--- a/NEWS Mon Dec 23 15:30:31 2013 +0100
+++ b/NEWS Mon Dec 23 16:59:56 2013 +0100
@@ -28,8 +28,11 @@
*** HOL ***
+* Word library: bit representations prefer type bool over type bit.
+INCOMPATIBILITY.
+
* Theorem disambiguation Inf_le_Sup (on finite sets) ~> Inf_fin_le_Sup_fin.
-INCOMPATBILITY.
+INCOMPATIBILITY.
* Code generations are provided for make, fields, extend and truncate
operations on records.
--- a/src/HOL/Word/Bit_Comparison.thy Mon Dec 23 15:30:31 2013 +0100
+++ b/src/HOL/Word/Bit_Comparison.thy Mon Dec 23 16:59:56 2013 +0100
@@ -20,10 +20,10 @@
proof (cases y rule: bin_exhaust)
case (1 bin' bit')
from 3 have "0 \<le> bin"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit) (simp_all add: Bit_def)
then have "0 \<le> bin AND bin'" by (rule 3)
with 1 show ?thesis
- by simp (simp add: Bit_def bitval_def split add: bit.split)
+ by simp (simp add: Bit_def)
qed
next
case 2
@@ -41,12 +41,12 @@
proof (cases y rule: bin_exhaust)
case (1 bin' bit')
from 3 have "0 \<le> bin"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit) (simp_all add: Bit_def)
moreover from 1 3 have "0 \<le> bin'"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit') (simp_all add: Bit_def)
ultimately have "0 \<le> bin OR bin'" by (rule 3)
with 1 show ?thesis
- by simp (simp add: Bit_def bitval_def split add: bit.split)
+ by simp (simp add: Bit_def)
qed
qed simp_all
@@ -61,12 +61,12 @@
proof (cases y rule: bin_exhaust)
case (1 bin' bit')
from 3 have "0 \<le> bin"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit) (simp_all add: Bit_def)
moreover from 1 3 have "0 \<le> bin'"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit') (simp_all add: Bit_def)
ultimately have "0 \<le> bin XOR bin'" by (rule 3)
with 1 show ?thesis
- by simp (simp add: Bit_def bitval_def split add: bit.split)
+ by simp (simp add: Bit_def)
qed
next
case 2
@@ -84,10 +84,10 @@
proof (cases y rule: bin_exhaust)
case (1 bin' bit')
from 3 have "0 \<le> bin"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit) (simp_all add: Bit_def)
then have "bin AND bin' \<le> bin" by (rule 3)
with 1 show ?thesis
- by simp (simp add: Bit_def bitval_def split add: bit.split)
+ by simp (simp add: Bit_def)
qed
next
case 2
@@ -108,10 +108,10 @@
proof (cases x rule: bin_exhaust)
case (1 bin' bit')
from 3 have "0 \<le> bin"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit) (simp_all add: Bit_def)
then have "bin' AND bin \<le> bin" by (rule 3)
with 1 show ?thesis
- by simp (simp add: Bit_def bitval_def split add: bit.split)
+ by simp (simp add: Bit_def)
qed
next
case 2
@@ -135,21 +135,21 @@
proof (cases n)
case 0
with 3 have "bin BIT bit = 0" by simp
- then have "bin = 0" "bit = 0"
- by (auto simp add: Bit_def bitval_def split add: bit.split_asm) arith
+ then have "bin = 0" and "\<not> bit"
+ by (auto simp add: Bit_def split: if_splits) arith
then show ?thesis using 0 1 `y < 2 ^ n`
by simp
next
case (Suc m)
from 3 have "0 \<le> bin"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit) (simp_all add: Bit_def)
moreover from 3 Suc have "bin < 2 ^ m"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit) (simp_all add: Bit_def)
moreover from 1 3 Suc have "bin' < 2 ^ m"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit') (simp_all add: Bit_def)
ultimately have "bin OR bin' < 2 ^ m" by (rule 3)
with 1 Suc show ?thesis
- by simp (simp add: Bit_def bitval_def split add: bit.split)
+ by simp (simp add: Bit_def)
qed
qed
qed simp_all
@@ -168,21 +168,21 @@
proof (cases n)
case 0
with 3 have "bin BIT bit = 0" by simp
- then have "bin = 0" "bit = 0"
- by (auto simp add: Bit_def bitval_def split add: bit.split_asm) arith
+ then have "bin = 0" and "\<not> bit"
+ by (auto simp add: Bit_def split: if_splits) arith
then show ?thesis using 0 1 `y < 2 ^ n`
by simp
next
case (Suc m)
from 3 have "0 \<le> bin"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit) (simp_all add: Bit_def)
moreover from 3 Suc have "bin < 2 ^ m"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit) (simp_all add: Bit_def)
moreover from 1 3 Suc have "bin' < 2 ^ m"
- by (simp add: Bit_def bitval_def split add: bit.split_asm)
+ by (cases bit') (simp_all add: Bit_def)
ultimately have "bin XOR bin' < 2 ^ m" by (rule 3)
with 1 Suc show ?thesis
- by simp (simp add: Bit_def bitval_def split add: bit.split)
+ by simp (simp add: Bit_def)
qed
qed
next
--- a/src/HOL/Word/Bit_Int.thy Mon Dec 23 15:30:31 2013 +0100
+++ b/src/HOL/Word/Bit_Int.thy Mon Dec 23 16:59:56 2013 +0100
@@ -9,7 +9,7 @@
header {* Bitwise Operations on Binary Integers *}
theory Bit_Int
-imports Bit_Representation Bit_Bit
+imports Bit_Representation Bit_Operations
begin
subsection {* Logical operations *}
@@ -25,7 +25,7 @@
function bitAND_int where
"bitAND_int x y =
(if x = 0 then 0 else if x = -1 then y else
- (bin_rest x AND bin_rest y) BIT (bin_last x AND bin_last y))"
+ (bin_rest x AND bin_rest y) BIT (bin_last x \<and> bin_last y))"
by pat_completeness simp
termination
@@ -46,7 +46,7 @@
subsubsection {* Basic simplification rules *}
lemma int_not_BIT [simp]:
- "NOT (w BIT b) = (NOT w) BIT (NOT b)"
+ "NOT (w BIT b) = (NOT w) BIT (\<not> b)"
unfolding int_not_def Bit_def by (cases b, simp_all)
lemma int_not_simps [simp]:
@@ -68,7 +68,7 @@
by (simp add: bitAND_int.simps)
lemma int_and_Bits [simp]:
- "(x BIT b) AND (y BIT c) = (x AND y) BIT (b AND c)"
+ "(x BIT b) AND (y BIT c) = (x AND y) BIT (b \<and> c)"
by (subst bitAND_int.simps, simp add: Bit_eq_0_iff Bit_eq_m1_iff)
lemma int_or_zero [simp]: "(0::int) OR x = x"
@@ -78,40 +78,40 @@
unfolding int_or_def by simp
lemma int_or_Bits [simp]:
- "(x BIT b) OR (y BIT c) = (x OR y) BIT (b OR c)"
- unfolding int_or_def bit_or_def by simp
+ "(x BIT b) OR (y BIT c) = (x OR y) BIT (b \<or> c)"
+ unfolding int_or_def by simp
lemma int_xor_zero [simp]: "(0::int) XOR x = x"
unfolding int_xor_def by simp
lemma int_xor_Bits [simp]:
- "(x BIT b) XOR (y BIT c) = (x XOR y) BIT (b XOR c)"
- unfolding int_xor_def bit_xor_def by simp
+ "(x BIT b) XOR (y BIT c) = (x XOR y) BIT ((b \<or> c) \<and> \<not> (b \<and> c))"
+ unfolding int_xor_def by auto
subsubsection {* Binary destructors *}
lemma bin_rest_NOT [simp]: "bin_rest (NOT x) = NOT (bin_rest x)"
by (cases x rule: bin_exhaust, simp)
-lemma bin_last_NOT [simp]: "bin_last (NOT x) = NOT (bin_last x)"
+lemma bin_last_NOT [simp]: "bin_last (NOT x) \<longleftrightarrow> \<not> bin_last x"
by (cases x rule: bin_exhaust, simp)
lemma bin_rest_AND [simp]: "bin_rest (x AND y) = bin_rest x AND bin_rest y"
by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp)
-lemma bin_last_AND [simp]: "bin_last (x AND y) = bin_last x AND bin_last y"
+lemma bin_last_AND [simp]: "bin_last (x AND y) \<longleftrightarrow> bin_last x \<and> bin_last y"
by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp)
lemma bin_rest_OR [simp]: "bin_rest (x OR y) = bin_rest x OR bin_rest y"
by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp)
-lemma bin_last_OR [simp]: "bin_last (x OR y) = bin_last x OR bin_last y"
+lemma bin_last_OR [simp]: "bin_last (x OR y) \<longleftrightarrow> bin_last x \<or> bin_last y"
by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp)
lemma bin_rest_XOR [simp]: "bin_rest (x XOR y) = bin_rest x XOR bin_rest y"
by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp)
-lemma bin_last_XOR [simp]: "bin_last (x XOR y) = bin_last x XOR bin_last y"
+lemma bin_last_XOR [simp]: "bin_last (x XOR y) \<longleftrightarrow> (bin_last x \<or> bin_last y) \<and> \<not> (bin_last x \<and> bin_last y)"
by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp)
