author  wenzelm 
Sat, 07 Apr 2012 16:41:59 +0200  
changeset 47389  e8552cba702d 
parent 45653  63ed1be524eb 
child 49521  06cb12198b92 
permissions  rwrr 
42151  1 
(* Title: HOL/HOLCF/FOCUS/Buffer_adm.thy 
17293  2 
Author: David von Oheimb, TU Muenchen 
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added FOCUS including the OneElement Buffer by Manfred Broy
oheimb
parents:
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*) 
4c55b020d6ee
added FOCUS including the OneElement Buffer by Manfred Broy
oheimb
parents:
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header {* Oneelement buffer, proof of Buf_Eq_imp_AC by induction + admissibility *} 
11350
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oheimb
parents:
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theory Buffer_adm 
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imports Buffer Stream_adm 

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begin 

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declare enat_0 [simp] 
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lemma BufAC_Asm_d2: "a\<leadsto>s:BufAC_Asm ==> ? d. a=Md d" 

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by (drule BufAC_Asm_unfold [THEN iffD1], auto) 

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lemma BufAC_Asm_d3: 

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"a\<leadsto>b\<leadsto>s:BufAC_Asm ==> ? d. a=Md d & b=\<bullet> & s:BufAC_Asm" 

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by (drule BufAC_Asm_unfold [THEN iffD1], auto) 

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lemma BufAC_Asm_F_def3: 

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"(s:BufAC_Asm_F A) = (s=<>  

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(? d. ft\<cdot>s=Def(Md d)) & (rt\<cdot>s=<>  ft\<cdot>(rt\<cdot>s)=Def \<bullet> & rt\<cdot>(rt\<cdot>s):A))" 

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by (unfold BufAC_Asm_F_def, auto) 

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lemma cont_BufAC_Asm_F: "down_cont BufAC_Asm_F" 

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by (auto simp add: down_cont_def BufAC_Asm_F_def3) 

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lemma BufAC_Cmt_F_def3: 

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"((s,t):BufAC_Cmt_F C) = (!d x. 

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(s = <> > t = <> ) & 

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(s = Md d\<leadsto><> > t = <> ) & 

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(s = Md d\<leadsto>\<bullet>\<leadsto>x > ft\<cdot>t = Def d & (x,rt\<cdot>t):C))" 

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apply (unfold BufAC_Cmt_F_def) 

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apply (subgoal_tac "!d x. (s = Md d\<leadsto>\<bullet>\<leadsto>x > (? y. t = d\<leadsto>y & (x,y):C)) = 

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(s = Md d\<leadsto>\<bullet>\<leadsto>x > ft\<cdot>t = Def d & (x,rt\<cdot>t):C)") 

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apply (simp) 

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apply (auto intro: surjectiv_scons [symmetric]) 

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done 

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lemma cont_BufAC_Cmt_F: "down_cont BufAC_Cmt_F" 

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by (auto simp add: down_cont_def BufAC_Cmt_F_def3) 

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(**** adm_BufAC_Asm ***********************************************************) 

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lemma BufAC_Asm_F_stream_monoP: "stream_monoP BufAC_Asm_F" 

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apply (unfold BufAC_Asm_F_def stream_monoP_def) 

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apply (rule_tac x="{x. (? d. x = Md d\<leadsto>\<bullet>\<leadsto><>)}" in exI) 

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apply (rule_tac x="Suc (Suc 0)" in exI) 

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apply (clarsimp) 

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done 

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lemma adm_BufAC_Asm: "adm (%x. x:BufAC_Asm)" 

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apply (unfold BufAC_Asm_def) 

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apply (rule cont_BufAC_Asm_F [THEN BufAC_Asm_F_stream_monoP [THEN fstream_gfp_admI]]) 

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done 

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(**** adm_non_BufAC_Asm *******************************************************) 

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lemma BufAC_Asm_F_stream_antiP: "stream_antiP BufAC_Asm_F" 

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apply (unfold stream_antiP_def BufAC_Asm_F_def) 

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apply (intro strip) 

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apply (rule_tac x="{x. (? d. x = Md d\<leadsto>\<bullet>\<leadsto><>)}" in exI) 

