succ(i))" |
|
767 |
apply (simp add: Lset_succ) |
|
46823 | 768 |
apply (subst Ords_of_Lset_eq [symmetric], assumption, |
769 |
rule Ords_in_DPow [OF Transset_Lset]) |
|
13223 | 770 |
done |
771 |
||
13651
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents:
13647
diff
changeset
|
772 |
lemma Ord_in_L: "Ord(i) ==> L(i)" |
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents:
13647
diff
changeset
|
773 |
by (simp add: L_def, blast intro: Ord_in_Lset) |
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents:
13647
diff
changeset
|
774 |
|
60770 | 775 |
subsubsection\<open>Unions\<close> |
13223 | 776 |
|
777 |
lemma Union_in_Lset: |
|
46823 | 778 |
"X \<in> Lset(i) ==> \<Union>(X) \<in> Lset(succ(i))" |
13223 | 779 |
apply (insert Transset_Lset) |
780 |
apply (rule LsetI [OF succI1]) |
|
46823 | 781 |
apply (simp add: Transset_def DPow_def) |
13223 | 782 |
apply (intro conjI, blast) |
60770 | 783 |
txt\<open>Now to create the formula @{term "\<exists>y. y \<in> X \<and> x \<in> y"}\<close> |
46823 | 784 |
apply (rule_tac x="Cons(X,Nil)" in bexI) |
785 |
apply (rule_tac x="Exists(And(Member(0,2), Member(1,0)))" in bexI) |
|
13223 | 786 |
apply typecheck |
46823 | 787 |
apply (simp add: succ_Un_distrib [symmetric], blast) |
13223 | 788 |
done |
789 |
||
46823 | 790 |
theorem Union_in_L: "L(X) ==> L(\<Union>(X))" |
791 |
by (simp add: L_def, blast dest: Union_in_Lset) |
|
13651
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents:
13647
diff
changeset
|
792 |
|
60770 | 793 |
subsubsection\<open>Finite sets and ordered pairs\<close> |
13223 | 794 |
|
46953 | 795 |
lemma singleton_in_Lset: "a \<in> Lset(i) ==> {a} \<in> Lset(succ(i))" |
46823 | 796 |
by (simp add: Lset_succ singleton_in_DPow) |
13223 | 797 |
|
798 |
lemma doubleton_in_Lset: |
|
46953 | 799 |
"[| a \<in> Lset(i); b \<in> Lset(i) |] ==> {a,b} \<in> Lset(succ(i))" |
46823 | 800 |
by (simp add: Lset_succ empty_in_DPow cons_in_DPow) |
13223 | 801 |
|
802 |
lemma Pair_in_Lset: |
|
46953 | 803 |
"[| a \<in> Lset(i); b \<in> Lset(i); Ord(i) |] ==> <a,b> \<in> Lset(succ(succ(i)))" |
13223 | 804 |
apply (unfold Pair_def) |
46823 | 805 |
apply (blast intro: doubleton_in_Lset) |
13223 | 806 |
done |
807 |
||
45602 | 808 |
lemmas Lset_UnI1 = Un_upper1 [THEN Lset_mono [THEN subsetD]] |
809 |
lemmas Lset_UnI2 = Un_upper2 [THEN Lset_mono [THEN subsetD]] |
|
13223 | 810 |
|
60770 | 811 |
text\<open>Hard work is finding a single @{term"j \<in> i"} such that @{term"{a,b} \<subseteq> Lset(j)"}\<close> |
13223 | 812 |
lemma doubleton_in_LLimit: |
46953 | 813 |
"[| a \<in> Lset(i); b \<in> Lset(i); Limit(i) |] ==> {a,b} \<in> Lset(i)" |
13223 | 814 |
apply (erule Limit_LsetE, assumption) |
815 |
apply (erule Limit_LsetE, assumption) |
|
13269 | 816 |
apply (blast intro: lt_LsetI [OF doubleton_in_Lset] |
817 |
Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt) |
|
13223 | 818 |
done |
819 |
||
13651
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents:
13647
diff
changeset
|
820 |
theorem doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})" |
46823 | 821 |
apply (simp add: L_def, clarify) |
822 |
apply (drule Ord2_imp_greater_Limit, assumption) |
|
823 |
apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord) |
|
13651
