(* Author: Florian Haftmann, TU Muenchen *)
header {* Operations on lists beyond the standard List theory *}
theory More_List
imports Main Multiset
begin
hide_const (open) Finite_Set.fold
text {* Repairing code generator setup *}
declare (in lattice) Inf_fin_set_fold [code_unfold del]
declare (in lattice) Sup_fin_set_fold [code_unfold del]
declare (in linorder) Min_fin_set_fold [code_unfold del]
declare (in linorder) Max_fin_set_fold [code_unfold del]
declare (in complete_lattice) Inf_set_fold [code_unfold del]
declare (in complete_lattice) Sup_set_fold [code_unfold del]
text {* Fold combinator with canonical argument order *}
primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
"fold f [] = id"
| "fold f (x # xs) = fold f xs \<circ> f x"
lemma foldl_fold:
"foldl f s xs = fold (\<lambda>x s. f s x) xs s"
by (induct xs arbitrary: s) simp_all
lemma foldr_fold_rev:
"foldr f xs = fold f (rev xs)"
by (simp add: foldr_foldl foldl_fold fun_eq_iff)
lemma fold_rev_conv [code_unfold]:
"fold f (rev xs) = foldr f xs"
by (simp add: foldr_fold_rev)
lemma fold_cong [fundef_cong]:
"a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
\<Longrightarrow> fold f xs a = fold g ys b"
by (induct ys arbitrary: a b xs) simp_all
lemma fold_id:
assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
shows "fold f xs = id"
using assms by (induct xs) simp_all
lemma fold_commute:
assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
shows "h \<circ> fold g xs = fold f xs \<circ> h"
using assms by (induct xs) (simp_all add: fun_eq_iff)
lemma fold_commute_apply:
assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
shows "h (fold g xs s) = fold f xs (h s)"
proof -
from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
then show ?thesis by (simp add: fun_eq_iff)
qed
lemma fold_invariant:
assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
shows "P (fold f xs s)"
using assms by (induct xs arbitrary: s) simp_all
lemma fold_weak_invariant:
assumes "P s"
and "\<And>s x. x \<in> set xs \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
shows "P (fold f xs s)"
using assms by (induct xs arbitrary: s) simp_all
lemma fold_append [simp]:
"fold f (xs @ ys) = fold f ys \<circ> fold f xs"
by (induct xs) simp_all
lemma fold_map [code_unfold]:
"fold g (map f xs) = fold (g o f) xs"
by (induct xs) simp_all
lemma fold_remove1_split:
assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
and x: "x \<in> set xs"
shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
using assms by (induct xs) (auto simp add: o_assoc [symmetric])
lemma fold_multiset_equiv:
assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
and equiv: "multiset_of xs = multiset_of ys"
shows "fold f xs = fold f ys"
using f equiv [symmetric] proof (induct xs arbitrary: ys)
case Nil then show ?case by simp
next
case (Cons x xs)
then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
by (rule Cons.prems(1)) (simp_all add: *)
moreover from * have "x \<in> set ys" by simp
ultimately have "fold f ys = fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
moreover from Cons.prems have "fold f xs = fold f (remove1 x ys)" by (auto intro: Cons.hyps)
ultimately show ?case by simp
qed
lemma fold_rev:
assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
shows "fold f (rev xs) = fold f xs"
by (rule fold_multiset_equiv, rule assms) (simp_all add: in_multiset_in_set)
lemma foldr_fold:
assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
shows "foldr f xs = fold f xs"
using assms unfolding foldr_fold_rev by (rule fold_rev)
lemma fold_Cons_rev:
"fold Cons xs = append (rev xs)"
by (induct xs) simp_all
lemma rev_conv_fold [code]:
"rev xs = fold Cons xs []"
by (simp add: fold_Cons_rev)
lemma fold_append_concat_rev:
"fold append xss = append (concat (rev xss))"
by (induct xss) simp_all
lemma concat_conv_foldr [code]:
"concat xss = foldr append xss []"
by (simp add: fold_append_concat_rev foldr_fold_rev)
lemma fold_plus_listsum_rev:
"fold plus xs = plus (listsum (rev xs))"
by (induct xs) (simp_all add: add.assoc)
lemma (in monoid_add) listsum_conv_fold [code]:
"listsum xs = fold (\<lambda>x y. y + x) xs 0"
by (auto simp add: listsum_foldl foldl_fold fun_eq_iff)
lemma (in linorder) sort_key_conv_fold:
assumes "inj_on f (set xs)"
shows "sort_key f xs = fold (insort_key f) xs []"
proof -
have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
proof (rule fold_rev, rule ext)
fix zs
fix x y
assume "x \<in> set xs" "y \<in> set xs"
with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
have **: "x = y \<longleftrightarrow> y = x" by auto
show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
by (induct zs) (auto intro: * simp add: **)
