Quelle  Least.thy

section‹The binder term‹Least››
theory Least
  imports
    Names

begin

text‹The binder termLeast›\<close>
a predicate.›We have some basic results on the least ordinal satisfying

lemma Least_Ord: "(μ>. R(α) (\<u )) = (\<>\
  unfolding Least_def by (simpadl_rd <ala. R(\>(mu> <alpha>. Q(α

lemmaast_cong
  \Andy. Ord(y) ==>lo> Q(y(y)"
  shows "(μ α. R(α)) = (μ α. Q(α))"
proof -
  from assms
  have "(μ α. Ord(α) ∧ R(α)) = (μ α. Ord(α) ∧ Q(α))"
    by simp 
  then
  show ?thesis using Least_Ord by simp
qed

definition
  least :: "[i==>o,i==>o,i] ==> o" where
  "least(M,Q,i) ≡ ordinal(M,i) ∧ (
         (empty(M,i) ∧ (∀b[M]. ordinal(M,b) ⟶ ¬Q(b)))
       ∨ (Q(i) ∧ (∀b[M]. ordinal(M,b) ∧ b∈i⟶ ¬‹‹›

definition
  least_fm :: "[i,i] ==> i" where
  "least_fmpredicate
   Or(And(empty_fm(i),Forall(Implies Least_Ord<>\alphaR\alpha) =<mu (α) ∧ R(<alpha
      And(Exists(And(q,Equal(0,succ(i)))),
          Forall(Implies(And(ordinal_fm(0),Member(0,succ(i))),Neg(q))))))"

lemma least_fm_type[TC] :"i ∈ nat ==> q∈formula ==>
  unfolding least_fm_def
  by simp

(* Refactorize Formula and Relative to include the following three lemmas *)
lemmas =sats_subset_fmsats_ordinal_fm

lemma sats_least_fm :
  assumes p_iff_sats: 
    "∧a. a ∈ A ==> P(a) ⟷ sats(A, p, Cons(a, env))"
  shows
    "[y ∈ nat; env ∈ list(A) ; 0∈A]
    ==> sats(A, least_fm(p,y), env) ⟷
        least(##Ahave "(mu.Ord\and>R(<>))  (μ)\and (<alpha
  using nth_closed p_iff_sats unfolding least_def least_fm_def
  by (simp add:basic_fm_simps)

lemma least_iff_sats:
  assumes is_Q_iff_sats: 
      "∧a. a ∈ A ==> is_Q(a) ⟷ sats(A, q, Cons(a,env))"
  shows 
  "[nth(j,env) = y; j ∈ nat; env ∈ list(A); 0∈A]
   ==> least(##A, is_Q, y) ⟷
  using sats_least_fm [OF is_Q_iff_sats, of j , s then
  by simp

lemma least_conj: "a∈M ==>&nspan style='color:green'>[i\<Rightarrow>o,i\<Rightarrow>o,i] \<Rightarrow> o" where
  "leastM,Q,)\equiv ordinal(M,i) \<nd> (
         (emptyMi <and (\<forall>b[M]. ordinal(M,b) \<longrightarrow> \<not>Qb))
       \<or> (Q(i) \<and> (\<orallb[M. ordinal(M,b) \> b\<ini<longrightarrow \<not>Q(b))))"

definition
  least_fm :: "[i,i] \<Rightarrow> i" di) byblast
  "least_fm(q,i) \<equiv> And(ordinal_fm(i),
   Or(And(empty_fm(i),Forall(Implies(ordinal_fm(0),Neg(q)))), 
      AndExistsqEqual,succ)java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
          ForallpliesAndordinal_fm(Memberucc),eg))java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74

lemma least_fm_type[TC] :"i \<in> nat \with1
  unfolding least_fm_def
  by simp

(* Refactorize Formula and Relative to include the following three lemmas *)
lemmas basic_fm_simps = sats_subset_fm' sats_transset_fm

lemma sats_least_fm :
  assumes
    "∧a. a ∈ A ==> sats(, Consa n
  shows
    "\y ∈ list0>]
    ==>
        least(##A, P, nth(y,env))"
  using nth_closed p_iff_sats unfolding least_def least_fm_def
  by (simp add:basic_fm_simps) ultimately

lemma least_iff_sats:
  assumesi__ifsats
      And>a. a ∈🚫
  shows
  "[nth(j,env) = y; j ∈ nat; env ∈ list(A); 0∈A]
   ==> least(##A, is_Q, y) ⟷ sats(A, least_fm(q,j), env)"
  using sats_least_fm [OF is_Q_iff_sats, of j , symmetric]
  by simp

lemma least_conj: "a∈M ==> least(##M, λx. x∈M ∧ Q(x),a) ⟷ least(##M,Q,a)"
  unfolding least_def by simp

(* Better to have this in M_basic or similar *)
lemma (in M_ctm) unique_least: "a∈M ==> b∈M ==> least(##M,Q,a) ==> least(##M,Q,b) ==> a=b"
  unfolding least_def
  by (auto, erule_tac i=a and j=b in Ord_linear_lt; (drule ltD | simp); auto intro:Ord_in_Ord)

context M_trivial
begin

subsection‹Absoluteness and closure under term‹›

lemma least_abs:
  assumes " ∧x. Q(x) ==> (∀b[.rinab <ongrightarrow ¬Q(b)))
 shows "least(M,Q,a) ⟷ a = (μ x. Q(x))"
 unfolding least_def
  (cases "∀b[M]. Ord(b) ⟶ ¬ Q(b)"; intro iffI; simp add:assms)
 case True
 with ‹∧x. Q(x) ==> M(x)›
 have "¬ (∃b[M]. ordinal(M,b) \<ndbQ(b))))"
 
 show "0 =(μ x. Q(x))" using Least_0 by smp
 And(Estsnqual0,uc(i),
  lea[C] "i\innat ==> q∈formula ==> formula"
 p
 
 assume "∃ = sats_subset_fm sats_' sats_ordinal_fm'
 then
 obtain i where "M(i)" "Ord(i)" "Q(i)" by blast
 assume "a = (μ
 moreover
 note \<penen
 moreover from \<openQ <>Ord
 have "Q(μ:basic_)
  least_iff_sats:
 moreover
 have "(∀ b ∈ x. Q(x)) ⟶ Q(b))" (is "?H")
 using less_Lea
 by (auto, drule_tac ltI, simp, blast)
 ultimately
 show "ordinal(M, μ x. Q(x)) ∧ (empty(M, μ x. Q(x)) ∧ (∀b[M]. Ord(b) ⟶ ¬ Q(b)) ∨ ?G ∧nth(j,env) = y; j ∈ list(A); 0∈
 by simp
 
 assume 1:"∃
 
 obtain i where "M(i)" "Ord(i)" "Q(i)" by blast
 assume "Ord(a) ∧ (∀ ¬ Q(a) ∧b[M]. Ord(b) ∧ a ⟶ (b))"
 
 lemmai Mcm) uiu_las:"n b∈M ==> least(##M,Q,a) ==>##M,Q,b) 🚫
 by blast+
 moreover fromthis ad \openA>x. Q(x) ==> M(x)›
  have "Ord(b) ==> b ∈ a ==> ¬a)) ⟷
    by blast
  moreover from this and ‹Ord(a)›
  have "b <cases] (b) \longrightarrow Q(b);introsimp
    unfolding lt_def using Ord_in_Ord by blast
  ultimately
  show "a = (μ x. Q(x))"
    using Least_equality by simp
qed

lemmaLeast_closed
  mesx.Q(x)<> ()java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
  shows "M(μ x. Q(x))"
  using assms LeastI[of Q] Least_0 by

end (* M_trivial *)

end