Theory Stream_adm

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theory Stream_adm
imports Stream Continuity

(*  Title:      HOLCF/ex/Stream_adm.thy
Author: David von Oheimb, TU Muenchen
*)


header {* Admissibility for streams *}

theory Stream_adm
imports Stream Continuity
begin


definition
stream_monoP :: "(('a stream) set => ('a stream) set) => bool" where
"stream_monoP F = (∃Q i. ∀P s. Fin i ≤ #s -->
(s ∈ F P) = (stream_take i·s ∈ Q ∧ iterate i·rt·s ∈ P))"


definition
stream_antiP :: "(('a stream) set => ('a stream) set) => bool" where
"stream_antiP F = (∀P x. ∃Q i.
(#x < Fin i --> (∀y. x \<sqsubseteq> y --> y ∈ F P --> x ∈ F P)) ∧
(Fin i <= #x --> (∀y. x \<sqsubseteq> y -->
(y ∈ F P) = (stream_take i·y ∈ Q ∧ iterate i·rt·y ∈ P))))"


definition
antitonP :: "'a set => bool" where
"antitonP P = (∀x y. x \<sqsubseteq> y --> y∈P --> x∈P)"



(* ----------------------------------------------------------------------- *)

section "admissibility"

lemma flatstream_adm_lemma:
assumes 1: "Porder.chain Y"
assumes 2: "!i. P (Y i)"
assumes 3: "(!!Y. [| Porder.chain Y; !i. P (Y i); !k. ? j. Fin k < #((Y j)::'a::flat stream)|]
==> P(LUB i. Y i))"

shows "P(LUB i. Y i)"

apply (rule increasing_chain_adm_lemma [of _ P, OF 1 2])
apply (erule 3, assumption)
apply (erule thin_rl)
apply (rule allI)
apply (case_tac "!j. stream_finite (Y j)")
apply ( rule chain_incr)
apply ( rule allI)
apply ( drule spec)
apply ( safe)
apply ( rule exI)
apply ( rule slen_strict_mono)
apply ( erule spec)
apply ( assumption)
apply ( assumption)
apply (metis inat_ord_code(4) slen_infinite)
done

(* should be without reference to stream length? *)
lemma flatstream_admI: "[|(!!Y. [| Porder.chain Y; !i. P (Y i);
!k. ? j. Fin k < #((Y j)::'a::flat stream)|] ==> P(LUB i. Y i))|]==> adm P"

apply (unfold adm_def)
apply (intro strip)
apply (erule (1) flatstream_adm_lemma)
apply (fast)
done


(* context (theory "Nat_InFinity");*)
lemma ile_lemma: "Fin (i + j) <= x ==> Fin i <= x"
by (rule order_trans) auto

lemma stream_monoP2I:
"!!X. stream_monoP F ==> !i. ? l. !x y.
Fin l <= #x --> (x::'a::flat stream) << y --> x:down_iterate F i --> y:down_iterate F i"

apply (unfold stream_monoP_def)
apply (safe)
apply (rule_tac x="i*ia" in exI)
apply (induct_tac "ia")
apply ( simp)
apply (simp)
apply (intro strip)
apply (erule allE, erule all_dupE, drule mp, erule ile_lemma)
apply (drule_tac P="%x. x" in subst, assumption)
apply (erule allE, drule mp, rule ile_lemma) back
apply ( erule order_trans)
apply ( erule slen_mono)
apply (erule ssubst)
apply (safe)
apply ( erule (2) ile_lemma [THEN slen_take_lemma3, THEN subst])
apply (erule allE)
apply (drule mp)
apply ( erule slen_rt_mult)
apply (erule allE)
apply (drule mp)
apply (erule monofun_rt_mult)
apply (drule (1) mp)
apply (assumption)
done

lemma stream_monoP2_gfp_admI: "[| !i. ? l. !x y.
Fin l <= #x --> (x::'a::flat stream) << y --> x:down_iterate F i --> y:down_iterate F i;
down_cont F |] ==> adm (%x. x:gfp F)"

