Theory Formula (Isabelle2005: October 2005)

# Theory Formula

Up to index of Isabelle/ZF/Constructible

theory Formula
imports Main
begin

    (*  Title:      ZF/Constructible/Formula.thy
        ID: $Id: Formula.thy,v 1.22 2005/06/17 14:15:10 haftmann Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    *)

    header {* First-Order Formulas and the Definition of the Class L *}

    theory Formula imports Main begin

    subsection{*Internalized formulas of FOL*}

    text{*De Bruijn representation.
      Unbound variables get their denotations from an environment.*}

    consts   formula :: i
    datatype
      "formula" = Member ("x: nat", "y: nat")
                | Equal  ("x: nat", "y: nat")
                | Nand ("p: formula", "q: formula")
                | Forall ("p: formula")

    declare formula.intros [TC]

    constdefs Neg :: "i=>i"
        "Neg(p) == Nand(p,p)"

    constdefs And :: "[i,i]=>i"
        "And(p,q) == Neg(Nand(p,q))"

    constdefs Or :: "[i,i]=>i"
        "Or(p,q) == Nand(Neg(p),Neg(q))"

    constdefs Implies :: "[i,i]=>i"
        "Implies(p,q) == Nand(p,Neg(q))"

    constdefs Iff :: "[i,i]=>i"
        "Iff(p,q) == And(Implies(p,q), Implies(q,p))"

    constdefs Exists :: "i=>i"
        "Exists(p) == Neg(Forall(Neg(p)))";

    lemma Neg_type [TC]: "p ∈ formula ==> Neg(p) ∈ formula"
    by (simp add: Neg_def)

    lemma And_type [TC]: "[| p ∈ formula; q ∈ formula |] ==> And(p,q) ∈ formula"
    by (simp add: And_def)

    lemma Or_type [TC]: "[| p ∈ formula; q ∈ formula |] ==> Or(p,q) ∈ formula"
    by (simp add: Or_def)

    lemma Implies_type [TC]:
         "[| p ∈ formula; q ∈ formula |] ==> Implies(p,q) ∈ formula"
    by (simp add: Implies_def)

    lemma Iff_type [TC]:
         "[| p ∈ formula; q ∈ formula |] ==> Iff(p,q) ∈ formula"
    by (simp add: Iff_def)

    lemma Exists_type [TC]: "p ∈ formula ==> Exists(p) ∈ formula"
    by (simp add: Exists_def)

    consts   satisfies :: "[i,i]=>i"
    primrec (*explicit lambda is required because the environment varies*)
      "satisfies(A,Member(x,y)) =
          (λenv ∈ list(A). bool_of_o (nth(x,env) ∈ nth(y,env)))"

      "satisfies(A,Equal(x,y)) =
          (λenv ∈ list(A). bool_of_o (nth(x,env) = nth(y,env)))"

      "satisfies(A,Nand(p,q)) =
          (λenv ∈ list(A). not ((satisfies(A,p)`env) and (satisfies(A,q)`env)))"

      "satisfies(A,Forall(p)) =
          (λenv ∈ list(A). bool_of_o (∀x∈A. satisfies(A,p) ` (Cons(x,env)) = 1))"

    lemma "p ∈ formula ==> satisfies(A,p) ∈ list(A) -> bool"
    by (induct_tac p, simp_all)

    syntax sats :: "[i,i,i] => o"
    translations "sats(A,p,env)" == "satisfies(A,p)`env = 1"

    lemma [simp]:
      "env ∈ list(A)
       ==> sats(A, Member(x,y), env) <-> nth(x,env) ∈ nth(y,env)"
    by simp

    lemma [simp]:
      "env ∈ list(A)
       ==> sats(A, Equal(x,y), env) <-> nth(x,env) = nth(y,env)"
    by simp

    lemma sats_Nand_iff [simp]:
      "env ∈ list(A)
       ==> (sats(A, Nand(p,q), env)) <-> ~ (sats(A,p,env) & sats(A,q,env))"
    by (simp add: Bool.and_def Bool.not_def cond_def)

    lemma sats_Forall_iff [simp]:
      "env ∈ list(A)
       ==> sats(A, Forall(p), env) <-> (∀x∈A. sats(A, p, Cons(x,env)))"
    by simp

    declare satisfies.simps [simp del];

    subsection{*Dividing line between primitive and derived connectives*}

    lemma sats_Neg_iff [simp]:
      "env ∈ list(A)
       ==> sats(A, Neg(p), env) <-> ~ sats(A,p,env)"
    by (simp add: Neg_def)

    lemma sats_And_iff [simp]:
      "env ∈ list(A)
       ==> (sats(A, And(p,q), env)) <-> sats(A,p,env) & sats(A,q,env)"
    by (simp add: And_def)

    lemma sats_Or_iff [simp]:
      "env ∈ list(A)
       ==> (sats(A, Or(p,q), env)) <-> sats(A,p,env) | sats(A,q,env)"
    by (simp add: Or_def)

    lemma sats_Implies_iff [simp]:
      "env ∈ list(A)
       ==> (sats(A, Implies(p,q), env)) <-> (sats(A,p,env) --> sats(A,q,env))"
    by (simp add: Implies_def, blast)

    lemma sats_Iff_iff [simp]:
      "env ∈ list(A)
       ==> (sats(A, Iff(p,q), env)) <-> (sats(A,p,env) <-> sats(A,q,env))"
    by (simp add: Iff_def, blast)

    lemma sats_Exists_iff [simp]:
      "env ∈ list(A)
       ==> sats(A, Exists(p), env) <-> (∃x∈A. sats(A, p, Cons(x,env)))"
    by (simp add: Exists_def)

    subsubsection{*Derived rules to help build up formulas*}

    lemma mem_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; env ∈ list(A)|]
           ==> (x∈y) <-> sats(A, Member(i,j), env)"
    by (simp add: satisfies.simps)

    lemma equal_iff_sats:
          "[| nth(i,env) = x; nth(j,env) = y; env ∈ list(A)|]
           ==> (x=y) <-> sats(A, Equal(i,j), env)"
    by (simp add: satisfies.simps)

    lemma not_iff_sats:
          "[| P <-> sats(A,p,env); env ∈ list(A)|]
           ==> (~P) <-> sats(A, Neg(p), env)"
    by simp

    lemma conj_iff_sats:
          "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env ∈ list(A)|]
           ==> (P & Q) <-> sats(A, And(p,q), env)"
    by (simp add: sats_And_iff)

    lemma disj_iff_sats:
          "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env ∈ list(A)|]
           ==> (P | Q) <-> sats(A, Or(p,q), env)"
    by (simp add: sats_Or_iff)

    lemma iff_iff_sats:
          "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env ∈ list(A)|]
           ==> (P <-> Q) <-> sats(A, Iff(p,q), env)"
    by (simp add: sats_Forall_iff)

    lemma imp_iff_sats:
          "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env ∈ list(A)|]
           ==> (P --> Q) <-> sats(A, Implies(p,q), env)"
    by (simp add: sats_Forall_iff)

    lemma ball_iff_sats:
          "[| !!x. x∈A ==> P(x) <-> sats(A, p, Cons(x, env)); env ∈ list(A)|]
           ==> (∀x∈A. P(x)) <-> sats(A, Forall(p), env)"
    by (simp add: sats_Forall_iff)

    lemma bex_iff_sats:
          "[| !!x. x∈A ==> P(x) <-> sats(A, p, Cons(x, env)); env ∈ list(A)|]
           ==> (∃x∈A. P(x)) <-> sats(A, Exists(p), env)"
    by (simp add: sats_Exists_iff)

    lemmas FOL_iff_sats =
            mem_iff_sats equal_iff_sats not_iff_sats conj_iff_sats
            disj_iff_sats imp_iff_sats iff_iff_sats imp_iff_sats ball_iff_sats
            bex_iff_sats

    subsection{*Arity of a Formula: Maximum Free de Bruijn Index*}

    consts   arity :: "i=>i"
    primrec
      "arity(Member(x,y)) = succ(x) ∪ succ(y)"

      "arity(Equal(x,y)) = succ(x) ∪ succ(y)"

      "arity(Nand(p,q)) = arity(p) ∪ arity(q)"

