(* Title: HOL/Datatype_Examples/Process.thy
Author: Andrei Popescu, TU Muenchen
Copyright 2012
Processes.
*)
header {* Processes *}
theory Process
imports "~~/src/HOL/Library/Stream"
begin
codatatype 'a process =
isAction: Action (prefOf: 'a) (contOf: "'a process") |
isChoice: Choice (ch1Of: "'a process") (ch2Of: "'a process")
(* Read: prefix of, continuation of, choice 1 of, choice 2 of *)
section {* Customization *}
subsection {* Basic properties *}
declare
rel_pre_process_def[simp]
rel_sum_def[simp]
rel_prod_def[simp]
(* Constructors versus discriminators *)
theorem isAction_isChoice:
"isAction p \<or> isChoice p"
by (rule process.exhaust_disc) auto
theorem not_isAction_isChoice: "\<not> (isAction p \<and> isChoice p)"
by (cases rule: process.exhaust[of p]) auto
subsection{* Coinduction *}
theorem process_coind[elim, consumes 1, case_names iss Action Choice, induct pred: "HOL.eq"]:
assumes phi: "\<phi> p p'" and
iss: "\<And>p p'. \<phi> p p' \<Longrightarrow> (isAction p \<longleftrightarrow> isAction p') \<and> (isChoice p \<longleftrightarrow> isChoice p')" and
Act: "\<And> a a' p p'. \<phi> (Action a p) (Action a' p') \<Longrightarrow> a = a' \<and> \<phi> p p'" and
Ch: "\<And> p q p' q'. \<phi> (Choice p q) (Choice p' q') \<Longrightarrow> \<phi> p p' \<and> \<phi> q q'"
shows "p = p'"
using assms
by (coinduct rule: process.coinduct) (metis process.collapse(1,2) process.disc(3))
(* Stronger coinduction, up to equality: *)
theorem process_strong_coind[elim, consumes 1, case_names iss Action Choice]:
assumes phi: "\<phi> p p'" and
iss: "\<And>p p'. \<phi> p p' \<Longrightarrow> (isAction p \<longleftrightarrow> isAction p') \<and> (isChoice p \<longleftrightarrow> isChoice p')" and
Act: "\<And> a a' p p'. \<phi> (Action a p) (Action a' p') \<Longrightarrow> a = a' \<and> (\<phi> p p' \<or> p = p')" and
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')"
shows "p = p'"
using assms
by (coinduct rule: process.coinduct_strong) (metis process.collapse(1,2) process.disc(3))
subsection {* Coiteration (unfold) *}
section{* Coinductive definition of the notion of trace *}
coinductive trace where
"trace p as \<Longrightarrow> trace (Action a p) (a ## as)"
|
"trace p as \<or> trace q as \<Longrightarrow> trace (Choice p q) as"
section{* Examples of corecursive definitions: *}
subsection{* Single-guard fixpoint definition *}
primcorec BX where
"isAction BX"
| "prefOf BX = ''a''"
| "contOf BX = BX"
subsection{* Multi-guard fixpoint definitions, simulated with auxiliary arguments *}
datatype x_y_ax = x | y | ax
primcorec F :: "x_y_ax \<Rightarrow> char list process" where
"xyax = x \<Longrightarrow> isChoice (F xyax)"
| "ch1Of (F xyax) = F ax"
| "ch2Of (F xyax) = F y"
| "prefOf (F xyax) = (if xyax = y then ''b'' else ''a'')"
| "contOf (F xyax) = F x"
definition "X = F x" definition "Y = F y" definition "AX = F ax"
lemma X_Y_AX: "X = Choice AX Y" "Y = Action ''b'' X" "AX = Action ''a'' X"
unfolding X_def Y_def AX_def by (subst F.code, simp)+
(* end product: *)
lemma X_AX:
"X = Choice AX (Action ''b'' X)"
"AX = Action ''a'' X"
using X_Y_AX by simp_all
section{* Case study: Multi-guard fixpoint definitions, without auxiliary arguments *}
hide_const x y ax X Y AX
(* Process terms *)
datatype ('a,'pvar) process_term =
VAR 'pvar |
PROC "'a process" |
ACT 'a "('a,'pvar) process_term" | CH "('a,'pvar) process_term" "('a,'pvar) process_term"
(* below, sys represents a system of equations *)
fun isACT where
"isACT sys (VAR X) =
(case sys X of ACT a T \<Rightarrow> True |PROC p \<Rightarrow> isAction p |_ \<Rightarrow> False)"
|
"isACT sys (PROC p) = isAction p"
|
"isACT sys (ACT a T) = True"
|
"isACT sys (CH T1 T2) = False"
fun PREF where
"PREF sys (VAR X) =
(case sys X of ACT a T \<Rightarrow> a | PROC p \<Rightarrow> prefOf p)"
|
"PREF sys (PROC p) = prefOf p"
|
"PREF sys (ACT a T) = a"
fun CONT where
"CONT sys (VAR X) =
(case sys X of ACT a T \<Rightarrow> T | PROC p \<Rightarrow> PROC (contOf p))"
|
"CONT sys (PROC p) = PROC (contOf p)"
|
"CONT sys (ACT a T) = T"
fun CH1 where
"CH1 sys (VAR X) =
(case sys X of CH T1 T2 \<Rightarrow> T1 |PROC p \<Rightarrow> PROC (ch1Of p))"
|
"CH1 sys (PROC p) = PROC (ch1Of p)"
|
"CH1 sys (CH T1 T2) = T1"
fun CH2 where
"CH2 sys (VAR X) =
(case sys X of CH T1 T2 \<Rightarrow> T2 |PROC p \<Rightarrow> PROC (ch2Of p))"
|
"CH2 sys (PROC p) = PROC (ch2Of p)"
|
"CH2 sys (CH T1 T2) = T2"
definition "guarded sys \<equiv> \<forall> X Y. sys X \<noteq> VAR Y"
primcorec solution where
"isACT sys T \<Longrightarrow> solution sys T = Action (PREF sys T) (solution sys (CONT sys T))"
| "_ \<Longrightarrow> solution sys T = Choice (solution sys (CH1 sys T)) (solution sys (CH2 sys T))"
lemma isACT_VAR:
assumes g: "guarded sys"
shows "isACT sys (VAR X) \<longleftrightarrow> isACT sys (sys X)"
using g unfolding guarded_def by (cases "sys X") auto
lemma solution_VAR:
assumes g: "guarded sys"
shows "solution sys (VAR X) = solution sys (sys X)"
proof(cases "isACT sys (VAR X)")
case True
hence T: "isACT sys (sys X)" unfolding isACT_VAR[OF g] .