lemma bin_nth_ops:
@@ -225,36 +225,36 @@
other simp rules. *}
lemma bin_rl_eqI: "\<lbrakk>bin_rest x = bin_rest y; bin_last x = bin_last y\<rbrakk> \<Longrightarrow> x = y"
- by (metis bin_rl_simp)
+ by (metis (mono_tags) BIT_eq_iff bin_ex_rl bin_last_BIT bin_rest_BIT)
lemma bin_rest_neg_numeral_BitM [simp]:
"bin_rest (- numeral (Num.BitM w)) = - numeral w"
by (simp only: BIT_bin_simps [symmetric] bin_rest_BIT)
lemma bin_last_neg_numeral_BitM [simp]:
- "bin_last (- numeral (Num.BitM w)) = 1"
+ "bin_last (- numeral (Num.BitM w))"
by (simp only: BIT_bin_simps [symmetric] bin_last_BIT)
text {* FIXME: The rule sets below are very large (24 rules for each
operator). Is there a simpler way to do this? *}
lemma int_and_numerals [simp]:
- "numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (numeral x AND numeral y) BIT 0"
- "numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT 0"
- "numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (numeral x AND numeral y) BIT 0"
- "numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT 1"
- "numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (numeral x AND - numeral y) BIT 0"
- "numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (numeral x AND - numeral (y + Num.One)) BIT 0"
- "numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (numeral x AND - numeral y) BIT 0"
- "numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = (numeral x AND - numeral (y + Num.One)) BIT 1"
- "- numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (- numeral x AND numeral y) BIT 0"
- "- numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (- numeral x AND numeral y) BIT 0"
- "- numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (- numeral (x + Num.One) AND numeral y) BIT 0"
- "- numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (- numeral (x + Num.One) AND numeral y) BIT 1"
- "- numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (- numeral x AND - numeral y) BIT 0"
- "- numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (- numeral x AND - numeral (y + Num.One)) BIT 0"
- "- numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (- numeral (x + Num.One) AND - numeral y) BIT 0"
- "- numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = (- numeral (x + Num.One) AND - numeral (y + Num.One)) BIT 1"
+ "numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (numeral x AND numeral y) BIT False"
+ "numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT False"
+ "numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (numeral x AND numeral y) BIT False"
+ "numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT True"
+ "numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (numeral x AND - numeral y) BIT False"
+ "numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (numeral x AND - numeral (y + Num.One)) BIT False"
+ "numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (numeral x AND - numeral y) BIT False"
+ "numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = (numeral x AND - numeral (y + Num.One)) BIT True"
+ "- numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (- numeral x AND numeral y) BIT False"
+ "- numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (- numeral x AND numeral y) BIT False"
+ "- numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (- numeral (x + Num.One) AND numeral y) BIT False"
+ "- numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (- numeral (x + Num.One) AND numeral y) BIT True"
+ "- numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (- numeral x AND - numeral y) BIT False"
+ "- numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (- numeral x AND - numeral (y + Num.One)) BIT False"
+ "- numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (- numeral (x + Num.One) AND - numeral y) BIT False"
+ "- numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = (- numeral (x + Num.One) AND - numeral (y + Num.One)) BIT True"
"(1::int) AND numeral (Num.Bit0 y) = 0"
"(1::int) AND numeral (Num.Bit1 y) = 1"
"(1::int) AND - numeral (Num.Bit0 y) = 0"
@@ -266,22 +266,22 @@
by (rule bin_rl_eqI, simp, simp)+
lemma int_or_numerals [simp]:
- "numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (numeral x OR numeral y) BIT 0"
- "numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT 1"
- "numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (numeral x OR numeral y) BIT 1"
- "numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT 1"
- "numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (numeral x OR - numeral y) BIT 0"
- "numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = (numeral x OR - numeral (y + Num.One)) BIT 1"
- "numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = (numeral x OR - numeral y) BIT 1"
- "numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = (numeral x OR - numeral (y + Num.One)) BIT 1"
- "- numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (- numeral x OR numeral y) BIT 0"
- "- numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (- numeral x OR numeral y) BIT 1"
- "- numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (- numeral (x + Num.One) OR numeral y) BIT 1"
- "- numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (- numeral (x + Num.One) OR numeral y) BIT 1"
- "- numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (- numeral x OR - numeral y) BIT 0"
- "- numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = (- numeral x OR - numeral (y + Num.One)) BIT 1"
- "- numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = (- numeral (x + Num.One) OR - numeral y) BIT 1"
- "- numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = (- numeral (x + Num.One) OR - numeral (y + Num.One)) BIT 1"
+ "numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (numeral x OR numeral y) BIT False"
+ "numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT True"
+ "numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (numeral x OR numeral y) BIT True"
+ "numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT True"
+ "numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (numeral x OR - numeral y) BIT False"
+ "numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = (numeral x OR - numeral (y + Num.One)) BIT True"
+ "numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = (numeral x OR - numeral y) BIT True"
+ "numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = (numeral x OR - numeral (y + Num.One)) BIT True"
+ "- numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (- numeral x OR numeral y) BIT False"
+ "- numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (- numeral x OR numeral y) BIT True"
+ "- numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (- numeral (x + Num.One) OR numeral y) BIT True"
+ "- numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (- numeral (x + Num.One) OR numeral y) BIT True"
+ "- numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (- numeral x OR - numeral y) BIT False"
+ "- numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = (- numeral x OR - numeral (y + Num.One)) BIT True"
+ "- numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = (- numeral (x + Num.One) OR - numeral y) BIT True"
+ "- numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = (- numeral (x + Num.One) OR - numeral (y + Num.One)) BIT True"
"(1::int) OR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)"
"(1::int) OR numeral (Num.Bit1 y) = numeral (Num.Bit1 y)"
"(1::int) OR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)"
