author | blanchet |
Fri, 21 Sep 2012 16:34:40 +0200 | |
changeset 49509 | 163914705f8d |
parent 49508 | 1e205327f059 |
permissions | -rw-r--r-- |
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1 |
(* Title: HOL/BNF/Examples/Process.thy |
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2 |
Author: Andrei Popescu, TU Muenchen |
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3 |
Copyright 2012 |
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4 |
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5 |
Processes. |
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6 |
*) |
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header {* Processes *} |
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theory Process |
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imports "../BNF" |
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begin |
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hide_fact (open) Quotient_Product.prod_rel_def |
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49237 | 16 |
codata 'a process = |
17 |
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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40 |
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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43 |
assumes phi: "\<phi> p p'" and |
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44 |
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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47 |
shows "p = p'" |
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proof(intro mp[OF process.rel_coinduct, of \<phi>, OF _ phi], clarify) |
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fix p p' assume \<phi>: "\<phi> p p'" |
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show "pre_process_rel (op =) \<phi> (process_dtor p) (process_dtor p')" |
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proof(cases rule: process.exhaust[of p]) |
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52 |
case (Action a q) note p = Action |
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hence "isAction p'" using iss[OF \<phi>] by (cases rule: process.exhaust[of p'], auto) |
54 |
then obtain a' q' where p': "p' = Action a' q'" by (cases rule: process.exhaust[of p'], auto) |
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55 |
have 0: "a = a' \<and> \<phi> q q'" using Act[OF \<phi>[unfolded p p']] . |
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have dtor: "process_dtor p = Inl (a,q)" "process_dtor p' = Inl (a',q')" |
57 |
unfolding p p' Action_def process.dtor_ctor by simp_all |
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58 |
show ?thesis using 0 unfolding dtor by simp |
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59 |
next |
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case (Choice p1 p2) note p = Choice |
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hence "isChoice p'" using iss[OF \<phi>] by (cases rule: process.exhaust[of p'], auto) |
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62 |
then obtain p1' p2' where p': "p' = Choice p1' p2'" |
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by (cases rule: process.exhaust[of p'], auto) |
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64 |
have 0: "\<phi> p1 p1' \<and> \<phi> p2 p2'" using Ch[OF \<phi>[unfolded p p']] . |
49508 | 65 |
have dtor: "process_dtor p = Inr (p1,p2)" "process_dtor p' = Inr (p1',p2')" |
66 |
unfolding p p' Choice_def process.dtor_ctor by simp_all |
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show ?thesis using 0 unfolding dtor by simp |
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68 |
qed |
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69 |
qed |
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70 |
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71 |
(* 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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73 |
assumes phi: "\<phi> p p'" and |
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74 |
iss: "\<And>p p'. \<phi> p p' \<Longrightarrow> (isAction p \<longleftrightarrow> isAction p') \<and> (isChoice p \<longleftrightarrow> isChoice p')" and |
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75 |
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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76 |
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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77 |
shows "p = p'" |
49508 | 78 |
proof(intro mp[OF process.rel_strong_coinduct, of \<phi>, OF _ phi], clarify) |
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79 |
fix p p' assume \<phi>: "\<phi> p p'" |
49508 | 80 |
show "pre_process_rel (op =) (\<lambda>a b. \<phi> a b \<or> a = b) (process_dtor p) (process_dtor p')" |
49301 | 81 |
proof(cases rule: process.exhaust[of p]) |
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82 |
case (Action a q) note p = Action |
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hence "isAction p'" using iss[OF \<phi>] by (cases rule: process.exhaust[of p'], auto) |
84 |
then obtain a' q' where p': "p' = Action a' q'" by (cases rule: process.exhaust[of p'], auto) |
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85 |
have 0: "a = a' \<and> (\<phi> q q' \<or> q = q')" using Act[OF \<phi>[unfolded p p']] . |
49508 | 86 |
have dtor: "process_dtor p = Inl (a,q)" "process_dtor p' = Inl (a',q')" |
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unfolding p p' Action_def process.dtor_ctor by simp_all |
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88 |
show ?thesis using 0 unfolding dtor by simp |
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89 |
next |
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90 |
case (Choice p1 p2) note p = Choice |
49301 | 91 |
hence "isChoice p'" using iss[OF \<phi>] by (cases rule: process.exhaust[of p'], auto) |
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92 |
then obtain p1' p2' where p': "p' = Choice p1' p2'" |
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by (cases rule: process.exhaust[of p'], auto) |
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94 |
have 0: "(\<phi> p1 p1' \<or> p1 = p1') \<and> (\<phi> p2 p2' \<or> p2 = p2')" using Ch[OF \<phi>[unfolded p p']] . |
49508 | 95 |
have dtor: "process_dtor p = Inr (p1,p2)" "process_dtor p' = Inr (p1',p2')" |
96 |
unfolding p p' Choice_def process.dtor_ctor by simp_all |
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97 |
show ?thesis using 0 unfolding dtor by simp |
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98 |
qed |
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99 |
qed |
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100 |
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101 |
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49508 | 102 |
subsection {* Coiteration (unfold) *} |
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103 |