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents:
13647
diff
changeset
|
824 |
done |
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents:
13647
diff
changeset
|
825 |
|
13223 | 826 |
lemma Pair_in_LLimit: |
46953 | 827 |
"[| a \<in> Lset(i); b \<in> Lset(i); Limit(i) |] ==> <a,b> \<in> Lset(i)" |
60770 | 828 |
txt\<open>Infer that a, b occur at ordinals x,xa < i.\<close> |
13223 | 829 |
apply (erule Limit_LsetE, assumption) |
830 |
apply (erule Limit_LsetE, assumption) |
|
60770 | 831 |
txt\<open>Infer that @{term"succ(succ(x \<union> xa)) < i"}\<close> |
13223 | 832 |
apply (blast intro: lt_Ord lt_LsetI [OF Pair_in_Lset] |
833 |
Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt) |
|
834 |
done |
|
835 |
||
836 |
||
837 |
||
60770 | 838 |
text\<open>The rank function for the constructible universe\<close> |
21233 | 839 |
definition |
61798 | 840 |
lrank :: "i=>i" where \<comment>\<open>Kunen's definition VI 1.7\<close> |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
841 |
"lrank(x) == \<mu> i. x \<in> Lset(succ(i))" |
13223 | 842 |
|
843 |
lemma L_I: "[|x \<in> Lset(i); Ord(i)|] ==> L(x)" |
|
844 |
by (simp add: L_def, blast) |
|
845 |
||
846 |
lemma L_D: "L(x) ==> \<exists>i. Ord(i) & x \<in> Lset(i)" |
|
847 |
by (simp add: L_def) |
|
848 |
||
849 |
lemma Ord_lrank [simp]: "Ord(lrank(a))" |
|
850 |
by (simp add: lrank_def) |
|
851 |
||
46823 | 852 |
lemma Lset_lrank_lt [rule_format]: "Ord(i) ==> x \<in> Lset(i) \<longrightarrow> lrank(x) < i" |
13223 | 853 |
apply (erule trans_induct3) |
46823 | 854 |
apply simp |
855 |
apply (simp only: lrank_def) |
|
856 |
apply (blast intro: Least_le) |
|
857 |
apply (simp_all add: Limit_Lset_eq) |
|
858 |
apply (blast intro: ltI Limit_is_Ord lt_trans) |
|
13223 | 859 |
done |
860 |
||
60770 | 861 |
text\<open>Kunen's VI 1.8. The proof is much harder than the text would |
13651
ac80e101306a
Cosmetic changes suggested by writing the paper. Deleted some
paulson
parents:
13647
diff
changeset
|
862 |
suggest. For a start, it needs the previous lemma, which is proved by |
60770 | 863 |
induction.\<close> |
46823 | 864 |
lemma Lset_iff_lrank_lt: "Ord(i) ==> x \<in> Lset(i) \<longleftrightarrow> L(x) & lrank(x) < i" |
865 |
apply (simp add: L_def, auto) |
|
866 |
apply (blast intro: Lset_lrank_lt) |
|
867 |
apply (unfold lrank_def) |
|
868 |
apply (drule succI1 [THEN Lset_mono_mem, THEN subsetD]) |
|
869 |
apply (drule_tac P="\<lambda>i. x \<in> Lset(succ(i))" in LeastI, assumption) |
|
870 |
apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) |
|
13223 | 871 |
done |
872 |
||
46823 | 873 |
lemma Lset_succ_lrank_iff [simp]: "x \<in> Lset(succ(lrank(x))) \<longleftrightarrow> L(x)" |
13223 | 874 |
by (simp add: Lset_iff_lrank_lt) |
875 |
||
60770 | 876 |
text\<open>Kunen's VI 1.9 (a)\<close> |
13223 | 877 |
lemma lrank_of_Ord: "Ord(i) ==> lrank(i) = i" |
46823 | 878 |
apply (unfold lrank_def) |
879 |
apply (rule Least_equality) |
|
880 |
apply (erule Ord_in_Lset) |
|
13223 | 881 |
apply assumption |
46823 | 882 |
apply (insert notin_Lset [of i]) |
883 |
apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) |
|
13223 | 884 |
done |
885 |
||
13245 | 886 |
|
60770 | 887 |
text\<open>This is lrank(lrank(a)) = lrank(a)\<close> |