@@ -293,22 +293,22 @@
by (rule bin_rl_eqI, simp, simp)+
lemma int_xor_numerals [simp]:
- "numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (numeral x XOR numeral y) BIT 0"
- "numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT 1"
- "numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (numeral x XOR numeral y) BIT 1"
- "numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT 0"
- "numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (numeral x XOR - numeral y) BIT 0"
- "numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = (numeral x XOR - numeral (y + Num.One)) BIT 1"
- "numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = (numeral x XOR - numeral y) BIT 1"
- "numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (numeral x XOR - numeral (y + Num.One)) BIT 0"
- "- numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (- numeral x XOR numeral y) BIT 0"
- "- numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (- numeral x XOR numeral y) BIT 1"
- "- numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (- numeral (x + Num.One) XOR numeral y) BIT 1"
- "- numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (- numeral (x + Num.One) XOR numeral y) BIT 0"
- "- numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (- numeral x XOR - numeral y) BIT 0"
- "- numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = (- numeral x XOR - numeral (y + Num.One)) BIT 1"
- "- numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = (- numeral (x + Num.One) XOR - numeral y) BIT 1"
- "- numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (- numeral (x + Num.One) XOR - numeral (y + Num.One)) BIT 0"
+ "numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (numeral x XOR numeral y) BIT False"
+ "numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT True"
+ "numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (numeral x XOR numeral y) BIT True"
+ "numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT False"
+ "numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (numeral x XOR - numeral y) BIT False"
+ "numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = (numeral x XOR - numeral (y + Num.One)) BIT True"
+ "numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = (numeral x XOR - numeral y) BIT True"
+ "numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (numeral x XOR - numeral (y + Num.One)) BIT False"
+ "- numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (- numeral x XOR numeral y) BIT False"
+ "- numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (- numeral x XOR numeral y) BIT True"
+ "- numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (- numeral (x + Num.One) XOR numeral y) BIT True"
+ "- numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (- numeral (x + Num.One) XOR numeral y) BIT False"
+ "- numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (- numeral x XOR - numeral y) BIT False"
+ "- numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = (- numeral x XOR - numeral (y + Num.One)) BIT True"
+ "- numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = (- numeral (x + Num.One) XOR - numeral y) BIT True"
+ "- numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (- numeral (x + Num.One) XOR - numeral (y + Num.One)) BIT False"
"(1::int) XOR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)"
"(1::int) XOR numeral (Num.Bit1 y) = numeral (Num.Bit0 y)"
"(1::int) XOR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)"
@@ -332,7 +332,7 @@
apply (unfold Bit_def)
apply clarsimp
apply (erule_tac x = "x" in allE)
- apply (simp add: bitval_def split: bit.split)
+ apply simp
done
lemma le_int_or:
@@ -385,7 +385,7 @@
subsection {* Setting and clearing bits *}
primrec
- bin_sc :: "nat => bit => int => int"
+ bin_sc :: "nat => bool => int => int"
where
Z: "bin_sc 0 b w = bin_rest w BIT b"
| Suc: "bin_sc (Suc n) b w = bin_sc n b (bin_rest w) BIT bin_last w"
@@ -393,7 +393,7 @@
(** nth bit, set/clear **)
lemma bin_nth_sc [simp]:
- "bin_nth (bin_sc n b w) n = (b = 1)"
+ "bin_nth (bin_sc n b w) n \<longleftrightarrow> b"
by (induct n arbitrary: w) auto
lemma bin_sc_sc_same [simp]:
@@ -409,11 +409,11 @@
done
lemma bin_nth_sc_gen:
- "bin_nth (bin_sc n b w) m = (if m = n then b = 1 else bin_nth w m)"
+ "bin_nth (bin_sc n b w) m = (if m = n then b else bin_nth w m)"
by (induct n arbitrary: w m) (case_tac [!] m, auto)
lemma bin_sc_nth [simp]:
- "(bin_sc n (If (bin_nth w n) 1 0) w) = w"
+ "(bin_sc n (bin_nth w n) w) = w"
by (induct n arbitrary: w) auto
lemma bin_sign_sc [simp]:
@@ -428,21 +428,21 @@
done
lemma bin_clr_le:
- "bin_sc n 0 w <= w"
+ "bin_sc n False w <= w"
apply (induct n arbitrary: w)
apply (case_tac [!] w rule: bin_exhaust)
apply (auto simp: le_Bits)
done
lemma bin_set_ge:
- "bin_sc n 1 w >= w"
+ "bin_sc n True w >= w"
apply (induct n arbitrary: w)
apply (case_tac [!] w rule: bin_exhaust)
apply (auto simp: le_Bits)
done
lemma bintr_bin_clr_le:
- "bintrunc n (bin_sc m 0 w) <= bintrunc n w"
+ "bintrunc n (bin_sc m False w) <= bintrunc n w"
apply (induct n arbitrary: w m)
apply simp
apply (case_tac w rule: bin_exhaust)
@@ -451,7 +451,7 @@
done
lemma bintr_bin_set_ge:
- "bintrunc n (bin_sc m 1 w) >= bintrunc n w"
+ "bintrunc n (bin_sc m True w) >= bintrunc n w"
apply (induct n arbitrary: w m)
apply simp
apply (case_tac w rule: bin_exhaust)
@@ -459,10 +459,10 @@
apply (auto simp: le_Bits)
done
-lemma bin_sc_FP [simp]: "bin_sc n 0 0 = 0"
+lemma bin_sc_FP [simp]: "bin_sc n False 0 = 0"
by (induct n) auto
-lemma bin_sc_TM [simp]: "bin_sc n 1 -1 = -1"
+lemma bin_sc_TM [simp]: "bin_sc n True -1 = -1"
by (induct n) auto
lemmas bin_sc_simps = bin_sc.Z bin_sc.Suc bin_sc_TM bin_sc_FP
@@ -482,25 +482,30 @@
subsection {* Splitting and concatenation *}
-definition bin_rcat :: "nat \<Rightarrow> int list \<Rightarrow> int" where
+definition bin_rcat :: "nat \<Rightarrow> int list \<Rightarrow> int"
+where
"bin_rcat n = foldl (\<lambda>u v. bin_cat u n v) 0"
-fun bin_rsplit_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where
+fun bin_rsplit_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list"
+where
"bin_rsplit_aux n m c bs =
(if m = 0 | n = 0 then bs else
let (a, b) = bin_split n c
in bin_rsplit_aux n (m - n) a (b # bs))"
-definition bin_rsplit :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list" where
+definition bin_rsplit :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list"
+where
"bin_rsplit n w = bin_rsplit_aux n (fst w) (snd w) []"
-fun bin_rsplitl_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where
+fun bin_rsplitl_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list"
+where
"bin_rsplitl_aux n m c bs =
(if m = 0 | n = 0 then bs else
let (a, b) = bin_split (min m n) c
in bin_rsplitl_aux n (m - n) a (b # bs))"
-definition bin_rsplitl :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list" where
+definition bin_rsplitl :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list"
+where
"bin_rsplitl n w = bin_rsplitl_aux n (fst w) (snd w) []"
declare bin_rsplit_aux.simps [simp del]
@@ -611,8 +616,7 @@
apply (simp add: bin_rest_def zdiv_zmult2_eq)
apply (case_tac b rule: bin_exhaust)
apply simp
- apply (simp add: Bit_def mod_mult_mult1 p1mod22k bitval_def
- split: bit.split)
+ apply (simp add: Bit_def mod_mult_mult1 p1mod22k)
done
subsection {* Miscellaneous lemmas *}
@@ -632,7 +636,7 @@
"(EX m. n = Suc m & (m = k | P m)) = (n = Suc k | (EX m. n = Suc m & P m))"
by auto
-lemma power_BIT: "2 ^ (Suc n) - 1 = (2 ^ n - 1) BIT 1"
+lemma power_BIT: "2 ^ (Suc n) - 1 = (2 ^ n - 1) BIT True"
