```(*  Title:      HOL/HOLCF/FOCUS/Buffer_adm.thy
Author:     David von Oheimb, TU Muenchen
*)

section ‹One-element buffer, proof of Buf_Eq_imp_AC by induction + admissibility›

begin

declare enat_0 [simp]

lemma BufAC_Asm_d2: "a↝s ∈ BufAC_Asm ⟹ ∃d. a=Md d"
by (drule BufAC_Asm_unfold [THEN iffD1], auto)

lemma BufAC_Asm_d3:
"a↝b↝s ∈ BufAC_Asm ⟹ ∃d. a=Md d ∧ b=∙ ∧ s ∈ BufAC_Asm"
by (drule BufAC_Asm_unfold [THEN iffD1], auto)

lemma BufAC_Asm_F_def3:
"(s ∈ BufAC_Asm_F A) = (s=<> ∨
(∃d. ft⋅s=Def(Md d)) ∧ (rt⋅s=<> ∨ ft⋅(rt⋅s)=Def ∙ ∧ rt⋅(rt⋅s) ∈ A))"
by (unfold BufAC_Asm_F_def, auto)

lemma cont_BufAC_Asm_F: "inf_continuous BufAC_Asm_F"
by (auto simp add: inf_continuous_def BufAC_Asm_F_def3)

lemma BufAC_Cmt_F_def3:
"((s,t) ∈ BufAC_Cmt_F C) = (∀d x.
(s = <>       ⟶ t = <>                   ) ∧
(s = Md d↝<>  ⟶ t = <>                   ) ∧
(s = Md d↝∙↝x ⟶ ft⋅t = Def d & (x,rt⋅t) ∈ C))"
apply (unfold BufAC_Cmt_F_def)
apply (subgoal_tac "∀d x. (s = Md d↝∙↝x ⟶ (∃y. t = d↝y ∧ (x,y) ∈ C)) =
(s = Md d↝∙↝x ⟶ ft⋅t = Def d ∧ (x,rt⋅t) ∈ C)")
apply (simp)
apply (auto intro: surjectiv_scons [symmetric])
done

lemma cont_BufAC_Cmt_F: "inf_continuous BufAC_Cmt_F"
by (auto simp add: inf_continuous_def BufAC_Cmt_F_def3)

lemma BufAC_Asm_F_stream_monoP: "stream_monoP BufAC_Asm_F"
apply (unfold BufAC_Asm_F_def stream_monoP_def)
apply (rule_tac x="{x. (∃d. x = Md d↝∙↝<>)}" in exI)
apply (rule_tac x="Suc (Suc 0)" in exI)
apply (clarsimp)
done

apply (unfold BufAC_Asm_def)
apply (rule cont_BufAC_Asm_F [THEN BufAC_Asm_F_stream_monoP [THEN fstream_gfp_admI]])
done

lemma BufAC_Asm_F_stream_antiP: "stream_antiP BufAC_Asm_F"
apply (unfold stream_antiP_def BufAC_Asm_F_def)
apply (intro strip)
apply (rule_tac x="{x. (∃d. x = Md d↝∙↝<>)}" in exI)
apply (rule_tac x="Suc (Suc 0)" in exI)
apply (rule conjI)
prefer 2
apply ( intro strip)
apply ( drule slen_mono)
apply ( drule (1) order_trans)
apply (force)+
done

apply (unfold BufAC_Asm_def)
apply (rule cont_BufAC_Asm_F [THEN BufAC_Asm_F_stream_antiP [THEN fstream_non_gfp_admI]])
done

lemma BufAC_Asm_cong [rule_format]: "∀f ff. f ∈ BufEq ⟶ ff ∈ BufEq ⟶ s ∈ BufAC_Asm ⟶ f⋅s = ff⋅s"
