section ‹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 ⟹ ∃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 -
define k where "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: "∀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_invariant:
assumes Invariant: "⋀s. P s ⟹ b s ⟹ P (c s)"
assumes TestCommute: "⋀s. P s ⟹ b s = b' (f s)"
assumes BodyCommute: "⋀s. P s ⟹ b s ⟹ f (c s) = c' (f s)"
assumes Initial: "P s"
shows "map_option 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)"
with Initial show "∃k. ¬ b' ((c' ^^ k) (f s))"
proof (induction k arbitrary: s)
case 0 thus ?case by (auto simp: TestCommute intro: exI[of _ 0])
next
case (Suc k) thus ?case
proof (cases "b s")
assume "b s"
with Suc.IH[of "c s"] Suc.prems show ?thesis
by (metis BodyCommute Invariant comp_apply funpow.simps(2) funpow_swap1)
next
assume "¬ b s"
with Suc show ?thesis by (auto simp: TestCommute intro: exI [of _ 0])
qed
qed
next
fix k
assume "¬ b' ((c' ^^ k) (f s))"
with Initial show "∃k. ¬ b ((c ^^ k) s)"
proof (induction k arbitrary: s)
case 0 thus ?case by (auto simp: TestCommute intro: exI[of _ 0])
next
case (Suc k) thus ?case
proof (cases "b s")
assume "b s"
with Suc.IH[of "c s"] Suc.prems show ?thesis
by (metis BodyCommute Invariant comp_apply funpow.simps(2) funpow_swap1)
next
assume "¬ b s"
with Suc show ?thesis by (auto simp: TestCommute intro: exI [of _ 0])
qed
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" by (auto simp: TestCommute Initial)
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)"
and "b ((c ^^ k) s) = b' ((c' ^^ k) (f s))"
and "P ((c ^^ k) s)"
by (induct k) (auto simp: b' assms)
with ‹k ≤ k'›
have "b ((c ^^ k) s)"
and "f ((c ^^ k) s) = (c' ^^ k) (f s)"
and "P ((c ^^ k) s)"
by (auto simp: b')
}
note b = this(1) and body = this(2) and inv = this(3)
hence k': "f ((c ^^ k') s) = (c' ^^ k') (f s)" by auto
ultimately show ?thesis unfolding Suc using b
proof (intro Least_equality[symmetric], goal_cases)
case 1
hence Test: "¬ b' (f ((c ^^ Suc k') s))"
by (auto simp: BodyCommute inv b)
have "P ((c ^^ Suc k') s)" by (auto simp: Invariant inv b)
with Test show ?case by (auto simp: TestCommute)
next
case 2
thus ?case by (metis not_less_eq_eq)
qed
qed
have "f ((c ^^ ?k) s) = (c' ^^ ?k') (f s)" unfolding *
proof (rule funpow_commute, clarify)
fix k assume "k < ?k"
hence TestTrue: "b ((c ^^ k) s)" by (auto dest: not_less_Least)
from ‹k < ?k› have "P ((c ^^ k) s)"
proof (induct k)
case 0 thus ?case by (auto simp: assms)
next
case (Suc h)
hence "P ((c ^^ h) s)" by auto
with Suc show ?case
by (auto, metis (lifting, no_types) Invariant Suc_lessD not_less_Least)
qed
with TestTrue show "f (c ((c ^^ k) s)) = c' (f ((c ^^ k) s))"
by (metis BodyCommute)
qed
thus "∃z. (c ^^ ?k) s = z ∧ f z = (c' ^^ ?k') (f s)" by blast
qed
lemma while_option_commute:
assumes "⋀s. b s = b' (f s)" "⋀s. ⟦b s⟧ ⟹ f (c s) = c' (f s)"
shows "map_option f (while_option b c s) = while_option b' c' (f s)"
by(rule while_option_commute_invariant[where P = "λ_. True"])
(auto simp add: assms)
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 "∃t. while_option b c s = Some t"
using assms(1,3)
proof (induction s)
case less thus ?case using assms(2)