unfolding Bit_B1
by (induct n) simp_all
@@ -645,8 +649,8 @@
by (rule mult_left_mono) simp
then have "2 * (bin mod 2 ^ n) + 1 < 2 * 2 ^ n" by simp
then show ?thesis
- by (auto simp add: Bit_def bitval_def mod_mult_mult1 mod_add_left_eq [of "2 * bin"]
- mod_pos_pos_trivial split add: bit.split)
+ by (auto simp add: Bit_def mod_mult_mult1 mod_add_left_eq [of "2 * bin"]
+ mod_pos_pos_trivial)
qed
lemma AND_mod:
@@ -665,7 +669,7 @@
show ?case
proof (cases n)
case 0
- then show ?thesis by (simp add: int_and_extra_simps)
+ then show ?thesis by simp
next
case (Suc m)
with 3 show ?thesis
--- a/src/HOL/Word/Bit_Operations.thy Mon Dec 23 15:30:31 2013 +0100
+++ b/src/HOL/Word/Bit_Operations.thy Mon Dec 23 16:59:56 2013 +0100
@@ -5,7 +5,7 @@
header {* Syntactic classes for bitwise operations *}
theory Bit_Operations
-imports "~~/src/HOL/Library/Bit"
+imports Main
begin
class bit =
--- a/src/HOL/Word/Bit_Representation.thy Mon Dec 23 15:30:31 2013 +0100
+++ b/src/HOL/Word/Bit_Representation.thy Mon Dec 23 16:59:56 2013 +0100
@@ -5,40 +5,39 @@
header {* Integers as implict bit strings *}
theory Bit_Representation
-imports "~~/src/HOL/Library/Bit" Misc_Numeric
+imports Misc_Numeric
begin
subsection {* Constructors and destructors for binary integers *}
-definition bitval :: "bit \<Rightarrow> 'a\<Colon>zero_neq_one" where
- "bitval = bit_case 0 1"
-
-lemma bitval_simps [simp]:
- "bitval 0 = 0"
- "bitval 1 = 1"
- by (simp_all add: bitval_def)
-
-definition Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where
- "k BIT b = bitval b + k + k"
+definition Bit :: "int \<Rightarrow> bool \<Rightarrow> int" (infixl "BIT" 90)
+where
+ "k BIT b = (if b then 1 else 0) + k + k"
lemma Bit_B0:
- "k BIT (0::bit) = k + k"
+ "k BIT False = k + k"
by (unfold Bit_def) simp
lemma Bit_B1:
- "k BIT (1::bit) = k + k + 1"
+ "k BIT True = k + k + 1"
by (unfold Bit_def) simp
-lemma Bit_B0_2t: "k BIT (0::bit) = 2 * k"
+lemma Bit_B0_2t: "k BIT False = 2 * k"
by (rule trans, rule Bit_B0) simp
-lemma Bit_B1_2t: "k BIT (1::bit) = 2 * k + 1"
+lemma Bit_B1_2t: "k BIT True = 2 * k + 1"
by (rule trans, rule Bit_B1) simp
-definition bin_last :: "int \<Rightarrow> bit" where
- "bin_last w = (if w mod 2 = 0 then (0::bit) else (1::bit))"
+definition bin_last :: "int \<Rightarrow> bool"
+where
+ "bin_last w \<longleftrightarrow> w mod 2 = 1"
-definition bin_rest :: "int \<Rightarrow> int" where
+lemma bin_last_odd:
+ "bin_last = odd"
+ by (rule ext) (simp add: bin_last_def even_def)
+
+definition bin_rest :: "int \<Rightarrow> int"
+where
"bin_rest w = w div 2"
lemma bin_rl_simp [simp]:
@@ -56,48 +55,55 @@
by (cases b, simp_all add: z1pmod2)
lemma BIT_eq_iff [iff]: "u BIT b = v BIT c \<longleftrightarrow> u = v \<and> b = c"
- by (metis bin_rest_BIT bin_last_BIT)
+ apply (auto simp add: Bit_def)
+ apply arith
+ apply arith
+ done
lemma BIT_bin_simps [simp]:
- "numeral k BIT 0 = numeral (Num.Bit0 k)"
- "numeral k BIT 1 = numeral (Num.Bit1 k)"
- "(- numeral k) BIT 0 = - numeral (Num.Bit0 k)"
- "(- numeral k) BIT 1 = - numeral (Num.BitM k)"
+ "numeral k BIT False = numeral (Num.Bit0 k)"
+ "numeral k BIT True = numeral (Num.Bit1 k)"
+ "(- numeral k) BIT False = - numeral (Num.Bit0 k)"
+ "(- numeral k) BIT True = - numeral (Num.BitM k)"
unfolding numeral.simps numeral_BitM
- unfolding Bit_def bitval_simps
+ unfolding Bit_def
by (simp_all del: arith_simps add_numeral_special diff_numeral_special)
lemma BIT_special_simps [simp]:
- shows "0 BIT 0 = 0" and "0 BIT 1 = 1"
- and "1 BIT 0 = 2" and "1 BIT 1 = 3"
- and "(- 1) BIT 0 = - 2" and "(- 1) BIT 1 = - 1"
+ shows "0 BIT False = 0" and "0 BIT True = 1"
+ and "1 BIT False = 2" and "1 BIT True = 3"
+ and "(- 1) BIT False = - 2" and "(- 1) BIT True = - 1"
unfolding Bit_def by simp_all
-lemma Bit_eq_0_iff: "w BIT b = 0 \<longleftrightarrow> w = 0 \<and> b = 0"
- by (subst BIT_eq_iff [symmetric], simp)
+lemma Bit_eq_0_iff: "w BIT b = 0 \<longleftrightarrow> w = 0 \<and> \<not> b"
+ apply (auto simp add: Bit_def)
+ apply arith
+ done
-lemma Bit_eq_m1_iff: "w BIT b = - 1 \<longleftrightarrow> w = - 1 \<and> b = 1"
- by (cases b) (auto simp add: Bit_def, arith)
+lemma Bit_eq_m1_iff: "w BIT b = -1 \<longleftrightarrow> w = -1 \<and> b"
+ apply (auto simp add: Bit_def)
+ apply arith
+ done
lemma BitM_inc: "Num.BitM (Num.inc w) = Num.Bit1 w"
by (induct w, simp_all)
lemma expand_BIT:
- "numeral (Num.Bit0 w) = numeral w BIT 0"
- "numeral (Num.Bit1 w) = numeral w BIT 1"
- "- numeral (Num.Bit0 w) = - numeral w BIT 0"
- "- numeral (Num.Bit1 w) = (- numeral (w + Num.One)) BIT 1"
+ "numeral (Num.Bit0 w) = numeral w BIT False"
+ "numeral (Num.Bit1 w) = numeral w BIT True"
+ "- numeral (Num.Bit0 w) = (- numeral w) BIT False"
+ "- numeral (Num.Bit1 w) = (- numeral (w + Num.One)) BIT True"
unfolding add_One by (simp_all add: BitM_inc)
lemma bin_last_numeral_simps [simp]:
- "bin_last 0 = 0"
- "bin_last 1 = 1"
- "bin_last -1 = 1"
- "bin_last Numeral1 = 1"
- "bin_last (numeral (Num.Bit0 w)) = 0"
- "bin_last (numeral (Num.Bit1 w)) = 1"
- "bin_last (- numeral (Num.Bit0 w)) = 0"
- "bin_last (- numeral (Num.Bit1 w)) = 1"
+ "\<not> bin_last 0"
+ "bin_last 1"
+ "bin_last -1"
+ "bin_last Numeral1"
+ "\<not> bin_last (numeral (Num.Bit0 w))"
+ "bin_last (numeral (Num.Bit1 w))"
+ "\<not> bin_last (- numeral (Num.Bit0 w))"
+ "bin_last (- numeral (Num.Bit1 w))"
unfolding expand_BIT bin_last_BIT by (simp_all add: bin_last_def zmod_zminus1_eq_if)
lemma bin_rest_numeral_simps [simp]:
@@ -112,51 +118,42 @@
unfolding expand_BIT bin_rest_BIT by (simp_all add: bin_rest_def zdiv_zminus1_eq_if)
lemma less_Bits:
- "(v BIT b < w BIT c) = (v < w | v <= w & b = (0::bit) & c = (1::bit))"
- unfolding Bit_def by (auto simp add: bitval_def split: bit.split)
+ "v BIT b < w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> \<not> b \<and> c"
+ unfolding Bit_def by auto
lemma le_Bits:
- "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= (1::bit) | c ~= (0::bit)))"
- unfolding Bit_def by (auto simp add: bitval_def split: bit.split)
+ "v BIT b \<le> w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> (\<not> b \<or> c)"
+ unfolding Bit_def by auto
lemma pred_BIT_simps [simp]:
- "x BIT 0 - 1 = (x - 1) BIT 1"
- "x BIT 1 - 1 = x BIT 0"
+ "x BIT False - 1 = (x - 1) BIT True"
+ "x BIT True - 1 = x BIT False"
by (simp_all add: Bit_B0_2t Bit_B1_2t)
lemma succ_BIT_simps [simp]:
- "x BIT 0 + 1 = x BIT 1"
- "x BIT 1 + 1 = (x + 1) BIT 0"
+ "x BIT False + 1 = x BIT True"
+ "x BIT True + 1 = (x + 1) BIT False"
by (simp_all add: Bit_B0_2t Bit_B1_2t)
lemma add_BIT_simps [simp]:
- "x BIT 0 + y BIT 0 = (x + y) BIT 0"
- "x BIT 0 + y BIT 1 = (x + y) BIT 1"
- "x BIT 1 + y BIT 0 = (x + y) BIT 1"
- "x BIT 1 + y BIT 1 = (x + y + 1) BIT 0"
+ "x BIT False + y BIT False = (x + y) BIT False"
+ "x BIT False + y BIT True = (x + y) BIT True"
+ "x BIT True + y BIT False = (x + y) BIT True"
+ "x BIT True + y BIT True = (x + y + 1) BIT False"
by (simp_all add: Bit_B0_2t Bit_B1_2t)
lemma mult_BIT_simps [simp]:
- "x BIT 0 * y = (x * y) BIT 0"
- "x * y BIT 0 = (x * y) BIT 0"
- "x BIT 1 * y = (x * y) BIT 0 + y"
+ "x BIT False * y = (x * y) BIT False"
+ "x * y BIT False = (x * y) BIT False"
+ "x BIT True * y = (x * y) BIT False + y"
by (simp_all add: Bit_B0_2t Bit_B1_2t algebra_simps)
lemma B_mod_2':
- "X = 2 ==> (w BIT (1::bit)) mod X = 1 & (w BIT (0::bit)) mod X = 0"
+ "X = 2 ==> (w BIT True) mod X = 1 & (w BIT False) mod X = 0"
apply (simp (no_asm) only: Bit_B0 Bit_B1)
apply (simp add: z1pmod2)
done
-lemma neB1E [elim!]:
