--- a/src/HOL/List.thy Sun Aug 31 09:10:41 2014 +0200
+++ b/src/HOL/List.thy Sun Aug 31 09:10:42 2014 +0200
@@ -105,9 +105,6 @@
"concat [] = []" |
"concat (x # xs) = x @ concat xs"
-definition (in monoid_add) listsum :: "'a list \<Rightarrow> 'a" where
-"listsum xs = foldr plus xs 0"
-
primrec drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
drop_Nil: "drop n [] = []" |
drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
@@ -313,8 +310,7 @@
@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
-@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
-@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
+@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}
\end{tabular}}
\caption{Characteristic examples}
\label{fig:Characteristic}
@@ -3490,149 +3486,6 @@
auto simp add: injD[OF assms])
-subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
-
-lemma (in monoid_add) listsum_simps [simp]:
- "listsum [] = 0"
- "listsum (x # xs) = x + listsum xs"
- by (simp_all add: listsum_def)
-
-lemma (in monoid_add) listsum_append [simp]:
- "listsum (xs @ ys) = listsum xs + listsum ys"
- by (induct xs) (simp_all add: add.assoc)
-
-lemma (in comm_monoid_add) listsum_rev [simp]:
- "listsum (rev xs) = listsum xs"
- by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff ac_simps)
-
-lemma (in monoid_add) fold_plus_listsum_rev:
- "fold plus xs = plus (listsum (rev xs))"
-proof
- fix x
- have "fold plus xs x = fold plus xs (x + 0)" by simp
- also have "\<dots> = fold plus (x # xs) 0" by simp
- also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
- also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
- also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
- finally show "fold plus xs x = listsum (rev xs) + x" by simp
-qed
-
-text{* Some syntactic sugar for summing a function over a list: *}
-
-syntax
- "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<-_. _)" [0, 51, 10] 10)
-syntax (xsymbols)
- "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
-syntax (HTML output)
- "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
-
-translations -- {* Beware of argument permutation! *}
- "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
- "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
-
-lemma (in comm_monoid_add) listsum_map_remove1:
- "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
- by (induct xs) (auto simp add: ac_simps)
-
-lemma (in monoid_add) size_list_conv_listsum:
- "size_list f xs = listsum (map f xs) + size xs"
- by (induct xs) auto
-
-lemma (in monoid_add) length_concat:
- "length (concat xss) = listsum (map length xss)"
- by (induct xss) simp_all
-
-lemma (in monoid_add) length_product_lists:
- "length (product_lists xss) = foldr op * (map length xss) 1"
-proof (induct xss)
- case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)
-qed simp
-
-lemma (in monoid_add) listsum_map_filter:
- assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
- shows "listsum (map f (filter P xs)) = listsum (map f xs)"
- using assms by (induct xs) auto
-
-lemma (in comm_monoid_add) distinct_listsum_conv_Setsum:
- "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
- by (induct xs) simp_all
-
-lemma listsum_eq_0_nat_iff_nat [simp]:
- "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
- by (induct ns) simp_all
-
-lemma member_le_listsum_nat:
- "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
- by (induct ns) auto
-
-lemma elem_le_listsum_nat:
- "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
- by (rule member_le_listsum_nat) simp
-
-lemma listsum_update_nat:
- "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
-apply(induct ns arbitrary:k)
- apply (auto split:nat.split)
-apply(drule elem_le_listsum_nat)
-apply arith
-done
-
-lemma (in monoid_add) listsum_triv:
- "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
- by (induct xs) (simp_all add: distrib_right)
-
-lemma (in monoid_add) listsum_0 [simp]:
- "(\<Sum>x\<leftarrow>xs. 0) = 0"
- by (induct xs) (simp_all add: distrib_right)
-
-text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
-lemma (in ab_group_add) uminus_listsum_map:
- "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
- by (induct xs) simp_all
-
-lemma (in comm_monoid_add) listsum_addf:
- "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