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apply (rule_tac x="Suc (Suc 0)" in exI) 

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apply (rule conjI) 

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prefer 2 

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apply ( intro strip) 

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apply ( drule slen_mono) 

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apply ( drule (1) order_trans) 
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apply (force)+ 
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done 

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lemma adm_non_BufAC_Asm: "adm (%u. u~:BufAC_Asm)" 

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apply (unfold BufAC_Asm_def) 

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apply (rule cont_BufAC_Asm_F [THEN BufAC_Asm_F_stream_antiP [THEN fstream_non_gfp_admI]]) 

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done 

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(**** adm_BufAC ***************************************************************) 

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(*adm_non_BufAC_Asm*) 

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lemma BufAC_Asm_cong [rule_format]: "!f ff. f:BufEq > ff:BufEq > s:BufAC_Asm > f\<cdot>s = ff\<cdot>s" 

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apply (rule fstream_ind2) 

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apply (simp add: adm_non_BufAC_Asm) 

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apply (force dest: Buf_f_empty) 

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apply (force dest!: BufAC_Asm_d2 

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dest: Buf_f_d elim: ssubst) 

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apply (safe dest!: BufAC_Asm_d3) 

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apply (drule Buf_f_d_req)+ 

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apply (fast elim: ssubst) 

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done 

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(*adm_non_BufAC_Asm,BufAC_Asm_cong*) 

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lemma BufAC_Cmt_d_req: 

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"!!X. [f:BufEq; s:BufAC_Asm; (s, f\<cdot>s):BufAC_Cmt] ==> (a\<leadsto>b\<leadsto>s, f\<cdot>(a\<leadsto>b\<leadsto>s)):BufAC_Cmt" 

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apply (rule BufAC_Cmt_unfold [THEN iffD2]) 

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apply (intro strip) 

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apply (frule Buf_f_d_req) 

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apply (auto elim: BufAC_Asm_cong [THEN subst]) 

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done 

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(*adm_BufAC_Asm*) 

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lemma BufAC_Asm_antiton: "antitonP BufAC_Asm" 

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apply (rule antitonPI) 

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apply (rule allI) 

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apply (rule fstream_ind2) 

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apply ( rule adm_lemmas)+ 

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apply ( rule cont_id) 

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apply ( rule adm_BufAC_Asm) 

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apply ( safe) 

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apply ( rule BufAC_Asm_empty) 

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apply ( force dest!: fstream_prefix 

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dest: BufAC_Asm_d2 intro: BufAC_Asm_d) 

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apply ( force dest!: fstream_prefix 

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dest: BufAC_Asm_d3 intro!: BufAC_Asm_d_req) 

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done 

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(*adm_BufAC_Asm,BufAC_Asm_antiton,adm_non_BufAC_Asm,BufAC_Asm_cong*) 

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lemma BufAC_Cmt_2stream_monoP: "f:BufEq ==> ? l. !i x s. s:BufAC_Asm > x << s > enat (l i) < #x > 
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(x,f\<cdot>x):down_iterate BufAC_Cmt_F i > 
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(s,f\<cdot>s):down_iterate BufAC_Cmt_F i" 

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apply (rule_tac x="%i. 2*i" in exI) 

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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 add: add_commute) 

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apply (intro strip) 

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apply (subst BufAC_Cmt_F_def3) 

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apply (drule_tac P="%x. x" in BufAC_Cmt_F_def3 [THEN subst]) 

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apply safe 

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apply ( erule Buf_f_empty) 

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apply ( erule Buf_f_d) 

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apply ( drule Buf_f_d_req) 

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apply ( safe, erule ssubst, simp) 

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apply clarsimp 

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apply (rename_tac i d xa ya t) 

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(* 

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1. \<And>i d xa ya t. 