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105 |
section{* Coinductive definition of the notion of trace *} |
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106 |
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107 |
(* Say we have a type of streams: *) |
49238 | 108 |
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109 |
typedecl 'a stream |
49238 | 110 |
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111 |
consts Ccons :: "'a \<Rightarrow> 'a stream \<Rightarrow> 'a stream" |
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112 |
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113 |
(* Use the existing coinductive package (distinct from our |
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new codatatype package, but highly compatible with it): *) |
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115 |
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116 |
coinductive trace where |
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117 |
"trace p as \<Longrightarrow> trace (Action a p) (Ccons a as)" |
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118 |
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119 |
"trace p as \<or> trace q as \<Longrightarrow> trace (Choice p q) as" |
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122 |
section{* Examples of corecursive definitions: *} |
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123 |
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124 |
subsection{* Single-guard fixpoint definition *} |
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125 |
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126 |
definition |
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127 |
"BX \<equiv> |
49508 | 128 |
process_unfold |
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129 |
(\<lambda> P. True) |
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130 |
(\<lambda> P. ''a'') |
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131 |
(\<lambda> P. P) |
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132 |
undefined |
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133 |
undefined |
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134 |
()" |
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135 |
|
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136 |
lemma BX: "BX = Action ''a'' BX" |
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137 |
unfolding BX_def |
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using process.unfolds(1)[of "\<lambda> P. True" "()" "\<lambda> P. ''a''" "\<lambda> P. P"] by simp |
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139 |
|
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140 |
|
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141 |
subsection{* Multi-guard fixpoint definitions, simulated with auxiliary arguments *} |
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142 |
|
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143 |
datatype x_y_ax = x | y | ax |
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144 |
|
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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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146 |
definition "pr \<equiv> \<lambda> K. case K of x \<Rightarrow> undefined |y \<Rightarrow> ''b'' |ax \<Rightarrow> ''a''" |
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147 |
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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149 |
|
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150 |
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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153 |
|
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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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156 |
|
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lemma X_Y_AX: "X = Choice AX Y" "Y = Action ''b'' X" "AX = Action ''a'' X" |
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158 |
unfolding X_def Y_def AX_def F_def |
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using process.unfolds(2)[of isA x "pr" co c1 c2] |
160 |
process.unfolds(1)[of isA y "pr" co c1 c2] |
|
161 |
process.unfolds(1)[of isA ax "pr" co c1 c2] |
|
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162 |
unfolding Action_defs Choice_defs by simp_all |
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163 |
|
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164 |
(* end product: *) |
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165 |
lemma X_AX: |
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166 |
"X = Choice AX (Action ''b'' X)" |
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167 |
"AX = Action ''a'' X" |
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168 |
using X_Y_AX by simp_all |
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169 |
|
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170 |
|
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171 |
|
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172 |
section{* Case study: Multi-guard fixpoint definitions, without auxiliary arguments *} |
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|
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174 |
hide_const x y ax X Y AX |
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175 |
|
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176 |
(* Process terms *) |
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datatype ('a,'pvar) process_term = |
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178 |
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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181 |
|
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182 |
(* below, sys represents a system of equations *) |
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183 |
fun isACT where |
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184 |
"isACT sys (VAR X) = |
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185 |
(case sys X of ACT a T \<Rightarrow> True |PROC p \<Rightarrow> isAction p |_ \<Rightarrow> False)" |
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186 |
| |
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187 |
"isACT sys (PROC p) = isAction p" |
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188 |
| |
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189 |
"isACT sys (ACT a T) = True" |
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190 |
| |
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191 |
"isACT sys (CH T1 T2) = False" |
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192 |
|
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193 |
fun PREF where |
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194 |
"PREF sys (VAR X) = |
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195 |