by (subst while_option_unfold) simp
qed
lemma wf_rel_while_option_Some:
assumes wf: "wf R"
assumes smaller: "⋀s. P s ∧ b s ⟹ (c s, s) ∈ R"
assumes inv: "⋀s. P s ∧ b s ⟹ P(c s)"
assumes init: "P s"
shows "∃t. while_option b c s = Some t"
proof -
from smaller have "{(t,s). P s ∧ b s ∧ t = c s} ⊆ R" by auto
with wf have "wf {(t,s). P s ∧ b s ∧ t = c s}" by (auto simp: wf_subset)
with inv init show ?thesis by (auto simp: wf_while_option_Some)
qed
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 ⟹ ∃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
lemma wf_finite_less:
assumes "finite (C :: 'a::order set)"
shows "wf {(x, y). {x, y} ⊆ C ∧ x < y}"
by (rule wf_measure[where f="λb. card {a. a ∈ C ∧ a < b}", THEN wf_subset])
(fastforce simp: less_eq assms intro: psubset_card_mono)
lemma wf_finite_greater:
assumes "finite (C :: 'a::order set)"
shows "wf {(x, y). {x, y} ⊆ C ∧ y < x}"
by (rule wf_measure[where f="λb. card {a. a ∈ C ∧ b < a}", THEN wf_subset])
(fastforce simp: less_eq assms intro: psubset_card_mono)
lemma while_option_finite_increasing_Some:
fixes f :: "'a::order ⇒ 'a"
assumes "mono f" and "finite (UNIV :: 'a set)" and "s ≤ f s"
shows "∃P. while_option (λA. f A ≠ A) f s = Some P"
by (rule wf_rel_while_option_Some[where R="{(x, y). y < x}" and P="λA. A ≤ f A" and s="s"])
(auto simp: assms monoD intro: wf_finite_greater[where C="UNIV::'a set", simplified])
lemma lfp_the_while_option_lattice:
fixes f :: "'a::complete_lattice ⇒ 'a"
assumes "mono f" and "finite (UNIV :: 'a set)"
shows "lfp f = the (while_option (λA. f A ≠ A) f bot)"
proof -
obtain P where "while_option (λA. f A ≠ A) f bot = Some P"
using while_option_finite_increasing_Some[OF assms, where s=bot] by simp blast
with while_option_stop2[OF this] lfp_Kleene_iter[OF assms(1)]
show ?thesis by auto
qed
lemma lfp_while_lattice:
fixes f :: "'a::complete_lattice ⇒ 'a"
assumes "mono f" and "finite (UNIV :: 'a set)"
shows "lfp f = while (λA. f A ≠ A) f bot"
unfolding while_def using assms by (rule lfp_the_while_option_lattice)
lemma while_option_finite_decreasing_Some:
fixes f :: "'a::order ⇒ 'a"
assumes "mono f" and "finite (UNIV :: 'a set)" and "f s ≤ s"
shows "∃P. while_option (λA. f A ≠ A) f s = Some P"
by (rule wf_rel_while_option_Some[where R="{(x, y). x < y}" and P="λA. f A ≤ A" and s="s"])
(auto simp add: assms monoD intro: wf_finite_less[where C="UNIV::'a set", simplified])
lemma gfp_the_while_option_lattice:
fixes f :: "'a::complete_lattice ⇒ 'a"
assumes "mono f" and "finite (UNIV :: 'a set)"
shows "gfp f = the(while_option (λA. f A ≠ A) f top)"
proof -
obtain P where "while_option (λA. f A ≠ A) f top = Some P"
using while_option_finite_decreasing_Some[OF assms, where s=top] by simp blast
with while_option_stop2[OF this] gfp_Kleene_iter[OF assms(1)]
show ?thesis by auto
qed
lemma gfp_while_lattice:
fixes f :: "'a::complete_lattice ⇒ 'a"
assumes "mono f" and "finite (UNIV :: 'a set)"
shows "gfp f = while (λA. f A ≠ A) f top"
unfolding while_def using assms by (rule gfp_the_while_option_lattice)
text‹Computing the reflexive, transitive closure by iterating a successor
function. Stops when an element is found that dos not satisfy the test.
