prefer "Bits" as theory name for abstract bit operations, similar to "Orderings", "Lattices", "Groups" etc.
--- a/src/HOL/Word/Bit_Bit.thy Mon Dec 23 16:29:43 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,73 +0,0 @@
-(* Title: HOL/Word/Bit_Bit.thy
- Author: Author: Brian Huffman, PSU and Gerwin Klein, NICTA
-*)
-
-header {* Bit operations in $\cal Z_2$ *}
-
-theory Bit_Bit
-imports Bit_Operations "~~/src/HOL/Library/Bit"
-begin
-
-instantiation bit :: bit
-begin
-
-primrec bitNOT_bit where
- "NOT 0 = (1::bit)"
- | "NOT 1 = (0::bit)"
-
-primrec bitAND_bit where
- "0 AND y = (0::bit)"
- | "1 AND y = (y::bit)"
-
-primrec bitOR_bit where
- "0 OR y = (y::bit)"
- | "1 OR y = (1::bit)"
-
-primrec bitXOR_bit where
- "0 XOR y = (y::bit)"
- | "1 XOR y = (NOT y :: bit)"
-
-instance ..
-
-end
-
-lemmas bit_simps =
- bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simps
-
-lemma bit_extra_simps [simp]:
- "x AND 0 = (0::bit)"
- "x AND 1 = (x::bit)"
- "x OR 1 = (1::bit)"
- "x OR 0 = (x::bit)"
- "x XOR 1 = NOT (x::bit)"
- "x XOR 0 = (x::bit)"
- by (cases x, auto)+
-
-lemma bit_ops_comm:
- "(x::bit) AND y = y AND x"
- "(x::bit) OR y = y OR x"
- "(x::bit) XOR y = y XOR x"
- by (cases y, auto)+
-
-lemma bit_ops_same [simp]:
- "(x::bit) AND x = x"
- "(x::bit) OR x = x"
- "(x::bit) XOR x = 0"
- by (cases x, auto)+
-
-lemma bit_not_not [simp]: "NOT (NOT (x::bit)) = x"
- by (cases x) auto
-
-lemma bit_or_def: "(b::bit) OR c = NOT (NOT b AND NOT c)"
- by (induct b, simp_all)
-
-lemma bit_xor_def: "(b::bit) XOR c = (b AND NOT c) OR (NOT b AND c)"
- by (induct b, simp_all)
-
-lemma bit_NOT_eq_1_iff [simp]: "NOT (b::bit) = 1 \<longleftrightarrow> b = 0"
- by (induct b, simp_all)
-
-lemma bit_AND_eq_1_iff [simp]: "(a::bit) AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1"
- by (induct a, simp_all)
-
-end
--- a/src/HOL/Word/Bit_Comparison.thy Mon Dec 23 16:29:43 2013 +0100
+++ b/src/HOL/Word/Bit_Comparison.thy Mon Dec 23 18:37:51 2013 +0100
@@ -6,7 +6,7 @@
*)
theory Bit_Comparison
-imports Type_Length Bit_Operations Bit_Int
+imports Bits_Int
begin
lemma AND_lower [simp]:
--- a/src/HOL/Word/Bit_Int.thy Mon Dec 23 16:29:43 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,681 +0,0 @@
-(*
- Author: Jeremy Dawson and Gerwin Klein, NICTA
-
- Definitions and basic theorems for bit-wise logical operations
- for integers expressed using Pls, Min, BIT,
- and converting them to and from lists of bools.
-*)
-
-header {* Bitwise Operations on Binary Integers *}
-
-theory Bit_Int
-imports Bit_Representation Bit_Operations
-begin
-
-subsection {* Logical operations *}
-
-text "bit-wise logical operations on the int type"
-
-instantiation int :: bit
-begin
-
-definition int_not_def:
- "bitNOT = (\<lambda>x::int. - x - 1)"
-
-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))"
- by pat_completeness simp
-
-termination
- by (relation "measure (nat o abs o fst)", simp_all add: bin_rest_def)
-
-declare bitAND_int.simps [simp del]
-
-definition int_or_def:
- "bitOR = (\<lambda>x y::int. NOT (NOT x AND NOT y))"
-
-definition int_xor_def:
- "bitXOR = (\<lambda>x y::int. (x AND NOT y) OR (NOT x AND y))"
-
-instance ..