      "arity(Forall(p)) = Arith.pred(arity(p))"

    lemma arity_type [TC]: "p ∈ formula ==> arity(p) ∈ nat"
    by (induct_tac p, simp_all)

    lemma arity_Neg [simp]: "arity(Neg(p)) = arity(p)"
    by (simp add: Neg_def)

    lemma arity_And [simp]: "arity(And(p,q)) = arity(p) ∪ arity(q)"
    by (simp add: And_def)

    lemma arity_Or [simp]: "arity(Or(p,q)) = arity(p) ∪ arity(q)"
    by (simp add: Or_def)

    lemma arity_Implies [simp]: "arity(Implies(p,q)) = arity(p) ∪ arity(q)"
    by (simp add: Implies_def)

    lemma arity_Iff [simp]: "arity(Iff(p,q)) = arity(p) ∪ arity(q)"
    by (simp add: Iff_def, blast)

    lemma arity_Exists [simp]: "arity(Exists(p)) = Arith.pred(arity(p))"
    by (simp add: Exists_def)

    lemma arity_sats_iff [rule_format]:
      "[| p ∈ formula; extra ∈ list(A) |]
       ==> ∀env ∈ list(A).
               arity(p) ≤ length(env) -->
               sats(A, p, env @ extra) <-> sats(A, p, env)"
    apply (induct_tac p)
    apply (simp_all add: Arith.pred_def nth_append Un_least_lt_iff nat_imp_quasinat
                    split: split_nat_case, auto)
    done

    lemma arity_sats1_iff:
      "[| arity(p) ≤ succ(length(env)); p ∈ formula; x ∈ A; env ∈ list(A);
          extra ∈ list(A) |]
       ==> sats(A, p, Cons(x, env @ extra)) <-> sats(A, p, Cons(x, env))"
    apply (insert arity_sats_iff [of p extra A "Cons(x,env)"])
    apply simp
    done

    subsection{*Renaming Some de Bruijn Variables*}

    constdefs incr_var :: "[i,i]=>i"
        "incr_var(x,nq) == if x<nq then x else succ(x)"

    lemma incr_var_lt: "x<nq ==> incr_var(x,nq) = x"
    by (simp add: incr_var_def)

    lemma incr_var_le: "nq≤x ==> incr_var(x,nq) = succ(x)"
    apply (simp add: incr_var_def)
    apply (blast dest: lt_trans1)
    done

    consts   incr_bv :: "i=>i"
    primrec
      "incr_bv(Member(x,y)) =
          (λnq ∈ nat. Member (incr_var(x,nq), incr_var(y,nq)))"

      "incr_bv(Equal(x,y)) =
          (λnq ∈ nat. Equal (incr_var(x,nq), incr_var(y,nq)))"

      "incr_bv(Nand(p,q)) =
          (λnq ∈ nat. Nand (incr_bv(p)`nq, incr_bv(q)`nq))"

      "incr_bv(Forall(p)) =
          (λnq ∈ nat. Forall (incr_bv(p) ` succ(nq)))"

    lemma [TC]: "x ∈ nat ==> incr_var(x,nq) ∈ nat"
    by (simp add: incr_var_def)

    lemma incr_bv_type [TC]: "p ∈ formula ==> incr_bv(p) ∈ nat -> formula"
    by (induct_tac p, simp_all)

    text{*Obviously, @{term DPow} is closed under complements and finite
    intersections and unions.  Needs an inductive lemma to allow two lists of
    parameters to be combined.*}

    lemma sats_incr_bv_iff [rule_format]:
      "[| p ∈ formula; env ∈ list(A); x ∈ A |]
       ==> ∀bvs ∈ list(A).
               sats(A, incr_bv(p) ` length(bvs), bvs @ Cons(x,env)) <->
               sats(A, p, bvs@env)"
    apply (induct_tac p)
    apply (simp_all add: incr_var_def nth_append succ_lt_iff length_type)
    apply (auto simp add: diff_succ not_lt_iff_le)
    done

    (*the following two lemmas prevent huge case splits in arity_incr_bv_lemma*)
    lemma incr_var_lemma:
         "[| x ∈ nat; y ∈ nat; nq ≤ x |]
          ==> succ(x) ∪ incr_var(y,nq) = succ(x ∪ y)"
    apply (simp add: incr_var_def Ord_Un_if, auto)
      apply (blast intro: leI)
     apply (simp add: not_lt_iff_le)
     apply (blast intro: le_anti_sym)
    apply (blast dest: lt_trans2)
    done

    lemma incr_And_lemma:
         "y < x ==> y ∪ succ(x) = succ(x ∪ y)"
    apply (simp add: Ord_Un_if lt_Ord lt_Ord2 succ_lt_iff)
    apply (blast dest: lt_asym)
    done

    lemma arity_incr_bv_lemma [rule_format]:
      "p ∈ formula
       ==> ∀n ∈ nat. arity (incr_bv(p) ` n) =
                     (if n < arity(p) then succ(arity(p)) else arity(p))"
    apply (induct_tac p)
    apply (simp_all add: imp_disj not_lt_iff_le Un_least_lt_iff lt_Un_iff le_Un_iff
                         succ_Un_distrib [symmetric] incr_var_lt incr_var_le
                         Un_commute incr_var_lemma Arith.pred_def nat_imp_quasinat
                split: split_nat_case)
     txt{*the Forall case reduces to linear arithmetic*}
     prefer 2
     apply clarify
     apply (blast dest: lt_trans1)
    txt{*left with the And case*}
    apply safe
     apply (blast intro: incr_And_lemma lt_trans1)
    apply (subst incr_And_lemma)
     apply (blast intro: lt_trans1)
    apply (simp add: Un_commute)
    done

    subsection{*Renaming all but the First de Bruijn Variable*}

    constdefs incr_bv1 :: "i => i"
        "incr_bv1(p) == incr_bv(p)`1"

    lemma incr_bv1_type [TC]: "p ∈ formula ==> incr_bv1(p) ∈ formula"
    by (simp add: incr_bv1_def)

    (*For renaming all but the bound variable at level 0*)
    lemma sats_incr_bv1_iff:
      "[| p ∈ formula; env ∈ list(A); x ∈ A; y ∈ A |]
       ==> sats(A, incr_bv1(p), Cons(x, Cons(y, env))) <->
           sats(A, p, Cons(x,env))"
    apply (insert sats_incr_bv_iff [of p env A y "Cons(x,Nil)"])
    apply (simp add: incr_bv1_def)
    done

    lemma formula_add_params1 [rule_format]:
      "[| p ∈ formula; n ∈ nat; x ∈ A |]
       ==> ∀bvs ∈ list(A). ∀env ∈ list(A).
              length(bvs) = n -->
              sats(A, iterates(incr_bv1, n, p), Cons(x, bvs@env)) <->
              sats(A, p, Cons(x,env))"
    apply (induct_tac n, simp, clarify)
    apply (erule list.cases)
    apply (simp_all add: sats_incr_bv1_iff)
    done

    lemma arity_incr_bv1_eq:
      "p ∈ formula
       ==> arity(incr_bv1(p)) =
            (if 1 < arity(p) then succ(arity(p)) else arity(p))"
    apply (insert arity_incr_bv_lemma [of p 1])
    apply (simp add: incr_bv1_def)
    done

    lemma arity_iterates_incr_bv1_eq:
      "[| p ∈ formula; n ∈ nat |]
       ==> arity(incr_bv1^n(p)) =
             (if 1 < arity(p) then n #+ arity(p) else arity(p))"
    apply (induct_tac n)
    apply (simp_all add: arity_incr_bv1_eq)
    apply (simp add: not_lt_iff_le)
    apply (blast intro: le_trans add_le_self2 arity_type)
    done

    subsection{*Definable Powerset*}

    text{*The definable powerset operation: Kunen's definition VI 1.1, page 165.*}
    constdefs DPow :: "i => i"
      "DPow(A) == {X ∈ Pow(A).
                   ∃env ∈ list(A). ∃p ∈ formula.
                     arity(p) ≤ succ(length(env)) &
                     X = {x∈A. sats(A, p, Cons(x,env))}}"

    lemma DPowI:
      "[|env ∈ list(A);  p ∈ formula;  arity(p) ≤ succ(length(env))|]
       ==> {x∈A. sats(A, p, Cons(x,env))} ∈ DPow(A)"
    by (simp add: DPow_def, blast)

    text{*With this rule we can specify @{term p} later.*}
    lemma DPowI2 [rule_format]:
      "[|∀x∈A. P(x) <-> sats(A, p, Cons(x,env));
         env ∈ list(A);  p ∈ formula;  arity(p) ≤ succ(length(env))|]
       ==> {x∈A. P(x)} ∈ DPow(A)"
    by (simp add: DPow_def, blast)

    lemma DPowD:
      "X ∈ DPow(A)
       ==> X <= A &
           (∃env ∈ list(A).
            ∃p ∈ formula. arity(p) ≤ succ(length(env)) &
                          X = {x∈A. sats(A, p, Cons(x,env))})"
    by (simp add: DPow_def)

    lemmas DPow_imp_subset = DPowD [THEN conjunct1]