- assumes ne: "y \<noteq> (1::bit)"
- assumes y: "y = (0::bit) \<Longrightarrow> P"
- shows "P"
- apply (rule y)
- apply (cases y rule: bit.exhaust, simp)
- apply (simp add: ne)
- done
-
lemma bin_ex_rl: "EX w b. w BIT b = bin"
by (metis bin_rl_simp)
@@ -170,8 +167,10 @@
done
primrec bin_nth where
- Z: "bin_nth w 0 = (bin_last w = (1::bit))"
- | Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n"
+ Z: "bin_nth w 0 \<longleftrightarrow> bin_last w"
+ | Suc: "bin_nth w (Suc n) \<longleftrightarrow> bin_nth (bin_rest w) n"
+
+find_theorems "bin_rest _ = _"
lemma bin_abs_lem:
"bin = (w BIT b) ==> bin ~= -1 --> bin ~= 0 -->
@@ -248,7 +247,7 @@
lemma bin_nth_minus1 [simp]: "bin_nth -1 n"
by (induct n) auto
-lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = (1::bit))"
+lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 \<longleftrightarrow> b"
by auto
lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n"
@@ -275,7 +274,8 @@
subsection {* Truncating binary integers *}
-definition bin_sign :: "int \<Rightarrow> int" where
+definition bin_sign :: "int \<Rightarrow> int"
+where
bin_sign_def: "bin_sign k = (if k \<ge> 0 then 0 else - 1)"
lemma bin_sign_simps [simp]:
@@ -285,8 +285,8 @@
"bin_sign (numeral k) = 0"
"bin_sign (- numeral k) = -1"
"bin_sign (w BIT b) = bin_sign w"
- unfolding bin_sign_def Bit_def bitval_def
- by (simp_all split: bit.split)
+ unfolding bin_sign_def Bit_def
+ by simp_all
lemma bin_sign_rest [simp]:
"bin_sign (bin_rest w) = bin_sign w"
@@ -297,7 +297,7 @@
| Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"
primrec sbintrunc :: "nat => int => int" where
- Z : "sbintrunc 0 bin = (case bin_last bin of (1::bit) \<Rightarrow> -1 | (0::bit) \<Rightarrow> 0)"
+ Z : "sbintrunc 0 bin = (if bin_last bin then -1 else 0)"
| Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
lemma sign_bintr: "bin_sign (bintrunc n w) = 0"
@@ -313,7 +313,8 @@
apply simp
apply (subst mod_add_left_eq)
apply (simp add: bin_last_def)
- apply (simp add: bin_last_def bin_rest_def Bit_def)
+ apply arith
+ apply (simp add: bin_last_def bin_rest_def Bit_def mod_2_neq_1_eq_eq_0)
apply (clarsimp simp: mod_mult_mult1 [symmetric]
zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]])
apply (rule trans [symmetric, OF _ emep1])
@@ -334,13 +335,13 @@
lemma bintrunc_Suc_numeral:
"bintrunc (Suc n) 1 = 1"
- "bintrunc (Suc n) -1 = bintrunc n -1 BIT 1"
- "bintrunc (Suc n) (numeral (Num.Bit0 w)) = bintrunc n (numeral w) BIT 0"
- "bintrunc (Suc n) (numeral (Num.Bit1 w)) = bintrunc n (numeral w) BIT 1"
+ "bintrunc (Suc n) -1 = bintrunc n -1 BIT True"
+ "bintrunc (Suc n) (numeral (Num.Bit0 w)) = bintrunc n (numeral w) BIT False"
+ "bintrunc (Suc n) (numeral (Num.Bit1 w)) = bintrunc n (numeral w) BIT True"
"bintrunc (Suc n) (- numeral (Num.Bit0 w)) =
- bintrunc n (- numeral w) BIT 0"
+ bintrunc n (- numeral w) BIT False"
"bintrunc (Suc n) (- numeral (Num.Bit1 w)) =
- bintrunc n (- numeral (w + Num.One)) BIT 1"
+ bintrunc n (- numeral (w + Num.One)) BIT True"
by simp_all
lemma sbintrunc_0_numeral [simp]:
@@ -354,21 +355,15 @@
lemma sbintrunc_Suc_numeral:
"sbintrunc (Suc n) 1 = 1"
"sbintrunc (Suc n) (numeral (Num.Bit0 w)) =
- sbintrunc n (numeral w) BIT 0"
+ sbintrunc n (numeral w) BIT False"
"sbintrunc (Suc n) (numeral (Num.Bit1 w)) =
- sbintrunc n (numeral w) BIT 1"
+ sbintrunc n (numeral w) BIT True"
"sbintrunc (Suc n) (- numeral (Num.Bit0 w)) =
- sbintrunc n (- numeral w) BIT 0"
+ sbintrunc n (- numeral w) BIT False"
"sbintrunc (Suc n) (- numeral (Num.Bit1 w)) =
- sbintrunc n (- numeral (w + Num.One)) BIT 1"
+ sbintrunc n (- numeral (w + Num.One)) BIT True"
by simp_all
-lemma bit_bool:
- "(b = (b' = (1::bit))) = (b' = (if b then (1::bit) else (0::bit)))"
- by (cases b') auto
-
-lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric]
-
lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = -1) = bin_nth bin n"
apply (induct n arbitrary: bin)
apply (case_tac bin rule: bin_exhaust, case_tac b, auto)
@@ -384,23 +379,25 @@
"bin_nth (sbintrunc m w) n =
(if n < m then bin_nth w n else bin_nth w m)"
apply (induct n arbitrary: w m)
- apply (case_tac m, simp_all split: bit.splits)[1]
- apply (case_tac m, simp_all split: bit.splits)[1]
+ apply (case_tac m)
+ apply simp_all
+ apply (case_tac m)
+ apply simp_all
done
lemma bin_nth_Bit:
- "bin_nth (w BIT b) n = (n = 0 & b = (1::bit) | (EX m. n = Suc m & bin_nth w m))"
+ "bin_nth (w BIT b) n = (n = 0 & b | (EX m. n = Suc m & bin_nth w m))"
by (cases n) auto
lemma bin_nth_Bit0:
"bin_nth (numeral (Num.Bit0 w)) n \<longleftrightarrow>
(\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)"
- using bin_nth_Bit [where w="numeral w" and b="(0::bit)"] by simp
+ using bin_nth_Bit [where w="numeral w" and b="False"] by simp
lemma bin_nth_Bit1:
"bin_nth (numeral (Num.Bit1 w)) n \<longleftrightarrow>
n = 0 \<or> (\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)"
- using bin_nth_Bit [where w="numeral w" and b="(1::bit)"] by simp
+ using bin_nth_Bit [where w="numeral w" and b="True"] by simp
lemma bintrunc_bintrunc_l:
"n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)"
@@ -452,19 +449,19 @@
lemmas sbintrunc_Pls =
sbintrunc.Z [where bin="0",
- simplified bin_last_numeral_simps bin_rest_numeral_simps bit.simps]
+ simplified bin_last_numeral_simps bin_rest_numeral_simps]
lemmas sbintrunc_Min =
sbintrunc.Z [where bin="-1",
- simplified bin_last_numeral_simps bin_rest_numeral_simps bit.simps]
+ simplified bin_last_numeral_simps bin_rest_numeral_simps]
lemmas sbintrunc_0_BIT_B0 [simp] =
- sbintrunc.Z [where bin="w BIT (0::bit)",
- simplified bin_last_numeral_simps bin_rest_numeral_simps bit.simps] for w
+ sbintrunc.Z [where bin="w BIT False",
+ simplified bin_last_numeral_simps bin_rest_numeral_simps] for w
lemmas sbintrunc_0_BIT_B1 [simp] =
- sbintrunc.Z [where bin="w BIT (1::bit)",
- simplified bin_last_BIT bin_rest_numeral_simps bit.simps] for w
+ sbintrunc.Z [where bin="w BIT True",
+ simplified bin_last_BIT bin_rest_numeral_simps] for w
lemmas sbintrunc_0_simps =
sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1
@@ -583,25 +580,25 @@
lemma bintrunc_numeral_simps [simp]:
"bintrunc (numeral k) (numeral (Num.Bit0 w)) =
- bintrunc (pred_numeral k) (numeral w) BIT 0"
+ bintrunc (pred_numeral k) (numeral w) BIT False"
"bintrunc (numeral k) (numeral (Num.Bit1 w)) =
- bintrunc (pred_numeral k) (numeral w) BIT 1"
+ bintrunc (pred_numeral k) (numeral w) BIT True"
"bintrunc (numeral k) (- numeral (Num.Bit0 w)) =
- bintrunc (pred_numeral k) (- numeral w) BIT 0"
+ bintrunc (pred_numeral k) (- numeral w) BIT False"
"bintrunc (numeral k) (- numeral (Num.Bit1 w)) =
- bintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT 1"
+ bintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT True"
"bintrunc (numeral k) 1 = 1"
by (simp_all add: bintrunc_numeral)
lemma sbintrunc_numeral_simps [simp]:
"sbintrunc (numeral k) (numeral (Num.Bit0 w)) =
- sbintrunc (pred_numeral k) (numeral w) BIT 0"
+ sbintrunc (pred_numeral k) (numeral w) BIT False"
"sbintrunc (numeral k) (numeral (Num.Bit1 w)) =
- sbintrunc (pred_numeral k) (numeral w) BIT 1"
+ sbintrunc (pred_numeral k) (numeral w) BIT True"
"sbintrunc (numeral k) (- numeral (Num.Bit0 w)) =
- sbintrunc (pred_numeral k) (- numeral w) BIT 0"
+ sbintrunc (pred_numeral k) (- numeral w) BIT False"
"sbintrunc (numeral k) (- numeral (Num.Bit1 w)) =
- sbintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT 1"
+ sbintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT True"
"sbintrunc (numeral k) 1 = 1"
by (simp_all add: sbintrunc_numeral)
