src/HOL/Library/While_Combinator.thy
 author wenzelm Sat, 03 Feb 2001 17:43:05 +0100 changeset 11047 10c51288b00d parent 10997 e14029f92770 child 11284 981ea92a86dd permissions -rw-r--r--
tuned;
```
(*  Title:      HOL/Library/While.thy
ID:         \$Id\$
Author:     Tobias Nipkow
*)

\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
*}

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)"

recdef_tc while_aux_tc: while_aux
apply (rule wf_same_fst)
apply (rule wf_same_fst)
apply blast
done

constdefs
while :: "('a => bool) => ('a => 'a) => 'a => 'a"
"while b c s == while_aux (b, c, s)"

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 (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

hide const while_aux

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)
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

text {*
\medskip An application: computation of the @{term lfp} on finite
sets via iteration.
*}

theorem lfp_conv_while:
"[| mono f; finite U; f U = U |] ==>
lfp f = fst (while (\<lambda>(A, fA). A \<noteq> fA) (\<lambda>(A, fA). (fA, f fA)) ({}, f {}))"
apply (rule_tac P = "\<lambda>(A, B). (A \<subseteq> U \<and> B = f A \<and> A \<subseteq> B \<and> B \<subseteq> lfp f)" and
r = "((Pow U \<times> UNIV) \<times> (Pow U \<times> UNIV)) \<inter>
inv_image finite_psubset (op - U o fst)" in while_rule)
apply (subst lfp_unfold)
apply assumption
apply (subst lfp_unfold)
apply assumption
apply clarsimp
apply (blast dest: monoD)
apply (fastsimp intro!: lfp_lowerbound)
apply (blast intro: wf_finite_psubset Int_lower2 [THEN [2] wf_subset])
apply (clarsimp simp add: inv_image_def finite_psubset_def order_less_le)
apply (blast intro!: finite_Diff dest: monoD)
done

text {*
An example of using the @{term while} combinator.\footnote{It is safe
to keep the example here, since there is no effect on the current
theory.}
*}

theorem "P (lfp (\<lambda>N::int set. {#0} \<union> {(n + #2) mod #6 | n. n \<in> N})) =
P {#0, #4, #2}"
proof -
have aux: "!!f A B. {f n | n. A n \<or> B n} = {f n | n. A n} \<union> {f n | n. B n}"
apply blast
done
show ?thesis
apply (subst lfp_conv_while [where ?U = "{#0, #1, #2, #3, #4, #5}"])
apply (rule monoI)
apply blast
apply simp