author | traytel |
Tue, 20 Aug 2013 16:10:58 +0200 | |
changeset 53103 | c0217c4a6b2d |
parent 53026 | e1a548c11845 |
child 53694 | 7b453b619b5f |
permissions | -rw-r--r-- |
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(* Title: HOL/BNF/Examples/Process.thy |
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Author: Andrei Popescu, TU Muenchen |
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Copyright 2012 |
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Processes. |
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*) |
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header {* Processes *} |
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theory Process |
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imports Stream |
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begin |
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codatatype 'a process = |
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isAction: Action (prefOf: 'a) (contOf: "'a process") | |
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isChoice: Choice (ch1Of: "'a process") (ch2Of: "'a process") |
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(* Read: prefix of, continuation of, choice 1 of, choice 2 of *) |
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section {* Customization *} |
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subsection {* Basic properties *} |
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declare |
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pre_process_rel_def[simp] |
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sum_rel_def[simp] |
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prod_rel_def[simp] |
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(* Constructors versus discriminators *) |
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theorem isAction_isChoice: |
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"isAction p \<or> isChoice p" |
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by (rule process.disc_exhaust) auto |
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theorem not_isAction_isChoice: "\<not> (isAction p \<and> isChoice p)" |
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by (cases rule: process.exhaust[of p]) auto |
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subsection{* Coinduction *} |
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theorem process_coind[elim, consumes 1, case_names iss Action Choice, induct pred: "HOL.eq"]: |
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assumes phi: "\<phi> p p'" and |
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iss: "\<And>p p'. \<phi> p p' \<Longrightarrow> (isAction p \<longleftrightarrow> isAction p') \<and> (isChoice p \<longleftrightarrow> isChoice p')" and |
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Act: "\<And> a a' p p'. \<phi> (Action a p) (Action a' p') \<Longrightarrow> a = a' \<and> \<phi> p p'" and |
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Ch: "\<And> p q p' q'. \<phi> (Choice p q) (Choice p' q') \<Longrightarrow> \<phi> p p' \<and> \<phi> q q'" |
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shows "p = p'" |
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using assms |
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by (coinduct rule: process.coinduct) (metis process.collapse(1,2) process.discs(3)) |
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(* Stronger coinduction, up to equality: *) |
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theorem process_strong_coind[elim, consumes 1, case_names iss Action Choice]: |
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assumes phi: "\<phi> p p'" and |
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iss: "\<And>p p'. \<phi> p p' \<Longrightarrow> (isAction p \<longleftrightarrow> isAction p') \<and> (isChoice p \<longleftrightarrow> isChoice p')" and |
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Act: "\<And> a a' p p'. \<phi> (Action a p) (Action a' p') \<Longrightarrow> a = a' \<and> (\<phi> p p' \<or> p = p')" and |
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Ch: "\<And> p q p' q'. \<phi> (Choice p q) (Choice p' q') \<Longrightarrow> (\<phi> p p' \<or> p = p') \<and> (\<phi> q q' \<or> q = q')" |
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shows "p = p'" |
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using assms |
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by (coinduct rule: process.strong_coinduct) (metis process.collapse(1,2) process.discs(3)) |
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subsection {* Coiteration (unfold) *} |
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section{* Coinductive definition of the notion of trace *} |
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coinductive trace where |
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"trace p as \<Longrightarrow> trace (Action a p) (a ## as)" |
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"trace p as \<or> trace q as \<Longrightarrow> trace (Choice p q) as" |
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section{* Examples of corecursive definitions: *} |
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subsection{* Single-guard fixpoint definition *} |
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definition |
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"BX \<equiv> |
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process_unfold |
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(\<lambda> P. True) |
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(\<lambda> P. ''a'') |
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(\<lambda> P. P) |
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undefined |
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undefined |
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()" |
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lemma BX: "BX = Action ''a'' BX" |
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unfolding BX_def |
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using process.unfold(1)[of "\<lambda> P. True" "()" "\<lambda> P. ''a''" "\<lambda> P. P"] by simp |
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subsection{* Multi-guard fixpoint definitions, simulated with auxiliary arguments *} |
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datatype x_y_ax = x | y | ax |
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definition "isA \<equiv> \<lambda> K. case K of x \<Rightarrow> False |y \<Rightarrow> True |ax \<Rightarrow> True" |
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definition "pr \<equiv> \<lambda> K. case K of x \<Rightarrow> undefined |y \<Rightarrow> ''b'' |ax \<Rightarrow> ''a''" |
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definition "co \<equiv> \<lambda> K. case K of x \<Rightarrow> undefined |y \<Rightarrow> x |ax \<Rightarrow> x" |