qed
then show ?thesis by (simp add: sort_key_def foldr_fold_rev)
qed
lemma (in linorder) sort_conv_fold:
"sort xs = fold insort xs []"
by (rule sort_key_conv_fold) simp
text {* @{const Finite_Set.fold} and @{const fold} *}
lemma (in comp_fun_commute) fold_set_remdups:
"Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
lemma (in comp_fun_idem) fold_set:
"Finite_Set.fold f y (set xs) = fold f xs y"
by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
lemma (in ab_semigroup_idem_mult) fold1_set:
assumes "xs \<noteq> []"
shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
proof -
interpret comp_fun_idem times by (fact comp_fun_idem)
from assms obtain y ys where xs: "xs = y # ys"
by (cases xs) auto
show ?thesis
proof (cases "set ys = {}")
case True with xs show ?thesis by simp
next
case False
then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
by (simp only: finite_set fold1_eq_fold_idem)
with xs show ?thesis by (simp add: fold_set mult_commute)
qed
qed
lemma (in lattice) Inf_fin_set_fold:
"Inf_fin (set (x # xs)) = fold inf xs x"
proof -
interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
by (fact ab_semigroup_idem_mult_inf)
show ?thesis
by (simp add: Inf_fin_def fold1_set del: set.simps)
qed
lemma (in lattice) Inf_fin_set_foldr [code_unfold]:
"Inf_fin (set (x # xs)) = foldr inf xs x"
by (simp add: Inf_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
lemma (in lattice) Sup_fin_set_fold:
"Sup_fin (set (x # xs)) = fold sup xs x"
proof -
interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
by (fact ab_semigroup_idem_mult_sup)
show ?thesis
by (simp add: Sup_fin_def fold1_set del: set.simps)
qed
lemma (in lattice) Sup_fin_set_foldr [code_unfold]:
"Sup_fin (set (x # xs)) = foldr sup xs x"
by (simp add: Sup_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
lemma (in linorder) Min_fin_set_fold:
"Min (set (x # xs)) = fold min xs x"
proof -
interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
by (fact ab_semigroup_idem_mult_min)
show ?thesis
by (simp add: Min_def fold1_set del: set.simps)
qed
lemma (in linorder) Min_fin_set_foldr [code_unfold]:
"Min (set (x # xs)) = foldr min xs x"
by (simp add: Min_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
lemma (in linorder) Max_fin_set_fold:
"Max (set (x # xs)) = fold max xs x"
proof -
interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
by (fact ab_semigroup_idem_mult_max)
show ?thesis
by (simp add: Max_def fold1_set del: set.simps)
qed
lemma (in linorder) Max_fin_set_foldr [code_unfold]:
"Max (set (x # xs)) = foldr max xs x"
by (simp add: Max_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
lemma (in complete_lattice) Inf_set_fold:
"Inf (set xs) = fold inf xs top"
proof -
interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
by (fact comp_fun_idem_inf)
show ?thesis by (simp add: Inf_fold_inf fold_set inf_commute)
qed
lemma (in complete_lattice) Inf_set_foldr [code_unfold]:
"Inf (set xs) = foldr inf xs top"
by (simp add: Inf_set_fold ac_simps foldr_fold fun_eq_iff)
lemma (in complete_lattice) Sup_set_fold:
"Sup (set xs) = fold sup xs bot"
proof -
interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
by (fact comp_fun_idem_sup)
show ?thesis by (simp add: Sup_fold_sup fold_set sup_commute)
qed
lemma (in complete_lattice) Sup_set_foldr [code_unfold]:
"Sup (set xs) = foldr sup xs bot"
by (simp add: Sup_set_fold ac_simps foldr_fold fun_eq_iff)
lemma (in complete_lattice) INFI_set_fold:
"INFI (set xs) f = fold (inf \<circ> f) xs top"
unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..
lemma (in complete_lattice) SUPR_set_fold:
"SUPR (set xs) f = fold (sup \<circ> f) xs bot"
unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map ..
text {* @{text nth_map} *}
definition nth_map :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
"nth_map n f xs = (if n < length xs then
take n xs @ [f (xs ! n)] @ drop (Suc n) xs
else xs)"
lemma nth_map_id:
"n \<ge> length xs \<Longrightarrow> nth_map n f xs = xs"
by (simp add: nth_map_def)
lemma nth_map_unfold:
"n < length xs \<Longrightarrow> nth_map n f xs = take n xs @ [f (xs ! n)] @ drop (Suc n) xs"
by (simp add: nth_map_def)
lemma nth_map_Nil [simp]:
"nth_map n f [] = []"
by (simp add: nth_map_def)
lemma nth_map_zero [simp]:
"nth_map 0 f (x # xs) = f x # xs"
by (simp add: nth_map_def)
lemma nth_map_Suc [simp]:
"nth_map (Suc n) f (x # xs) = x # nth_map n f xs"
by (simp add: nth_map_def)
text {* monad operation *}
definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
"bind xs f = concat (map f xs)"
lemma bind_simps [simp]:
"bind [] f = []"
"bind (x # xs) f = f x @ bind xs f"
by (simp_all add: bind_def)
end