    (*Kunen's Lemma VI 1.2*)
    lemma "[| p ∈ formula; env ∈ list(A); arity(p) ≤ succ(length(env)) |]
           ==> {x∈A. sats(A, p, Cons(x,env))} ∈ DPow(A)"
    by (blast intro: DPowI)

    lemma DPow_subset_Pow: "DPow(A) <= Pow(A)"
    by (simp add: DPow_def, blast)

    lemma empty_in_DPow: "0 ∈ DPow(A)"
    apply (simp add: DPow_def)
    apply (rule_tac x=Nil in bexI)
     apply (rule_tac x="Neg(Equal(0,0))" in bexI)
      apply (auto simp add: Un_least_lt_iff)
    done

    lemma Compl_in_DPow: "X ∈ DPow(A) ==> (A-X) ∈ DPow(A)"
    apply (simp add: DPow_def, clarify, auto)
    apply (rule bexI)
     apply (rule_tac x="Neg(p)" in bexI)
      apply auto
    done

    lemma Int_in_DPow: "[| X ∈ DPow(A); Y ∈ DPow(A) |] ==> X Int Y ∈ DPow(A)"
    apply (simp add: DPow_def, auto)
    apply (rename_tac envp p envq q)
    apply (rule_tac x="envp@envq" in bexI)
     apply (rule_tac x="And(p, iterates(incr_bv1,length(envp),q))" in bexI)
      apply typecheck
    apply (rule conjI)
    (*finally check the arity!*)
     apply (simp add: arity_iterates_incr_bv1_eq length_app Un_least_lt_iff)
     apply (force intro: add_le_self le_trans)
    apply (simp add: arity_sats1_iff formula_add_params1, blast)
    done

    lemma Un_in_DPow: "[| X ∈ DPow(A); Y ∈ DPow(A) |] ==> X Un Y ∈ DPow(A)"
    apply (subgoal_tac "X Un Y = A - ((A-X) Int (A-Y))")
    apply (simp add: Int_in_DPow Compl_in_DPow)
    apply (simp add: DPow_def, blast)
    done

    lemma singleton_in_DPow: "a ∈ A ==> {a} ∈ DPow(A)"
    apply (simp add: DPow_def)
    apply (rule_tac x="Cons(a,Nil)" in bexI)
     apply (rule_tac x="Equal(0,1)" in bexI)
      apply typecheck
    apply (force simp add: succ_Un_distrib [symmetric])
    done

    lemma cons_in_DPow: "[| a ∈ A; X ∈ DPow(A) |] ==> cons(a,X) ∈ DPow(A)"
    apply (rule cons_eq [THEN subst])
    apply (blast intro: singleton_in_DPow Un_in_DPow)
    done

    (*Part of Lemma 1.3*)
    lemma Fin_into_DPow: "X ∈ Fin(A) ==> X ∈ DPow(A)"
    apply (erule Fin.induct)
     apply (rule empty_in_DPow)
    apply (blast intro: cons_in_DPow)
    done

    text{*@{term DPow} is not monotonic.  For example, let @{term A} be some
    non-constructible set of natural numbers, and let @{term B} be @{term nat}.
    Then @{term "A<=B"} and obviously @{term "A ∈ DPow(A)"} but @{term "A ~:
    DPow(B)"}.*}

    (*This may be true but the proof looks difficult, requiring relativization
    lemma DPow_insert: "DPow (cons(a,A)) = DPow(A) Un {cons(a,X) . X: DPow(A)}"
    apply (rule equalityI, safe)
    oops
    *)

    lemma Finite_Pow_subset_Pow: "Finite(A) ==> Pow(A) <= DPow(A)"
    by (blast intro: Fin_into_DPow Finite_into_Fin Fin_subset)

    lemma Finite_DPow_eq_Pow: "Finite(A) ==> DPow(A) = Pow(A)"
    apply (rule equalityI)
    apply (rule DPow_subset_Pow)
    apply (erule Finite_Pow_subset_Pow)
    done

    subsection{*Internalized Formulas for the Ordinals*}

    text{*The @{text sats} theorems below differ from the usual form in that they
    include an element of absoluteness.  That is, they relate internalized
    formulas to real concepts such as the subset relation, rather than to the
    relativized concepts defined in theory @{text Relative}.  This lets us prove
    the theorem as @{text Ords_in_DPow} without first having to instantiate the
    locale @{text M_trivial}.  Note that the present theory does not even take
    @{text Relative} as a parent.*}

    subsubsection{*The subset relation*}

    constdefs subset_fm :: "[i,i]=>i"
        "subset_fm(x,y) == Forall(Implies(Member(0,succ(x)), Member(0,succ(y))))"

    lemma subset_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> subset_fm(x,y) ∈ formula"
    by (simp add: subset_fm_def)

    lemma arity_subset_fm [simp]:
         "[| x ∈ nat; y ∈ nat |] ==> arity(subset_fm(x,y)) = succ(x) ∪ succ(y)"
    by (simp add: subset_fm_def succ_Un_distrib [symmetric])

    lemma sats_subset_fm [simp]:
       "[|x < length(env); y ∈ nat; env ∈ list(A); Transset(A)|]
        ==> sats(A, subset_fm(x,y), env) <-> nth(x,env) ⊆ nth(y,env)"
    apply (frule lt_length_in_nat, assumption)
    apply (simp add: subset_fm_def Transset_def)
    apply (blast intro: nth_type)
    done

    subsubsection{*Transitive sets*}

    constdefs transset_fm :: "i=>i"
       "transset_fm(x) == Forall(Implies(Member(0,succ(x)), subset_fm(0,succ(x))))"

    lemma transset_type [TC]: "x ∈ nat ==> transset_fm(x) ∈ formula"
    by (simp add: transset_fm_def)

    lemma arity_transset_fm [simp]:
         "x ∈ nat ==> arity(transset_fm(x)) = succ(x)"
    by (simp add: transset_fm_def succ_Un_distrib [symmetric])

    lemma sats_transset_fm [simp]:
       "[|x < length(env); env ∈ list(A); Transset(A)|]
        ==> sats(A, transset_fm(x), env) <-> Transset(nth(x,env))"
    apply (frule lt_nat_in_nat, erule length_type)
    apply (simp add: transset_fm_def Transset_def)
    apply (blast intro: nth_type)
    done

    subsubsection{*Ordinals*}

    constdefs ordinal_fm :: "i=>i"
       "ordinal_fm(x) ==
          And(transset_fm(x), Forall(Implies(Member(0,succ(x)), transset_fm(0))))"

    lemma ordinal_type [TC]: "x ∈ nat ==> ordinal_fm(x) ∈ formula"
    by (simp add: ordinal_fm_def)

    lemma arity_ordinal_fm [simp]:
         "x ∈ nat ==> arity(ordinal_fm(x)) = succ(x)"
    by (simp add: ordinal_fm_def succ_Un_distrib [symmetric])

    lemma sats_ordinal_fm:
       "[|x < length(env); env ∈ list(A); Transset(A)|]
        ==> sats(A, ordinal_fm(x), env) <-> Ord(nth(x,env))"
    apply (frule lt_nat_in_nat, erule length_type)
    apply (simp add: ordinal_fm_def Ord_def Transset_def)
    apply (blast intro: nth_type)
    done

    text{*The subset consisting of the ordinals is definable.  Essential lemma for
    @{text Ord_in_Lset}.  This result is the objective of the present subsection.*}
    theorem Ords_in_DPow: "Transset(A) ==> {x ∈ A. Ord(x)} ∈ DPow(A)"
    apply (simp add: DPow_def Collect_subset)
    apply (rule_tac x=Nil in bexI)
     apply (rule_tac x="ordinal_fm(0)" in bexI)
    apply (simp_all add: sats_ordinal_fm)
    done

    subsection{* Constant Lset: Levels of the Constructible Universe *}

    constdefs
      Lset :: "i=>i"
        "Lset(i) == transrec(i, %x f. \<Union>y∈x. DPow(f`y))"