apply (rule fstream_ind2)
apply   (force dest: Buf_f_empty)
apply  (force dest!: BufAC_Asm_d2
dest: Buf_f_d elim: ssubst)
apply (safe dest!: BufAC_Asm_d3)
apply (drule Buf_f_d_req)+
apply (fast elim: ssubst)
done

lemma BufAC_Cmt_d_req:
"⋀X. [|f ∈ BufEq; s ∈ BufAC_Asm; (s, f⋅s) ∈ BufAC_Cmt|] ==> (a↝b↝s, f⋅(a↝b↝s)) ∈ BufAC_Cmt"
apply (rule BufAC_Cmt_unfold [THEN iffD2])
apply (intro strip)
apply (frule Buf_f_d_req)
apply (auto elim: BufAC_Asm_cong [THEN subst])
done

lemma BufAC_Asm_antiton: "antitonP BufAC_Asm"
apply (rule antitonPI)
apply (rule allI)
apply (rule fstream_ind2)
apply (   rule cont_id)
apply (  safe)
apply (  rule BufAC_Asm_empty)
apply ( force dest!: fstream_prefix
dest: BufAC_Asm_d2 intro: BufAC_Asm_d)
apply ( force dest!: fstream_prefix
dest: BufAC_Asm_d3 intro!: BufAC_Asm_d_req)
done

lemma BufAC_Cmt_2stream_monoP: "f ∈ BufEq ⟹ ∃l. ∀i x s. s ∈ BufAC_Asm ⟶ x << s ⟶ enat (l i) < #x ⟶
(x,f⋅x) ∈ (BufAC_Cmt_F ^^ i) top ⟶
(s,f⋅s) ∈ (BufAC_Cmt_F ^^ i) top"
apply (rule_tac x="%i. 2*i" in exI)
apply (rule allI)
apply (induct_tac "i")
apply ( simp)
apply (intro strip)
apply (subst BufAC_Cmt_F_def3)
apply (drule_tac P="%x. x" in BufAC_Cmt_F_def3 [THEN subst])
apply safe
apply (   erule Buf_f_empty)
apply (  erule Buf_f_d)
apply ( drule Buf_f_d_req)
apply ( safe, erule ssubst, simp)
apply clarsimp
apply (rename_tac i d xa ya t)
(*
1. ⋀i d xa ya t.
⟦f ∈ BufEq;
∀x s. s ∈ BufAC_Asm ⟶
x ⊑ s ⟶
enat (2 * i) < #x ⟶
(x, f⋅x) ∈ down_iterate BufAC_Cmt_F i ⟶
(s, f⋅s) ∈ down_iterate BufAC_Cmt_F i;
Md d↝∙↝xa ∈ BufAC_Asm; enat (2 * i) < #ya; f⋅(Md d↝∙↝ya) = d↝t;
(ya, t) ∈ down_iterate BufAC_Cmt_F i; ya ⊑ xa⟧
⟹ (xa, rt⋅(f⋅(Md d↝∙↝xa))) ∈ down_iterate BufAC_Cmt_F i
*)
apply (rotate_tac 2)
apply (drule BufAC_Asm_prefix2)
apply (frule Buf_f_d_req, erule exE, erule conjE, rotate_tac -1, erule ssubst)
apply (frule Buf_f_d_req, erule exE, erule conjE)
apply (            subgoal_tac "f⋅(Md d↝∙↝ya) = d↝ffa⋅ya")
prefer 2
apply ( assumption)
apply (            rotate_tac -1)
apply (            simp)
apply (erule subst)
(*
1. ⋀i d xa ya t ff ffa.
