(* Title: HOL/Library/While.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 2000 TU Muenchen
*)
header {*
\title{A general ``while'' combinator}
\author{Tobias Nipkow}
*}
theory While_Combinator = Main:
text {*
We define a while-combinator @{term while} and prove: (a) an
unrestricted unfolding law (even if while diverges!) (I got this
idea from Wolfgang Goerigk), and (b) the invariant rule for reasoning
about @{term while}.
*}
consts while_aux :: "('a => bool) \<times> ('a => 'a) \<times> 'a => 'a"
recdef while_aux
"same_fst (\<lambda>b. True) (\<lambda>b. same_fst (\<lambda>c. True) (\<lambda>c.
{(t, s). b s \<and> c s = t \<and>
\<not> (\<exists>f. f 0 = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1)))}))"
"while_aux (b, c, s) =
(if (\<exists>f. f 0 = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1)))
then arbitrary
else if b s then while_aux (b, c, c s)
else s)"
constdefs
while :: "('a => bool) => ('a => 'a) => 'a => 'a"
"while b c s == while_aux (b, c, s)"
ML_setup {*
goalw_cterm [] (cterm_of (sign_of (the_context ()))
(HOLogic.mk_Trueprop (hd (RecdefPackage.tcs_of (the_context ()) "while_aux"))));
br wf_same_fst 1;
br wf_same_fst 1;
by (asm_full_simp_tac (simpset() addsimps [wf_iff_no_infinite_down_chain]) 1);
by (Blast_tac 1);
qed "while_aux_tc";
*} (* FIXME cannot access recdef tcs in Isar yet! *)
lemma while_aux_unfold:
"while_aux (b, c, s) =
(if \<exists>f. f 0 = s \<and> (\<forall>i. b (f i) \<and> c (f i) = f (i + 1))
then arbitrary
else if b s then while_aux (b, c, c s)
else s)"
apply (rule while_aux_tc [THEN while_aux.simps [THEN trans]])
apply (simp add: same_fst_def)
apply (rule refl)
done
text {*
The recursion equation for @{term while}: directly executable!
*}
theorem while_unfold:
"while b c s = (if b s then while b c (c s) else s)"
apply (unfold while_def)
apply (rule while_aux_unfold [THEN trans])
apply auto
apply (subst while_aux_unfold)
apply simp
apply clarify
apply (erule_tac x = "\<lambda>i. f (Suc i)" in allE)
apply blast
done
text {*
The proof rule for @{term while}, where @{term P} is the invariant.
*}
theorem while_rule_lemma[rule_format]:
"(!!s. P s ==> b s ==> P (c s)) ==>
(!!s. P s ==> \<not> b s ==> Q s) ==>
wf {(t, s). P s \<and> b s \<and> t = c s} ==>
P s --> Q (while b c s)"
proof -
case antecedent
assume wf: "wf {(t, s). P s \<and> b s \<and> t = c s}"
show ?thesis
apply (induct s rule: wf [THEN wf_induct])
apply simp
apply clarify
apply (subst while_unfold)
apply (simp add: antecedent)
done
qed
theorem while_rule:
"[| P s; !!s. [| P s; b s |] ==> P (c s);
!!s. [| P s; \<not> b s |] ==> Q s;
wf r; !!s. [| P s; b s |] ==> (c s, s) \<in> r |] ==>
Q (while b c s)"
apply (rule while_rule_lemma)
prefer 4 apply assumption
apply blast
apply blast
apply(erule wf_subset)
apply blast
done
hide const while_aux
end