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lemmas Action_defs = isA_def pr_def co_def |
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definition "c1 \<equiv> \<lambda> K. case K of x \<Rightarrow> ax |y \<Rightarrow> undefined |ax \<Rightarrow> undefined" |
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definition "c2 \<equiv> \<lambda> K. case K of x \<Rightarrow> y |y \<Rightarrow> undefined |ax \<Rightarrow> undefined" |
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lemmas Choice_defs = c1_def c2_def |
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definition "F \<equiv> process_unfold isA pr co c1 c2" |
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definition "X = F x" definition "Y = F y" definition "AX = F ax" |
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lemma X_Y_AX: "X = Choice AX Y" "Y = Action ''b'' X" "AX = Action ''a'' X" |
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unfolding X_def Y_def AX_def F_def |
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using process.unfold(2)[of isA x "pr" co c1 c2] |
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process.unfold(1)[of isA y "pr" co c1 c2] |
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process.unfold(1)[of isA ax "pr" co c1 c2] |
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unfolding Action_defs Choice_defs by simp_all |
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(* end product: *) |
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lemma X_AX: |
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"X = Choice AX (Action ''b'' X)" |
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"AX = Action ''a'' X" |
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using X_Y_AX by simp_all |
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section{* Case study: Multi-guard fixpoint definitions, without auxiliary arguments *} |
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hide_const x y ax X Y AX |
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(* Process terms *) |
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datatype ('a,'pvar) process_term = |
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VAR 'pvar | |
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PROC "'a process" | |
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ACT 'a "('a,'pvar) process_term" | CH "('a,'pvar) process_term" "('a,'pvar) process_term" |
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129 |
|
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130 |
(* below, sys represents a system of equations *) |
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131 |
fun isACT where |
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132 |
"isACT sys (VAR X) = |
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133 |
(case sys X of ACT a T \<Rightarrow> True |PROC p \<Rightarrow> isAction p |_ \<Rightarrow> False)" |
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|
134 |
| |
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|
135 |
"isACT sys (PROC p) = isAction p" |
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|
136 |
| |
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137 |
"isACT sys (ACT a T) = True" |
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|
138 |
| |
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|
139 |
"isACT sys (CH T1 T2) = False" |
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|
140 |
|
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|
141 |
fun PREF where |
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142 |
"PREF sys (VAR X) = |
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143 |
(case sys X of ACT a T \<Rightarrow> a | PROC p \<Rightarrow> prefOf p)" |
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|
144 |
| |
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|
145 |
"PREF sys (PROC p) = prefOf p" |
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|
146 |
| |
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|
147 |
"PREF sys (ACT a T) = a" |
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|
148 |
|
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|
149 |
fun CONT where |
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150 |
"CONT sys (VAR X) = |
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151 |
(case sys X of ACT a T \<Rightarrow> T | PROC p \<Rightarrow> PROC (contOf p))" |
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|
152 |
| |
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|
153 |
"CONT sys (PROC p) = PROC (contOf p)" |
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|
154 |
| |
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155 |
"CONT sys (ACT a T) = T" |
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|
156 |
|
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|
157 |
fun CH1 where |
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|
158 |
"CH1 sys (VAR X) = |
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|
159 |
(case sys X of CH T1 T2 \<Rightarrow> T1 |PROC p \<Rightarrow> PROC (ch1Of p))" |
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|
160 |
| |
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|
161 |
"CH1 sys (PROC p) = PROC (ch1Of p)" |
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|
162 |
| |
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|
163 |
"CH1 sys (CH T1 T2) = T1" |
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|
164 |
|
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|
165 |
fun CH2 where |
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166 |
"CH2 sys (VAR X) = |
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|
167 |
(case sys X of CH T1 T2 \<Rightarrow> T2 |PROC p \<Rightarrow> PROC (ch2Of p))" |
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|
168 |
| |
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|
169 |
"CH2 sys (PROC p) = PROC (ch2Of p)" |
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|
170 |
| |
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|
171 |
"CH2 sys (CH T1 T2) = T2" |
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|
172 |
|
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|
173 |
definition "guarded sys \<equiv> \<forall> X Y. sys X \<noteq> VAR Y" |
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|
174 |
|
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|
175 |
definition |
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|
176 |
"solution sys \<equiv> |
49508 | 177 |
process_unfold |
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|
178 |
(isACT sys) |
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|
179 |
(PREF sys) |
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|
180 |
(CONT sys) |
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181 |
