header {* A general ``while'' combinator *}
theory While_Combinator
imports Main
begin
subsection {* Partial version *}
definition while_option :: "('a => bool) => ('a => 'a) => 'a => 'a option" where
"while_option b c s = (if (∃k. ~ b ((c ^^ k) s))
then Some ((c ^^ (LEAST k. ~ b ((c ^^ k) s))) s)
else None)"
theorem while_option_unfold[code]:
"while_option b c s = (if b s then while_option b c (c s) else Some s)"
proof cases
assume "b s"
show ?thesis
proof (cases "∃k. ~ b ((c ^^ k) s)")
case True
then obtain k where 1: "~ b ((c ^^ k) s)" ..
with `b s` obtain l where "k = Suc l" by (cases k) auto
with 1 have "~ b ((c ^^ l) (c s))" by (auto simp: funpow_swap1)
then have 2: "∃l. ~ b ((c ^^ l) (c s))" ..
from 1
have "(LEAST k. ~ b ((c ^^ k) s)) = Suc (LEAST l. ~ b ((c ^^ Suc l) s))"
by (rule Least_Suc) (simp add: `b s`)
also have "... = Suc (LEAST l. ~ b ((c ^^ l) (c s)))"
by (simp add: funpow_swap1)
finally
show ?thesis
using True 2 `b s` by (simp add: funpow_swap1 while_option_def)
next
case False
then have "~ (∃l. ~ b ((c ^^ Suc l) s))" by blast
then have "~ (∃l. ~ b ((c ^^ l) (c s)))"
by (simp add: funpow_swap1)
with False `b s` show ?thesis by (simp add: while_option_def)
qed
next
assume [simp]: "~ b s"
have least: "(LEAST k. ~ b ((c ^^ k) s)) = 0"
by (rule Least_equality) auto
moreover
have "∃k. ~ b ((c ^^ k) s)" by (rule exI[of _ "0::nat"]) auto
ultimately show ?thesis unfolding while_option_def by auto
qed
lemma while_option_stop2:
"while_option b c s = Some t ==> EX k. t = (c^^k) s ∧ ¬ b t"
apply(simp add: while_option_def split: if_splits)
by (metis (lifting) LeastI_ex)
lemma while_option_stop: "while_option b c s = Some t ==> ~ b t"
by(metis while_option_stop2)
theorem while_option_rule:
assumes step: "!!s. P s ==> b s ==> P (c s)"
and result: "while_option b c s = Some t"
and init: "P s"
shows "P t"
proof -
def k == "LEAST k. ~ b ((c ^^ k) s)"
from assms have t: "t = (c ^^ k) s"
by (simp add: while_option_def k_def split: if_splits)
have 1: "ALL i<k. b ((c ^^ i) s)"
by (auto simp: k_def dest: not_less_Least)
{ fix i assume "i <= k" then have "P ((c ^^ i) s)"
by (induct i) (auto simp: init step 1) }
thus "P t" by (auto simp: t)
qed
lemma funpow_commute:
"[|∀k' < k. f (c ((c^^k') s)) = c' (f ((c^^k') s))|] ==> f ((c^^k) s) = (c'^^k) (f s)"
by (induct k arbitrary: s) auto
lemma while_option_commute:
assumes "!!s. b s = b' (f s)" "!!s. [|b s|] ==> f (c s) = c' (f s)"
shows "Option.map f (while_option b c s) = while_option b' c' (f s)"
unfolding while_option_def
proof (rule trans[OF if_distrib if_cong], safe, unfold option.inject)
fix k assume "¬ b ((c ^^ k) s)"
thus "∃k. ¬ b' ((c' ^^ k) (f s))"
proof (induction k arbitrary: s)
case 0 thus ?case by (auto simp: assms(1) intro: exI[of _ 0])
next
case (Suc k)
hence "¬ b ((c^^k) (c s))" by (auto simp: funpow_swap1)
then guess k by (rule exE[OF Suc.IH[of "c s"]])
with assms show ?case by (cases "b s") (auto simp: funpow_swap1 intro: exI[of _ "Suc k"] exI[of _ "0"])
qed
next
fix k assume "¬ b' ((c' ^^ k) (f s))"
thus "∃k. ¬ b ((c ^^ k) s)"
proof (induction k arbitrary: s)
case 0 thus ?case by (auto simp: assms(1) intro: exI[of _ 0])
next
case (Suc k)
hence *: "¬ b' ((c'^^k) (c' (f s)))" by (auto simp: funpow_swap1)
show ?case
proof (cases "b s")
case True
with assms(2) * have "¬ b' ((c'^^k) (f (c s)))" by simp
then guess k by (rule exE[OF Suc.IH[of "c s"]])
thus ?thesis by (auto simp: funpow_swap1 intro: exI[of _ "Suc k"])
qed (auto intro: exI[of _ "0"])
qed
next
fix k assume k: "¬ b' ((c' ^^ k) (f s))"
have *: "(LEAST k. ¬ b' ((c' ^^ k) (f s))) = (LEAST k. ¬ b ((c ^^ k) s))" (is "?k' = ?k")
proof (cases ?k')
case 0
have "¬ b' ((c'^^0) (f s))" unfolding 0[symmetric] by (rule LeastI[of _ k]) (rule k)
hence "¬ b s" unfolding assms(1) by simp