-
-end
-
-subsubsection {* Basic simplification rules *}
-
-lemma int_not_BIT [simp]:
- "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]:
- "NOT (0::int) = -1"
- "NOT (1::int) = -2"
- "NOT (- 1::int) = 0"
- "NOT (numeral w::int) = - numeral (w + Num.One)"
- "NOT (- numeral (Num.Bit0 w)::int) = numeral (Num.BitM w)"
- "NOT (- numeral (Num.Bit1 w)::int) = numeral (Num.Bit0 w)"
- unfolding int_not_def by simp_all
-
-lemma int_not_not [simp]: "NOT (NOT (x::int)) = x"
- unfolding int_not_def by simp
-
-lemma int_and_0 [simp]: "(0::int) AND x = 0"
- by (simp add: bitAND_int.simps)
-
-lemma int_and_m1 [simp]: "(-1::int) AND x = x"
- 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)"
- 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"
- unfolding int_or_def by simp
-
-lemma int_or_minus1 [simp]: "(-1::int) OR x = -1"
- 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 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 \<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) \<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) \<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) \<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) \<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:
- "!!x y. bin_nth (x AND y) n = (bin_nth x n & bin_nth y n)"
- "!!x y. bin_nth (x OR y) n = (bin_nth x n | bin_nth y n)"
- "!!x y. bin_nth (x XOR y) n = (bin_nth x n ~= bin_nth y n)"
- "!!x. bin_nth (NOT x) n = (~ bin_nth x n)"
- by (induct n) auto
-
-subsubsection {* Derived properties *}
-
-lemma int_xor_minus1 [simp]: "(-1::int) XOR x = NOT x"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma int_xor_extra_simps [simp]:
- "w XOR (0::int) = w"
- "w XOR (-1::int) = NOT w"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma int_or_extra_simps [simp]:
- "w OR (0::int) = w"
- "w OR (-1::int) = -1"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma int_and_extra_simps [simp]:
- "w AND (0::int) = 0"
- "w AND (-1::int) = w"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-(* commutativity of the above *)
-lemma bin_ops_comm:
- shows
- int_and_comm: "!!y::int. x AND y = y AND x" and
- int_or_comm: "!!y::int. x OR y = y OR x" and
- int_xor_comm: "!!y::int. x XOR y = y XOR x"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma bin_ops_same [simp]:
- "(x::int) AND x = x"
- "(x::int) OR x = x"
- "(x::int) XOR x = 0"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemmas bin_log_esimps =
- int_and_extra_simps int_or_extra_simps int_xor_extra_simps
- int_and_0 int_and_m1 int_or_zero int_or_minus1 int_xor_zero int_xor_minus1
-
-(* basic properties of logical (bit-wise) operations *)
-
-lemma bbw_ao_absorb:
- "!!y::int. x AND (y OR x) = x & x OR (y AND x) = x"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma bbw_ao_absorbs_other:
- "x AND (x OR y) = x \<and> (y AND x) OR x = (x::int)"
- "(y OR x) AND x = x \<and> x OR (x AND y) = (x::int)"
- "(x OR y) AND x = x \<and> (x AND y) OR x = (x::int)"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemmas bbw_ao_absorbs [simp] = bbw_ao_absorb bbw_ao_absorbs_other
-
-lemma int_xor_not:
- "!!y::int. (NOT x) XOR y = NOT (x XOR y) &
- x XOR (NOT y) = NOT (x XOR y)"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma int_and_assoc:
- "(x AND y) AND (z::int) = x AND (y AND z)"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma int_or_assoc:
- "(x OR y) OR (z::int) = x OR (y OR z)"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma int_xor_assoc:
- "(x XOR y) XOR (z::int) = x XOR (y XOR z)"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc
-
-(* BH: Why are these declared as simp rules??? *)
-lemma bbw_lcs [simp]:
- "(y::int) AND (x AND z) = x AND (y AND z)"
- "(y::int) OR (x OR z) = x OR (y OR z)"
- "(y::int) XOR (x XOR z) = x XOR (y XOR z)"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma bbw_not_dist:
- "!!y::int. NOT (x OR y) = (NOT x) AND (NOT y)"
- "!!y::int. NOT (x AND y) = (NOT x) OR (NOT y)"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma bbw_oa_dist:
- "!!y z::int. (x AND y) OR z =
- (x OR z) AND (y OR z)"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-lemma bbw_ao_dist:
- "!!y z::int. (x OR y) AND z =
- (x AND z) OR (y AND z)"
- by (auto simp add: bin_eq_iff bin_nth_ops)
-
-(*
-Why were these declared simp???