      L :: "i=>o" --{*Kunen's definition VI 1.5, page 167*}
        "L(x) == ∃i. Ord(i) & x ∈ Lset(i)"

    text{*NOT SUITABLE FOR REWRITING -- RECURSIVE!*}
    lemma Lset: "Lset(i) = (UN j:i. DPow(Lset(j)))"
    by (subst Lset_def [THEN def_transrec], simp)

    lemma LsetI: "[|y∈x; A ∈ DPow(Lset(y))|] ==> A ∈ Lset(x)";
    by (subst Lset, blast)

    lemma LsetD: "A ∈ Lset(x) ==> ∃y∈x. A ∈ DPow(Lset(y))";
    apply (insert Lset [of x])
    apply (blast intro: elim: equalityE)
    done

    subsubsection{* Transitivity *}

    lemma elem_subset_in_DPow: "[|X ∈ A; X ⊆ A|] ==> X ∈ DPow(A)"
    apply (simp add: Transset_def DPow_def)
    apply (rule_tac x="[X]" in bexI)
     apply (rule_tac x="Member(0,1)" in bexI)
      apply (auto simp add: Un_least_lt_iff)
    done

    lemma Transset_subset_DPow: "Transset(A) ==> A <= DPow(A)"
    apply clarify
    apply (simp add: Transset_def)
    apply (blast intro: elem_subset_in_DPow)
    done

    lemma Transset_DPow: "Transset(A) ==> Transset(DPow(A))"
    apply (simp add: Transset_def)
    apply (blast intro: elem_subset_in_DPow dest: DPowD)
    done

    text{*Kunen's VI 1.6 (a)*}
    lemma Transset_Lset: "Transset(Lset(i))"
    apply (rule_tac a=i in eps_induct)
    apply (subst Lset)
    apply (blast intro!: Transset_Union_family Transset_Un Transset_DPow)
    done

    lemma mem_Lset_imp_subset_Lset: "a ∈ Lset(i) ==> a ⊆ Lset(i)"
    apply (insert Transset_Lset)
    apply (simp add: Transset_def)
    done

    subsubsection{* Monotonicity *}

    text{*Kunen's VI 1.6 (b)*}
    lemma Lset_mono [rule_format]:
         "ALL j. i<=j --> Lset(i) <= Lset(j)"
    proof (induct i rule: eps_induct, intro allI impI)
      fix x j
      assume "∀y∈x. ∀j. y ⊆ j --> Lset(y) ⊆ Lset(j)"
         and "x ⊆ j"
      thus "Lset(x) ⊆ Lset(j)"
        by (force simp add: Lset [of x] Lset [of j])
    qed

    text{*This version lets us remove the premise @{term "Ord(i)"} sometimes.*}
    lemma Lset_mono_mem [rule_format]:
         "ALL j. i:j --> Lset(i) <= Lset(j)"
    proof (induct i rule: eps_induct, intro allI impI)
      fix x j
      assume "∀y∈x. ∀j. y ∈ j --> Lset(y) ⊆ Lset(j)"
         and "x ∈ j"
      thus "Lset(x) ⊆ Lset(j)"
        by (force simp add: Lset [of j]
                  intro!: bexI intro: elem_subset_in_DPow dest: LsetD DPowD)
    qed

    text{*Useful with Reflection to bump up the ordinal*}
    lemma subset_Lset_ltD: "[|A ⊆ Lset(i); i < j|] ==> A ⊆ Lset(j)"
    by (blast dest: ltD [THEN Lset_mono_mem])

    subsubsection{* 0, successor and limit equations for Lset *}

    lemma Lset_0 [simp]: "Lset(0) = 0"
    by (subst Lset, blast)

    lemma Lset_succ_subset1: "DPow(Lset(i)) <= Lset(succ(i))"
    by (subst Lset, rule succI1 [THEN RepFunI, THEN Union_upper])

    lemma Lset_succ_subset2: "Lset(succ(i)) <= DPow(Lset(i))"
    apply (subst Lset, rule UN_least)
    apply (erule succE)
     apply blast
    apply clarify
    apply (rule elem_subset_in_DPow)
     apply (subst Lset)
     apply blast
    apply (blast intro: dest: DPowD Lset_mono_mem)
    done

    lemma Lset_succ: "Lset(succ(i)) = DPow(Lset(i))"
    by (intro equalityI Lset_succ_subset1 Lset_succ_subset2)

    lemma Lset_Union [simp]: "Lset(\<Union>(X)) = (\<Union>y∈X. Lset(y))"
    apply (subst Lset)
    apply (rule equalityI)
     txt{*first inclusion*}
     apply (rule UN_least)
     apply (erule UnionE)
     apply (rule subset_trans)
      apply (erule_tac [2] UN_upper, subst Lset, erule UN_upper)
    txt{*opposite inclusion*}
    apply (rule UN_least)
    apply (subst Lset, blast)
    done

    subsubsection{* Lset applied to Limit ordinals *}

    lemma Limit_Lset_eq:
        "Limit(i) ==> Lset(i) = (\<Union>y∈i. Lset(y))"
    by (simp add: Lset_Union [symmetric] Limit_Union_eq)

    lemma lt_LsetI: "[| a: Lset(j);  j<i |] ==> a ∈ Lset(i)"
    by (blast dest: Lset_mono [OF le_imp_subset [OF leI]])

    lemma Limit_LsetE:
        "[| a: Lset(i);  ~R ==> Limit(i);
            !!x. [| x<i;  a: Lset(x) |] ==> R
         |] ==> R"
    apply (rule classical)
    apply (rule Limit_Lset_eq [THEN equalityD1, THEN subsetD, THEN UN_E])
      prefer 2 apply assumption
     apply blast
    apply (blast intro: ltI  Limit_is_Ord)
    done

    subsubsection{* Basic closure properties *}

    lemma zero_in_Lset: "y:x ==> 0 ∈ Lset(x)"
    by (subst Lset, blast intro: empty_in_DPow)

    lemma notin_Lset: "x ∉ Lset(x)"
    apply (rule_tac a=x in eps_induct)
    apply (subst Lset)
    apply (blast dest: DPowD)
    done

    subsection{*Constructible Ordinals: Kunen's VI 1.9 (b)*}

    lemma Ords_of_Lset_eq: "Ord(i) ==> {x∈Lset(i). Ord(x)} = i"
    apply (erule trans_induct3)
      apply (simp_all add: Lset_succ Limit_Lset_eq Limit_Union_eq)
    txt{*The successor case remains.*}
    apply (rule equalityI)
    txt{*First inclusion*}
     apply clarify
     apply (erule Ord_linear_lt, assumption)
       apply (blast dest: DPow_imp_subset ltD notE [OF notin_Lset])
      apply blast
     apply (blast dest: ltD)
    txt{*Opposite inclusion, @{term "succ(x) ⊆ DPow(Lset(x)) ∩ ON"}*}
    apply auto
    txt{*Key case: *}
      apply (erule subst, rule Ords_in_DPow [OF Transset_Lset])
     apply (blast intro: elem_subset_in_DPow dest: OrdmemD elim: equalityE)
    apply (blast intro: Ord_in_Ord)
    done

    lemma Ord_subset_Lset: "Ord(i) ==> i ⊆ Lset(i)"
    by (subst Ords_of_Lset_eq [symmetric], assumption, fast)

    lemma Ord_in_Lset: "Ord(i) ==> i ∈ Lset(succ(i))"
    apply (simp add: Lset_succ)
    apply (subst Ords_of_Lset_eq [symmetric], assumption,
           rule Ords_in_DPow [OF Transset_Lset])
    done

    lemma Ord_in_L: "Ord(i) ==> L(i)"
    by (simp add: L_def, blast intro: Ord_in_Lset)

    subsubsection{* Unions *}

    lemma Union_in_Lset:
         "X ∈ Lset(i) ==> Union(X) ∈ Lset(succ(i))"
    apply (insert Transset_Lset)
    apply (rule LsetI [OF succI1])
    apply (simp add: Transset_def DPow_def)
    apply (intro conjI, blast)
    txt{*Now to create the formula @{term "∃y. y ∈ X ∧ x ∈ y"} *}
    apply (rule_tac x="Cons(X,Nil)" in bexI)
     apply (rule_tac x="Exists(And(Member(0,2), Member(1,0)))" in bexI)
      apply typecheck
    apply (simp add: succ_Un_distrib [symmetric], blast)
    done

