
diff -r 3ed399935d7c -r 38298c04c12e src/HOL/Parity.thy
--- 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