@@ -728,7 +725,7 @@
"sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
apply (induct n arbitrary: bin, simp)
apply (case_tac bin rule: bin_exhaust)
- apply (auto simp: bintrunc_bintrunc_l split: bit.splits)
+ apply (auto simp: bintrunc_bintrunc_l split: bool.splits)
done
lemma bintrunc_rest':
--- a/src/HOL/Word/Bool_List_Representation.thy Mon Dec 23 15:30:31 2013 +0100
+++ b/src/HOL/Word/Bool_List_Representation.thy Mon Dec 23 16:59:56 2013 +0100
@@ -31,27 +31,33 @@
subsection {* Operations on lists of booleans *}
-primrec bl_to_bin_aux :: "bool list \<Rightarrow> int \<Rightarrow> int" where
+primrec bl_to_bin_aux :: "bool list \<Rightarrow> int \<Rightarrow> int"
+where
Nil: "bl_to_bin_aux [] w = w"
| Cons: "bl_to_bin_aux (b # bs) w =
- bl_to_bin_aux bs (w BIT (if b then 1 else 0))"
+ bl_to_bin_aux bs (w BIT b)"
-definition bl_to_bin :: "bool list \<Rightarrow> int" where
+definition bl_to_bin :: "bool list \<Rightarrow> int"
+where
bl_to_bin_def: "bl_to_bin bs = bl_to_bin_aux bs 0"
-primrec bin_to_bl_aux :: "nat \<Rightarrow> int \<Rightarrow> bool list \<Rightarrow> bool list" where
+primrec bin_to_bl_aux :: "nat \<Rightarrow> int \<Rightarrow> bool list \<Rightarrow> bool list"
+where
Z: "bin_to_bl_aux 0 w bl = bl"
| Suc: "bin_to_bl_aux (Suc n) w bl =
- bin_to_bl_aux n (bin_rest w) ((bin_last w = 1) # bl)"
+ bin_to_bl_aux n (bin_rest w) ((bin_last w) # bl)"
-definition bin_to_bl :: "nat \<Rightarrow> int \<Rightarrow> bool list" where
+definition bin_to_bl :: "nat \<Rightarrow> int \<Rightarrow> bool list"
+where
bin_to_bl_def : "bin_to_bl n w = bin_to_bl_aux n w []"
-primrec bl_of_nth :: "nat \<Rightarrow> (nat \<Rightarrow> bool) \<Rightarrow> bool list" where
+primrec bl_of_nth :: "nat \<Rightarrow> (nat \<Rightarrow> bool) \<Rightarrow> bool list"
+where
Suc: "bl_of_nth (Suc n) f = f n # bl_of_nth n f"
| Z: "bl_of_nth 0 f = []"
-primrec takefill :: "'a \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+primrec takefill :: "'a \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
Z: "takefill fill 0 xs = []"
| Suc: "takefill fill (Suc n) xs = (
case xs of [] => fill # takefill fill n xs
@@ -65,21 +71,25 @@
assuming input list(s) the same length, and don't extend them.
*}
-primrec rbl_succ :: "bool list => bool list" where
+primrec rbl_succ :: "bool list => bool list"
+where
Nil: "rbl_succ Nil = Nil"
| Cons: "rbl_succ (x # xs) = (if x then False # rbl_succ xs else True # xs)"
-primrec rbl_pred :: "bool list => bool list" where
+primrec rbl_pred :: "bool list => bool list"
+where
Nil: "rbl_pred Nil = Nil"
| Cons: "rbl_pred (x # xs) = (if x then False # xs else True # rbl_pred xs)"
-primrec rbl_add :: "bool list => bool list => bool list" where
+primrec rbl_add :: "bool list => bool list => bool list"
+where
-- "result is length of first arg, second arg may be longer"
Nil: "rbl_add Nil x = Nil"
| Cons: "rbl_add (y # ys) x = (let ws = rbl_add ys (tl x) in
(y ~= hd x) # (if hd x & y then rbl_succ ws else ws))"
-primrec rbl_mult :: "bool list => bool list => bool list" where
+primrec rbl_mult :: "bool list => bool list => bool list"
+where
-- "result is length of first arg, second arg may be longer"
Nil: "rbl_mult Nil x = Nil"
| Cons: "rbl_mult (y # ys) x = (let ws = False # rbl_mult ys x in
@@ -106,7 +116,7 @@
lemma bin_to_bl_aux_Bit_minus_simp [simp]:
"0 < n ==> bin_to_bl_aux n (w BIT b) bl =
- bin_to_bl_aux (n - 1) w ((b = 1) # bl)"
+ bin_to_bl_aux (n - 1) w (b # bl)"
by (cases n) auto
lemma bin_to_bl_aux_Bit0_minus_simp [simp]:
@@ -253,8 +263,7 @@
apply (induct n arbitrary: w bs)
apply clarsimp
apply (cases w rule: bin_exhaust)
- apply (simp split add : bit.split)
- apply clarsimp
+ apply simp
done
lemma bl_sbin_sign:
@@ -317,7 +326,6 @@
apply (induct bs arbitrary: w)
apply clarsimp
apply clarsimp
- apply safe
apply (drule meta_spec, erule xtrans(8) [rotated], simp add: Bit_def)+
done
@@ -334,9 +342,9 @@
apply (induct bs arbitrary: w)
apply clarsimp
apply clarsimp
- apply safe
apply (drule meta_spec, erule order_trans [rotated],
simp add: Bit_B0_2t Bit_B1_2t algebra_simps)+
+ apply (simp add: Bit_def)
done
lemma bl_to_bin_ge0: "bl_to_bin bs >= 0"
@@ -422,15 +430,15 @@
by (cases xs) auto
lemma last_bin_last':
- "size xs > 0 \<Longrightarrow> last xs = (bin_last (bl_to_bin_aux xs w) = 1)"
+ "size xs > 0 \<Longrightarrow> last xs \<longleftrightarrow> bin_last (bl_to_bin_aux xs w)"
by (induct xs arbitrary: w) auto
lemma last_bin_last:
- "size xs > 0 ==> last xs = (bin_last (bl_to_bin xs) = 1)"
+ "size xs > 0 ==> last xs \<longleftrightarrow> bin_last (bl_to_bin xs)"
unfolding bl_to_bin_def by (erule last_bin_last')
lemma bin_last_last:
- "bin_last w = (if last (bin_to_bl (Suc n) w) then 1 else 0)"
+ "bin_last w \<longleftrightarrow> last (bin_to_bl (Suc n) w)"
apply (unfold bin_to_bl_def)
apply simp
apply (auto simp add: bin_to_bl_aux_alt)
@@ -447,7 +455,7 @@
apply (case_tac w rule: bin_exhaust)
apply clarsimp
apply (case_tac b)
- apply (case_tac ba, safe, simp_all)+
+ apply auto
done
lemma bl_or_aux_bin:
@@ -458,8 +466,6 @@
apply (case_tac v rule: bin_exhaust)
apply (case_tac w rule: bin_exhaust)
apply clarsimp
- apply (case_tac b)
- apply (case_tac ba, safe, simp_all)+
done
lemma bl_and_aux_bin:
@@ -846,7 +852,7 @@
lemmas rbl_mult_Suc = lessI [THEN rbl_mult_gt]
lemma rbbl_Cons:
- "b # rev (bin_to_bl n x) = rev (bin_to_bl (Suc n) (x BIT If b 1 0))"
+ "b # rev (bin_to_bl n x) = rev (bin_to_bl (Suc n) (x BIT b))"
apply (unfold bin_to_bl_def)
apply simp
apply (simp add: bin_to_bl_aux_alt)
--- a/src/HOL/Word/Misc_Numeric.thy Mon Dec 23 15:30:31 2013 +0100
+++ b/src/HOL/Word/Misc_Numeric.thy Mon Dec 23 16:59:56 2013 +0100
@@ -22,6 +22,11 @@
lemmas min_minus' [simp] = trans [OF min_max.inf_commute min_minus]
+lemma mod_2_neq_1_eq_eq_0:
+ fixes k :: int
+ shows "k mod 2 \<noteq> 1 \<longleftrightarrow> k mod 2 = 0"
+ by arith
+
lemma z1pmod2:
"(2 * b + 1) mod 2 = (1::int)" by arith
--- a/src/HOL/Word/Word.thy Mon Dec 23 15:30:31 2013 +0100
+++ b/src/HOL/Word/Word.thy Mon Dec 23 16:59:56 2013 +0100
@@ -35,7 +35,8 @@
shows "uint w mod 2 ^ len_of TYPE('a) = uint w"
using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial)
-definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" where
+definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word"
+where
-- {* representation of words using unsigned or signed bins,
only difference in these is the type class *}
"word_of_int k = Abs_word (k mod 2 ^ len_of TYPE('a))"
@@ -72,7 +73,8 @@
instantiation word :: (len0) equal
begin
-definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where
+definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
+where
"equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"
instance proof
@@ -101,31 +103,39 @@
subsection {* Type conversions and casting *}
-definition sint :: "'a :: len word => int" where
+definition sint :: "'a :: len word => int"
+where
-- {* treats the most-significant-bit as a sign bit *}
sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)"
-definition unat :: "'a :: len0 word => nat" where
+definition unat :: "'a :: len0 word => nat"
+where
"unat w = nat (uint w)"
-definition uints :: "nat => int set" where
+definition uints :: "nat => int set"
+where
-- "the sets of integers representing the words"
"uints n = range (bintrunc n)"
-definition sints :: "nat => int set" where
+definition sints :: "nat => int set"
+where
"sints n = range (sbintrunc (n - 1))"
-definition unats :: "nat => nat set" where
+definition unats :: "nat => nat set"
+where