    theorem Union_in_L: "L(X) ==> L(Union(X))"
    by (simp add: L_def, blast dest: Union_in_Lset)

    subsubsection{* Finite sets and ordered pairs *}

    lemma singleton_in_Lset: "a: Lset(i) ==> {a} ∈ Lset(succ(i))"
    by (simp add: Lset_succ singleton_in_DPow)

    lemma doubleton_in_Lset:
         "[| a: Lset(i);  b: Lset(i) |] ==> {a,b} ∈ Lset(succ(i))"
    by (simp add: Lset_succ empty_in_DPow cons_in_DPow)

    lemma Pair_in_Lset:
        "[| a: Lset(i);  b: Lset(i); Ord(i) |] ==> <a,b> ∈ Lset(succ(succ(i)))"
    apply (unfold Pair_def)
    apply (blast intro: doubleton_in_Lset)
    done

    lemmas Lset_UnI1 = Un_upper1 [THEN Lset_mono [THEN subsetD], standard]
    lemmas Lset_UnI2 = Un_upper2 [THEN Lset_mono [THEN subsetD], standard]

    text{*Hard work is finding a single j:i such that {a,b}<=Lset(j)*}
    lemma doubleton_in_LLimit:
        "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> {a,b} ∈ Lset(i)"
    apply (erule Limit_LsetE, assumption)
    apply (erule Limit_LsetE, assumption)
    apply (blast intro: lt_LsetI [OF doubleton_in_Lset]
                        Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
    done

    theorem doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})"
    apply (simp add: L_def, clarify)
    apply (drule Ord2_imp_greater_Limit, assumption)
    apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord)
    done

    lemma Pair_in_LLimit:
        "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> <a,b> ∈ Lset(i)"
    txt{*Infer that a, b occur at ordinals x,xa < i.*}
    apply (erule Limit_LsetE, assumption)
    apply (erule Limit_LsetE, assumption)
    txt{*Infer that succ(succ(x Un xa)) < i *}
    apply (blast intro: lt_Ord lt_LsetI [OF Pair_in_Lset]
                        Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
    done

    text{*The rank function for the constructible universe*}
    constdefs
      lrank :: "i=>i" --{*Kunen's definition VI 1.7*}
        "lrank(x) == μ i. x ∈ Lset(succ(i))"

    lemma L_I: "[|x ∈ Lset(i); Ord(i)|] ==> L(x)"
    by (simp add: L_def, blast)

    lemma L_D: "L(x) ==> ∃i. Ord(i) & x ∈ Lset(i)"
    by (simp add: L_def)

    lemma Ord_lrank [simp]: "Ord(lrank(a))"
    by (simp add: lrank_def)

    lemma Lset_lrank_lt [rule_format]: "Ord(i) ==> x ∈ Lset(i) --> lrank(x) < i"
    apply (erule trans_induct3)
      apply simp
     apply (simp only: lrank_def)
     apply (blast intro: Least_le)
    apply (simp_all add: Limit_Lset_eq)
    apply (blast intro: ltI Limit_is_Ord lt_trans)
    done

    text{*Kunen's VI 1.8.  The proof is much harder than the text would
    suggest.  For a start, it needs the previous lemma, which is proved by
    induction.*}
    lemma Lset_iff_lrank_lt: "Ord(i) ==> x ∈ Lset(i) <-> L(x) & lrank(x) < i"
    apply (simp add: L_def, auto)
     apply (blast intro: Lset_lrank_lt)
     apply (unfold lrank_def)
    apply (drule succI1 [THEN Lset_mono_mem, THEN subsetD])
    apply (drule_tac P="λi. x ∈ Lset(succ(i))" in LeastI, assumption)
    apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD])
    done

    lemma Lset_succ_lrank_iff [simp]: "x ∈ Lset(succ(lrank(x))) <-> L(x)"
    by (simp add: Lset_iff_lrank_lt)

    text{*Kunen's VI 1.9 (a)*}
    lemma lrank_of_Ord: "Ord(i) ==> lrank(i) = i"
    apply (unfold lrank_def)
    apply (rule Least_equality)
      apply (erule Ord_in_Lset)
     apply assumption
    apply (insert notin_Lset [of i])
    apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD])
    done

    text{*This is lrank(lrank(a)) = lrank(a) *}
    declare Ord_lrank [THEN lrank_of_Ord, simp]

    text{*Kunen's VI 1.10 *}
    lemma Lset_in_Lset_succ: "Lset(i) ∈ Lset(succ(i))";
    apply (simp add: Lset_succ DPow_def)
    apply (rule_tac x=Nil in bexI)
     apply (rule_tac x="Equal(0,0)" in bexI)
    apply auto
    done

    lemma lrank_Lset: "Ord(i) ==> lrank(Lset(i)) = i"
    apply (unfold lrank_def)
    apply (rule Least_equality)
      apply (rule Lset_in_Lset_succ)
     apply assumption
    apply clarify
    apply (subgoal_tac "Lset(succ(ia)) <= Lset(i)")
     apply (blast dest: mem_irrefl)
    apply (blast intro!: le_imp_subset Lset_mono)
    done

    text{*Kunen's VI 1.11 *}
    lemma Lset_subset_Vset: "Ord(i) ==> Lset(i) <= Vset(i)";
    apply (erule trans_induct)
    apply (subst Lset)
    apply (subst Vset)
    apply (rule UN_mono [OF subset_refl])
    apply (rule subset_trans [OF DPow_subset_Pow])
    apply (rule Pow_mono, blast)
    done

    text{*Kunen's VI 1.12 *}
    lemma Lset_subset_Vset': "i ∈ nat ==> Lset(i) = Vset(i)";
    apply (erule nat_induct)
     apply (simp add: Vfrom_0)
    apply (simp add: Lset_succ Vset_succ Finite_Vset Finite_DPow_eq_Pow)
    done

    text{*Every set of constructible sets is included in some @{term Lset}*}
    lemma subset_Lset:
         "(∀x∈A. L(x)) ==> ∃i. Ord(i) & A ⊆ Lset(i)"
    by (rule_tac x = "\<Union>x∈A. succ(lrank(x))" in exI, force)

    lemma subset_LsetE:
         "[|∀x∈A. L(x);
            !!i. [|Ord(i); A ⊆ Lset(i)|] ==> P|]
          ==> P"
    by (blast dest: subset_Lset)

    subsubsection{*For L to satisfy the Powerset axiom *}

    lemma LPow_env_typing:
        "[| y ∈ Lset(i); Ord(i); y ⊆ X |]
         ==> ∃z ∈ Pow(X). y ∈ Lset(succ(lrank(z)))"
    by (auto intro: L_I iff: Lset_succ_lrank_iff)

    lemma LPow_in_Lset:
         "[|X ∈ Lset(i); Ord(i)|] ==> ∃j. Ord(j) & {y ∈ Pow(X). L(y)} ∈ Lset(j)"
    apply (rule_tac x="succ(\<Union>y ∈ Pow(X). succ(lrank(y)))" in exI)
    apply simp
    apply (rule LsetI [OF succI1])
    apply (simp add: DPow_def)
    apply (intro conjI, clarify)
     apply (rule_tac a=x in UN_I, simp+)
    txt{*Now to create the formula @{term "y ⊆ X"} *}
    apply (rule_tac x="Cons(X,Nil)" in bexI)
     apply (rule_tac x="subset_fm(0,1)" in bexI)
      apply typecheck
     apply (rule conjI)
    apply (simp add: succ_Un_distrib [symmetric])
    apply (rule equality_iffI)
    apply (simp add: Transset_UN [OF Transset_Lset] LPow_env_typing)
    apply (auto intro: L_I iff: Lset_succ_lrank_iff)
    done

    theorem LPow_in_L: "L(X) ==> L({y ∈ Pow(X). L(y)})"
    by (blast intro: L_I dest: L_D LPow_in_Lset)

    subsection{*Eliminating @{term arity} from the Definition of @{term Lset}*}

    lemma nth_zero_eq_0: "n ∈ nat ==> nth(n,[0]) = 0"
    by (induct_tac n, auto)

