(* Title: CCL/ex/Flag.thy
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section {* Dutch national flag program -- except that the point of Dijkstra's example was to use
arrays and this uses lists. *}
theory Flag
imports List
begin
definition Colour :: "i set"
where "Colour == Unit + Unit + Unit"
definition red :: "i"
where "red == inl(one)"
definition white :: "i"
where "white == inr(inl(one))"
definition blue :: "i"
where "blue == inr(inr(one))"
definition ccase :: "[i,i,i,i]\<Rightarrow>i"
where "ccase(c,r,w,b) == when(c, \<lambda>x. r, \<lambda>wb. when(wb, \<lambda>x. w, \<lambda>x. b))"
definition flag :: "i"
where
"flag == lam l. letrec
flagx l be lcase(l,<[],<[],[]>>,
\<lambda>h t. split(flagx(t), \<lambda>lr p. split(p, \<lambda>lw lb.
ccase(h, <red$lr,<lw,lb>>,
<lr,<white$lw,lb>>,
<lr,<lw,blue$lb>>))))
in flagx(l)"
axiomatization Perm :: "i \<Rightarrow> i \<Rightarrow> o"
definition Flag :: "i \<Rightarrow> i \<Rightarrow> o" where
"Flag(l,x) == ALL lr:List(Colour).ALL lw:List(Colour).ALL lb:List(Colour).
x = <lr,<lw,lb>> \<longrightarrow>
(ALL c:Colour.(c mem lr = true \<longrightarrow> c=red) \<and>
(c mem lw = true \<longrightarrow> c=white) \<and>
(c mem lb = true \<longrightarrow> c=blue)) \<and>
Perm(l,lr @ lw @ lb)"
lemmas flag_defs = Colour_def red_def white_def blue_def ccase_def
lemma ColourXH: "a : Colour \<longleftrightarrow> (a=red | a=white | a=blue)"
unfolding simp_type_defs flag_defs by blast
lemma redT: "red : Colour"
and whiteT: "white : Colour"
and blueT: "blue : Colour"
unfolding ColourXH by blast+
lemma ccaseT:
"\<lbrakk>c:Colour; c=red \<Longrightarrow> r : C(red); c=white \<Longrightarrow> w : C(white); c=blue \<Longrightarrow> b : C(blue)\<rbrakk>
\<Longrightarrow> ccase(c,r,w,b) : C(c)"
unfolding flag_defs by ncanT
lemma "flag : List(Colour)->List(Colour)*List(Colour)*List(Colour)"
apply (unfold flag_def)
apply (typechk redT whiteT blueT ccaseT)
apply clean_ccs
apply (erule ListPRI [THEN ListPR_wf [THEN wfI]])
apply assumption
done
lemma "flag : PROD l:List(Colour).{x:List(Colour)*List(Colour)*List(Colour).Flag(x,l)}"
apply (unfold flag_def)
apply (gen_ccs redT whiteT blueT ccaseT)
oops
end