"unats n = {i. i < 2 ^ n}"
-definition norm_sint :: "nat => int => int" where
+definition norm_sint :: "nat => int => int"
+where
"norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)"
-definition scast :: "'a :: len word => 'b :: len word" where
+definition scast :: "'a :: len word => 'b :: len word"
+where
-- "cast a word to a different length"
"scast w = word_of_int (sint w)"
-definition ucast :: "'a :: len0 word => 'b :: len0 word" where
+definition ucast :: "'a :: len0 word => 'b :: len0 word"
+where
"ucast w = word_of_int (uint w)"
instantiation word :: (len0) size
@@ -138,29 +148,37 @@
end
-definition source_size :: "('a :: len0 word => 'b) => nat" where
+definition source_size :: "('a :: len0 word => 'b) => nat"
+where
-- "whether a cast (or other) function is to a longer or shorter length"
"source_size c = (let arb = undefined ; x = c arb in size arb)"
-definition target_size :: "('a => 'b :: len0 word) => nat" where
+definition target_size :: "('a => 'b :: len0 word) => nat"
+where
"target_size c = size (c undefined)"
-definition is_up :: "('a :: len0 word => 'b :: len0 word) => bool" where
+definition is_up :: "('a :: len0 word => 'b :: len0 word) => bool"
+where
"is_up c \<longleftrightarrow> source_size c <= target_size c"
-definition is_down :: "('a :: len0 word => 'b :: len0 word) => bool" where
+definition is_down :: "('a :: len0 word => 'b :: len0 word) => bool"
+where
"is_down c \<longleftrightarrow> target_size c <= source_size c"
-definition of_bl :: "bool list => 'a :: len0 word" where
+definition of_bl :: "bool list => 'a :: len0 word"
+where
"of_bl bl = word_of_int (bl_to_bin bl)"
-definition to_bl :: "'a :: len0 word => bool list" where
+definition to_bl :: "'a :: len0 word => bool list"
+where
"to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)"
-definition word_reverse :: "'a :: len0 word => 'a word" where
+definition word_reverse :: "'a :: len0 word => 'a word"
+where
"word_reverse w = of_bl (rev (to_bl w))"
-definition word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b" where
+definition word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b"
+where
"word_int_case f w = f (uint w)"
translations
@@ -232,7 +250,8 @@
subsection {* Correspondence relation for theorem transfer *}
definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool"
- where "cr_word \<equiv> (\<lambda>x y. word_of_int x = y)"
+where
+ "cr_word = (\<lambda>x y. word_of_int x = y)"
lemma Quotient_word:
"Quotient (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y)
@@ -341,7 +360,8 @@
apply (simp add: word_of_nat wi_hom_sub)
done
-definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where
+definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50)
+where
"a udvd b = (EX n>=0. uint b = n * uint a)"
@@ -361,10 +381,12 @@
end
-definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where
+definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50)
+where
"a <=s b = (sint a <= sint b)"
-definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where
+definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50)
+where
"(x <s y) = (x <=s y & x ~= y)"
@@ -390,18 +412,20 @@
definition
word_set_bit_def: "set_bit a n x =
- word_of_int (bin_sc n (If x 1 0) (uint a))"
+ word_of_int (bin_sc n x (uint a))"
definition
word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"
definition
- word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1"
-
-definition shiftl1 :: "'a word \<Rightarrow> 'a word" where
- "shiftl1 w = word_of_int (uint w BIT 0)"
-
-definition shiftr1 :: "'a word \<Rightarrow> 'a word" where
+ word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a)"
+
+definition shiftl1 :: "'a word \<Rightarrow> 'a word"
+where
+ "shiftl1 w = word_of_int (uint w BIT False)"
+
+definition shiftr1 :: "'a word \<Rightarrow> 'a word"
+where
-- "shift right as unsigned or as signed, ie logical or arithmetic"
"shiftr1 w = word_of_int (bin_rest (uint w))"
@@ -434,76 +458,95 @@
end
-definition setBit :: "'a :: len0 word => nat => 'a word" where
+definition setBit :: "'a :: len0 word => nat => 'a word"
+where
"setBit w n = set_bit w n True"
-definition clearBit :: "'a :: len0 word => nat => 'a word" where
+definition clearBit :: "'a :: len0 word => nat => 'a word"
+where
"clearBit w n = set_bit w n False"
subsection "Shift operations"
-definition sshiftr1 :: "'a :: len word => 'a word" where
+definition sshiftr1 :: "'a :: len word => 'a word"
+where
"sshiftr1 w = word_of_int (bin_rest (sint w))"
-definition bshiftr1 :: "bool => 'a :: len word => 'a word" where
+definition bshiftr1 :: "bool => 'a :: len word => 'a word"
+where
"bshiftr1 b w = of_bl (b # butlast (to_bl w))"
-definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where
+definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55)
+where
"w >>> n = (sshiftr1 ^^ n) w"
-definition mask :: "nat => 'a::len word" where
+definition mask :: "nat => 'a::len word"
+where
"mask n = (1 << n) - 1"
-definition revcast :: "'a :: len0 word => 'b :: len0 word" where
+definition revcast :: "'a :: len0 word => 'b :: len0 word"
+where
"revcast w = of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
-definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where
+definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word"
+where
"slice1 n w = of_bl (takefill False n (to_bl w))"
-definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where
+definition slice :: "nat => 'a :: len0 word => 'b :: len0 word"
+where
"slice n w = slice1 (size w - n) w"
subsection "Rotation"
-definition rotater1 :: "'a list => 'a list" where
+definition rotater1 :: "'a list => 'a list"
+where
"rotater1 ys =
(case ys of [] => [] | x # xs => last ys # butlast ys)"
-definition rotater :: "nat => 'a list => 'a list" where
+definition rotater :: "nat => 'a list => 'a list"
+where
"rotater n = rotater1 ^^ n"
-definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where
+definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word"
+where
"word_rotr n w = of_bl (rotater n (to_bl w))"
-definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where
+definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word"
+where
"word_rotl n w = of_bl (rotate n (to_bl w))"
-definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where
+definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word"
+where
"word_roti i w = (if i >= 0 then word_rotr (nat i) w
else word_rotl (nat (- i)) w)"
subsection "Split and cat operations"
-definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where
+definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word"
+where
"word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
-definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where
+definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)"
+where
"word_split a =
(case bin_split (len_of TYPE ('c)) (uint a) of
(u, v) => (word_of_int u, word_of_int v))"
-definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where
+definition word_rcat :: "'a :: len0 word list => 'b :: len0 word"
+where
"word_rcat ws =
word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
-definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where
+definition word_rsplit :: "'a :: len0 word => 'b :: len word list"
+where
"word_rsplit w =
map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
-definition max_word :: "'a::len word" -- "Largest representable machine integer." where
+definition max_word :: "'a::len word" -- "Largest representable machine integer."