- by (induct xs) (simp_all add: algebra_simps)
-
-lemma (in ab_group_add) listsum_subtractf:
- "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
- by (induct xs) (simp_all add: algebra_simps)
-
-lemma (in semiring_0) listsum_const_mult:
- "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
- by (induct xs) (simp_all add: algebra_simps)
-
-lemma (in semiring_0) listsum_mult_const:
- "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
- by (induct xs) (simp_all add: algebra_simps)
-
-lemma (in ordered_ab_group_add_abs) listsum_abs:
- "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
- by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
-
-lemma listsum_mono:
- fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
- shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"
- by (induct xs) (simp, simp add: add_mono)
-
-lemma (in monoid_add) listsum_distinct_conv_setsum_set:
- "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
- by (induct xs) simp_all
-
-lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
- "listsum (map f [m..<n]) = setsum f (set [m..<n])"
- by (simp add: listsum_distinct_conv_setsum_set)
-
-lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
- "listsum (map f [k..l]) = setsum f (set [k..l])"
- by (simp add: listsum_distinct_conv_setsum_set)
-
-text {* General equivalence between @{const listsum} and @{const setsum} *}
-lemma (in monoid_add) listsum_setsum_nth:
- "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
- using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
-
-
subsubsection {* @{const insert} *}
lemma in_set_insert [simp]:
@@ -4314,9 +4167,6 @@
lemma n_lists_Nil [simp]: "List.n_lists n [] = (if n = 0 then [[]] else [])"
by (induct n) simp_all
-lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n"
- by (induct n) (auto simp add: length_concat o_def listsum_triv)
-
lemma length_n_lists_elem: "ys \<in> set (List.n_lists n xs) \<Longrightarrow> length ys = n"
by (induct n arbitrary: ys) auto
@@ -4335,28 +4185,6 @@
qed
qed
-lemma distinct_n_lists:
- assumes "distinct xs"
- shows "distinct (List.n_lists n xs)"
-proof (rule card_distinct)
- from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
- have "card (set (List.n_lists n xs)) = card (set xs) ^ n"
- proof (induct n)
- case 0 then show ?case by simp
- next
- case (Suc n)
- moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs)
- = (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
- by (rule card_UN_disjoint) auto
- moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
- by (rule card_image) (simp add: inj_on_def)
- ultimately show ?case by auto
- qed
- also have "\<dots> = length xs ^ n" by (simp add: card_length)
- finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)"
- by (simp add: length_n_lists)
-qed
-
subsubsection {* @{const splice} *}
@@ -6264,10 +6092,6 @@
"(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
by auto
-lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
- "setsum f (set [m..<n]) = listsum (map f [m..<n])"
- by (simp add: interv_listsum_conv_setsum_set_nat)
-
text{* Bounded @{text LEAST} operator: *}
definition "Bleast S P = (LEAST x. x \<in> S \<and> P x)"
@@ -6315,10 +6139,6 @@
lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
-lemma setsum_set_upto_conv_listsum_int [code_unfold]:
- "setsum f (set [i..j::int]) = listsum (map f [i..j])"
- by (simp add: interv_listsum_conv_setsum_set_int)
-
subsubsection {* Optimizing by rewriting *}
@@ -6629,10 +6449,6 @@
"Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
by (simp_all add: Pow_insert Let_def)
-lemma setsum_code [code]:
- "setsum f (set xs) = listsum (map f (remdups xs))"
-by (simp add: listsum_distinct_conv_setsum_set)
-
definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
"map_project f A = {b. \<exists> a \<in> A. f a = Some b}"
@@ -6883,13 +6699,6 @@
apply (erule_tac xs=x in list_all2_induct, simp, simp add: rel_fun_def)
done
-lemma listsum_transfer[transfer_rule]:
- assumes [transfer_rule]: "A 0 0"
- assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
- shows "(list_all2 A ===> A) listsum listsum"
- unfolding listsum_def[abs_def]
- by transfer_prover
-
lemma rtrancl_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_unique A" "bi_total A"
shows "(rel_set (rel_prod A A) ===> rel_set (rel_prod A A)) rtrancl rtrancl"