-declare bin_ops_comm [simp] bbw_assocs [simp]
-*)
-
-subsubsection {* Simplification with numerals *}
-
-text {* Cases for @{text "0"} and @{text "-1"} are already covered by
- 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 (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))"
- 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 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"
- "(1::int) AND - numeral (Num.Bit1 y) = 1"
- "numeral (Num.Bit0 x) AND (1::int) = 0"
- "numeral (Num.Bit1 x) AND (1::int) = 1"
- "- numeral (Num.Bit0 x) AND (1::int) = 0"
- "- numeral (Num.Bit1 x) AND (1::int) = 1"
- 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 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)"
- "(1::int) OR - numeral (Num.Bit1 y) = - numeral (Num.Bit1 y)"
- "numeral (Num.Bit0 x) OR (1::int) = numeral (Num.Bit1 x)"
- "numeral (Num.Bit1 x) OR (1::int) = numeral (Num.Bit1 x)"
- "- numeral (Num.Bit0 x) OR (1::int) = - numeral (Num.BitM x)"
- "- numeral (Num.Bit1 x) OR (1::int) = - numeral (Num.Bit1 x)"
- 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 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)"
- "(1::int) XOR - numeral (Num.Bit1 y) = - numeral (Num.Bit0 (y + Num.One))"
- "numeral (Num.Bit0 x) XOR (1::int) = numeral (Num.Bit1 x)"
- "numeral (Num.Bit1 x) XOR (1::int) = numeral (Num.Bit0 x)"
- "- numeral (Num.Bit0 x) XOR (1::int) = - numeral (Num.BitM x)"
- "- numeral (Num.Bit1 x) XOR (1::int) = - numeral (Num.Bit0 (x + Num.One))"
- by (rule bin_rl_eqI, simp, simp)+
-
-subsubsection {* Interactions with arithmetic *}
-
-lemma plus_and_or [rule_format]:
- "ALL y::int. (x AND y) + (x OR y) = x + y"
- apply (induct x rule: bin_induct)
- apply clarsimp
- apply clarsimp
- apply clarsimp
- apply (case_tac y rule: bin_exhaust)
- apply clarsimp
- apply (unfold Bit_def)
- apply clarsimp
- apply (erule_tac x = "x" in allE)
- apply simp
- done
-
-lemma le_int_or:
- "bin_sign (y::int) = 0 ==> x <= x OR y"
- apply (induct y arbitrary: x rule: bin_induct)
- apply clarsimp
- apply clarsimp
- apply (case_tac x rule: bin_exhaust)
- apply (case_tac b)
- apply (case_tac [!] bit)
- apply (auto simp: le_Bits)
- done
-
-lemmas int_and_le =
- xtrans(3) [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or]
-
-(* interaction between bit-wise and arithmetic *)
-(* good example of bin_induction *)
-lemma bin_add_not: "x + NOT x = (-1::int)"
- apply (induct x rule: bin_induct)
- apply clarsimp
- apply clarsimp
- apply (case_tac bit, auto)
- done
-
-subsubsection {* Truncating results of bit-wise operations *}
-
-lemma bin_trunc_ao:
- "!!x y. (bintrunc n x) AND (bintrunc n y) = bintrunc n (x AND y)"
- "!!x y. (bintrunc n x) OR (bintrunc n y) = bintrunc n (x OR y)"
- by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr)
-
-lemma bin_trunc_xor:
- "!!x y. bintrunc n (bintrunc n x XOR bintrunc n y) =
- bintrunc n (x XOR y)"
- by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr)
-
-lemma bin_trunc_not:
- "!!x. bintrunc n (NOT (bintrunc n x)) = bintrunc n (NOT x)"
- by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr)
-
-(* want theorems of the form of bin_trunc_xor *)
-lemma bintr_bintr_i:
- "x = bintrunc n y ==> bintrunc n x = bintrunc n y"
- by auto
-
-lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i]
-lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i]
-
-subsection {* Setting and clearing bits *}
-
-primrec
- 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"
-
-(** nth bit, set/clear **)
-
-lemma bin_nth_sc [simp]:
- "bin_nth (bin_sc n b w) n \<longleftrightarrow> b"
- by (induct n arbitrary: w) auto
-
-lemma bin_sc_sc_same [simp]:
- "bin_sc n c (bin_sc n b w) = bin_sc n c w"
- by (induct n arbitrary: w) auto
-
-lemma bin_sc_sc_diff:
- "m ~= n ==>
- bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)"
- apply (induct n arbitrary: w m)
- apply (case_tac [!] m)
- apply auto
- done
-
-lemma bin_nth_sc_gen:
- "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 (bin_nth w n) w) = w"
- by (induct n arbitrary: w) auto
-
-lemma bin_sign_sc [simp]:
- "bin_sign (bin_sc n b w) = bin_sign w"
- by (induct n arbitrary: w) auto
-
-lemma bin_sc_bintr [simp]:
- "bintrunc m (bin_sc n x (bintrunc m (w))) = bintrunc m (bin_sc n x w)"
- apply (induct n arbitrary: w m)
- apply (case_tac [!] w rule: bin_exhaust)
- apply (case_tac [!] m, auto)
- done
-
-lemma bin_clr_le:
- "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 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 False w) <= bintrunc n w"
- apply (induct n arbitrary: w m)
- apply simp
- apply (case_tac w rule: bin_exhaust)
- apply (case_tac m)
- apply (auto simp: le_Bits)
- done
-
-lemma bintr_bin_set_ge:
- "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)
- apply (case_tac m)
- apply (auto simp: le_Bits)
- done
-
-lemma bin_sc_FP [simp]: "bin_sc n False 0 = 0"
- by (induct n) auto
-
-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
-
-lemma bin_sc_minus:
- "0 < n ==> bin_sc (Suc (n - 1)) b w = bin_sc n b w"
- by auto
-
-lemmas bin_sc_Suc_minus =
- trans [OF bin_sc_minus [symmetric] bin_sc.Suc]
-
-lemma bin_sc_numeral [simp]:
- "bin_sc (numeral k) b w =
- bin_sc (pred_numeral k) b (bin_rest w) BIT bin_last w"
- by (simp add: numeral_eq_Suc)
-
-
-subsection {* Splitting and concatenation *}
-
-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
- "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
- "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
- "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
- "bin_rsplitl n w = bin_rsplitl_aux n (fst w) (snd w) []"
-
-declare bin_rsplit_aux.simps [simp del]
-declare bin_rsplitl_aux.simps [simp del]
-
-lemma bin_sign_cat:
- "bin_sign (bin_cat x n y) = bin_sign x"
- by (induct n arbitrary: y) auto
-
-lemma bin_cat_Suc_Bit:
- "bin_cat w (Suc n) (v BIT b) = bin_cat w n v BIT b"
- by auto
-
-lemma bin_nth_cat:
- "bin_nth (bin_cat x k y) n =
- (if n < k then bin_nth y n else bin_nth x (n - k))"
- apply (induct k arbitrary: n y)
- apply clarsimp
- apply (case_tac n, auto)
- done
-
-lemma bin_nth_split:
- "bin_split n c = (a, b) ==>
- (ALL k. bin_nth a k = bin_nth c (n + k)) &
- (ALL k. bin_nth b k = (k < n & bin_nth c k))"
- apply (induct n arbitrary: b c)
- apply clarsimp
- apply (clarsimp simp: Let_def split: prod.split_asm)
- apply (case_tac k)
- apply auto
- done
-
-lemma bin_cat_assoc:
- "bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)"
- by (induct n arbitrary: z) auto
-
-lemma bin_cat_assoc_sym:
- "bin_cat x m (bin_cat y n z) = bin_cat (bin_cat x (m - n) y) (min m n) z"
- apply (induct n arbitrary: z m, clarsimp)
- apply (case_tac m, auto)
- done
-
-lemma bin_cat_zero [simp]: "bin_cat 0 n w = bintrunc n w"
- by (induct n arbitrary: w) auto
-
-lemma bintr_cat1:
- "bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b"
- by (induct n arbitrary: b) auto
-
-lemma bintr_cat: "bintrunc m (bin_cat a n b) =
- bin_cat (bintrunc (m - n) a) n (bintrunc (min m n) b)"
- by (rule bin_eqI) (auto simp: bin_nth_cat nth_bintr)
-
-lemma bintr_cat_same [simp]:
- "bintrunc n (bin_cat a n b) = bintrunc n b"
- by (auto simp add : bintr_cat)
-
-lemma cat_bintr [simp]:
- "bin_cat a n (bintrunc n b) = bin_cat a n b"
- by (induct n arbitrary: b) auto
-
-lemma split_bintrunc:
- "bin_split n c = (a, b) ==> b = bintrunc n c"
- by (induct n arbitrary: b c) (auto simp: Let_def split: prod.split_asm)
-
-lemma bin_cat_split:
- "bin_split n w = (u, v) ==> w = bin_cat u n v"
- by (induct n arbitrary: v w) (auto simp: Let_def split: prod.split_asm)
-
-lemma bin_split_cat:
- "bin_split n (bin_cat v n w) = (v, bintrunc n w)"
- by (induct n arbitrary: w) auto
-
-lemma bin_split_zero [simp]: "bin_split n 0 = (0, 0)"
- by (induct n) auto
-
-lemma bin_split_minus1 [simp]:
- "bin_split n -1 = (-1, bintrunc n -1)"
- by (induct n) auto
-
-lemma bin_split_trunc:
- "bin_split (min m n) c = (a, b) ==>
- bin_split n (bintrunc m c) = (bintrunc (m - n) a, b)"
- apply (induct n arbitrary: m b c, clarsimp)
- apply (simp add: bin_rest_trunc Let_def split: prod.split_asm)
- apply (case_tac m)
- apply (auto simp: Let_def split: prod.split_asm)
- done
-
-lemma bin_split_trunc1:
- "bin_split n c = (a, b) ==>
- bin_split n (bintrunc m c) = (bintrunc (m - n) a, bintrunc m b)"
- apply (induct n arbitrary: m b c, clarsimp)
- apply (simp add: bin_rest_trunc Let_def split: prod.split_asm)
- apply (case_tac m)
- apply (auto simp: Let_def split: prod.split_asm)
- done
-
-lemma bin_cat_num:
- "bin_cat a n b = a * 2 ^ n + bintrunc n b"
- apply (induct n arbitrary: b, clarsimp)
- apply (simp add: Bit_def)
- done
-
-lemma bin_split_num:
- "bin_split n b = (b div 2 ^ n, b mod 2 ^ n)"
- apply (induct n arbitrary: b, simp)
- 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)
- done
-
-subsection {* Miscellaneous lemmas *}
-
-lemma nth_2p_bin:
- "bin_nth (2 ^ n) m = (m = n)"
- apply (induct n arbitrary: m)
- apply clarsimp
- apply safe
- apply (case_tac m)
- apply (auto simp: Bit_B0_2t [symmetric])
- done
-
-(* for use when simplifying with bin_nth_Bit *)
-
-lemma ex_eq_or:
- "(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 True"
- unfolding Bit_B1
- by (induct n) simp_all
-
-lemma mod_BIT:
- "bin BIT bit mod 2 ^ Suc n = (bin mod 2 ^ n) BIT bit"
-proof -
- have "bin mod 2 ^ n < 2 ^ n" by simp
- then have "bin mod 2 ^ n \<le> 2 ^ n - 1" by simp
- then have "2 * (bin mod 2 ^ n) \<le> 2 * (2 ^ n - 1)"
- 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 mod_mult_mult1 mod_add_left_eq [of "2 * bin"]
- mod_pos_pos_trivial)
-qed
-
-lemma AND_mod:
- fixes x :: int
- shows "x AND 2 ^ n - 1 = x mod 2 ^ n"
-proof (induct x arbitrary: n rule: bin_induct)
- case 1
- then show ?case
- by simp
-next
- case 2
- then show ?case
- by (simp, simp add: m1mod2k)
-next
- case (3 bin bit)
- show ?case
- proof (cases n)
- case 0
- then show ?thesis by simp
- next
- case (Suc m)
- with 3 show ?thesis
- by (simp only: power_BIT mod_BIT int_and_Bits) simp
- qed
-qed
-
-end
-
--- a/src/HOL/Word/Bit_Operations.thy Mon Dec 23 16:29:43 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,39 +0,0 @@
-(* Title: HOL/Word/Bit_Operations.thy
- Author: Author: Brian Huffman, PSU and Gerwin Klein, NICTA
-*)
-
-header {* Syntactic classes for bitwise operations *}
-
-theory Bit_Operations
-imports Main
-begin
-
-class bit =
- fixes bitNOT :: "'a \<Rightarrow> 'a" ("NOT _" [70] 71)
- and bitAND :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "AND" 64)
- and bitOR :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "OR" 59)
- and bitXOR :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "XOR" 59)
-
-text {*
- We want the bitwise operations to bind slightly weaker
- than @{text "+"} and @{text "-"}, but @{text "~~"} to
- bind slightly stronger than @{text "*"}.
-*}
-
-text {*
- Testing and shifting operations.
-*}
-
-class bits = bit +
- fixes test_bit :: "'a \<Rightarrow> nat \<Rightarrow> bool" (infixl "!!" 100)
- and lsb :: "'a \<Rightarrow> bool"
- and set_bit :: "'a \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"
- and set_bits :: "(nat \<Rightarrow> bool) \<Rightarrow> 'a" (binder "BITS " 10)
- and shiftl :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixl "<<" 55)
- and shiftr :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixl ">>" 55)
-
-class bitss = bits +
- fixes msb :: "'a \<Rightarrow> bool"
-
-end
-
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Word/Bits.thy Mon Dec 23 18:37:51 2013 +0100
@@ -0,0 +1,39 @@
+(* Title: HOL/Word/Bit_Operations.thy
+ Author: Author: Brian Huffman, PSU and Gerwin Klein, NICTA
+*)
+
+header {* Syntactic classes for bitwise operations *}
+
+theory Bits
+imports Main
+begin
+
+class bit =
+ fixes bitNOT :: "'a \<Rightarrow> 'a" ("NOT _" [70] 71)
+ and bitAND :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "AND" 64)
+ and bitOR :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "OR" 59)
+ and bitXOR :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "XOR" 59)
+
+text {*
+ We want the bitwise operations to bind slightly weaker
+ than @{text "+"} and @{text "-"}, but @{text "~~"} to
+ bind slightly stronger than @{text "*"}.
+*}
+
+text {*
+ Testing and shifting operations.