(case sys X of ACT a T \<Rightarrow> a | PROC p \<Rightarrow> prefOf p)" |
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196 |
| |
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197 |
"PREF sys (PROC p) = prefOf p" |
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198 |
| |
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199 |
"PREF sys (ACT a T) = a" |
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200 |
|
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201 |
fun CONT where |
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202 |
"CONT sys (VAR X) = |
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203 |
(case sys X of ACT a T \<Rightarrow> T | PROC p \<Rightarrow> PROC (contOf p))" |
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204 |
| |
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205 |
"CONT sys (PROC p) = PROC (contOf p)" |
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206 |
| |
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207 |
"CONT sys (ACT a T) = T" |
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208 |
|
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209 |
fun CH1 where |
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210 |
"CH1 sys (VAR X) = |
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211 |
(case sys X of CH T1 T2 \<Rightarrow> T1 |PROC p \<Rightarrow> PROC (ch1Of p))" |
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212 |
| |
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213 |
"CH1 sys (PROC p) = PROC (ch1Of p)" |
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214 |
| |
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215 |
"CH1 sys (CH T1 T2) = T1" |
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216 |
|
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217 |
fun CH2 where |
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218 |
"CH2 sys (VAR X) = |
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219 |
(case sys X of CH T1 T2 \<Rightarrow> T2 |PROC p \<Rightarrow> PROC (ch2Of p))" |
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220 |
| |
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221 |
"CH2 sys (PROC p) = PROC (ch2Of p)" |
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222 |
| |
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223 |
"CH2 sys (CH T1 T2) = T2" |
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224 |
|
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225 |
definition "guarded sys \<equiv> \<forall> X Y. sys X \<noteq> VAR Y" |
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226 |
|
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227 |
definition |
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228 |
"solution sys \<equiv> |
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process_unfold |
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230 |
(isACT sys) |
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231 |
(PREF sys) |
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232 |
(CONT sys) |
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233 |
(CH1 sys) |
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234 |
(CH2 sys)" |
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235 |
|
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236 |
lemma solution_Action: |
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237 |
assumes "isACT sys T" |
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238 |
shows "solution sys T = Action (PREF sys T) (solution sys (CONT sys T))" |
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239 |
unfolding solution_def |
49508 | 240 |
using process.unfolds(1)[of "isACT sys" T "PREF sys" "CONT sys" "CH1 sys" "CH2 sys"] |
49238 | 241 |
assms by simp |
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242 |
|
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243 |
lemma solution_Choice: |
49238 | 244 |
assumes "\<not> isACT sys T" |
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245 |
shows "solution sys T = Choice (solution sys (CH1 sys T)) (solution sys (CH2 sys T))" |
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246 |
unfolding solution_def |
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using process.unfolds(2)[of "isACT sys" T "PREF sys" "CONT sys" "CH1 sys" "CH2 sys"] |
49238 | 248 |
assms by simp |
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249 |
|
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250 |
lemma isACT_VAR: |
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251 |
assumes g: "guarded sys" |
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added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
252 |
shows "isACT sys (VAR X) \<longleftrightarrow> isACT sys (sys X)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
253 |
using g unfolding guarded_def by (cases "sys X") auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
254 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
255 |
lemma solution_VAR: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
256 |
assumes g: "guarded sys" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
257 |
shows "solution sys (VAR X) = solution sys (sys X)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
258 |
proof(cases "isACT sys (VAR X)") |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
259 |
case True |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
260 |
hence T: "isACT sys (sys X)" unfolding isACT_VAR[OF g] . |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
261 |
show ?thesis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
262 |
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
|
263 |
unfolding guarded_def by (cases "sys X", auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
264 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
265 |
case False note FFalse = False |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
266 |
hence TT: "\<not> isACT sys (sys X)" unfolding isACT_VAR[OF g] . |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
267 |
show ?thesis |
49238 | 268 |
unfolding solution_Choice[OF TT] using solution_Choice[of sys "VAR X"] FFalse g |
269 |
unfolding guarded_def by (cases "sys X", auto) |
|
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
270 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
271 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
272 |
lemma solution_PROC[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
273 |
"solution sys (PROC p) = p" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
274 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
275 |
{fix q assume "q = solution sys (PROC p)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
276 |
hence "p = q" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
277 |
proof(induct rule: process_coind) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
278 |
case (iss p p') |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
