src/HOL/HOLCF/FOCUS/Stream_adm.thy
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explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Title:      HOL/HOLCF/FOCUS/Stream_adm.thy
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    Author:     David von Oheimb, TU Muenchen
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*)
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header {* Admissibility for streams *}
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theory Stream_adm
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imports "~~/src/HOL/HOLCF/Library/Stream" "~~/src/HOL/Library/Continuity"
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begin
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definition
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  stream_monoP  :: "(('a stream) set \<Rightarrow> ('a stream) set) \<Rightarrow> bool" where
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  "stream_monoP F = (\<exists>Q i. \<forall>P s. enat i \<le> #s \<longrightarrow>
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                    (s \<in> F P) = (stream_take i\<cdot>s \<in> Q \<and> iterate i\<cdot>rt\<cdot>s \<in> P))"
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definition
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  stream_antiP  :: "(('a stream) set \<Rightarrow> ('a stream) set) \<Rightarrow> bool" where
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  "stream_antiP F = (\<forall>P x. \<exists>Q i.
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                (#x  < enat i \<longrightarrow> (\<forall>y. x \<sqsubseteq> y \<longrightarrow> y \<in> F P \<longrightarrow> x \<in> F P)) \<and>
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                (enat i <= #x \<longrightarrow> (\<forall>y. x \<sqsubseteq> y \<longrightarrow>
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                (y \<in> F P) = (stream_take i\<cdot>y \<in> Q \<and> iterate i\<cdot>rt\<cdot>y \<in> P))))"
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definition
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  antitonP :: "'a set => bool" where
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  "antitonP P = (\<forall>x y. x \<sqsubseteq> y \<longrightarrow> y\<in>P \<longrightarrow> x\<in>P)"
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(* ----------------------------------------------------------------------- *)
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section "admissibility"
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lemma infinite_chain_adm_lemma:
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  "\<lbrakk>Porder.chain Y; \<forall>i. P (Y i);  
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    \<And>Y. \<lbrakk>Porder.chain Y; \<forall>i. P (Y i); \<not> finite_chain Y\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)\<rbrakk>
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      \<Longrightarrow> P (\<Squnion>i. Y i)"
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apply (case_tac "finite_chain Y")
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prefer 2 apply fast
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apply (unfold finite_chain_def)
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apply safe
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apply (erule lub_finch1 [THEN lub_eqI, THEN ssubst])
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apply assumption
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apply (erule spec)
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done
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lemma increasing_chain_adm_lemma:
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  "\<lbrakk>Porder.chain Y;  \<forall>i. P (Y i); \<And>Y. \<lbrakk>Porder.chain Y; \<forall>i. P (Y i);
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    \<forall>i. \<exists>j>i. Y i \<noteq> Y j \<and> Y i \<sqsubseteq> Y j\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)\<rbrakk>
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      \<Longrightarrow> P (\<Squnion>i. Y i)"
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apply (erule infinite_chain_adm_lemma)
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apply assumption
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apply (erule thin_rl)
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apply (unfold finite_chain_def)
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apply (unfold max_in_chain_def)
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apply (fast dest: le_imp_less_or_eq elim: chain_mono_less)
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done
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lemma flatstream_adm_lemma:
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  assumes 1: "Porder.chain Y"
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  assumes 2: "!i. P (Y i)"
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  assumes 3: "(!!Y. [| Porder.chain Y; !i. P (Y i); !k. ? j. enat k < #((Y j)::'a::flat stream)|]
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  ==> P(LUB i. Y i))"
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  shows "P(LUB i. Y i)"
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apply (rule increasing_chain_adm_lemma [OF 1 2])
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apply (erule 3, assumption)
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apply (erule thin_rl)
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apply (rule allI)
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apply (case_tac "!j. stream_finite (Y j)")
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apply ( rule chain_incr)
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apply ( rule allI)
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apply ( drule spec)
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apply ( safe)
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apply ( rule exI)
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apply ( rule slen_strict_mono)
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apply (   erule spec)
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apply (  assumption)
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apply ( assumption)
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apply (metis enat_ord_code(4) slen_infinite)
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done
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(* should be without reference to stream length? *)
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lemma flatstream_admI: "[|(!!Y. [| Porder.chain Y; !i. P (Y i); 
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 !k. ? j. enat k < #((Y j)::'a::flat stream)|] ==> P(LUB i. Y i))|]==> adm P"
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apply (unfold adm_def)
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apply (intro strip)
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apply (erule (1) flatstream_adm_lemma)
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apply (fast)
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done
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(* context (theory "Extended_Nat");*)
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lemma ile_lemma: "enat (i + j) <= x ==> enat i <= x"
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  by (rule order_trans) auto
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lemma stream_monoP2I:
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"!!X. stream_monoP F ==> !i. ? l. !x y. 
