Quelle ZF.thy Sprache: Isabelle

section\<open>Main ZF Theory: Everything Except AC\<close>

theory ZF imports List IntDiv CardinalArith begin

(*The theory of "iterates" logically belongs to Nat, but can't go there because
  primrec isn't available into after Datatype.*)

subsection\<open>Iteration of the function \<^term>\<open>F\<close>\<close>

consts  iterates :: "[i\i,i,i] \ i" (\(\notation=\mixfix iterates\\_^_ '(_'))\ [60,1000,1000] 60)

primrec
    "F^0 (x) = x"
    "F^(succ(n)) (x) = F(F^n (x))"

definition
  iterates_omega :: "[i\i,i] \ i" (\(\notation=\mixfix iterates_omega\\_^\ '(_'))\ [60,1000] 60) where
    "F^\ (x) \ \n\nat. F^n (x)"

lemma iterates_triv:
     "\n\nat; F(x) = x\ \ F^n (x) = x"
by (induct n rule: nat_induct, simp_all)

lemma iterates_type [TC]:
     "\n \ nat; a \ A; \x. x \ A \ F(x) \ A\
      \<Longrightarrow> F^n (a) \<in> A"
by (induct n rule: nat_induct, simp_all)

lemma iterates_omega_triv:
    "F(x) = x \ F^\ (x) = x"
by (simp add: iterates_omega_def iterates_triv)

lemma Ord_iterates [simp]:
     "\n\nat; \i. Ord(i) \ Ord(F(i)); Ord(x)\
      \<Longrightarrow> Ord(F^n (x))"
by (induct n rule: nat_induct, simp_all)

lemma iterates_commute: "n \ nat \ F(F^n (x)) = F^n (F(x))"
by (induct_tac n, simp_all)

subsection\<open>Transfinite Recursion\<close>

text\<open>Transfinite recursion for definitions based on the
    three cases of ordinals\<close>

definition
  transrec3 :: "[i, i, [i,i]\i, [i,i]\i] \i" where
    "transrec3(k, a, b, c) \
       transrec(k, \<lambda>x r.
         if x=0 then a
         else if Limit(x) then c(x, \<lambda>y\<in>x. r`y)
         else b(Arith.pred(x), r ` Arith.pred(x)))"

lemma transrec3_0 [simp]: "transrec3(0,a,b,c) = a"
by (rule transrec3_def [THEN def_transrec, THEN trans], simp)

lemma transrec3_succ [simp]:
     "transrec3(succ(i),a,b,c) = b(i, transrec3(i,a,b,c))"
by (rule transrec3_def [THEN def_transrec, THEN trans], simp)

lemma transrec3_Limit:
     "Limit(i) \
      transrec3(i,a,b,c) = c(i, \<lambda>j\<in>i. transrec3(j,a,b,c))"
by (rule transrec3_def [THEN def_transrec, THEN trans], force)

declaration \<open>fn _ =>
  Simplifier.map_ss (Simplifier.set_mksimps (fn ctxt =>
    map mk_eq o Ord_atomize o Variable.gen_all ctxt))
\<close>

end

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