More refined (and hence more efficient) versions can be found in ITP 2011 paper
by Nipkow (the theories are in the AFP entry Flyspeck by Nipkow)
and the AFP article Executable Transitive Closures by René Thiemann.›
context
fixes p :: "'a ⇒ bool"
and f :: "'a ⇒ 'a list"
and x :: 'a
begin
qualified fun rtrancl_while_test :: "'a list × 'a set ⇒ bool"
where "rtrancl_while_test (ws,_) = (ws ≠ [] ∧ p(hd ws))"
qualified fun rtrancl_while_step :: "'a list × 'a set ⇒ 'a list × 'a set"
where "rtrancl_while_step (ws, Z) =
(let x = hd ws; new = remdups (filter (λy. y ∉ Z) (f x))
in (new @ tl ws, set new ∪ Z))"
definition rtrancl_while :: "('a list * 'a set) option"
where "rtrancl_while = while_option rtrancl_while_test rtrancl_while_step ([x],{x})"
qualified fun rtrancl_while_invariant :: "'a list × 'a set ⇒ bool"
where "rtrancl_while_invariant (ws, Z) =
(x ∈ Z ∧ set ws ⊆ Z ∧ distinct ws ∧ {(x,y). y ∈ set(f x)} `` (Z - set ws) ⊆ Z ∧
Z ⊆ {(x,y). y ∈ set(f x)}⇧* `` {x} ∧ (∀z∈Z - set ws. p z))"
qualified lemma rtrancl_while_invariant:
assumes inv: "rtrancl_while_invariant st" and test: "rtrancl_while_test st"
shows "rtrancl_while_invariant (rtrancl_while_step st)"
proof (cases st)
fix ws Z assume st: "st = (ws, Z)"
with test obtain h t where "ws = h # t" "p h" by (cases ws) auto
with inv st show ?thesis by (auto intro: rtrancl.rtrancl_into_rtrancl)
qed
lemma rtrancl_while_Some: assumes "rtrancl_while = Some(ws,Z)"
shows "if ws = []
then Z = {(x,y). y ∈ set(f x)}⇧* `` {x} ∧ (∀z∈Z. p z)
else ¬p(hd ws) ∧ hd ws ∈ {(x,y). y ∈ set(f x)}⇧* `` {x}"
proof -
have "rtrancl_while_invariant ([x],{x})" by simp
with rtrancl_while_invariant have I: "rtrancl_while_invariant (ws,Z)"
by (rule while_option_rule[OF _ assms[unfolded rtrancl_while_def]])
{ assume "ws = []"
hence ?thesis using I
by (auto simp del:Image_Collect_case_prod dest: Image_closed_trancl)
} moreover
{ assume "ws ≠ []"
hence ?thesis using I while_option_stop[OF assms[unfolded rtrancl_while_def]]
by (simp add: subset_iff)
}
ultimately show ?thesis by simp
qed
lemma rtrancl_while_finite_Some:
assumes "finite ({(x, y). y ∈ set (f x)}⇧* `` {x})" (is "finite ?Cl")
shows "∃y. rtrancl_while = Some y"
proof -
let ?R = "(λ(_, Z). card (?Cl - Z)) <*mlex*> (λ(ws, _). length ws) <*mlex*> {}"
have "wf ?R" by (blast intro: wf_mlex)
then show ?thesis unfolding rtrancl_while_def
proof (rule wf_rel_while_option_Some[of ?R rtrancl_while_invariant])
fix st assume *: "rtrancl_while_invariant st ∧ rtrancl_while_test st"
hence I: "rtrancl_while_invariant (rtrancl_while_step st)"
by (blast intro: rtrancl_while_invariant)
show "(rtrancl_while_step st, st) ∈ ?R"
proof (cases st)
fix ws Z let ?ws = "fst (rtrancl_while_step st)" and ?Z = "snd (rtrancl_while_step st)"
assume st: "st = (ws, Z)"
with * obtain h t where ws: "ws = h # t" "p h" by (cases ws) auto
{ assume "remdups (filter (λy. y ∉ Z) (f h)) ≠ []"
then obtain z where "z ∈ set (remdups (filter (λy. y ∉ Z) (f h)))" by fastforce
with st ws I have "Z ⊂ ?Z" "Z ⊆ ?Cl" "?Z ⊆ ?Cl" by auto
with assms have "card (?Cl - ?Z) < card (?Cl - Z)" by (blast intro: psubset_card_mono)
with st ws have ?thesis unfolding mlex_prod_def by simp
}
moreover
{ assume "remdups (filter (λy. y ∉ Z) (f h)) = []"
with st ws have "?Z = Z" "?ws = t" by (auto simp: filter_empty_conv)
with st ws have ?thesis unfolding mlex_prod_def by simp
}
ultimately show ?thesis by blast
qed
qed (simp_all add: rtrancl_while_invariant)
qed
end
end