279 |
from isAction_isChoice[of p] show ?case |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
280 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
281 |
assume p: "isAction p" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
282 |
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
|
283 |
thus ?thesis using iss not_isAction_isChoice |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
284 |
unfolding solution_Action[OF 0] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
285 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
286 |
assume "isChoice p" |
49238 | 287 |
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
|
288 |
using not_isAction_isChoice by auto |
49238 | 289 |
thus ?thesis using iss isAction_isChoice |
290 |
unfolding solution_Choice[OF 0] by auto |
|
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
291 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
292 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
293 |
case (Action a a' p p') |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
294 |
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
|
295 |
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
|
296 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
297 |
case (Choice p q p' q') |
49238 | 298 |
hence 0: "\<not> isACT sys (PROC (Choice p q))" using not_isAction_isChoice by auto |
299 |
show ?case using Choice unfolding solution_Choice[OF 0] by simp |
|
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
300 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
301 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
302 |
thus ?thesis by metis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
303 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
304 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
305 |
lemma solution_ACT[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
306 |
"solution sys (ACT a T) = Action a (solution sys T)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
307 |
by (metis CONT.simps(3) PREF.simps(3) isACT.simps(3) solution_Action) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
308 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
309 |
lemma solution_CH[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
310 |
"solution sys (CH T1 T2) = Choice (solution sys T1) (solution sys T2)" |
49238 | 311 |
by (metis CH1.simps(3) CH2.simps(3) isACT.simps(4) solution_Choice) |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
312 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
313 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
314 |
(* Example: *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
315 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
316 |
fun sys where |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
317 |
"sys 0 = CH (VAR (Suc 0)) (ACT ''b'' (VAR 0))" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
318 |
| |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
319 |
"sys (Suc 0) = ACT ''a'' (VAR 0)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
320 |
| (* dummy guarded term for variables outside the system: *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
321 |
"sys X = ACT ''a'' (VAR 0)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
322 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
323 |
lemma guarded_sys: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
324 |
"guarded sys" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
325 |
unfolding guarded_def proof (intro allI) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
326 |
fix X Y show "sys X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
327 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
328 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
329 |
(* the actual processes: *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
330 |
definition "x \<equiv> solution sys (VAR 0)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
331 |
definition "ax \<equiv> solution sys (VAR (Suc 0))" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
332 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
333 |
(* end product: *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
334 |
lemma x_ax: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
335 |
"x = Choice ax (Action ''b'' x)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
336 |
"ax = Action ''a'' x" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
337 |
unfolding x_def ax_def by (subst solution_VAR[OF guarded_sys], simp)+ |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
338 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
339 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
340 |
(* Thanks to the inclusion of processes as process terms, one can |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
341 |
also consider parametrized systems of equations---here, x is a (semantic) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
342 |
process parameter: *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
343 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
344 |
fun sys' where |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
345 |
"sys' 0 = CH (PROC x) (ACT ''b'' (VAR 0))" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
346 |
| |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
347 |
"sys' (Suc 0) = CH (ACT ''a'' (VAR 0)) (PROC x)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
348 |
| (* dummy guarded term : *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
349 |
"sys' X = ACT ''a'' (VAR 0)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
350 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
351 |
lemma guarded_sys': |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
352 |
"guarded sys'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
353 |
unfolding guarded_def proof (intro allI) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
354 |
fix X Y show "sys' X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
355 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
356 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
357 |
(* the actual processes: *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
358 |
definition "y \<equiv> solution sys' (VAR 0)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
359 |
definition "ay \<equiv> solution sys' (VAR (Suc 0))" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
360 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
361 |
(* end product: *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
362 |
lemma y_ay: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
363 |
"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
|
364 |
"ay = Choice (Action ''a'' y) x" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
365 |
unfolding y_def ay_def by (subst solution_VAR[OF guarded_sys'], simp)+ |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
366 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
367 |
end |