1.1 --- a/src/HOL/Library/More_List.thy Mon Dec 26 22:17:10 2011 +0100
1.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,312 +0,0 @@
1.4 -(* Author: Florian Haftmann, TU Muenchen *)
1.5 -
1.6 -header {* Operations on lists beyond the standard List theory *}
1.7 -
1.8 -theory More_List
1.9 -imports Main Multiset
1.10 -begin
1.11 -
1.12 -hide_const (open) Finite_Set.fold
1.13 -
1.14 -text {* Repairing code generator setup *}
1.15 -
1.16 -declare (in lattice) Inf_fin_set_fold [code_unfold del]
1.17 -declare (in lattice) Sup_fin_set_fold [code_unfold del]
1.18 -declare (in linorder) Min_fin_set_fold [code_unfold del]
1.19 -declare (in linorder) Max_fin_set_fold [code_unfold del]
1.20 -declare (in complete_lattice) Inf_set_fold [code_unfold del]
1.21 -declare (in complete_lattice) Sup_set_fold [code_unfold del]
1.22 -
1.23 -
1.24 -text {* Fold combinator with canonical argument order *}
1.25 -
1.26 -primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
1.27 - "fold f [] = id"
1.28 - | "fold f (x # xs) = fold f xs \<circ> f x"
1.29 -
1.30 -lemma foldl_fold:
1.31 - "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
1.32 - by (induct xs arbitrary: s) simp_all
1.33 -
1.34 -lemma foldr_fold_rev:
1.35 - "foldr f xs = fold f (rev xs)"
1.36 - by (simp add: foldr_foldl foldl_fold fun_eq_iff)
1.37 -
1.38 -lemma fold_rev_conv [code_unfold]:
1.39 - "fold f (rev xs) = foldr f xs"
1.40 - by (simp add: foldr_fold_rev)
1.41 -
1.42 -lemma fold_cong [fundef_cong]:
1.43 - "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
1.44 - \<Longrightarrow> fold f xs a = fold g ys b"
1.45 - by (induct ys arbitrary: a b xs) simp_all
1.46 -
1.47 -lemma fold_id:
1.48 - assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
1.49 - shows "fold f xs = id"
1.50 - using assms by (induct xs) simp_all
1.51 -
1.52 -lemma fold_commute:
1.53 - assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
1.54 - shows "h \<circ> fold g xs = fold f xs \<circ> h"
1.55 - using assms by (induct xs) (simp_all add: fun_eq_iff)
1.56 -
1.57 -lemma fold_commute_apply:
1.58 - assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
1.59 - shows "h (fold g xs s) = fold f xs (h s)"
1.60 -proof -
1.61 - from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
1.62 - then show ?thesis by (simp add: fun_eq_iff)
1.63 -qed
1.64 -
1.65 -lemma fold_invariant:
1.66 - assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
1.67 - and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
1.68 - shows "P (fold f xs s)"
1.69 - using assms by (induct xs arbitrary: s) simp_all
1.70 -
1.71 -lemma fold_weak_invariant:
1.72 - assumes "P s"
1.73 - and "\<And>s x. x \<in> set xs \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
1.74 - shows "P (fold f xs s)"
1.75 - using assms by (induct xs arbitrary: s) simp_all
1.76 -
1.77 -lemma fold_append [simp]:
1.78 - "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
1.79 - by (induct xs) simp_all
1.80 -
1.81 -lemma fold_map [code_unfold]:
1.82 - "fold g (map f xs) = fold (g o f) xs"
1.83 - by (induct xs) simp_all
1.84 -
1.85 -lemma fold_remove1_split:
1.86 - 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"
1.87 - and x: "x \<in> set xs"
1.88 - shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
1.89 - using assms by (induct xs) (auto simp add: o_assoc [symmetric])
1.90 -
1.91 -lemma fold_multiset_equiv:
1.92 - 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"
1.93 - and equiv: "multiset_of xs = multiset_of ys"
1.94 - shows "fold f xs = fold f ys"
1.95 -using f equiv [symmetric] proof (induct xs arbitrary: ys)
1.96 - case Nil then show ?case by simp
1.97 -next
1.98 - case (Cons x xs)
1.99 - then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
1.100 - have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
1.101 - by (rule Cons.prems(1)) (simp_all add: *)
1.102 - moreover from * have "x \<in> set ys" by simp
1.103 - ultimately have "fold f ys = fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