(CH1 sys) |
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|
182 |
(CH2 sys)" |
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|
183 |
|
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|
184 |
lemma solution_Action: |
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|
185 |
assumes "isACT sys T" |
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|
186 |
shows "solution sys T = Action (PREF sys T) (solution sys (CONT sys T))" |
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|
187 |
unfolding solution_def |
49594
55e798614c45
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blanchet
parents:
49588
diff
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|
188 |
using process.unfold(1)[of "isACT sys" T "PREF sys" "CONT sys" "CH1 sys" "CH2 sys"] |
49238 | 189 |
assms by simp |
48975
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changeset
|
190 |
|
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diff
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|
191 |
lemma solution_Choice: |
49238 | 192 |
assumes "\<not> isACT sys T" |
48975
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changeset
|
193 |
shows "solution sys T = Choice (solution sys (CH1 sys T)) (solution sys (CH2 sys T))" |
7f79f94a432c
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parents:
diff
changeset
|
194 |
unfolding solution_def |
49594
55e798614c45
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blanchet
parents:
49588
diff
changeset
|
195 |
using process.unfold(2)[of "isACT sys" T "PREF sys" "CONT sys" "CH1 sys" "CH2 sys"] |
49238 | 196 |
assms by simp |
48975
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diff
changeset
|
197 |
|
7f79f94a432c
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parents:
diff
changeset
|
198 |
lemma isACT_VAR: |
7f79f94a432c
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parents:
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changeset
|
199 |
assumes g: "guarded sys" |
7f79f94a432c
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parents:
diff
changeset
|
200 |
shows "isACT sys (VAR X) \<longleftrightarrow> isACT sys (sys X)" |
7f79f94a432c
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parents:
diff
changeset
|
201 |
using g unfolding guarded_def by (cases "sys X") auto |
7f79f94a432c
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parents:
diff
changeset
|
202 |
|
7f79f94a432c
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blanchet
parents:
diff
changeset
|
203 |
lemma solution_VAR: |
7f79f94a432c
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parents:
diff
changeset
|
204 |
assumes g: "guarded sys" |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
205 |
shows "solution sys (VAR X) = solution sys (sys X)" |
7f79f94a432c
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parents:
diff
changeset
|
206 |
proof(cases "isACT sys (VAR X)") |
7f79f94a432c
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parents:
diff
changeset
|
207 |
case True |
7f79f94a432c
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diff
changeset
|
208 |
hence T: "isACT sys (sys X)" unfolding isACT_VAR[OF g] . |
7f79f94a432c
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parents:
diff
changeset
|
209 |
show ?thesis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
changeset
|
210 |
unfolding solution_Action[OF T] using solution_Action[of sys "VAR X"] True g |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
211 |
unfolding guarded_def by (cases "sys X", auto) |
7f79f94a432c
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parents:
diff
changeset
|
212 |
next |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
213 |
case False note FFalse = False |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
214 |
hence TT: "\<not> isACT sys (sys X)" unfolding isACT_VAR[OF g] . |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
215 |
show ?thesis |
49238 | 216 |
unfolding solution_Choice[OF TT] using solution_Choice[of sys "VAR X"] FFalse g |
217 |
unfolding guarded_def by (cases "sys X", auto) |
|
48975
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|
218 |
qed |
7f79f94a432c
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parents:
diff
changeset
|
219 |
|
7f79f94a432c
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parents:
diff
changeset
|
220 |
lemma solution_PROC[simp]: |
7f79f94a432c
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parents:
diff
changeset
|
221 |
"solution sys (PROC p) = p" |
7f79f94a432c
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parents:
diff
changeset
|
222 |
proof- |
7f79f94a432c
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parents:
diff
changeset
|
223 |
{fix q assume "q = solution sys (PROC p)" |
7f79f94a432c
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parents:
diff
changeset
|
224 |
hence "p = q" |
7f79f94a432c
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parents:
diff
changeset
|
225 |
proof(induct rule: process_coind) |
7f79f94a432c
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parents:
diff
changeset
|
226 |
case (iss p p') |
7f79f94a432c
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parents:
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changeset
|
227 |
from isAction_isChoice[of p] show ?case |
7f79f94a432c
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parents:
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changeset
|
228 |
proof |
7f79f94a432c
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parents:
diff
changeset
|
229 |
assume p: "isAction p" |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
230 |
hence 0: "isACT sys (PROC p)" by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
231 |
thus ?thesis using iss not_isAction_isChoice |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
232 |
unfolding solution_Action[OF 0] by auto |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
233 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
234 |
assume "isChoice p" |
49238 | 235 |
hence 0: "\<not> isACT sys (PROC p)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
236 |
using not_isAction_isChoice by auto |
49238 | 237 |
thus ?thesis using iss isAction_isChoice |
238 |
unfolding solution_Choice[OF 0] by auto |
|
48975
7f79f94a432c
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blanchet
parents:
diff
changeset
|
239 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
240 |
next |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
241 |
case (Action a a' p p') |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
242 |
hence 0: "isACT sys (PROC (Action a p))" by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
243 |
show ?case using Action unfolding solution_Action[OF 0] by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
244 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
245 |
case (Choice p q p' q') |
49238 | 246 |