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\<lbrakk>f \<in> BufEq; 

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\<forall>x s. s \<in> BufAC_Asm \<longrightarrow> 

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x \<sqsubseteq> s \<longrightarrow> 

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enat (2 * i) < #x \<longrightarrow> 
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(x, f\<cdot>x) \<in> down_iterate BufAC_Cmt_F i \<longrightarrow> 
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(s, f\<cdot>s) \<in> down_iterate BufAC_Cmt_F i; 

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Md d\<leadsto>\<bullet>\<leadsto>xa \<in> BufAC_Asm; enat (2 * i) < #ya; f\<cdot>(Md d\<leadsto>\<bullet>\<leadsto>ya) = d\<leadsto>t; 
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(ya, t) \<in> down_iterate BufAC_Cmt_F i; ya \<sqsubseteq> xa\<rbrakk> 
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\<Longrightarrow> (xa, rt\<cdot>(f\<cdot>(Md d\<leadsto>\<bullet>\<leadsto>xa))) \<in> down_iterate BufAC_Cmt_F i 

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*) 

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apply (rotate_tac 2) 

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apply (drule BufAC_Asm_prefix2) 

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apply (frule Buf_f_d_req, erule exE, erule conjE, rotate_tac 1, erule ssubst) 

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apply (frule Buf_f_d_req, erule exE, erule conjE) 

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apply ( subgoal_tac "f\<cdot>(Md d\<leadsto>\<bullet>\<leadsto>ya) = d\<leadsto>ffa\<cdot>ya") 

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prefer 2 

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apply ( assumption) 

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apply ( rotate_tac 1) 

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apply ( simp) 

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apply (erule subst) 

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(* 

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1. \<And>i d xa ya t ff ffa. 

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\<lbrakk>f\<cdot>(Md d\<leadsto>\<bullet>\<leadsto>ya) = d\<leadsto>ffa\<cdot>ya; enat (2 * i) < #ya; 
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(ya, ffa\<cdot>ya) \<in> down_iterate BufAC_Cmt_F i; ya \<sqsubseteq> xa; f \<in> BufEq; 
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\<forall>x s. s \<in> BufAC_Asm \<longrightarrow> 

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x \<sqsubseteq> s \<longrightarrow> 

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enat (2 * i) < #x \<longrightarrow> 
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(x, f\<cdot>x) \<in> down_iterate BufAC_Cmt_F i \<longrightarrow> 
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(s, f\<cdot>s) \<in> down_iterate BufAC_Cmt_F i; 

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xa \<in> BufAC_Asm; ff \<in> BufEq; ffa \<in> BufEq\<rbrakk> 

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\<Longrightarrow> (xa, ff\<cdot>xa) \<in> down_iterate BufAC_Cmt_F i 

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*) 

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apply (drule spec, drule spec, drule (1) mp) 

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apply (drule (1) mp) 

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apply (drule (1) mp) 

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apply (erule impE) 

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apply ( subst BufAC_Asm_cong, assumption) 

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prefer 3 apply assumption 

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apply assumption 

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apply ( erule (1) BufAC_Asm_antiton [THEN antitonPD]) 

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apply (subst BufAC_Asm_cong, assumption) 

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prefer 3 apply assumption 

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apply assumption 

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apply assumption 

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done 

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lemma BufAC_Cmt_iterate_all: "(x\<in>BufAC_Cmt) = (\<forall>n. x\<in>down_iterate BufAC_Cmt_F n)" 

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apply (unfold BufAC_Cmt_def) 

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apply (subst cont_BufAC_Cmt_F [THEN INTER_down_iterate_is_gfp]) 

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apply (fast) 

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done 

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(*adm_BufAC_Asm,BufAC_Asm_antiton,adm_non_BufAC_Asm,BufAC_Asm_cong, 

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BufAC_Cmt_2stream_monoP*) 

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lemma adm_BufAC: "f:BufEq ==> adm (%s. s:BufAC_Asm > (s, f\<cdot>s):BufAC_Cmt)" 

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apply (rule flatstream_admI) 

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apply (subst BufAC_Cmt_iterate_all) 

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apply (drule BufAC_Cmt_2stream_monoP) 

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apply safe 

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apply (drule spec, erule exE) 

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apply (drule spec, erule impE) 

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apply (erule BufAC_Asm_antiton [THEN antitonPD]) 

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apply (erule is_ub_thelub) 