1.104 - moreover from Cons.prems have "fold f xs = fold f (remove1 x ys)" by (auto intro: Cons.hyps)
1.105 - ultimately show ?case by simp
1.106 -qed
1.107 -
1.108 -lemma fold_rev:
1.109 - assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
1.110 - shows "fold f (rev xs) = fold f xs"
1.111 - by (rule fold_multiset_equiv, rule assms) (simp_all add: in_multiset_in_set)
1.112 -
1.113 -lemma foldr_fold:
1.114 - assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
1.115 - shows "foldr f xs = fold f xs"
1.116 - using assms unfolding foldr_fold_rev by (rule fold_rev)
1.117 -
1.118 -lemma fold_Cons_rev:
1.119 - "fold Cons xs = append (rev xs)"
1.120 - by (induct xs) simp_all
1.121 -
1.122 -lemma rev_conv_fold [code]:
1.123 - "rev xs = fold Cons xs []"
1.124 - by (simp add: fold_Cons_rev)
1.125 -
1.126 -lemma fold_append_concat_rev:
1.127 - "fold append xss = append (concat (rev xss))"
1.128 - by (induct xss) simp_all
1.129 -
1.130 -lemma concat_conv_foldr [code]:
1.131 - "concat xss = foldr append xss []"
1.132 - by (simp add: fold_append_concat_rev foldr_fold_rev)
1.133 -
1.134 -lemma fold_plus_listsum_rev:
1.135 - "fold plus xs = plus (listsum (rev xs))"
1.136 - by (induct xs) (simp_all add: add.assoc)
1.137 -
1.138 -lemma (in monoid_add) listsum_conv_fold [code]:
1.139 - "listsum xs = fold (\<lambda>x y. y + x) xs 0"
1.140 - by (auto simp add: listsum_foldl foldl_fold fun_eq_iff)
1.141 -
1.142 -lemma (in linorder) sort_key_conv_fold:
1.143 - assumes "inj_on f (set xs)"
1.144 - shows "sort_key f xs = fold (insort_key f) xs []"
1.145 -proof -
1.146 - have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
1.147 - proof (rule fold_rev, rule ext)
1.148 - fix zs
1.149 - fix x y
1.150 - assume "x \<in> set xs" "y \<in> set xs"
1.151 - with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
1.152 - have **: "x = y \<longleftrightarrow> y = x" by auto
1.153 - show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
1.154 - by (induct zs) (auto intro: * simp add: **)
1.155 - qed
1.156 - then show ?thesis by (simp add: sort_key_def foldr_fold_rev)
1.157 -qed
1.158 -
1.159 -lemma (in linorder) sort_conv_fold:
1.160 - "sort xs = fold insort xs []"
1.161 - by (rule sort_key_conv_fold) simp
1.162 -
1.163 -
1.164 -text {* @{const Finite_Set.fold} and @{const fold} *}
1.165 -
1.166 -lemma (in comp_fun_commute) fold_set_remdups:
1.167 - "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
1.168 - by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
1.169 -
1.170 -lemma (in comp_fun_idem) fold_set:
1.171 - "Finite_Set.fold f y (set xs) = fold f xs y"
1.172 - by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
1.173 -
1.174 -lemma (in ab_semigroup_idem_mult) fold1_set:
1.175 - assumes "xs \<noteq> []"
1.176 - shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
1.177 -proof -
1.178 - interpret comp_fun_idem times by (fact comp_fun_idem)
1.179 - from assms obtain y ys where xs: "xs = y # ys"
1.180 - by (cases xs) auto
1.181 - show ?thesis
1.182 - proof (cases "set ys = {}")
1.183 - case True with xs show ?thesis by simp
1.184 - next
1.185 - case False
1.186 - then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
1.187 - by (simp only: finite_set fold1_eq_fold_idem)
1.188 - with xs show ?thesis by (simp add: fold_set mult_commute)
1.189 - qed
1.190 -qed
1.191 -
1.192 -lemma (in lattice) Inf_fin_set_fold:
1.193 - "Inf_fin (set (x # xs)) = fold inf xs x"
1.194 -proof -
1.195 - interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.196 - by (fact ab_semigroup_idem_mult_inf)
1.197 - show ?thesis
1.198 - by (simp add: Inf_fin_def fold1_set del: set.simps)
1.199 -qed
1.200 -
1.201 -lemma (in lattice) Inf_fin_set_foldr [code_unfold]:
1.202 - "Inf_fin (set (x # xs)) = foldr inf xs x"
1.203 - by (simp add: Inf_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
1.204 -
1.205 -lemma (in lattice) Sup_fin_set_fold:
1.206 - "Sup_fin (set (x # xs)) = fold sup xs x"
1.207 -proof -