hence 0: "\<not> isACT sys (PROC (Choice p q))" using not_isAction_isChoice by auto |
247 |
show ?case using Choice unfolding solution_Choice[OF 0] by simp |
|
48975
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248 |
qed |
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249 |
} |
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250 |
thus ?thesis by metis |
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251 |
qed |
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252 |
|
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253 |
lemma solution_ACT[simp]: |
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254 |
"solution sys (ACT a T) = Action a (solution sys T)" |
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255 |
by (metis CONT.simps(3) PREF.simps(3) isACT.simps(3) solution_Action) |
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256 |
|
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257 |
lemma solution_CH[simp]: |
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258 |
"solution sys (CH T1 T2) = Choice (solution sys T1) (solution sys T2)" |
49238 | 259 |
by (metis CH1.simps(3) CH2.simps(3) isACT.simps(4) solution_Choice) |
48975
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260 |
|
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261 |
|
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262 |
(* Example: *) |
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263 |
|
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264 |
fun sys where |
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265 |
"sys 0 = CH (VAR (Suc 0)) (ACT ''b'' (VAR 0))" |
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266 |
| |
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267 |
"sys (Suc 0) = ACT ''a'' (VAR 0)" |
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268 |
| (* dummy guarded term for variables outside the system: *) |
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269 |
"sys X = ACT ''a'' (VAR 0)" |
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270 |
|
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271 |
lemma guarded_sys: |
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272 |
"guarded sys" |
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273 |
unfolding guarded_def proof (intro allI) |
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274 |
fix X Y show "sys X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto) |
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275 |
qed |
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|
276 |
|
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277 |
(* the actual processes: *) |
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278 |
definition "x \<equiv> solution sys (VAR 0)" |
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definition "ax \<equiv> solution sys (VAR (Suc 0))" |
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280 |
|
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281 |
(* end product: *) |
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282 |
lemma x_ax: |
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283 |
"x = Choice ax (Action ''b'' x)" |
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284 |
"ax = Action ''a'' x" |
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|
285 |
unfolding x_def ax_def by (subst solution_VAR[OF guarded_sys], simp)+ |
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|
286 |
|
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|
287 |
|
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|
288 |
(* Thanks to the inclusion of processes as process terms, one can |
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|
289 |
also consider parametrized systems of equations---here, x is a (semantic) |
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290 |
process parameter: *) |
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|
291 |
|
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|
292 |
fun sys' where |
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|
293 |
"sys' 0 = CH (PROC x) (ACT ''b'' (VAR 0))" |
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|
294 |
| |
7f79f94a432c
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|
295 |
"sys' (Suc 0) = CH (ACT ''a'' (VAR 0)) (PROC x)" |
7f79f94a432c
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|
296 |
| (* dummy guarded term : *) |
7f79f94a432c
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|
297 |
"sys' X = ACT ''a'' (VAR 0)" |
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changeset
|
298 |
|
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diff
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|
299 |
lemma guarded_sys': |
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|
300 |
"guarded sys'" |
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parents:
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changeset
|
301 |
unfolding guarded_def proof (intro allI) |
7f79f94a432c
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parents:
diff
changeset
|
302 |
fix X Y show "sys' X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto) |
7f79f94a432c
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parents:
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|
303 |
qed |
7f79f94a432c
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parents:
diff
changeset
|
304 |
|
7f79f94a432c
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parents:
diff
changeset
|
305 |
(* the actual processes: *) |
7f79f94a432c
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parents:
diff
changeset
|
306 |
definition "y \<equiv> solution sys' (VAR 0)" |
7f79f94a432c
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changeset
|
307 |
definition "ay \<equiv> solution sys' (VAR (Suc 0))" |
7f79f94a432c
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parents:
diff
changeset
|
308 |
|
7f79f94a432c
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parents:
diff
changeset
|
309 |
(* end product: *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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changeset
|
310 |
lemma y_ay: |
7f79f94a432c
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parents:
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changeset
|
311 |
"y = Choice x (Action ''b'' y)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
312 |
"ay = Choice (Action ''a'' y) x" |
7f79f94a432c
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parents:
diff
changeset
|
313 |
unfolding y_def ay_def by (subst solution_VAR[OF guarded_sys'], simp)+ |
7f79f94a432c
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blanchet
parents:
diff
changeset
|
314 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
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diff
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|
315 |
end |