show ?thesis
unfolding solution.ctr(1)[OF T] using solution.ctr(1)[of sys "VAR X"] True g
unfolding guarded_def by (cases "sys X", auto)
next
case False note FFalse = False
hence TT: "\<not> isACT sys (sys X)" unfolding isACT_VAR[OF g] .
show ?thesis
unfolding solution.ctr(2)[OF TT] using solution.ctr(2)[of sys "VAR X"] FFalse g
unfolding guarded_def by (cases "sys X", auto)
qed
lemma solution_PROC[simp]:
"solution sys (PROC p) = p"
proof-
{fix q assume "q = solution sys (PROC p)"
hence "p = q"
proof (coinduct rule: process_coind)
case (iss p p')
from isAction_isChoice[of p] show ?case
proof
assume p: "isAction p"
hence 0: "isACT sys (PROC p)" by simp
thus ?thesis using iss not_isAction_isChoice by auto
next
assume "isChoice p"
hence 0: "\<not> isACT sys (PROC p)"
using not_isAction_isChoice by auto
thus ?thesis using iss isAction_isChoice by auto
qed
next
case (Action a a' p p')
hence 0: "isACT sys (PROC (Action a p))" by simp
show ?case using Action unfolding solution.ctr(1)[OF 0] by simp
next
case (Choice p q p' q')
hence 0: "\<not> isACT sys (PROC (Choice p q))" using not_isAction_isChoice by auto
show ?case using Choice unfolding solution.ctr(2)[OF 0] by simp
qed
}
thus ?thesis by metis
qed
lemma solution_ACT[simp]:
"solution sys (ACT a T) = Action a (solution sys T)"
by (metis CONT.simps(3) PREF.simps(3) isACT.simps(3) solution.ctr(1))
lemma solution_CH[simp]:
"solution sys (CH T1 T2) = Choice (solution sys T1) (solution sys T2)"
by (metis CH1.simps(3) CH2.simps(3) isACT.simps(4) solution.ctr(2))
(* Example: *)
fun sys where
"sys 0 = CH (VAR (Suc 0)) (ACT ''b'' (VAR 0))"
|
"sys (Suc 0) = ACT ''a'' (VAR 0)"
| (* dummy guarded term for variables outside the system: *)
"sys X = ACT ''a'' (VAR 0)"
lemma guarded_sys:
"guarded sys"
unfolding guarded_def proof (intro allI)
fix X Y show "sys X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto)
qed
(* the actual processes: *)
definition "x \<equiv> solution sys (VAR 0)"
definition "ax \<equiv> solution sys (VAR (Suc 0))"
(* end product: *)
lemma x_ax:
"x = Choice ax (Action ''b'' x)"
"ax = Action ''a'' x"
unfolding x_def ax_def by (subst solution_VAR[OF guarded_sys], simp)+
(* Thanks to the inclusion of processes as process terms, one can
also consider parametrized systems of equations---here, x is a (semantic)
process parameter: *)
fun sys' where
"sys' 0 = CH (PROC x) (ACT ''b'' (VAR 0))"
|
"sys' (Suc 0) = CH (ACT ''a'' (VAR 0)) (PROC x)"
| (* dummy guarded term : *)
"sys' X = ACT ''a'' (VAR 0)"
lemma guarded_sys':
"guarded sys'"
unfolding guarded_def proof (intro allI)
fix X Y show "sys' X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto)
qed
(* the actual processes: *)
definition "y \<equiv> solution sys' (VAR 0)"
definition "ay \<equiv> solution sys' (VAR (Suc 0))"
(* end product: *)
lemma y_ay:
"y = Choice x (Action ''b'' y)"
"ay = Choice (Action ''a'' y) x"
unfolding y_def ay_def by (subst solution_VAR[OF guarded_sys'], simp)+
end