Theory Flag

theory Flag
imports List
(*  Title:      CCL/ex/Flag.thy
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)


header {* 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]=>i"
where "ccase(c,r,w,b) == when(c,%x. r,%wb. when(wb,%x. w,%x. b))"

definition flag :: "i"
where
"flag == lam l. letrec
flagx l be lcase(l,<[],<[],[]>>,
%h t. split(flagx(t),%lr p. split(p,%lw lb.
ccase(h, <red$lr,<lw,lb>>,
<lr,<white$lw,lb>>,
<lr,<lw,blue$lb>>))))
in flagx(l)"


axiomatization Perm :: "i => i => o"
definition Flag :: "i => i => o" where
"Flag(l,x) == ALL lr:List(Colour).ALL lw:List(Colour).ALL lb:List(Colour).
x = <lr,<lw,lb>> -->
(ALL c:Colour.(c mem lr = true --> c=red) &
(c mem lw = true --> c=white) &
(c mem lb = true --> c=blue)) &
Perm(l,lr @ lw @ lb)"



lemmas flag_defs = Colour_def red_def white_def blue_def ccase_def

lemma ColourXH: "a : Colour <-> (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:
"[| c:Colour; c=red ==> r : C(red); c=white ==> w : C(white); c=blue ==> b : C(blue) |]
==> 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 (tactic {* typechk_tac @{context}
[@{thm redT}, @{thm whiteT}, @{thm blueT}, @{thm ccaseT}] 1 *}
)
apply (tactic "clean_ccs_tac @{context}")
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 (tactic {* gen_ccs_tac @{context}
[@{thm redT}, @{thm whiteT}, @{thm blueT}, @{thm ccaseT}] 1 *}
)
oops

end