hence "?k = 0" by (intro Least_equality) auto
with 0 show ?thesis by auto
next
case (Suc k')
have "¬ b' ((c'^^Suc k') (f s))" unfolding Suc[symmetric] by (rule LeastI) (rule k)
moreover
{ fix k assume "k ≤ k'"
hence "k < ?k'" unfolding Suc by simp
hence "b' ((c' ^^ k) (f s))" by (rule iffD1[OF not_not, OF not_less_Least])
} note b' = this
{ fix k assume "k ≤ k'"
hence "f ((c ^^ k) s) = (c'^^k) (f s)" by (induct k) (auto simp: b' assms)
with `k ≤ k'` have "b ((c^^k) s)"
proof (induct k)
case (Suc k) thus ?case unfolding assms(1) by (simp only: b')
qed (simp add: b'[of 0, simplified] assms(1))
} note b = this
hence k': "f ((c^^k') s) = (c'^^k') (f s)" by (induct k') (auto simp: assms(2))
ultimately show ?thesis unfolding Suc using b
by (intro sym[OF Least_equality])
(auto simp add: assms(1) assms(2)[OF b] k' not_less_eq_eq[symmetric])
qed
have "f ((c ^^ ?k) s) = (c' ^^ ?k') (f s)" unfolding *
by (auto intro: funpow_commute assms(2) dest: not_less_Least)
thus "∃z. (c ^^ ?k) s = z ∧ f z = (c' ^^ ?k') (f s)" by blast
qed
subsection {* Total version *}
definition while :: "('a => bool) => ('a => 'a) => 'a => 'a"
where "while b c s = the (while_option b c s)"
lemma while_unfold [code]:
"while b c s = (if b s then while b c (c s) else s)"
unfolding while_def by (subst while_option_unfold) simp
lemma def_while_unfold:
assumes fdef: "f == while test do"
shows "f x = (if test x then f(do x) else x)"
unfolding fdef by (fact while_unfold)
text {*
The proof rule for @{term while}, where @{term P} is the invariant.
*}
theorem while_rule_lemma:
assumes invariant: "!!s. P s ==> b s ==> P (c s)"
and terminate: "!!s. P s ==> ¬ b s ==> Q s"
and wf: "wf {(t, s). P s ∧ b s ∧ t = c s}"
shows "P s ==> Q (while b c s)"
using wf
apply (induct s)
apply simp
apply (subst while_unfold)
apply (simp add: invariant terminate)
done
theorem while_rule:
"[| P s;
!!s. [| P s; b s |] ==> P (c s);
!!s. [| P s; ¬ b s |] ==> Q s;
wf r;
!!s. [| P s; b s |] ==> (c s, s) ∈ 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{* Proving termination: *}
theorem wf_while_option_Some:
assumes "wf {(t, s). (P s ∧ b s) ∧ t = c s}"
and "!!s. P s ==> b s ==> P(c s)" and "P s"
shows "EX t. while_option b c s = Some t"
using assms(1,3)
apply (induct s)
using assms(2)
apply (subst while_option_unfold)
apply simp
done
theorem measure_while_option_Some: fixes f :: "'s => nat"
shows "(!!s. P s ==> b s ==> P(c s) ∧ f(c s) < f s)
==> P s ==> EX t. while_option b c s = Some t"
by(blast intro: wf_while_option_Some[OF wf_if_measure, of P b f])
text{* Kleene iteration starting from the empty set and assuming some finite
bounding set: *}
lemma while_option_finite_subset_Some: fixes C :: "'a set"
assumes "mono f" and "!!X. X ⊆ C ==> f X ⊆ C" and "finite C"
shows "∃P. while_option (λA. f A ≠ A) f {} = Some P"
proof(rule measure_while_option_Some[where
f= "%A::'a set. card C - card A" and P= "%A. A ⊆ C ∧ A ⊆ f A" and s= "{}"])
fix A assume A: "A ⊆ C ∧ A ⊆ f A" "f A ≠ A"
show "(f A ⊆ C ∧ f A ⊆ f (f A)) ∧ card C - card (f A) < card C - card A"
(is "?L ∧ ?R")
proof
show ?L by(metis A(1) assms(2) monoD[OF `mono f`])
show ?R by (metis A assms(2,3) card_seteq diff_less_mono2 equalityI linorder_le_less_linear rev_finite_subset)
qed
qed simp
lemma lfp_the_while_option:
assumes "mono f" and "!!X. X ⊆ C ==> f X ⊆ C" and "finite C"
shows "lfp f = the(while_option (λA. f A ≠ A) f {})"
proof-
obtain P where "while_option (λA. f A ≠ A) f {} = Some P"
using while_option_finite_subset_Some[OF assms] by blast
with while_option_stop2[OF this] lfp_Kleene_iter[OF assms(1)]
show ?thesis by auto
qed
lemma lfp_while:
assumes "mono f" and "!!X. X ⊆ C ==> f X ⊆ C" and "finite C"
shows "lfp f = while (λA. f A ≠ A) f {}"
unfolding while_def using assms by (rule lfp_the_while_option) blast
end