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  enat l <= #x --> (x::'a::flat stream) << y --> x:down_iterate F i --> y:down_iterate F i"
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apply (unfold stream_monoP_def)
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apply (safe)
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apply (rule_tac x="i*ia" in exI)
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apply (induct_tac "ia")
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apply ( simp)
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apply (simp)
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apply (intro strip)
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apply (erule allE, erule all_dupE, drule mp, erule ile_lemma)
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apply (drule_tac P="%x. x" in subst, assumption)
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apply (erule allE, drule mp, rule ile_lemma) back
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apply ( erule order_trans)
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apply ( erule slen_mono)
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apply (erule ssubst)
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apply (safe)
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apply ( erule (2) ile_lemma [THEN slen_take_lemma3, THEN subst])
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apply (erule allE)
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apply (drule mp)
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apply ( erule slen_rt_mult)
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apply (erule allE)
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apply (drule mp)
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apply (erule monofun_rt_mult)
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apply (drule (1) mp)
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apply (assumption)
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done
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lemma stream_monoP2_gfp_admI: "[| !i. ? l. !x y. 
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 enat l <= #x --> (x::'a::flat stream) << y --> x:down_iterate F i --> y:down_iterate F i;
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    down_cont F |] ==> adm (%x. x:gfp F)"
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apply (erule INTER_down_iterate_is_gfp [THEN ssubst]) (* cont *)
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apply (simp (no_asm))
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apply (rule adm_lemmas)
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apply (rule flatstream_admI)
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apply (erule allE)
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apply (erule exE)
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apply (erule allE, erule exE)
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apply (erule allE, erule allE, drule mp) (* stream_monoP *)
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apply ( drule ileI1)
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apply ( drule order_trans)
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apply (  rule ile_eSuc)
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apply ( drule eSuc_ile_mono [THEN iffD1])
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apply ( assumption)
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apply (drule mp)
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apply ( erule is_ub_thelub)
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apply (fast)
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done
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lemmas fstream_gfp_admI = stream_monoP2I [THEN stream_monoP2_gfp_admI]
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lemma stream_antiP2I:
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"!!X. [|stream_antiP (F::(('a::flat stream)set => ('a stream set)))|]
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  ==> !i x y. x << y --> y:down_iterate F i --> x:down_iterate F i"
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apply (unfold stream_antiP_def)
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apply (rule allI)
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apply (induct_tac "i")
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apply ( simp)
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apply (simp)
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apply (intro strip)
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apply (erule allE, erule all_dupE, erule exE, erule exE)
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apply (erule conjE)
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apply (case_tac "#x < enat i")
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apply ( fast)
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apply (unfold linorder_not_less)
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apply (drule (1) mp)
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apply (erule all_dupE, drule mp, rule below_refl)
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apply (erule ssubst)
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apply (erule allE, drule (1) mp)
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apply (drule_tac P="%x. x" in subst, assumption)
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apply (erule conjE, rule conjI)
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apply ( erule slen_take_lemma3 [THEN ssubst], assumption)
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apply ( assumption)
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apply (erule allE, erule allE, drule mp, erule monofun_rt_mult)
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apply (drule (1) mp)
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apply (assumption)
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done
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lemma stream_antiP2_non_gfp_admI:
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"!!X. [|!i x y. x << y --> y:down_iterate F i --> x:down_iterate F i; down_cont F |] 
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  ==> adm (%u. ~ u:gfp F)"
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apply (unfold adm_def)
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apply (simp add: INTER_down_iterate_is_gfp)
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apply (fast dest!: is_ub_thelub)
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done
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lemmas fstream_non_gfp_admI = stream_antiP2I [THEN stream_antiP2_non_gfp_admI]
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(**new approach for adm********************************************************)
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section "antitonP"
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lemma antitonPD: "[| antitonP P; y:P; x<<y |] ==> x:P"
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apply (unfold antitonP_def)
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apply auto
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done
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lemma antitonPI: "!x y. y:P --> x<<y --> x:P ==> antitonP P"
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apply (unfold antitonP_def)
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apply (fast)
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done
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lemma antitonP_adm_non_P: "antitonP P ==> adm (%u. u~:P)"
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apply (unfold adm_def)
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apply (auto dest: antitonPD elim: is_ub_thelub)
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done
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lemma def_gfp_adm_nonP: "P \<equiv> gfp F \<Longrightarrow> {y. \<exists>x::'a::pcpo. y \<sqsubseteq> x \<and> x \<in> P} \<subseteq> F {y. \<exists>x. y \<sqsubseteq> x \<and> x \<in> P} \<Longrightarrow> 
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  adm (\<lambda>u. u\<notin>P)"
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apply (simp)
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apply (rule antitonP_adm_non_P)
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apply (rule antitonPI)
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apply (drule gfp_upperbound)
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apply (fast)
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done
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lemma adm_set:
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"{\<Squnion>i. Y i |Y. Porder.chain Y & (\<forall>i. Y i \<in> P)} \<subseteq> P \<Longrightarrow> adm (\<lambda>x. x\<in>P)"
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apply (unfold adm_def)
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apply (fast)
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done
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lemma def_gfp_admI: "P \<equiv> gfp F \<Longrightarrow> {\<Squnion>i. Y i |Y. Porder.chain Y \<and> (\<forall>i. Y i \<in> P)} \<subseteq> 
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  F {\<Squnion>i. Y i |Y. Porder.chain Y \<and> (\<forall>i. Y i \<in> P)} \<Longrightarrow> adm (\<lambda>x. x\<in>P)"
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apply (simp)
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apply (rule adm_set)
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apply (erule gfp_upperbound)
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done
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end