+where
"max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
(* FIXME: only provide one theorem name *)
@@ -2524,15 +2567,15 @@
by (clarsimp simp add: nth_sbintr word_test_bit_def [symmetric])
lemma word_lsb_numeral [simp]:
- "lsb (numeral bin :: 'a :: len word) = (bin_last (numeral bin) = 1)"
+ "lsb (numeral bin :: 'a :: len word) \<longleftrightarrow> bin_last (numeral bin)"
unfolding word_lsb_alt test_bit_numeral by simp
lemma word_lsb_neg_numeral [simp]:
- "lsb (- numeral bin :: 'a :: len word) = (bin_last (- numeral bin) = 1)"
+ "lsb (- numeral bin :: 'a :: len word) \<longleftrightarrow> bin_last (- numeral bin)"
unfolding word_lsb_alt test_bit_neg_numeral by simp
lemma set_bit_word_of_int:
- "set_bit (word_of_int x) n b = word_of_int (bin_sc n (if b then 1 else 0) x)"
+ "set_bit (word_of_int x) n b = word_of_int (bin_sc n b x)"
unfolding word_set_bit_def
apply (rule word_eqI)
apply (simp add: word_size bin_nth_sc_gen word_ubin.eq_norm nth_bintr)
@@ -2540,28 +2583,28 @@
lemma word_set_numeral [simp]:
"set_bit (numeral bin::'a::len0 word) n b =
- word_of_int (bin_sc n (if b then 1 else 0) (numeral bin))"
+ word_of_int (bin_sc n b (numeral bin))"
unfolding word_numeral_alt by (rule set_bit_word_of_int)
lemma word_set_neg_numeral [simp]:
"set_bit (- numeral bin::'a::len0 word) n b =
- word_of_int (bin_sc n (if b then 1 else 0) (- numeral bin))"
+ word_of_int (bin_sc n b (- numeral bin))"
unfolding word_neg_numeral_alt by (rule set_bit_word_of_int)
lemma word_set_bit_0 [simp]:
- "set_bit 0 n b = word_of_int (bin_sc n (if b then 1 else 0) 0)"
+ "set_bit 0 n b = word_of_int (bin_sc n b 0)"
unfolding word_0_wi by (rule set_bit_word_of_int)
lemma word_set_bit_1 [simp]:
- "set_bit 1 n b = word_of_int (bin_sc n (if b then 1 else 0) 1)"
+ "set_bit 1 n b = word_of_int (bin_sc n b 1)"
unfolding word_1_wi by (rule set_bit_word_of_int)
lemma setBit_no [simp]:
- "setBit (numeral bin) n = word_of_int (bin_sc n 1 (numeral bin))"
+ "setBit (numeral bin) n = word_of_int (bin_sc n True (numeral bin))"
by (simp add: setBit_def)
lemma clearBit_no [simp]:
- "clearBit (numeral bin) n = word_of_int (bin_sc n 0 (numeral bin))"
+ "clearBit (numeral bin) n = word_of_int (bin_sc n False (numeral bin))"
by (simp add: clearBit_def)
lemma to_bl_n1:
@@ -2645,7 +2688,6 @@
fixes w :: "'a::len0 word"
shows "w >= set_bit w n False"
apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm)
- apply simp
apply (rule order_trans)
apply (rule bintr_bin_clr_le)
apply simp
@@ -2655,7 +2697,6 @@
fixes w :: "'a::len word"
shows "w <= set_bit w n True"
apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm)
- apply simp
apply (rule order_trans [OF _ bintr_bin_set_ge])
apply simp
done
@@ -2663,7 +2704,7 @@
subsection {* Shifting, Rotating, and Splitting Words *}
-lemma shiftl1_wi [simp]: "shiftl1 (word_of_int w) = word_of_int (w BIT 0)"
+lemma shiftl1_wi [simp]: "shiftl1 (word_of_int w) = word_of_int (w BIT False)"
unfolding shiftl1_def
apply (simp add: word_ubin.norm_eq_iff [symmetric] word_ubin.eq_norm)
apply (subst refl [THEN bintrunc_BIT_I, symmetric])
@@ -2682,10 +2723,10 @@
lemma shiftl1_0 [simp] : "shiftl1 0 = 0"
unfolding shiftl1_def by simp
-lemma shiftl1_def_u: "shiftl1 w = word_of_int (uint w BIT 0)"
+lemma shiftl1_def_u: "shiftl1 w = word_of_int (uint w BIT False)"
by (simp only: shiftl1_def) (* FIXME: duplicate *)
-lemma shiftl1_def_s: "shiftl1 w = word_of_int (sint w BIT 0)"
+lemma shiftl1_def_s: "shiftl1 w = word_of_int (sint w BIT False)"
unfolding shiftl1_def Bit_B0 wi_hom_syms by simp
lemma shiftr1_0 [simp]: "shiftr1 0 = 0"
@@ -3969,9 +4010,6 @@
subsubsection "map, map2, commuting with rotate(r)"
-lemma last_map: "xs ~= [] \<Longrightarrow> last (map f xs) = f (last xs)"
- by (induct xs) auto
-
lemma butlast_map:
"xs ~= [] \<Longrightarrow> butlast (map f xs) = map f (butlast xs)"
by (induct xs) auto
@@ -4538,7 +4576,8 @@
subsection {* Recursion combinator for words *}
-definition word_rec :: "'a \<Rightarrow> ('b::len word \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'b word \<Rightarrow> 'a" where
+definition word_rec :: "'a \<Rightarrow> ('b::len word \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'b word \<Rightarrow> 'a"
+where
"word_rec forZero forSuc n = nat_rec forZero (forSuc \<circ> of_nat) (unat n)"
lemma word_rec_0: "word_rec z s 0 = z"
--- a/src/HOL/Word/WordBitwise.thy Mon Dec 23 15:30:31 2013 +0100
+++ b/src/HOL/Word/WordBitwise.thy Mon Dec 23 16:59:56 2013 +0100
@@ -364,8 +364,7 @@
= (bl_to_bin (rev xs) < bl_to_bin (rev ys))"
apply (induct xs ys rule: list_induct2)
apply (simp_all add: rev_bl_order_simps bl_to_bin_app_cat)
- apply (simp add: bl_to_bin_def Bit_B0 Bit_B1 add1_zle_eq)
- apply arith?
+ apply (auto simp add: bl_to_bin_def Bit_B0 Bit_B1 add1_zle_eq Bit_def)
done
lemma word_le_rbl: