src/HOL/Library/Code_Bit_Shifts_for_Arithmetic.thy

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+++ b/src/HOL/Library/Code_Bit_Shifts_for_Arithmetic.thy	Thu Jan 16 09:26:56 2025 +0100
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+(*  Title:      HOL/Library/Code_Bit_Shifts_for_Arithmetic.thy
+    Author:     Florian Haftmann, TU Muenchen
+*)
+
+section \<open>Rewrite arithmetic operations to bit-shifts if feasible\<close>
+
+theory Code_Bit_Shifts_for_Arithmetic
+  imports Main
+begin
+
+context semiring_bit_operations
+begin
+
+context
+  includes bit_operations_syntax
+begin
+
+lemma [code_unfold]:
+  \<open>of_bool (odd a) = a AND 1\<close>
+  by (simp add: and_one_eq mod2_eq_if)
+
+lemma [code_unfold]:
+  \<open>even a \<longleftrightarrow> a AND 1 = 0\<close>
+  by (simp add: and_one_eq mod2_eq_if)
+
+lemma [code_unfold]:
+  \<open>2 * a = push_bit 1 a\<close>
+  by (simp add: ac_simps)
+
+lemma [code_unfold]:
+  \<open>a * 2 = push_bit 1 a\<close>
+  by simp
+
+lemma [code_unfold]:
+  \<open>2 ^ n * a = push_bit n a\<close>
+  by (simp add: push_bit_eq_mult ac_simps)
+
+lemma [code_unfold]:
+  \<open>a * 2 ^ n = push_bit n a\<close>
+  by (simp add: push_bit_eq_mult)
+
+lemma [code_unfold]:
+  \<open>a div 2 = drop_bit 1 a\<close>
+  by (simp add: drop_bit_eq_div)
+
+lemma [code_unfold]:
+  \<open>a div 2 ^ n = drop_bit n a\<close>
+  by (simp add: drop_bit_eq_div)
+
+lemma [code_unfold]:
+  \<open>a mod 2 = take_bit 1 a\<close>
+  by (simp add: take_bit_eq_mod)
+
+lemma [code_unfold]:
+  \<open>a mod 2 ^ n  = take_bit n a\<close>
+  by (simp add: take_bit_eq_mod)
+
+end
+
+end
+
+end