    lemma sats_app_0_iff [rule_format]:
      "[| p ∈ formula; 0 ∈ A |]
       ==> ∀env ∈ list(A). sats(A,p, env@[0]) <-> sats(A,p,env)"
    apply (induct_tac p)
    apply (simp_all del: app_Cons add: app_Cons [symmetric]
                    add: nth_zero_eq_0 nth_append not_lt_iff_le nth_eq_0)
    done

    lemma sats_app_zeroes_iff:
      "[| p ∈ formula; 0 ∈ A; env ∈ list(A); n ∈ nat |]
       ==> sats(A,p,env @ repeat(0,n)) <-> sats(A,p,env)"
    apply (induct_tac n, simp)
    apply (simp del: repeat.simps
                add: repeat_succ_app sats_app_0_iff app_assoc [symmetric])
    done

    lemma exists_bigger_env:
      "[| p ∈ formula; 0 ∈ A; env ∈ list(A) |]
       ==> ∃env' ∈ list(A). arity(p) ≤ succ(length(env')) &
                  (∀a∈A. sats(A,p,Cons(a,env')) <-> sats(A,p,Cons(a,env)))"
    apply (rule_tac x="env @ repeat(0,arity(p))" in bexI)
    apply (simp del: app_Cons add: app_Cons [symmetric]
                add: length_repeat sats_app_zeroes_iff, typecheck)
    done

    text{*A simpler version of @{term DPow}: no arity check!*}
    constdefs DPow' :: "i => i"
      "DPow'(A) == {X ∈ Pow(A).
                    ∃env ∈ list(A). ∃p ∈ formula.
                        X = {x∈A. sats(A, p, Cons(x,env))}}"

    lemma DPow_subset_DPow': "DPow(A) <= DPow'(A)";
    by (simp add: DPow_def DPow'_def, blast)

    lemma DPow'_0: "DPow'(0) = {0}"
    by (auto simp add: DPow'_def)

    lemma DPow'_subset_DPow: "0 ∈ A ==> DPow'(A) ⊆ DPow(A)"
    apply (auto simp add: DPow'_def DPow_def)
    apply (frule exists_bigger_env, assumption+, force)
    done

    lemma DPow_eq_DPow': "Transset(A) ==> DPow(A) = DPow'(A)"
    apply (drule Transset_0_disj)
    apply (erule disjE)
     apply (simp add: DPow'_0 Finite_DPow_eq_Pow)
    apply (rule equalityI)
     apply (rule DPow_subset_DPow')
    apply (erule DPow'_subset_DPow)
    done

    text{*And thus we can relativize @{term Lset} without bothering with
          @{term arity} and @{term length}*}
    lemma Lset_eq_transrec_DPow': "Lset(i) = transrec(i, %x f. \<Union>y∈x. DPow'(f`y))"
    apply (rule_tac a=i in eps_induct)
    apply (subst Lset)
    apply (subst transrec)
    apply (simp only: DPow_eq_DPow' [OF Transset_Lset], simp)
    done

    text{*With this rule we can specify @{term p} later and don't worry about
          arities at all!*}
    lemma DPow_LsetI [rule_format]:
      "[|∀x∈Lset(i). P(x) <-> sats(Lset(i), p, Cons(x,env));
         env ∈ list(Lset(i));  p ∈ formula|]
       ==> {x∈Lset(i). P(x)} ∈ DPow(Lset(i))"
    by (simp add: DPow_eq_DPow' [OF Transset_Lset] DPow'_def, blast)

    end

### Internalized formulas of FOL

lemma Neg_type:

      p ∈ formula ==> Neg(p) ∈ formula

lemma And_type:

      [| p ∈ formula; q ∈ formula |] ==> And(p, q) ∈ formula

lemma Or_type:

      [| p ∈ formula; q ∈ formula |] ==> Or(p, q) ∈ formula

lemma Implies_type:

      [| p ∈ formula; q ∈ formula |] ==> Implies(p, q) ∈ formula

lemma Iff_type:

      [| p ∈ formula; q ∈ formula |] ==> Iff(p, q) ∈ formula

lemma Exists_type:

      p ∈ formula ==> Exists(p) ∈ formula

lemma

      p ∈ formula ==> satisfies(A, p) ∈ list(A) -> bool

lemma

      env ∈ list(A) ==> sats(A, Member(x, y), env) <-> nth(x, env) ∈ nth(y, env)

lemma

      env ∈ list(A) ==> sats(A, Equal(x, y), env) <-> nth(x, env) = nth(y, env)

lemma sats_Nand_iff:

      env ∈ list(A)
      ==> sats(A, Nand(p, q), env) <-> ¬ (sats(A, p, env) ∧ sats(A, q, env))

lemma sats_Forall_iff:

      env ∈ list(A) ==> sats(A, Forall(p), env) <-> (∀x∈A. sats(A, p, Cons(x, env)))

### Dividing line between primitive and derived connectives

lemma sats_Neg_iff:

      env ∈ list(A) ==> sats(A, Neg(p), env) <-> satisfies(A, p) ` env ≠ 1

lemma sats_And_iff:

      env ∈ list(A) ==> sats(A, And(p, q), env) <-> sats(A, p, env) ∧ sats(A, q, env)

lemma sats_Or_iff:

      env ∈ list(A) ==> sats(A, Or(p, q), env) <-> sats(A, p, env) ∨ sats(A, q, env)

lemma sats_Implies_iff:

      env ∈ list(A)
      ==> sats(A, Implies(p, q), env) <-> sats(A, p, env) --> sats(A, q, env)

lemma sats_Iff_iff:

      env ∈ list(A)
      ==> sats(A, Iff(p, q), env) <-> sats(A, p, env) <-> sats(A, q, env)

lemma sats_Exists_iff:

      env ∈ list(A) ==> sats(A, Exists(p), env) <-> (∃x∈A. sats(A, p, Cons(x, env)))

#### Derived rules to help build up formulas

lemma mem_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; env ∈ list(A) |]
      ==> x ∈ y <-> sats(A, Member(i, j), env)

lemma equal_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; env ∈ list(A) |]
      ==> x = y <-> sats(A, Equal(i, j), env)

lemma not_iff_sats:

      [| P <-> sats(A, p, env); env ∈ list(A) |] ==> ¬ P <-> sats(A, Neg(p), env)

lemma conj_iff_sats:

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> P ∧ Q <-> sats(A, And(p, q), env)

lemma disj_iff_sats:

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> P ∨ Q <-> sats(A, Or(p, q), env)

lemma iff_iff_sats:

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> (P <-> Q) <-> sats(A, Iff(p, q), env)

lemma imp_iff_sats:

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> (P --> Q) <-> sats(A, Implies(p, q), env)

lemma ball_iff_sats:

      [| !!x. x ∈ A ==> P(x) <-> sats(A, p, Cons(x, env)); env ∈ list(A) |]
      ==> (∀x∈A. P(x)) <-> sats(A, Forall(p), env)

lemma bex_iff_sats:

      [| !!x. x ∈ A ==> P(x) <-> sats(A, p, Cons(x, env)); env ∈ list(A) |]
      ==> (∃x∈A. P(x)) <-> sats(A, Exists(p), env)

lemmas FOL_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; env ∈ list(A) |]
      ==> x ∈ y <-> sats(A, Member(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; env ∈ list(A) |]
      ==> x = y <-> sats(A, Equal(i, j), env)

      [| P <-> sats(A, p, env); env ∈ list(A) |] ==> ¬ P <-> sats(A, Neg(p), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> P ∧ Q <-> sats(A, And(p, q), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> P ∨ Q <-> sats(A, Or(p, q), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> (P --> Q) <-> sats(A, Implies(p, q), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> (P <-> Q) <-> sats(A, Iff(p, q), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> (P --> Q) <-> sats(A, Implies(p, q), env)

      [| !!x. x ∈ A ==> P(x) <-> sats(A, p, Cons(x, env)); env ∈ list(A) |]
      ==> (∀x∈A. P(x)) <-> sats(A, Forall(p), env)

      [| !!x. x ∈ A ==> P(x) <-> sats(A, p, Cons(x, env)); env ∈ list(A) |]
      ==> (∃x∈A. P(x)) <-> sats(A, Exists(p), env)

lemmas FOL_iff_sats:

      [| nth(i, env) = x; nth(j, env) = y; env ∈ list(A) |]
      ==> x ∈ y <-> sats(A, Member(i, j), env)

      [| nth(i, env) = x; nth(j, env) = y; env ∈ list(A) |]
      ==> x = y <-> sats(A, Equal(i, j), env)

      [| P <-> sats(A, p, env); env ∈ list(A) |] ==> ¬ P <-> sats(A, Neg(p), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> P ∧ Q <-> sats(A, And(p, q), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> P ∨ Q <-> sats(A, Or(p, q), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> (P --> Q) <-> sats(A, Implies(p, q), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> (P <-> Q) <-> sats(A, Iff(p, q), env)

      [| P <-> sats(A, p, env); Q <-> sats(A, q, env); env ∈ list(A) |]
      ==> (P --> Q) <-> sats(A, Implies(p, q), env)

      [| !!x. x ∈ A ==> P(x) <-> sats(A, p, Cons(x, env)); env ∈ list(A) |]
      ==> (∀x∈A. P(x)) <-> sats(A, Forall(p), env)

      [| !!x. x ∈ A ==> P(x) <-> sats(A, p, Cons(x, env)); env ∈ list(A) |]
      ==> (∃x∈A. P(x)) <-> sats(A, Exists(p), env)

### Arity of a Formula: Maximum Free de Bruijn Index

lemma arity_type:

      p ∈ formula ==> arity(p) ∈ nat

lemma arity_Neg:

      arity(Neg(p)) = arity(p)

lemma arity_And:

      arity(And(p, q)) = arity(p) ∪ arity(q)

lemma arity_Or:

      arity(Or(p, q)) = arity(p) ∪ arity(q)

lemma arity_Implies:

      arity(Implies(p, q)) = arity(p) ∪ arity(q)

lemma arity_Iff:

      arity(Iff(p, q)) = arity(p) ∪ arity(q)

lemma arity_Exists:

      arity(Exists(p)) = Arith.pred(arity(p))

lemma arity_sats_iff:

      [| p ∈ formula; extra ∈ list(A); env ∈ list(A); arity(p) ≤ length(env) |]
      ==> sats(A, p, env @ extra) <-> sats(A, p, env)

lemma arity_sats1_iff:

      [| arity(p) ≤ succ(length(env)); p ∈ formula; x ∈ A; env ∈ list(A);
         extra ∈ list(A) |]
      ==> sats(A, p, Cons(x, env @ extra)) <-> sats(A, p, Cons(x, env))

### Renaming Some de Bruijn Variables

lemma incr_var_lt:

      x < nq ==> incr_var(x, nq) = x

lemma incr_var_le:

      nq ≤ x ==> incr_var(x, nq) = succ(x)

lemma

      x ∈ nat ==> incr_var(x, nq) ∈ nat

lemma incr_bv_type:

      p ∈ formula ==> incr_bv(p) ∈ nat -> formula

lemma sats_incr_bv_iff:

      [| p ∈ formula; env ∈ list(A); x ∈ A; bvs ∈ list(A) |]
      ==> sats(A, incr_bv(p) ` length(bvs), bvs @ Cons(x, env)) <->
          sats(A, p, bvs @ env)

lemma incr_var_lemma:

      [| x ∈ nat; y ∈ nat; nq ≤ x |] ==> succ(x) ∪ incr_var(y, nq) = succ(x ∪ y)

lemma incr_And_lemma:

      y < x ==> y ∪ succ(x) = succ(x ∪ y)

lemma arity_incr_bv_lemma:

      [| p ∈ formula; n ∈ nat |]
      ==> arity(incr_bv(p) ` n) = (if n < arity(p) then succ(arity(p)) else arity(p))

### Renaming all but the First de Bruijn Variable

lemma incr_bv1_type:

      p ∈ formula ==> incr_bv1(p) ∈ formula

lemma sats_incr_bv1_iff:

      [| p ∈ formula; env ∈ list(A); x ∈ A; y ∈ A |]
      ==> sats(A, incr_bv1(p), Cons(x, Cons(y, env))) <-> sats(A, p, Cons(x, env))

lemma formula_add_params1:

      [| p ∈ formula; n ∈ nat; x ∈ A; bvs ∈ list(A); env ∈ list(A); length(bvs) = n |]
      ==> sats(A, incr_bv1^n (p), Cons(x, bvs @ env)) <-> sats(A, p, Cons(x, env))

lemma arity_incr_bv1_eq:

      p ∈ formula
      ==> arity(incr_bv1(p)) = (if 1 < arity(p) then succ(arity(p)) else arity(p))

lemma arity_iterates_incr_bv1_eq:

      [| p ∈ formula; n ∈ nat |]
      ==> arity(incr_bv1^n (p)) = (if 1 < arity(p) then n #+ arity(p) else arity(p))

### Definable Powerset

lemma DPowI:

      [| env ∈ list(A); p ∈ formula; arity(p) ≤ succ(length(env)) |]
      ==> {x ∈ A . sats(A, p, Cons(x, env))} ∈ DPow(A)

lemma DPowI2:

      [| !!x. x ∈ A ==> P(x) <-> sats(A, p, Cons(x, env)); env ∈ list(A); p ∈ formula;
         arity(p) ≤ succ(length(env)) |]
      ==> {x ∈ A . P(x)} ∈ DPow(A)

lemma DPowD:

      X ∈ DPow(A)
      ==> X ⊆ A ∧
          (∃env∈list(A).
              ∃p∈formula.
                 arity(p) ≤ succ(length(env)) ∧
                 X = {x ∈ A . sats(A, p, Cons(x, env))})

lemmas DPow_imp_subset:

      X1 ∈ DPow(A1) ==> X1 ⊆ A1

lemmas DPow_imp_subset:

      X1 ∈ DPow(A1) ==> X1 ⊆ A1

lemma

      [| p ∈ formula; env ∈ list(A); arity(p) ≤ succ(length(env)) |]
      ==> {x ∈ A . sats(A, p, Cons(x, env))} ∈ DPow(A)

lemma DPow_subset_Pow:

      DPow(A) ⊆ Pow(A)

lemma empty_in_DPow:

      0 ∈ DPow(A)

lemma Compl_in_DPow:

      X ∈ DPow(A) ==> A - X ∈ DPow(A)

lemma Int_in_DPow:

      [| X ∈ DPow(A); Y ∈ DPow(A) |] ==> X ∩ Y ∈ DPow(A)

lemma Un_in_DPow:

      [| X ∈ DPow(A); Y ∈ DPow(A) |] ==> X ∪ Y ∈ DPow(A)

lemma singleton_in_DPow:

      a ∈ A ==> {a} ∈ DPow(A)

lemma cons_in_DPow:

      [| a ∈ A; X ∈ DPow(A) |] ==> cons(a, X) ∈ DPow(A)

lemma Fin_into_DPow:

      X ∈ Fin(A) ==> X ∈ DPow(A)

lemma Finite_Pow_subset_Pow:

      Finite(A) ==> Pow(A) ⊆ DPow(A)

lemma Finite_DPow_eq_Pow:

      Finite(A) ==> DPow(A) = Pow(A)

### Internalized Formulas for the Ordinals

#### The subset relation

lemma subset_type:

      [| x ∈ nat; y ∈ nat |] ==> subset_fm(x, y) ∈ formula

lemma arity_subset_fm:

      [| x ∈ nat; y ∈ nat |] ==> arity(subset_fm(x, y)) = succ(x) ∪ succ(y)

lemma sats_subset_fm:

      [| x < length(env); y ∈ nat; env ∈ list(A); Transset(A) |]
      ==> sats(A, subset_fm(x, y), env) <-> nth(x, env) ⊆ nth(y, env)

#### Transitive sets

lemma transset_type:

      x ∈ nat ==> transset_fm(x) ∈ formula

lemma arity_transset_fm:

      x ∈ nat ==> arity(transset_fm(x)) = succ(x)

lemma sats_transset_fm:

      [| x < length(env); env ∈ list(A); Transset(A) |]
      ==> sats(A, transset_fm(x), env) <-> Transset(nth(x, env))

#### Ordinals

lemma ordinal_type:

      x ∈ nat ==> ordinal_fm(x) ∈ formula

lemma arity_ordinal_fm:

      x ∈ nat ==> arity(ordinal_fm(x)) = succ(x)

lemma sats_ordinal_fm:

      [| x < length(env); env ∈ list(A); Transset(A) |]
      ==> sats(A, ordinal_fm(x), env) <-> Ord(nth(x, env))

theorem Ords_in_DPow:

      Transset(A) ==> {x ∈ A . Ord(x)} ∈ DPow(A)

### Constant Lset: Levels of the Constructible Universe

lemma Lset:

      Lset(i) = (\<Union>j∈i. DPow(Lset(j)))

lemma LsetI:

      [| y ∈ x; A ∈ DPow(Lset(y)) |] ==> A ∈ Lset(x)

lemma LsetD:

      A ∈ Lset(x) ==> ∃y∈x. A ∈ DPow(Lset(y))

#### Transitivity

lemma elem_subset_in_DPow:

      [| X ∈ A; X ⊆ A |] ==> X ∈ DPow(A)

lemma Transset_subset_DPow:

      Transset(A) ==> A ⊆ DPow(A)

lemma Transset_DPow:

      Transset(A) ==> Transset(DPow(A))

lemma Transset_Lset:

      Transset(Lset(i))

lemma mem_Lset_imp_subset_Lset:

      a ∈ Lset(i) ==> a ⊆ Lset(i)

#### Monotonicity

lemma Lset_mono:

      i ⊆ j ==> Lset(i) ⊆ Lset(j)

lemma Lset_mono_mem:

      i ∈ j ==> Lset(i) ⊆ Lset(j)

lemma subset_Lset_ltD:

      [| A ⊆ Lset(i); i < j |] ==> A ⊆ Lset(j)

#### 0, successor and limit equations for Lset

lemma Lset_0:

      Lset(0) = 0

lemma Lset_succ_subset1:

      DPow(Lset(i)) ⊆ Lset(succ(i))

lemma Lset_succ_subset2:

      Lset(succ(i)) ⊆ DPow(Lset(i))

lemma Lset_succ:

      Lset(succ(i)) = DPow(Lset(i))

lemma Lset_Union:

      Lset(\<Union>X) = (\<Union>y∈X. Lset(y))

#### Lset applied to Limit ordinals

lemma Limit_Lset_eq:

      Limit(i) ==> Lset(i) = (\<Union>y∈i. Lset(y))

lemma lt_LsetI:

      [| a ∈ Lset(j); j < i |] ==> a ∈ Lset(i)

lemma Limit_LsetE:

      [| a ∈ Lset(i); ¬ R ==> Limit(i); !!x. [| x < i; a ∈ Lset(x) |] ==> R |] ==> R

#### Basic closure properties

lemma zero_in_Lset:

      y ∈ x ==> 0 ∈ Lset(x)

lemma notin_Lset:

      x ∉ Lset(x)

### Constructible Ordinals: Kunen's VI 1.9 (b)

lemma Ords_of_Lset_eq:

      Ord(i) ==> {x ∈ Lset(i) . Ord(x)} = i

lemma Ord_subset_Lset:

      Ord(i) ==> i ⊆ Lset(i)

lemma Ord_in_Lset:

      Ord(i) ==> i ∈ Lset(succ(i))

lemma Ord_in_L:

      Ord(i) ==> L(i)

#### Unions

lemma Union_in_Lset:

      X ∈ Lset(i) ==> \<Union>X ∈ Lset(succ(i))

theorem Union_in_L:

      L(X) ==> L(\<Union>X)

#### Finite sets and ordered pairs

lemma singleton_in_Lset:

      a ∈ Lset(i) ==> {a} ∈ Lset(succ(i))

lemma doubleton_in_Lset:

      [| a ∈ Lset(i); b ∈ Lset(i) |] ==> {a, b} ∈ Lset(succ(i))

lemma Pair_in_Lset:

      [| a ∈ Lset(i); b ∈ Lset(i); Ord(i) |] ==> ⟨a, b⟩ ∈ Lset(succ(succ(i)))

lemmas Lset_UnI1:

      c ∈ Lset(i) ==> c ∈ Lset(i ∪ B)

lemmas Lset_UnI1:

      c ∈ Lset(i) ==> c ∈ Lset(i ∪ B)

lemmas Lset_UnI2:

      c ∈ Lset(i) ==> c ∈ Lset(A ∪ i)

lemmas Lset_UnI2:

      c ∈ Lset(i) ==> c ∈ Lset(A ∪ i)

lemma doubleton_in_LLimit:

      [| a ∈ Lset(i); b ∈ Lset(i); Limit(i) |] ==> {a, b} ∈ Lset(i)

theorem doubleton_in_L:

      [| L(a); L(b) |] ==> L({a, b})

lemma Pair_in_LLimit:

      [| a ∈ Lset(i); b ∈ Lset(i); Limit(i) |] ==> ⟨a, b⟩ ∈ Lset(i)

lemma L_I:

      [| x ∈ Lset(i); Ord(i) |] ==> L(x)

lemma L_D:

      L(x) ==> ∃i. Ord(i) ∧ x ∈ Lset(i)

lemma Ord_lrank:

      Ord(lrank(a))

lemma Lset_lrank_lt:

      [| Ord(i); x ∈ Lset(i) |] ==> lrank(x) < i

lemma Lset_iff_lrank_lt:

      Ord(i) ==> x ∈ Lset(i) <-> L(x) ∧ lrank(x) < i

lemma Lset_succ_lrank_iff:

      x ∈ Lset(succ(lrank(x))) <-> L(x)

lemma lrank_of_Ord:

      Ord(i) ==> lrank(i) = i

lemma Lset_in_Lset_succ:

      Lset(i) ∈ Lset(succ(i))

lemma lrank_Lset:

      Ord(i) ==> lrank(Lset(i)) = i

lemma Lset_subset_Vset:

      Ord(i) ==> Lset(i) ⊆ Vset(i)

lemma Lset_subset_Vset':

      i ∈ nat ==> Lset(i) = Vset(i)

lemma subset_Lset:

      ∀x∈A. L(x) ==> ∃i. Ord(i) ∧ A ⊆ Lset(i)

lemma subset_LsetE:

      [| ∀x∈A. L(x); !!i. [| Ord(i); A ⊆ Lset(i) |] ==> P |] ==> P

#### For L to satisfy the Powerset axiom

lemma LPow_env_typing:

      [| y ∈ Lset(i); Ord(i); y ⊆ X |] ==> ∃z∈Pow(X). y ∈ Lset(succ(lrank(z)))

lemma LPow_in_Lset:

      [| X ∈ Lset(i); Ord(i) |] ==> ∃j. Ord(j) ∧ {y ∈ Pow(X) . L(y)} ∈ Lset(j)

theorem LPow_in_L:

      L(X) ==> L({y ∈ Pow(X) . L(y)})

### Eliminating @{term arity} from the Definition of @{term Lset}

lemma nth_zero_eq_0:

      n ∈ nat ==> nth(n, [0]) = 0

lemma sats_app_0_iff:

      [| p ∈ formula; 0 ∈ A; env ∈ list(A) |]
      ==> sats(A, p, env @ [0]) <-> sats(A, p, env)

lemma sats_app_zeroes_iff:

      [| p ∈ formula; 0 ∈ A; env ∈ list(A); n ∈ nat |]
      ==> sats(A, p, env @ repeat(0, n)) <-> sats(A, p, env)

lemma exists_bigger_env:

      [| p ∈ formula; 0 ∈ A; env ∈ list(A) |]
      ==> ∃env'∈list(A).
             arity(p) ≤ succ(length(env')) ∧
             (∀a∈A. sats(A, p, Cons(a, env')) <-> sats(A, p, Cons(a, env)))

lemma DPow_subset_DPow':

      DPow(A) ⊆ DPow'(A)

lemma DPow'_0:

      DPow'(0) = {0}

lemma DPow'_subset_DPow:

      0 ∈ A ==> DPow'(A) ⊆ DPow(A)

lemma DPow_eq_DPow':

      Transset(A) ==> DPow(A) = DPow'(A)

lemma Lset_eq_transrec_DPow':

      Lset(i) = transrec(i, %x f. \<Union>y∈x. DPow'(f ` y))

lemma DPow_LsetI:

      [| !!x. x ∈ Lset(i) ==> P(x) <-> sats(Lset(i), p, Cons(x, env));
         env ∈ list(Lset(i)); p ∈ formula |]
      ==> {x ∈ Lset(i) . P(x)} ∈ DPow(Lset(i))