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apply (tactic "smp_tac 3 1") 

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apply (drule is_ub_thelub) 

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apply (drule (1) mp) 

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apply (drule (1) mp) 

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apply (erule mp) 

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apply (drule BufAC_Cmt_iterate_all [THEN iffD1]) 

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apply (erule spec) 

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done 

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

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(*adm_BufAC_Asm,BufAC_Asm_antiton,adm_non_BufAC_Asm,BufAC_Asm_cong, 

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BufAC_Cmt_2stream_monoP,adm_BufAC,BufAC_Cmt_d_req*) 

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lemma Buf_Eq_imp_AC: "BufEq <= BufAC" 

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apply (unfold BufAC_def) 

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apply (rule subsetI) 

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apply (simp) 

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apply (rule allI) 

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apply (rule fstream_ind2) 

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back 

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apply ( erule adm_BufAC) 

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apply ( safe) 

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apply ( erule BufAC_Cmt_empty) 

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apply ( erule BufAC_Cmt_d) 

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apply ( drule BufAC_Asm_prefix2) 

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apply ( simp) 

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apply (fast intro: BufAC_Cmt_d_req BufAC_Asm_prefix2) 

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done 

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

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lemma adm_BufAC_Asm': "adm (\<lambda>x. x\<in>BufAC_Asm)" 
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apply (rule def_gfp_admI) 
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apply (rule BufAC_Asm_def [THEN eq_reflection]) 
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apply (safe) 
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apply (unfold BufAC_Asm_F_def) 

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apply (safe) 

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apply (erule contrapos_np) 

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apply (drule fstream_exhaust_eq [THEN iffD1]) 

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apply (clarsimp) 

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apply (drule (1) fstream_lub_lemma) 

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apply (clarify) 

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apply (erule_tac x="j" in all_dupE) 

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apply (simp) 

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apply (drule BufAC_Asm_d2) 

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apply (clarify) 

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apply (simp) 

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apply (rule disjCI) 

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apply (erule contrapos_np) 

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apply (drule fstream_exhaust_eq [THEN iffD1]) 

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apply (clarsimp) 

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apply (drule (1) fstream_lub_lemma) 

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apply (clarsimp) 

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apply (simp only: ex_simps [symmetric] all_simps [symmetric]) 
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apply (rule_tac x="Xa" in exI) 
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apply (rule allI) 

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apply (rotate_tac 1) 

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apply (erule_tac x="i" in allE) 

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apply (clarsimp) 

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apply (erule_tac x="jb" in allE) 

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apply (clarsimp) 

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apply (erule_tac x="jc" in allE) 

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apply (clarsimp dest!: BufAC_Asm_d3) 

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done 

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lemma adm_non_BufAC_Asm': "adm (\<lambda>u. u \<notin> BufAC_Asm)" (* uses antitonP *) 

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apply (rule def_gfp_adm_nonP) 

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apply (rule BufAC_Asm_def [THEN eq_reflection]) 
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apply (unfold BufAC_Asm_F_def) 
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apply (safe) 

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apply (erule contrapos_np) 

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apply (drule fstream_exhaust_eq [THEN iffD1]) 

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apply (clarsimp) 

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apply (frule fstream_prefix) 

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apply (clarsimp) 

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apply (frule BufAC_Asm_d2) 

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apply (clarsimp) 

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apply (rotate_tac 1) 

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apply (erule contrapos_pp) 

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apply (drule fstream_exhaust_eq [THEN iffD1]) 

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apply (clarsimp) 

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apply (frule fstream_prefix) 

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apply (clarsimp) 

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apply (frule BufAC_Asm_d3) 

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apply (force) 

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done 

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lemma adm_BufAC': "f \<in> BufEq \<Longrightarrow> adm (\<lambda>u. u \<in> BufAC_Asm \<longrightarrow> (u, f\<cdot>u) \<in> BufAC_Cmt)" 

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apply (rule triv_admI) 

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apply (clarify) 

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apply (erule (1) Buf_Eq_imp_AC_lemma) 

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(* this is what we originally aimed to show, using admissibilty :( *) 

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done 

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end 
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