13223 | 888 |
declare Ord_lrank [THEN lrank_of_Ord, simp] |
889 |
||
60770 | 890 |
text\<open>Kunen's VI 1.10\<close> |
58860 | 891 |
lemma Lset_in_Lset_succ: "Lset(i) \<in> Lset(succ(i))" |
46823 | 892 |
apply (simp add: Lset_succ DPow_def) |
893 |
apply (rule_tac x=Nil in bexI) |
|
894 |
apply (rule_tac x="Equal(0,0)" in bexI) |
|
895 |
apply auto |
|
13223 | 896 |
done |
897 |
||
898 |
lemma lrank_Lset: "Ord(i) ==> lrank(Lset(i)) = i" |
|
46823 | 899 |
apply (unfold lrank_def) |
900 |
apply (rule Least_equality) |
|
901 |
apply (rule Lset_in_Lset_succ) |
|
13223 | 902 |
apply assumption |
46823 | 903 |
apply clarify |
904 |
apply (subgoal_tac "Lset(succ(ia)) \<subseteq> Lset(i)") |
|
905 |
apply (blast dest: mem_irrefl) |
|
906 |
apply (blast intro!: le_imp_subset Lset_mono) |
|
13223 | 907 |
done |
908 |
||
60770 | 909 |
text\<open>Kunen's VI 1.11\<close> |
58860 | 910 |
lemma Lset_subset_Vset: "Ord(i) ==> Lset(i) \<subseteq> Vset(i)" |
13223 | 911 |
apply (erule trans_induct) |
46823 | 912 |
apply (subst Lset) |
913 |
apply (subst Vset) |
|
914 |
apply (rule UN_mono [OF subset_refl]) |
|
915 |
apply (rule subset_trans [OF DPow_subset_Pow]) |
|
916 |
apply (rule Pow_mono, blast) |
|
13223 | 917 |
done |
918 |
||
60770 | 919 |
text\<open>Kunen's VI 1.12\<close> |
58860 | 920 |
lemma Lset_subset_Vset': "i \<in> nat ==> Lset(i) = Vset(i)" |
13223 | 921 |
apply (erule nat_induct) |
46823 | 922 |
apply (simp add: Vfrom_0) |
923 |
apply (simp add: Lset_succ Vset_succ Finite_Vset Finite_DPow_eq_Pow) |
|
13223 | 924 |
done |
925 |
||
60770 | 926 |
text\<open>Every set of constructible sets is included in some @{term Lset}\<close> |
13291 | 927 |
lemma subset_Lset: |
928 |
"(\<forall>x\<in>A. L(x)) ==> \<exists>i. Ord(i) & A \<subseteq> Lset(i)" |
|
929 |
by (rule_tac x = "\<Union>x\<in>A. succ(lrank(x))" in exI, force) |
|
930 |
||
931 |
lemma subset_LsetE: |
|
932 |
"[|\<forall>x\<in>A. L(x); |
|
933 |
!!i. [|Ord(i); A \<subseteq> Lset(i)|] ==> P|] |
|
934 |
==> P" |
|
46823 | 935 |
by (blast dest: subset_Lset) |
13291 | 936 |
|
60770 | 937 |
subsubsection\<open>For L to satisfy the Powerset axiom\<close> |
13223 | 938 |
|
939 |
lemma LPow_env_typing: |
|
46823 | 940 |
"[| y \<in> Lset(i); Ord(i); y \<subseteq> X |] |
13511 | 941 |
==> \<exists>z \<in> Pow(X). y \<in> Lset(succ(lrank(z)))" |
46823 | 942 |
by (auto intro: L_I iff: Lset_succ_lrank_iff) |
13223 | 943 |
|
944 |
lemma LPow_in_Lset: |
|
945 |
"[|X \<in> Lset(i); Ord(i)|] ==> \<exists>j. Ord(j) & {y \<in> Pow(X). L(y)} \<in> Lset(j)" |
|
946 |
apply (rule_tac x="succ(\<Union>y \<in> Pow(X). succ(lrank(y)))" in exI) |
|
46823 | 947 |
apply simp |
13223 | 948 |
apply (rule LsetI [OF succI1]) |
46823 | 949 |
apply (simp add: DPow_def) |
950 |
apply (intro conjI, clarify) |
|
951 |
apply (rule_tac a=x in UN_I, simp+) |
|
60770 | 952 |
txt\<open>Now to create the formula @{term "y \<subseteq> X"}\<close> |
46823 | 953 |
apply (rule_tac x="Cons(X,Nil)" in bexI) |
954 |
apply (rule_tac x="subset_fm(0,1)" in bexI) |
|
13223 | 955 |
apply typecheck |
46823 | 956 |
apply (rule conjI) |
957 |
apply (simp add: succ_Un_distrib [symmetric]) |
|
958 |
apply (rule equality_iffI) |
|
13511 | 959 |
apply (simp add: Transset_UN [OF Transset_Lset] LPow_env_typing) |