1.208 - interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.209 - by (fact ab_semigroup_idem_mult_sup)
1.210 - show ?thesis
1.211 - by (simp add: Sup_fin_def fold1_set del: set.simps)
1.212 -qed
1.213 -
1.214 -lemma (in lattice) Sup_fin_set_foldr [code_unfold]:
1.215 - "Sup_fin (set (x # xs)) = foldr sup xs x"
1.216 - by (simp add: Sup_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
1.217 -
1.218 -lemma (in linorder) Min_fin_set_fold:
1.219 - "Min (set (x # xs)) = fold min xs x"
1.220 -proof -
1.221 - interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.222 - by (fact ab_semigroup_idem_mult_min)
1.223 - show ?thesis
1.224 - by (simp add: Min_def fold1_set del: set.simps)
1.225 -qed
1.226 -
1.227 -lemma (in linorder) Min_fin_set_foldr [code_unfold]:
1.228 - "Min (set (x # xs)) = foldr min xs x"
1.229 - by (simp add: Min_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
1.230 -
1.231 -lemma (in linorder) Max_fin_set_fold:
1.232 - "Max (set (x # xs)) = fold max xs x"
1.233 -proof -
1.234 - interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.235 - by (fact ab_semigroup_idem_mult_max)
1.236 - show ?thesis
1.237 - by (simp add: Max_def fold1_set del: set.simps)
1.238 -qed
1.239 -
1.240 -lemma (in linorder) Max_fin_set_foldr [code_unfold]:
1.241 - "Max (set (x # xs)) = foldr max xs x"
1.242 - by (simp add: Max_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
1.243 -
1.244 -lemma (in complete_lattice) Inf_set_fold:
1.245 - "Inf (set xs) = fold inf xs top"
1.246 -proof -
1.247 - interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.248 - by (fact comp_fun_idem_inf)
1.249 - show ?thesis by (simp add: Inf_fold_inf fold_set inf_commute)
1.250 -qed
1.251 -
1.252 -lemma (in complete_lattice) Inf_set_foldr [code_unfold]:
1.253 - "Inf (set xs) = foldr inf xs top"
1.254 - by (simp add: Inf_set_fold ac_simps foldr_fold fun_eq_iff)
1.255 -
1.256 -lemma (in complete_lattice) Sup_set_fold:
1.257 - "Sup (set xs) = fold sup xs bot"
1.258 -proof -
1.259 - interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.260 - by (fact comp_fun_idem_sup)
1.261 - show ?thesis by (simp add: Sup_fold_sup fold_set sup_commute)
1.262 -qed
1.263 -
1.264 -lemma (in complete_lattice) Sup_set_foldr [code_unfold]:
1.265 - "Sup (set xs) = foldr sup xs bot"
1.266 - by (simp add: Sup_set_fold ac_simps foldr_fold fun_eq_iff)
1.267 -
1.268 -lemma (in complete_lattice) INFI_set_fold:
1.269 - "INFI (set xs) f = fold (inf \<circ> f) xs top"
1.270 - unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..
1.271 -
1.272 -lemma (in complete_lattice) SUPR_set_fold:
1.273 - "SUPR (set xs) f = fold (sup \<circ> f) xs bot"
1.274 - unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map ..
1.275 -
1.276 -
1.277 -text {* @{text nth_map} *}
1.278 -
1.279 -definition nth_map :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
1.280 - "nth_map n f xs = (if n < length xs then
1.281 - take n xs @ [f (xs ! n)] @ drop (Suc n) xs
1.282 - else xs)"
1.283 -
1.284 -lemma nth_map_id:
1.285 - "n \<ge> length xs \<Longrightarrow> nth_map n f xs = xs"
1.286 - by (simp add: nth_map_def)
1.287 -
1.288 -lemma nth_map_unfold:
1.289 - "n < length xs \<Longrightarrow> nth_map n f xs = take n xs @ [f (xs ! n)] @ drop (Suc n) xs"
1.290 - by (simp add: nth_map_def)
1.291 -
1.292 -lemma nth_map_Nil [simp]:
1.293 - "nth_map n f [] = []"
1.294 - by (simp add: nth_map_def)
1.295 -
1.296 -lemma nth_map_zero [simp]:
1.297 - "nth_map 0 f (x # xs) = f x # xs"
1.298 - by (simp add: nth_map_def)
1.299 -
1.300 -lemma nth_map_Suc [simp]:
1.301 - "nth_map (Suc n) f (x # xs) = x # nth_map n f xs"
1.302 - by (simp add: nth_map_def)
1.303 -
1.304 -
1.305 -text {* monad operation *}
1.306 -
1.307 -definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
1.308 - "bind xs f = concat (map f xs)"
1.309 -
1.310 -lemma bind_simps [simp]:
1.311 - "bind [] f = []"
1.312 - "bind (x # xs) f = f x @ bind xs f"
1.313 - by (simp_all add: bind_def)
1.314 -
1.315 -end