⟦f⋅(Md d↝∙↝ya) = d↝ffa⋅ya; enat (2 * i) < #ya;
(ya, ffa⋅ya) ∈ down_iterate BufAC_Cmt_F i; ya ⊑ xa; f ∈ BufEq;
∀x s. s ∈ BufAC_Asm ⟶
x ⊑ s ⟶
enat (2 * i) < #x ⟶
(x, f⋅x) ∈ down_iterate BufAC_Cmt_F i ⟶
(s, f⋅s) ∈ down_iterate BufAC_Cmt_F i;
xa ∈ BufAC_Asm; ff ∈ BufEq; ffa ∈ BufEq⟧
⟹ (xa, ff⋅xa) ∈ down_iterate BufAC_Cmt_F i
*)
apply (drule spec, drule spec, drule (1) mp)
apply (drule (1) mp)
apply (drule (1) mp)
apply (erule impE)
apply ( subst BufAC_Asm_cong, assumption)
prefer 3 apply assumption
apply assumption
apply ( erule (1) BufAC_Asm_antiton [THEN antitonPD])
apply (subst BufAC_Asm_cong, assumption)
prefer 3 apply assumption
apply assumption
apply assumption
done

lemma BufAC_Cmt_iterate_all: "(x∈BufAC_Cmt) = (∀n. x∈(BufAC_Cmt_F ^^ n) top)"
apply (unfold BufAC_Cmt_def)
apply (subst cont_BufAC_Cmt_F [THEN inf_continuous_gfp])
apply (fast)
done

BufAC_Cmt_2stream_monoP*)
lemma adm_BufAC: "f ∈ BufEq ⟹ adm (λs. s ∈ BufAC_Asm ⟶ (s, f⋅s) ∈ BufAC_Cmt)"
apply (subst BufAC_Cmt_iterate_all)
apply (drule BufAC_Cmt_2stream_monoP)
apply safe
apply (drule spec, erule exE)
apply (drule spec, erule impE)
apply  (erule BufAC_Asm_antiton [THEN antitonPD])
apply  (erule is_ub_thelub)
apply (tactic "smp_tac @{context} 3 1")
apply (drule is_ub_thelub)
apply (drule (1) mp)
apply (drule (1) mp)
apply (erule mp)
apply (drule BufAC_Cmt_iterate_all [THEN iffD1])
apply (erule spec)
done

(**** Buf_Eq_imp_AC by induction **********************************************)

lemma Buf_Eq_imp_AC: "BufEq <= BufAC"
apply (unfold BufAC_def)
apply (rule subsetI)
apply (simp)
apply (rule allI)
apply (rule fstream_ind2)
back
apply (  safe)
apply (   erule BufAC_Cmt_empty)
apply (  erule BufAC_Cmt_d)
apply ( drule BufAC_Asm_prefix2)
apply ( simp)
apply (fast intro: BufAC_Cmt_d_req BufAC_Asm_prefix2)
done

(**** new approach for admissibility, reduces itself to absurdity *************)

apply (rule BufAC_Asm_def [THEN eq_reflection])
apply (safe)
apply (unfold BufAC_Asm_F_def)
apply (safe)
apply (erule contrapos_np)
apply (drule fstream_exhaust_eq [THEN iffD1])
apply (clarsimp)
apply (drule (1) fstream_lub_lemma)
apply (clarify)
apply (erule_tac x="j" in all_dupE)
apply (simp)
apply (drule BufAC_Asm_d2)
apply (clarify)
apply (simp)
apply (rule disjCI)
apply (erule contrapos_np)
apply (drule fstream_exhaust_eq [THEN iffD1])
apply (clarsimp)
apply (drule (1) fstream_lub_lemma)
apply (clarsimp)
apply (simp only: ex_simps [symmetric] all_simps [symmetric])
apply (rule_tac x="Xa" in exI)
apply (rule allI)
apply (rotate_tac -1)
apply (erule_tac x="i" in allE)
apply (clarsimp)
apply (erule_tac x="jb" in allE)
apply (clarsimp)
apply (erule_tac x="jc" in allE)
apply (clarsimp dest!: BufAC_Asm_d3)
done

apply (rule BufAC_Asm_def [THEN eq_reflection])
apply (unfold BufAC_Asm_F_def)
apply (safe)
apply (erule contrapos_np)
apply (drule fstream_exhaust_eq [THEN iffD1])
apply (clarsimp)
apply (frule fstream_prefix)
apply (clarsimp)
apply (frule BufAC_Asm_d2)
apply (clarsimp)
apply (rotate_tac -1)
apply (erule contrapos_pp)
apply (drule fstream_exhaust_eq [THEN iffD1])
apply (clarsimp)
apply (frule fstream_prefix)
apply (clarsimp)
apply (frule BufAC_Asm_d3)
apply (force)
done

lemma adm_BufAC': "f ∈ BufEq ⟹ adm (λu. u ∈ BufAC_Asm ⟶ (u, f⋅u) ∈ BufAC_Cmt)"