--- a/src/HOL/Parity.thy Tue Mar 20 09:27:40 2018 +0000
+++ b/src/HOL/Parity.thy Wed Mar 21 19:39:23 2018 +0100
@@ -681,29 +681,29 @@
text \<open>The primary purpose of the following operations is
to avoid ad-hoc simplification of concrete expressions @{term "2 ^ n"}\<close>
-definition bit_push :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
- where bit_push_eq_mult: "bit_push n a = a * 2 ^ n"
+definition push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
+ where push_bit_eq_mult: "push_bit n a = a * 2 ^ n"
-definition bit_take :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
- where bit_take_eq_mod: "bit_take n a = a mod of_nat (2 ^ n)"
+definition take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
+ where take_bit_eq_mod: "take_bit n a = a mod of_nat (2 ^ n)"
-definition bit_drop :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
- where bit_drop_eq_div: "bit_drop n a = a div of_nat (2 ^ n)"
+definition drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
+ where drop_bit_eq_div: "drop_bit n a = a div of_nat (2 ^ n)"
lemma bit_ident:
- "bit_push n (bit_drop n a) + bit_take n a = a"
- using div_mult_mod_eq by (simp add: bit_push_eq_mult bit_take_eq_mod bit_drop_eq_div)
+ "push_bit n (drop_bit n a) + take_bit n a = a"
+ using div_mult_mod_eq by (simp add: push_bit_eq_mult take_bit_eq_mod drop_bit_eq_div)
-lemma bit_take_bit_take [simp]:
- "bit_take n (bit_take n a) = bit_take n a"
- by (simp add: bit_take_eq_mod)
+lemma take_bit_take_bit [simp]:
+ "take_bit n (take_bit n a) = take_bit n a"
+ by (simp add: take_bit_eq_mod)
-lemma bit_take_0 [simp]:
- "bit_take 0 a = 0"
- by (simp add: bit_take_eq_mod)
+lemma take_bit_0 [simp]:
+ "take_bit 0 a = 0"
+ by (simp add: take_bit_eq_mod)
-lemma bit_take_Suc [simp]:
- "bit_take (Suc n) a = bit_take n (a div 2) * 2 + of_bool (odd a)"
+lemma take_bit_Suc [simp]:
+ "take_bit (Suc n) a = take_bit n (a div 2) * 2 + of_bool (odd a)"
proof -
have "1 + 2 * (a div 2) mod (2 * 2 ^ n) = (a div 2 * 2 + a mod 2) mod (2 * 2 ^ n)"
if "odd a"
@@ -712,99 +712,99 @@
also have "\<dots> = a mod (2 * 2 ^ n)"
by (simp only: div_mult_mod_eq)
finally show ?thesis
- by (simp add: bit_take_eq_mod algebra_simps mult_mod_right)
+ by (simp add: take_bit_eq_mod algebra_simps mult_mod_right)
qed
-lemma bit_take_of_0 [simp]:
- "bit_take n 0 = 0"
- by (simp add: bit_take_eq_mod)
+lemma take_bit_of_0 [simp]:
+ "take_bit n 0 = 0"
+ by (simp add: take_bit_eq_mod)
-lemma bit_take_plus:
- "bit_take n (bit_take n a + bit_take n b) = bit_take n (a + b)"
- by (simp add: bit_take_eq_mod mod_simps)
+lemma take_bit_plus:
+ "take_bit n (take_bit n a + take_bit n b) = take_bit n (a + b)"
+ by (simp add: take_bit_eq_mod mod_simps)
-lemma bit_take_of_1_eq_0_iff [simp]:
- "bit_take n 1 = 0 \<longleftrightarrow> n = 0"
- by (simp add: bit_take_eq_mod)
+lemma take_bit_of_1_eq_0_iff [simp]:
+ "take_bit n 1 = 0 \<longleftrightarrow> n = 0"
+ by (simp add: take_bit_eq_mod)
-lemma bit_push_eq_0_iff [simp]:
- "bit_push n a = 0 \<longleftrightarrow> a = 0"
- by (simp add: bit_push_eq_mult)
+lemma push_bit_eq_0_iff [simp]:
+ "push_bit n a = 0 \<longleftrightarrow> a = 0"
+ by (simp add: push_bit_eq_mult)
-lemma bit_drop_0 [simp]:
- "bit_drop 0 = id"
- by (simp add: fun_eq_iff bit_drop_eq_div)
+lemma drop_bit_0 [simp]:
+ "drop_bit 0 = id"
+ by (simp add: fun_eq_iff drop_bit_eq_div)
-lemma bit_drop_of_0 [simp]:
- "bit_drop n 0 = 0"
- by (simp add: bit_drop_eq_div)
+lemma drop_bit_of_0 [simp]:
+ "drop_bit n 0 = 0"
+ by (simp add: drop_bit_eq_div)
-lemma bit_drop_Suc [simp]:
- "bit_drop (Suc n) a = bit_drop n (a div 2)"
+lemma drop_bit_Suc [simp]:
+ "drop_bit (Suc n) a = drop_bit n (a div 2)"
proof (cases "even a")
case True
then obtain b where "a = 2 * b" ..
- moreover have "bit_drop (Suc n) (2 * b) = bit_drop n b"
- by (simp add: bit_drop_eq_div)
+ moreover have "drop_bit (Suc n) (2 * b) = drop_bit n b"
+ by (simp add: drop_bit_eq_div)
ultimately show ?thesis
by simp
next
case False
then obtain b where "a = 2 * b + 1" ..
- moreover have "bit_drop (Suc n) (2 * b + 1) = bit_drop n b"
+ moreover have "drop_bit (Suc n) (2 * b + 1) = drop_bit n b"
using div_mult2_eq' [of "1 + b * 2" 2 "2 ^ n"]
- by (auto simp add: bit_drop_eq_div ac_simps)
+ by (auto simp add: drop_bit_eq_div ac_simps)
ultimately show ?thesis
by simp
qed
-lemma bit_drop_half:
- "bit_drop n (a div 2) = bit_drop n a div 2"
+lemma drop_bit_half:
+ "drop_bit n (a div 2) = drop_bit n a div 2"
by (induction n arbitrary: a) simp_all
-lemma bit_drop_of_bool [simp]:
- "bit_drop n (of_bool d) = of_bool (n = 0 \<and> d)"
+lemma drop_bit_of_bool [simp]:
+ "drop_bit n (of_bool d) = of_bool (n = 0 \<and> d)"
by (cases n) simp_all
-lemma even_bit_take_eq [simp]:
- "even (bit_take n a) \<longleftrightarrow> n = 0 \<or> even a"
- by (cases n) (simp_all add: bit_take_eq_mod dvd_mod_iff)
+lemma even_take_bit_eq [simp]:
+ "even (take_bit n a) \<longleftrightarrow> n = 0 \<or> even a"
+ by (cases n) (simp_all add: take_bit_eq_mod dvd_mod_iff)
-lemma bit_push_0_id [simp]:
- "bit_push 0 = id"
- by (simp add: fun_eq_iff bit_push_eq_mult)
+lemma push_bit_0_id [simp]:
+ "push_bit 0 = id"
+ by (simp add: fun_eq_iff push_bit_eq_mult)
-lemma bit_push_of_0 [simp]:
- "bit_push n 0 = 0"
- by (simp add: bit_push_eq_mult)
+lemma push_bit_of_0 [simp]:
+ "push_bit n 0 = 0"
+ by (simp add: push_bit_eq_mult)
-lemma bit_push_of_1:
- "bit_push n 1 = 2 ^ n"
- by (simp add: bit_push_eq_mult)
+lemma push_bit_of_1:
+ "push_bit n 1 = 2 ^ n"
+ by (simp add: push_bit_eq_mult)
-lemma bit_push_Suc [simp]:
- "bit_push (Suc n) a = bit_push n (a * 2)"
- by (simp add: bit_push_eq_mult ac_simps)
+lemma push_bit_Suc [simp]:
+ "push_bit (Suc n) a = push_bit n (a * 2)"
+ by (simp add: push_bit_eq_mult ac_simps)
-lemma bit_push_double:
- "bit_push n (a * 2) = bit_push n a * 2"
- by (simp add: bit_push_eq_mult ac_simps)
+lemma push_bit_double:
+ "push_bit n (a * 2) = push_bit n a * 2"
+ by (simp add: push_bit_eq_mult ac_simps)
-lemma bit_drop_bit_take [simp]:
- "bit_drop n (bit_take n a) = 0"
- by (simp add: bit_drop_eq_div bit_take_eq_mod)
+lemma drop_bit_take_bit [simp]:
+ "drop_bit n (take_bit n a) = 0"
+ by (simp add: drop_bit_eq_div take_bit_eq_mod)
-lemma bit_take_bit_drop_commute:
- "bit_drop m (bit_take n a) = bit_take (n - m) (bit_drop m a)"
+lemma take_bit_drop_bit_commute:
+ "drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)"
for m n :: nat
proof (cases "n \<ge> m")
case True
moreover define q where "q = n - m"
ultimately have "n - m = q" and "n = m + q"
by simp_all
- moreover have "bit_drop m (bit_take (m + q) a) = bit_take q (bit_drop m a)"
+ moreover have "drop_bit m (take_bit (m + q) a) = take_bit q (drop_bit m a)"
using mod_mult2_eq' [of a "2 ^ m" "2 ^ q"]
- by (simp add: bit_drop_eq_div bit_take_eq_mod power_add)
+ by (simp add: drop_bit_eq_div take_bit_eq_mod power_add)
ultimately show ?thesis
by simp
next
@@ -812,9 +812,9 @@
moreover define q where "q = m - n"
ultimately have "m - n = q" and "m = n + q"
by simp_all
- moreover have "bit_drop (n + q) (bit_take n a) = 0"
+ moreover have "drop_bit (n + q) (take_bit n a) = 0"
using div_mult2_eq' [of "a mod 2 ^ n" "2 ^ n" "2 ^ q"]
- by (simp add: bit_drop_eq_div bit_take_eq_mod power_add div_mult2_eq)
+ by (simp add: drop_bit_eq_div take_bit_eq_mod power_add div_mult2_eq)
ultimately show ?thesis
by simp
qed