author | wenzelm |
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permissions | -rw-r--r-- |
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(* Title: HOL/Datatype_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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section {* Processes *} |
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theory Process |
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imports "~~/src/HOL/Library/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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rel_pre_process_def[simp] |
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rel_sum_def[simp] |
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rel_prod_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.exhaust_disc) 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.disc(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.coinduct_strong) (metis process.collapse(1,2) process.disc(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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primcorec BX where |
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"isAction BX" |
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| "prefOf BX = ''a''" |
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| "contOf BX = BX" |
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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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primcorec F :: "x_y_ax \<Rightarrow> char list process" where |
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"xyax = x \<Longrightarrow> isChoice (F xyax)" |
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| "ch1Of (F xyax) = F ax" |
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| "ch2Of (F xyax) = F y" |
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| "prefOf (F xyax) = (if xyax = y then ''b'' else ''a'')" |
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| "contOf (F xyax) = F x" |
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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 by (subst F.code, simp)+ |
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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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(* below, sys represents a system of equations *) |
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fun isACT where |
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"isACT sys (VAR X) = |
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(case sys X of ACT a T \<Rightarrow> True |PROC p \<Rightarrow> isAction p |_ \<Rightarrow> False)" |
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"isACT sys (PROC p) = isAction p" |
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"isACT sys (ACT a T) = True" |
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"isACT sys (CH T1 T2) = False" |
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fun PREF where |
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"PREF sys (VAR X) = |
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(case sys X of ACT a T \<Rightarrow> a | PROC p \<Rightarrow> prefOf p)" |
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"PREF sys (PROC p) = prefOf p" |
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"PREF sys (ACT a T) = a" |
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133 |
fun CONT where |
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134 |
"CONT sys (VAR X) = |
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135 |
(case sys X of ACT a T \<Rightarrow> T | PROC p \<Rightarrow> PROC (contOf p))" |
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136 |
| |
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137 |
"CONT sys (PROC p) = PROC (contOf p)" |
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138 |
| |
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139 |
"CONT sys (ACT a T) = T" |
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140 |
|
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141 |
fun CH1 where |
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142 |
"CH1 sys (VAR X) = |
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143 |
(case sys X of CH T1 T2 \<Rightarrow> T1 |PROC p \<Rightarrow> PROC (ch1Of p))" |
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144 |
| |
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145 |
"CH1 sys (PROC p) = PROC (ch1Of p)" |
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146 |
| |
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147 |
"CH1 sys (CH T1 T2) = T1" |
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148 |
|
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149 |
fun CH2 where |
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150 |
"CH2 sys (VAR X) = |
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151 |
(case sys X of CH T1 T2 \<Rightarrow> T2 |PROC p \<Rightarrow> PROC (ch2Of p))" |
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152 |
| |
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153 |
"CH2 sys (PROC p) = PROC (ch2Of p)" |
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154 |
| |
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155 |
"CH2 sys (CH T1 T2) = T2" |
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156 |
|
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157 |
definition "guarded sys \<equiv> \<forall> X Y. sys X \<noteq> VAR Y" |
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158 |
|
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159 |
primcorec solution where |
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160 |
"isACT sys T \<Longrightarrow> solution sys T = Action (PREF sys T) (solution sys (CONT sys T))" |
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161 |
| "_ \<Longrightarrow> solution sys T = Choice (solution sys (CH1 sys T)) (solution sys (CH2 sys T))" |
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162 |
|
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163 |
lemma isACT_VAR: |
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164 |
assumes g: "guarded sys" |
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165 |
shows "isACT sys (VAR X) \<longleftrightarrow> isACT sys (sys X)" |
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166 |
using g unfolding guarded_def by (cases "sys X") auto |
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167 |
|
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168 |
lemma solution_VAR: |
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169 |
assumes g: "guarded sys" |
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170 |
shows "solution sys (VAR X) = solution sys (sys X)" |
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171 |
proof(cases "isACT sys (VAR X)") |
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172 |
case True |
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173 |
hence T: "isACT sys (sys X)" unfolding isACT_VAR[OF g] . |
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174 |
show ?thesis |
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175 |
unfolding solution.ctr(1)[OF T] using solution.ctr(1)[of sys "VAR X"] True g |
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176 |
unfolding guarded_def by (cases "sys X", auto) |
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177 |
next |
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178 |
case False note FFalse = False |
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179 |
hence TT: "\<not> isACT sys (sys X)" unfolding isACT_VAR[OF g] . |
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180 |
show ?thesis |
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181 |
unfolding solution.ctr(2)[OF TT] using solution.ctr(2)[of sys "VAR X"] FFalse g |
49238 | 182 |
unfolding guarded_def by (cases "sys X", auto) |
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183 |
qed |
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184 |
|
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185 |
