--- a/src/HOL/ex/Word_Type.thy Sat Oct 08 14:09:53 2016 +0200
+++ b/src/HOL/ex/Word_Type.thy Sat Oct 08 14:09:55 2016 +0200
@@ -13,23 +13,21 @@
class semiring_bits = semiring_div_parity +
assumes semiring_bits: "(1 + 2 * a) mod of_nat (2 * n) = 1 + 2 * (a mod of_nat n)"
-
-context semiring_bits
begin
-definition bits :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
- where bits_eq_mod: "bits n a = a mod of_nat (2 ^ n)"
+definition bitrunc :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
+ where bitrunc_eq_mod: "bitrunc n a = a mod of_nat (2 ^ n)"
-lemma bits_bits [simp]:
- "bits n (bits n a) = bits n a"
- by (simp add: bits_eq_mod)
+lemma bitrunc_bitrunc [simp]:
+ "bitrunc n (bitrunc n a) = bitrunc n a"
+ by (simp add: bitrunc_eq_mod)
-lemma bits_0 [simp]:
- "bits 0 a = 0"
- by (simp add: bits_eq_mod)
+lemma bitrunc_0 [simp]:
+ "bitrunc 0 a = 0"
+ by (simp add: bitrunc_eq_mod)
-lemma bits_Suc [simp]:
- "bits (Suc n) a = bits n (a div 2) * 2 + a mod 2"
+lemma bitrunc_Suc [simp]:
+ "bitrunc (Suc n) a = bitrunc n (a div 2) * 2 + a mod 2"
proof -
define b and c
where "b = a div 2" and "c = a mod 2"
@@ -37,32 +35,32 @@
and "c = 0 \<or> c = 1"
by (simp_all add: mod_div_equality parity)
from \<open>c = 0 \<or> c = 1\<close>
- have "bits (Suc n) (b * 2 + c) = bits n b * 2 + c"
+ have "bitrunc (Suc n) (b * 2 + c) = bitrunc n b * 2 + c"
proof
assume "c = 0"
moreover have "(2 * b) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n)"
by (simp add: mod_mult_mult1)
ultimately show ?thesis
- by (simp add: bits_eq_mod ac_simps)
+ by (simp add: bitrunc_eq_mod ac_simps)
next
assume "c = 1"
with semiring_bits [of b "2 ^ n"] show ?thesis
- by (simp add: bits_eq_mod ac_simps)
+ by (simp add: bitrunc_eq_mod ac_simps)
qed
with a show ?thesis
by (simp add: b_def c_def)
qed
-lemma bits_of_0 [simp]:
- "bits n 0 = 0"
- by (simp add: bits_eq_mod)
+lemma bitrunc_of_0 [simp]:
+ "bitrunc n 0 = 0"
+ by (simp add: bitrunc_eq_mod)
-lemma bits_plus:
- "bits n (bits n a + bits n b) = bits n (a + b)"
- by (simp add: bits_eq_mod mod_add_eq [symmetric])
+lemma bitrunc_plus:
+ "bitrunc n (bitrunc n a + bitrunc n b) = bitrunc n (a + b)"
+ by (simp add: bitrunc_eq_mod mod_add_eq [symmetric])
-lemma bits_of_1_eq_0_iff [simp]:
- "bits n 1 = 0 \<longleftrightarrow> n = 0"
+lemma bitrunc_of_1_eq_0_iff [simp]:
+ "bitrunc n 1 = 0 \<longleftrightarrow> n = 0"
by (induct n) simp_all
end
@@ -73,32 +71,32 @@
instance int :: semiring_bits
by standard (simp add: pos_zmod_mult_2)
-lemma bits_uminus:
+lemma bitrunc_uminus:
fixes k :: int
- shows "bits n (- (bits n k)) = bits n (- k)"
- by (simp add: bits_eq_mod mod_minus_eq [symmetric])
+ shows "bitrunc n (- (bitrunc n k)) = bitrunc n (- k)"
+ by (simp add: bitrunc_eq_mod mod_minus_eq [symmetric])
-lemma bits_minus:
+lemma bitrunc_minus:
fixes k l :: int
- shows "bits n (bits n k - bits n l) = bits n (k - l)"
- by (simp add: bits_eq_mod mod_diff_eq [symmetric])
+ shows "bitrunc n (bitrunc n k - bitrunc n l) = bitrunc n (k - l)"
+ by (simp add: bitrunc_eq_mod mod_diff_eq [symmetric])
-lemma bits_nonnegative [simp]:
+lemma bitrunc_nonnegative [simp]:
fixes k :: int
- shows "bits n k \<ge> 0"
- by (simp add: bits_eq_mod)
+ shows "bitrunc n k \<ge> 0"
+ by (simp add: bitrunc_eq_mod)
-definition signed_bits :: "nat \<Rightarrow> int \<Rightarrow> int"
- where signed_bits_eq_bits:
- "signed_bits n k = bits (Suc n) (k + 2 ^ n) - 2 ^ n"
+definition signed_bitrunc :: "nat \<Rightarrow> int \<Rightarrow> int"
+ where signed_bitrunc_eq_bitrunc:
+ "signed_bitrunc n k = bitrunc (Suc n) (k + 2 ^ n) - 2 ^ n"
-lemma signed_bits_eq_bits':
+lemma signed_bitrunc_eq_bitrunc':
assumes "n > 0"
- shows "signed_bits (n - Suc 0) k = bits n (k + 2 ^ (n - 1)) - 2 ^ (n - 1)"
- using assms by (simp add: signed_bits_eq_bits)
+ shows "signed_bitrunc (n - Suc 0) k = bitrunc n (k + 2 ^ (n - 1)) - 2 ^ (n - 1)"
+ using assms by (simp add: signed_bitrunc_eq_bitrunc)
-lemma signed_bits_0 [simp]:
- "signed_bits 0 k = - (k mod 2)"
+lemma signed_bitrunc_0 [simp]:
+ "signed_bitrunc 0 k = - (k mod 2)"
proof (cases "even k")
case True
then have "odd (k + 1)"
@@ -106,54 +104,54 @@
then have "(k + 1) mod 2 = 1"
by (simp add: even_iff_mod_2_eq_zero)
with True show ?thesis
- by (simp add: signed_bits_eq_bits)
+ by (simp add: signed_bitrunc_eq_bitrunc)
next
case False
then show ?thesis
- by (simp add: signed_bits_eq_bits odd_iff_mod_2_eq_one)
+ by (simp add: signed_bitrunc_eq_bitrunc odd_iff_mod_2_eq_one)
qed
-lemma signed_bits_Suc [simp]:
- "signed_bits (Suc n) k = signed_bits n (k div 2) * 2 + k mod 2"
- using zero_not_eq_two by (simp add: signed_bits_eq_bits algebra_simps)
+lemma signed_bitrunc_Suc [simp]:
+ "signed_bitrunc (Suc n) k = signed_bitrunc n (k div 2) * 2 + k mod 2"
+ using zero_not_eq_two by (simp add: signed_bitrunc_eq_bitrunc algebra_simps)
-lemma signed_bits_of_0 [simp]:
- "signed_bits n 0 = 0"
- by (simp add: signed_bits_eq_bits bits_eq_mod)
+lemma signed_bitrunc_of_0 [simp]:
+ "signed_bitrunc n 0 = 0"
+ by (simp add: signed_bitrunc_eq_bitrunc bitrunc_eq_mod)
-lemma signed_bits_of_minus_1 [simp]:
- "signed_bits n (- 1) = - 1"
+lemma signed_bitrunc_of_minus_1 [simp]:
+ "signed_bitrunc n (- 1) = - 1"
by (induct n) simp_all
-lemma signed_bits_eq_iff_bits_eq:
+lemma signed_bitrunc_eq_iff_bitrunc_eq:
assumes "n > 0"
- shows "signed_bits (n - Suc 0) k = signed_bits (n - Suc 0) l \<longleftrightarrow> bits n k = bits n l" (is "?P \<longleftrightarrow> ?Q")
+ shows "signed_bitrunc (n - Suc 0) k = signed_bitrunc (n - Suc 0) l \<longleftrightarrow> bitrunc n k = bitrunc n l" (is "?P \<longleftrightarrow> ?Q")
proof -
from assms obtain m where m: "n = Suc m"
by (cases n) auto
show ?thesis
proof
assume ?Q
- have "bits (Suc m) (k + 2 ^ m) =
- bits (Suc m) (bits (Suc m) k + bits (Suc m) (2 ^ m))"
- by (simp only: bits_plus)
+ have "bitrunc (Suc m) (k + 2 ^ m) =
+ bitrunc (Suc m) (bitrunc (Suc m) k + bitrunc (Suc m) (2 ^ m))"
+ by (simp only: bitrunc_plus)
also have "\<dots> =
- bits (Suc m) (bits (Suc m) l + bits (Suc m) (2 ^ m))"
+ bitrunc (Suc m) (bitrunc (Suc m) l + bitrunc (Suc m) (2 ^ m))"
by (simp only: \<open>?Q\<close> m [symmetric])
- also have "\<dots> = bits (Suc m) (l + 2 ^ m)"
- by (simp only: bits_plus)
+ also have "\<dots> = bitrunc (Suc m) (l + 2 ^ m)"
+ by (simp only: bitrunc_plus)
finally show ?P
- by (simp only: signed_bits_eq_bits m) simp
+ by (simp only: signed_bitrunc_eq_bitrunc m) simp
next
assume ?P
with assms have "(k + 2 ^ (n - Suc 0)) mod 2 ^ n = (l + 2 ^ (n - Suc 0)) mod 2 ^ n"
- by (simp add: signed_bits_eq_bits' bits_eq_mod)
+ by (simp add: signed_bitrunc_eq_bitrunc' bitrunc_eq_mod)
then have "(i + (k + 2 ^ (n - Suc 0))) mod 2 ^ n = (i + (l + 2 ^ (n - Suc 0))) mod 2 ^ n" for i
by (metis mod_add_eq)
then have "k mod 2 ^ n = l mod 2 ^ n"
by (metis add_diff_cancel_right' uminus_add_conv_diff)
then show ?Q
- by (simp add: bits_eq_mod)
+ by (simp add: bitrunc_eq_mod)
qed
qed
@@ -162,7 +160,7 @@
subsubsection \<open>Basic properties\<close>
-quotient_type (overloaded) 'a word = int / "\<lambda>k l. bits LENGTH('a) k = bits LENGTH('a::len0) l"
+quotient_type (overloaded) 'a word = int / "\<lambda>k l. bitrunc LENGTH('a) k = bitrunc LENGTH('a::len0) l"
by (auto intro!: equivpI reflpI sympI transpI)
instantiation word :: (len0) "{semiring_numeral, comm_semiring_0, comm_ring}"
@@ -178,19 +176,19 @@
lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is plus
- by (subst bits_plus [symmetric]) (simp add: bits_plus)
+ by (subst bitrunc_plus [symmetric]) (simp add: bitrunc_plus)
lift_definition uminus_word :: "'a word \<Rightarrow> 'a word"
is uminus
- by (subst bits_uminus [symmetric]) (simp add: bits_uminus)
+ by (subst bitrunc_uminus [symmetric]) (simp add: bitrunc_uminus)
lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is minus
- by (subst bits_minus [symmetric]) (simp add: bits_minus)
+ by (subst bitrunc_minus [symmetric]) (simp add: bitrunc_minus)
lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is times
- by (auto simp add: bits_eq_mod intro: mod_mult_cong)
+ by (auto simp add: bitrunc_eq_mod intro: mod_mult_cong)
instance
by standard (transfer; simp add: algebra_simps)+
@@ -223,7 +221,7 @@
begin
lift_definition unsigned :: "'b::len0 word \<Rightarrow> 'a"
