imports Cont
```(*  Title:      HOL/HOLCF/Adm.thy
Author:     Franz Regensburger and Brian Huffman
*)

imports Cont
begin

default_sort cpo

subsection ‹Definitions›

definition
adm :: "('a::cpo ⇒ bool) ⇒ bool" where
"adm P = (∀Y. chain Y ⟶ (∀i. P (Y i)) ⟶ P (⨆i. Y i))"

"(⋀Y. ⟦chain Y; ∀i. P (Y i)⟧ ⟹ P (⨆i. Y i)) ⟹ adm P"

lemma admD: "⟦adm P; chain Y; ⋀i. P (Y i)⟧ ⟹ P (⨆i. Y i)"

lemma admD2: "⟦adm (λx. ¬ P x); chain Y; P (⨆i. Y i)⟧ ⟹ ∃i. P (Y i)"

text ‹For chain-finite (easy) types every formula is admissible.›

subsection ‹Admissibility of special formulae and propagation›

"(⋀y. adm (λx. P x y)) ⟹ adm (λx. ∀y. P x y)"

"(⋀y. y ∈ A ⟹ adm (λx. P x y)) ⟹ adm (λx. ∀y∈A. P x y)"

text ‹Admissibility for disjunction is hard to prove. It requires 2 lemmas.›

assumes chain: "chain Y"
assumes P: "∀i. ∃j≥i. P (Y j)"
shows "P (⨆i. Y i)"
proof -
define f where "f i = (LEAST j. i ≤ j ∧ P (Y j))" for i
have chain': "chain (λi. Y (f i))"
unfolding f_def
apply (rule chainI)
apply (rule chain_mono [OF chain])
apply (rule Least_le)
apply (rule LeastI2_ex)
done
have f1: "⋀i. i ≤ f i" and f2: "⋀i. P (Y (f i))"
using LeastI_ex [OF P [rule_format]] by (simp_all add: f_def)
have lub_eq: "(⨆i. Y i) = (⨆i. Y (f i))"
apply (rule below_antisym)
apply (rule lub_mono [OF chain chain'])
apply (rule chain_mono [OF chain f1])
apply (rule lub_range_mono [OF _ chain chain'])
apply clarsimp
done
show "P (⨆i. Y i)"
qed

"∀n::nat. P n ∨ Q n ⟹ (∀i. ∃j≥i. P j) ∨ (∀i. ∃j≥i. Q j)"
apply (erule contrapos_pp)
apply (clarsimp, rename_tac a b)
apply (rule_tac x="max a b" in exI)
apply simp
done

apply (erule (2) adm_disj_lemma1 [THEN disjI1])
apply (erule (2) adm_disj_lemma1 [THEN disjI2])
done

"⟦adm (λx. ¬ P x); adm (λx. Q x)⟧ ⟹ adm (λx. P x ⟶ Q x)"

"⟦adm (λx. P x ⟶ Q x); adm (λx. Q x ⟶ P x)⟧
⟹ adm (λx. P x = Q x)"

"⟦cont (λx. u x); cont (λx. v x)⟧ ⟹ adm (λx. u x ⊑ v x)"

"⟦cont (λx. u x); cont (λx. v x)⟧ ⟹ adm (λx. u x = v x)"

lemma adm_not_below [simp]: "cont (λx. t x) ⟹ adm (λx. t x \<notsqsubseteq> u)"

subsection ‹Compactness›

definition
compact :: "'a::cpo ⇒ bool" where
"compact k = adm (λx. k \<notsqsubseteq> x)"

lemma compactI: "adm (λx. k \<notsqsubseteq> x) ⟹ compact k"
unfolding compact_def .

lemma compactD: "compact k ⟹ adm (λx. k \<notsqsubseteq> x)"
unfolding compact_def .

lemma compactI2:
"(⋀Y. ⟦chain Y; x ⊑ (⨆i. Y i)⟧ ⟹ ∃i. x ⊑ Y i) ⟹ compact x"

lemma compactD2:
"⟦compact x; chain Y; x ⊑ (⨆i. Y i)⟧ ⟹ ∃i. x ⊑ Y i"

lemma compact_below_lub_iff:
"⟦compact x; chain Y⟧ ⟹ x ⊑ (⨆i. Y i) ⟷ (∃i. x ⊑ Y i)"
by (fast intro: compactD2 elim: below_lub)

lemma compact_chfin [simp]: "compact (x::'a::chfin)"

lemma compact_imp_max_in_chain:
"⟦chain Y; compact (⨆i. Y i)⟧ ⟹ ∃i. max_in_chain i Y"
apply (drule (1) compactD2, simp)
apply (erule exE, rule_tac x=i in exI)
apply (rule max_in_chainI)
apply (rule below_antisym)
apply (erule (1) chain_mono)
apply (erule (1) below_trans [OF is_ub_thelub])
done

"⟦compact k; cont (λx. t x)⟧ ⟹ adm (λx. k \<notsqsubseteq> t x)"

"⟦compact k; cont (λx. t x)⟧ ⟹ adm (λx. t x ≠ k)"

"⟦compact k; cont (λx. t x)⟧ ⟹ adm (λx. k ≠ t x)"

lemma compact_bottom [simp, intro]: "compact ⊥"
by (rule compactI, simp)

text ‹Any upward-closed predicate is admissible.›