46823 | 960 |
apply (auto intro: L_I iff: Lset_succ_lrank_iff) |
13223 | 961 |
done |
962 |
||
13245 | 963 |
theorem LPow_in_L: "L(X) ==> L({y \<in> Pow(X). L(y)})" |
13223 | 964 |
by (blast intro: L_I dest: L_D LPow_in_Lset) |
965 |
||
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
966 |
|
60770 | 967 |
subsection\<open>Eliminating @{term arity} from the Definition of @{term Lset}\<close> |
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
968 |
|
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
969 |
lemma nth_zero_eq_0: "n \<in> nat ==> nth(n,[0]) = 0" |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
970 |
by (induct_tac n, auto) |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
971 |
|
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
972 |
lemma sats_app_0_iff [rule_format]: |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
973 |
"[| p \<in> formula; 0 \<in> A |] |
46823 | 974 |
==> \<forall>env \<in> list(A). sats(A,p, env@[0]) \<longleftrightarrow> sats(A,p,env)" |
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
975 |
apply (induct_tac p) |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
976 |
apply (simp_all del: app_Cons add: app_Cons [symmetric] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
21404
diff
changeset
|
977 |
add: nth_zero_eq_0 nth_append not_lt_iff_le nth_eq_0) |
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
978 |
done |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
979 |
|
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
980 |
lemma sats_app_zeroes_iff: |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
981 |
"[| p \<in> formula; 0 \<in> A; env \<in> list(A); n \<in> nat |] |
46823 | 982 |
==> sats(A,p,env @ repeat(0,n)) \<longleftrightarrow> sats(A,p,env)" |
983 |
apply (induct_tac n, simp) |
|
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
984 |
apply (simp del: repeat.simps |
46823 | 985 |
add: repeat_succ_app sats_app_0_iff app_assoc [symmetric]) |
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
986 |
done |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
987 |
|
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
988 |
lemma exists_bigger_env: |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
989 |
"[| p \<in> formula; 0 \<in> A; env \<in> list(A) |] |
46823 | 990 |
==> \<exists>env' \<in> list(A). arity(p) \<le> succ(length(env')) & |
991 |
(\<forall>a\<in>A. sats(A,p,Cons(a,env')) \<longleftrightarrow> sats(A,p,Cons(a,env)))" |
|
992 |
apply (rule_tac x="env @ repeat(0,arity(p))" in bexI) |
|
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
993 |
apply (simp del: app_Cons add: app_Cons [symmetric] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
21404
diff
changeset
|
994 |
add: length_repeat sats_app_zeroes_iff, typecheck) |
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
995 |
done |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
996 |
|
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
997 |
|
60770 | 998 |
text\<open>A simpler version of @{term DPow}: no arity check!\<close> |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
999 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21233
diff
changeset
|
1000 |
DPow' :: "i => i" where |
