--- a/src/HOL/List.thy Sat Oct 19 16:16:24 2019 +0200
+++ b/src/HOL/List.thy Sat Oct 19 20:41:03 2019 +0200
@@ -2698,6 +2698,42 @@
with xs ys show ?thesis ..
qed
+lemma semilattice_map2:
+ "semilattice (map2 (\<^bold>*))" if "semilattice (\<^bold>*)"
+ for f (infixl "\<^bold>*" 70)
+proof -
+ from that interpret semilattice f .
+ show ?thesis
+ proof
+ show "map2 (\<^bold>*) (map2 (\<^bold>*) xs ys) zs = map2 (\<^bold>*) xs (map2 (\<^bold>*) ys zs)"
+ for xs ys zs :: "'a list"
+ proof (induction "zip xs (zip ys zs)" arbitrary: xs ys zs)
+ case Nil
+ from Nil [symmetric] show ?case
+ by (auto simp add: zip_eq_Nil_iff)
+ next
+ case (Cons xyz xyzs)
+ from Cons.hyps(2) [symmetric] show ?case
+ by (rule zip_eq_ConsE) (erule zip_eq_ConsE,
+ auto intro: Cons.hyps(1) simp add: ac_simps)
+ qed
+ show "map2 (\<^bold>*) xs ys = map2 (\<^bold>*) ys xs"
+ for xs ys :: "'a list"
+ proof (induction "zip xs ys" arbitrary: xs ys)
+ case Nil
+ then show ?case
+ by (auto simp add: zip_eq_Nil_iff dest: sym)
+ next
+ case (Cons xy xys)
+ from Cons.hyps(2) [symmetric] show ?case
+ by (rule zip_eq_ConsE) (auto intro: Cons.hyps(1) simp add: ac_simps)
+ qed
+ show "map2 (\<^bold>*) xs xs = xs"
+ for xs :: "'a list"
+ by (induction xs) simp_all
+ qed
+qed
+
lemma pair_list_eqI:
assumes "map fst xs = map fst ys" and "map snd xs = map snd ys"
shows "xs = ys"
--- a/src/HOL/Parity.thy Sat Oct 19 16:16:24 2019 +0200
+++ b/src/HOL/Parity.thy Sat Oct 19 20:41:03 2019 +0200
@@ -448,6 +448,38 @@
finally show ?thesis .
qed
+lemma numeral_Bit0_div_2:
+ "numeral (num.Bit0 n) div 2 = numeral n"
+proof -
+ have "numeral (num.Bit0 n) = numeral n + numeral n"
+ by (simp only: numeral.simps)
+ also have "\<dots> = numeral n * 2"
+ by (simp add: mult_2_right)
+ finally have "numeral (num.Bit0 n) div 2 = numeral n * 2 div 2"
+ by simp
+ also have "\<dots> = numeral n"
+ by (rule nonzero_mult_div_cancel_right) simp
+ finally show ?thesis .
+qed
+
+lemma numeral_Bit1_div_2:
+ "numeral (num.Bit1 n) div 2 = numeral n"
+proof -
+ have "numeral (num.Bit1 n) = numeral n + numeral n + 1"
+ by (simp only: numeral.simps)
+ also have "\<dots> = numeral n * 2 + 1"
+ by (simp add: mult_2_right)
+ finally have "numeral (num.Bit1 n) div 2 = (numeral n * 2 + 1) div 2"
+ by simp
+ also have "\<dots> = numeral n * 2 div 2 + 1 div 2"
+ using dvd_triv_right by (rule div_plus_div_distrib_dvd_left)
+ also have "\<dots> = numeral n * 2 div 2"
+ by simp
+ also have "\<dots> = numeral n"
+ by (rule nonzero_mult_div_cancel_right) simp
+ finally show ?thesis .
+qed
+
end
class unique_euclidean_ring_with_nat = ring + unique_euclidean_semiring_with_nat
@@ -1061,4 +1093,8 @@
"drop_bit n (Suc 0) = of_bool (n = 0)"
using drop_bit_of_1 [where ?'a = nat] by simp
+lemma push_bit_minus_one:
+ "push_bit n (- 1 :: int) = - (2 ^ n)"
+ by (simp add: push_bit_eq_mult)
+
end