- is "of_nat \<circ> nat \<circ> bits LENGTH('b)"
+ is "of_nat \<circ> nat \<circ> bitrunc LENGTH('b)"
by simp
lemma unsigned_0 [simp]:
@@ -245,8 +243,8 @@
begin
lift_definition signed :: "'b::len word \<Rightarrow> 'a"
- is "of_int \<circ> signed_bits (LENGTH('b) - 1)"
- by (simp add: signed_bits_eq_iff_bits_eq [symmetric])
+ is "of_int \<circ> signed_bitrunc (LENGTH('b) - 1)"
+ by (simp add: signed_bitrunc_eq_iff_bitrunc_eq [symmetric])
lemma signed_0 [simp]:
"signed 0 = 0"
@@ -255,8 +253,8 @@
end
lemma unsigned_of_nat [simp]:
- "unsigned (of_nat n :: 'a word) = bits LENGTH('a::len) n"
- by transfer (simp add: nat_eq_iff bits_eq_mod zmod_int)
+ "unsigned (of_nat n :: 'a word) = bitrunc LENGTH('a::len) n"
+ by transfer (simp add: nat_eq_iff bitrunc_eq_mod zmod_int)
lemma of_nat_unsigned [simp]:
"of_nat (unsigned a) = a"
@@ -271,17 +269,17 @@
lemma word_eq_iff_signed:
"a = b \<longleftrightarrow> signed a = signed b"
- by safe (transfer; auto simp add: signed_bits_eq_iff_bits_eq)
+ by safe (transfer; auto simp add: signed_bitrunc_eq_iff_bitrunc_eq)
end
lemma signed_of_int [simp]:
- "signed (of_int k :: 'a word) = signed_bits (LENGTH('a::len) - 1) k"
+ "signed (of_int k :: 'a word) = signed_bitrunc (LENGTH('a::len) - 1) k"
by transfer simp
lemma of_int_signed [simp]:
"of_int (signed a) = a"
- by transfer (simp add: signed_bits_eq_bits bits_eq_mod zdiff_zmod_left)
+ by transfer (simp add: signed_bitrunc_eq_bitrunc bitrunc_eq_mod zdiff_zmod_left)
subsubsection \<open>Properties\<close>
@@ -293,11 +291,11 @@
begin
lift_definition divide_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
- is "\<lambda>a b. bits LENGTH('a) a div bits LENGTH('a) b"
+ is "\<lambda>a b. bitrunc LENGTH('a) a div bitrunc LENGTH('a) b"
by simp
lift_definition modulo_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
- is "\<lambda>a b. bits LENGTH('a) a mod bits LENGTH('a) b"
+ is "\<lambda>a b. bitrunc LENGTH('a) a mod bitrunc LENGTH('a) b"
by simp
instance ..
@@ -311,11 +309,11 @@
begin
lift_definition less_eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
- is "\<lambda>a b. bits LENGTH('a) a \<le> bits LENGTH('a) b"
+ is "\<lambda>a b. bitrunc LENGTH('a) a \<le> bitrunc LENGTH('a) b"
by simp
lift_definition less_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
- is "\<lambda>a b. bits LENGTH('a) a < bits LENGTH('a) b"
+ is "\<lambda>a b. bitrunc LENGTH('a) a < bitrunc LENGTH('a) b"
by simp
instance
@@ -332,7 +330,7 @@
lemma word_less_iff_unsigned:
"a < b \<longleftrightarrow> unsigned a < unsigned b"
- by (transfer fixing: less) (auto dest: preorder_class.le_less_trans [OF bits_nonnegative])
+ by (transfer fixing: less) (auto dest: preorder_class.le_less_trans [OF bitrunc_nonnegative])
end