lemma solution_PROC[simp]: |
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186 |
"solution sys (PROC p) = p" |
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187 |
proof- |
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188 |
{fix q assume "q = solution sys (PROC p)" |
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189 |
hence "p = q" |
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190 |
proof (coinduct rule: process_coind) |
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191 |
case (iss p p') |
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192 |
from isAction_isChoice[of p] show ?case |
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193 |
proof |
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194 |
assume p: "isAction p" |
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195 |
hence 0: "isACT sys (PROC p)" by simp |
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196 |
thus ?thesis using iss not_isAction_isChoice by auto |
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197 |
next |
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198 |
assume "isChoice p" |
49238 | 199 |
hence 0: "\<not> isACT sys (PROC p)" |
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200 |
using not_isAction_isChoice by auto |
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201 |
thus ?thesis using iss isAction_isChoice by auto |
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202 |
qed |
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203 |
next |
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204 |
case (Action a a' p p') |
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205 |
hence 0: "isACT sys (PROC (Action a p))" by simp |
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206 |
show ?case using Action unfolding solution.ctr(1)[OF 0] by simp |
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207 |
next |
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208 |
case (Choice p q p' q') |
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hence 0: "\<not> isACT sys (PROC (Choice p q))" using not_isAction_isChoice by auto |
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210 |
show ?case using Choice unfolding solution.ctr(2)[OF 0] by simp |
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211 |
qed |
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212 |
} |
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213 |
thus ?thesis by metis |
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214 |
qed |
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215 |
|
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216 |
lemma solution_ACT[simp]: |
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217 |
"solution sys (ACT a T) = Action a (solution sys T)" |
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218 |
by (metis CONT.simps(3) PREF.simps(3) isACT.simps(3) solution.ctr(1)) |
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219 |
|
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220 |
lemma solution_CH[simp]: |
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221 |
"solution sys (CH T1 T2) = Choice (solution sys T1) (solution sys T2)" |
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222 |
by (metis CH1.simps(3) CH2.simps(3) isACT.simps(4) solution.ctr(2)) |
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223 |
|
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224 |
|
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225 |
(* Example: *) |
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226 |
|
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227 |
fun sys where |
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228 |
"sys 0 = CH (VAR (Suc 0)) (ACT ''b'' (VAR 0))" |
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229 |
| |
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230 |
"sys (Suc 0) = ACT ''a'' (VAR 0)" |
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231 |
| (* dummy guarded term for variables outside the system: *) |
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232 |
"sys X = ACT ''a'' (VAR 0)" |
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233 |
|
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234 |
lemma guarded_sys: |
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235 |
"guarded sys" |
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236 |
unfolding guarded_def proof (intro allI) |
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237 |
fix X Y show "sys X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto) |
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238 |
qed |
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239 |
|
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240 |
(* the actual processes: *) |
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241 |
definition "x \<equiv> solution sys (VAR 0)" |
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242 |
definition "ax \<equiv> solution sys (VAR (Suc 0))" |
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243 |
|
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244 |
(* end product: *) |
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245 |
lemma x_ax: |
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246 |
"x = Choice ax (Action ''b'' x)" |
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"ax = Action ''a'' x" |
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unfolding x_def ax_def by (subst solution_VAR[OF guarded_sys], simp)+ |
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249 |
|
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250 |
|
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251 |
(* Thanks to the inclusion of processes as process terms, one can |
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also consider parametrized systems of equations---here, x is a (semantic) |
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process parameter: *) |
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|
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fun sys' where |
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256 |
"sys' 0 = CH (PROC x) (ACT ''b'' (VAR 0))" |
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257 |
| |
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"sys' (Suc 0) = CH (ACT ''a'' (VAR 0)) (PROC x)" |
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| (* dummy guarded term : *) |
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"sys' X = ACT ''a'' (VAR 0)" |
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261 |
|
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262 |
lemma guarded_sys': |
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263 |
"guarded sys'" |
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unfolding guarded_def proof (intro allI) |
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fix X Y show "sys' X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto) |
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qed |
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267 |
|
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268 |
(* the actual processes: *) |
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definition "y \<equiv> solution sys' (VAR 0)" |
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definition "ay \<equiv> solution sys' (VAR (Suc 0))" |
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271 |
|
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272 |
(* end product: *) |
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lemma y_ay: |
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"y = Choice x (Action ''b'' y)" |
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275 |
"ay = Choice (Action ''a'' y) x" |
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276 |
unfolding y_def ay_def by (subst solution_VAR[OF guarded_sys'], simp)+ |
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277 |
|
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278 |
end |