46823 | 1001 |
"DPow'(A) == {X \<in> Pow(A). |
1002 |
\<exists>env \<in> list(A). \<exists>p \<in> formula. |
|
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1003 |
X = {x\<in>A. sats(A, p, Cons(x,env))}}" |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1004 |
|
58860 | 1005 |
lemma DPow_subset_DPow': "DPow(A) \<subseteq> DPow'(A)" |
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1006 |
by (simp add: DPow_def DPow'_def, blast) |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1007 |
|
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1008 |
lemma DPow'_0: "DPow'(0) = {0}" |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1009 |
by (auto simp add: DPow'_def) |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1010 |
|
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1011 |
lemma DPow'_subset_DPow: "0 \<in> A ==> DPow'(A) \<subseteq> DPow(A)" |
46823 | 1012 |
apply (auto simp add: DPow'_def DPow_def) |
1013 |
apply (frule exists_bigger_env, assumption+, force) |
|
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1014 |
done |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1015 |
|
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1016 |
lemma DPow_eq_DPow': "Transset(A) ==> DPow(A) = DPow'(A)" |
46823 | 1017 |
apply (drule Transset_0_disj) |
1018 |
apply (erule disjE) |
|
1019 |
apply (simp add: DPow'_0 Finite_DPow_eq_Pow) |
|
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1020 |
apply (rule equalityI) |
46823 | 1021 |
apply (rule DPow_subset_DPow') |
1022 |
apply (erule DPow'_subset_DPow) |
|
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1023 |
done |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1024 |
|
60770 | 1025 |
text\<open>And thus we can relativize @{term Lset} without bothering with |
1026 |
@{term arity} and @{term length}\<close> |
|
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1027 |
lemma Lset_eq_transrec_DPow': "Lset(i) = transrec(i, %x f. \<Union>y\<in>x. DPow'(f`y))" |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1028 |
apply (rule_tac a=i in eps_induct) |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1029 |
apply (subst Lset) |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1030 |
apply (subst transrec) |
46823 | 1031 |
apply (simp only: DPow_eq_DPow' [OF Transset_Lset], simp) |
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1032 |
done |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1033 |
|
60770 | 1034 |
text\<open>With this rule we can specify @{term p} later and don't worry about |
1035 |
arities at all!\<close> |
|
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1036 |
lemma DPow_LsetI [rule_format]: |
46823 | 1037 |
"[|\<forall>x\<in>Lset(i). P(x) \<longleftrightarrow> sats(Lset(i), p, Cons(x,env)); |
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1038 |
env \<in> list(Lset(i)); p \<in> formula|] |
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1039 |
==> {x\<in>Lset(i). P(x)} \<in> DPow(Lset(i))" |
46823 | 1040 |
by (simp add: DPow_eq_DPow' [OF Transset_Lset] DPow'_def, blast) |
13385
31df66ca0780
Expressing Lset and L without using length and arity; simplifies Separation
paulson
parents:
13339
diff
changeset
|
1041 |
|
13223 | 1042 |
end |