apply (erule INTER_down_iterate_is_gfp [THEN ssubst]) (* cont *)
apply (simp (no_asm))
apply (rule adm_lemmas)
apply (rule flatstream_admI)
apply (erule allE)
apply (erule exE)
apply (erule allE, erule exE)
apply (erule allE, erule allE, drule mp) (* stream_monoP *)
apply ( drule ileI1)
apply ( drule order_trans)
apply ( rule ile_iSuc)
apply ( drule iSuc_ile_mono [THEN iffD1])
apply ( assumption)
apply (drule mp)
apply ( erule is_ub_thelub)
apply (fast)
done

lemmas fstream_gfp_admI = stream_monoP2I [THEN stream_monoP2_gfp_admI]

lemma stream_antiP2I:
"!!X. [|stream_antiP (F::(('a::flat stream)set => ('a stream set)))|]
==> !i x y. x << y --> y:down_iterate F i --> x:down_iterate F i"

apply (unfold stream_antiP_def)
apply (rule allI)
apply (induct_tac "i")
apply ( simp)
apply (simp)
apply (intro strip)
apply (erule allE, erule all_dupE, erule exE, erule exE)
apply (erule conjE)
apply (case_tac "#x < Fin i")
apply ( fast)
apply (unfold linorder_not_less)
apply (drule (1) mp)
apply (erule all_dupE, drule mp, rule refl_less)
apply (erule ssubst)
apply (erule allE, drule (1) mp)
apply (drule_tac P="%x. x" in subst, assumption)
apply (erule conjE, rule conjI)
apply ( erule slen_take_lemma3 [THEN ssubst], assumption)
apply ( assumption)
apply (erule allE, erule allE, drule mp, erule monofun_rt_mult)
apply (drule (1) mp)
apply (assumption)
done

lemma stream_antiP2_non_gfp_admI:
"!!X. [|!i x y. x << y --> y:down_iterate F i --> x:down_iterate F i; down_cont F |]
==> adm (%u. ~ u:gfp F)"

apply (unfold adm_def)
apply (simp add: INTER_down_iterate_is_gfp)
apply (fast dest!: is_ub_thelub)
done

lemmas fstream_non_gfp_admI = stream_antiP2I [THEN stream_antiP2_non_gfp_admI]



(**new approach for adm********************************************************)

section "antitonP"

lemma antitonPD: "[| antitonP P; y:P; x<<y |] ==> x:P"
apply (unfold antitonP_def)
apply auto
done

lemma antitonPI: "!x y. y:P --> x<<y --> x:P ==> antitonP P"
apply (unfold antitonP_def)
apply (fast)
done

lemma antitonP_adm_non_P: "antitonP P ==> adm (%u. u~:P)"
apply (unfold adm_def)
apply (auto dest: antitonPD elim: is_ub_thelub)
done

lemma def_gfp_adm_nonP: "P ≡ gfp F ==> {y. ∃x::'a::pcpo. y \<sqsubseteq> x ∧ x ∈ P} ⊆ F {y. ∃x. y \<sqsubseteq> x ∧ x ∈ P} ==>
adm (λu. u∉P)"

apply (simp)
apply (rule antitonP_adm_non_P)
apply (rule antitonPI)
apply (drule gfp_upperbound)
apply (fast)
done

lemma adm_set:
"{\<Squnion>i. Y i |Y. Porder.chain Y & (∀i. Y i ∈ P)} ⊆ P ==> adm (λx. x∈P)"

apply (unfold adm_def)
apply (fast)
done

lemma def_gfp_admI: "P ≡ gfp F ==> {\<Squnion>i. Y i |Y. Porder.chain Y ∧ (∀i. Y i ∈ P)} ⊆
F {\<Squnion>i. Y i |Y. Porder.chain Y ∧ (∀i. Y i ∈ P)} ==> adm (λx. x∈P)"

apply (simp)
apply (rule adm_set)
apply (erule gfp_upperbound)
done

end