+*}
+
+class bits = bit +
+ fixes test_bit :: "'a \<Rightarrow> nat \<Rightarrow> bool" (infixl "!!" 100)
+ and lsb :: "'a \<Rightarrow> bool"
+ and set_bit :: "'a \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"
+ and set_bits :: "(nat \<Rightarrow> bool) \<Rightarrow> 'a" (binder "BITS " 10)
+ and shiftl :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixl "<<" 55)
+ and shiftr :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixl ">>" 55)
+
+class bitss = bits +
+ fixes msb :: "'a \<Rightarrow> bool"
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Word/Bits_Bit.thy Mon Dec 23 18:37:51 2013 +0100
@@ -0,0 +1,73 @@
+(* Title: HOL/Word/Bit_Bit.thy
+ Author: Author: Brian Huffman, PSU and Gerwin Klein, NICTA
+*)
+
+header {* Bit operations in $\cal Z_2$ *}
+
+theory Bits_Bit
+imports Bits "~~/src/HOL/Library/Bit"
+begin
+
+instantiation bit :: bit
+begin
+
+primrec bitNOT_bit where
+ "NOT 0 = (1::bit)"
+ | "NOT 1 = (0::bit)"
+
+primrec bitAND_bit where
+ "0 AND y = (0::bit)"
+ | "1 AND y = (y::bit)"
+
+primrec bitOR_bit where
+ "0 OR y = (y::bit)"
+ | "1 OR y = (1::bit)"
+
+primrec bitXOR_bit where
+ "0 XOR y = (y::bit)"
+ | "1 XOR y = (NOT y :: bit)"
+
+instance ..
+
+end
+
+lemmas bit_simps =
+ bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simps
+
+lemma bit_extra_simps [simp]:
+ "x AND 0 = (0::bit)"
+ "x AND 1 = (x::bit)"
+ "x OR 1 = (1::bit)"
+ "x OR 0 = (x::bit)"
+ "x XOR 1 = NOT (x::bit)"
+ "x XOR 0 = (x::bit)"
+ by (cases x, auto)+
+
+lemma bit_ops_comm:
+ "(x::bit) AND y = y AND x"
+ "(x::bit) OR y = y OR x"
+ "(x::bit) XOR y = y XOR x"
+ by (cases y, auto)+
+
+lemma bit_ops_same [simp]:
+ "(x::bit) AND x = x"
+ "(x::bit) OR x = x"
+ "(x::bit) XOR x = 0"
+ by (cases x, auto)+
+
+lemma bit_not_not [simp]: "NOT (NOT (x::bit)) = x"
+ by (cases x) auto
+
+lemma bit_or_def: "(b::bit) OR c = NOT (NOT b AND NOT c)"
+ by (induct b, simp_all)
+
+lemma bit_xor_def: "(b::bit) XOR c = (b AND NOT c) OR (NOT b AND c)"
+ by (induct b, simp_all)
+
+lemma bit_NOT_eq_1_iff [simp]: "NOT (b::bit) = 1 \<longleftrightarrow> b = 0"
+ by (induct b, simp_all)
+
+lemma bit_AND_eq_1_iff [simp]: "(a::bit) AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1"
+ by (induct a, simp_all)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Word/Bits_Int.thy Mon Dec 23 18:37:51 2013 +0100
@@ -0,0 +1,681 @@
+(*
+ Author: Jeremy Dawson and Gerwin Klein, NICTA
+
+ Definitions and basic theorems for bit-wise logical operations
+ for integers expressed using Pls, Min, BIT,
+ and converting them to and from lists of bools.
+*)
+
+header {* Bitwise Operations on Binary Integers *}
+
+theory Bits_Int
+imports Bits Bit_Representation
+begin
+
+subsection {* Logical operations *}
+
+text "bit-wise logical operations on the int type"
+
+instantiation int :: bit
+begin
+
+definition int_not_def:
+ "bitNOT = (\<lambda>x::int. - x - 1)"
+
+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))"
+ by pat_completeness simp
+
+termination
+ by (relation "measure (nat o abs o fst)", simp_all add: bin_rest_def)
+
+declare bitAND_int.simps [simp del]
+
+definition int_or_def:
+ "bitOR = (\<lambda>x y::int. NOT (NOT x AND NOT y))"
+
+definition int_xor_def:
+ "bitXOR = (\<lambda>x y::int. (x AND NOT y) OR (NOT x AND y))"
+
+instance ..
+
+end
+
+subsubsection {* Basic simplification rules *}
+
+lemma int_not_BIT [simp]:
+ "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]:
+ "NOT (0::int) = -1"
+ "NOT (1::int) = -2"
+ "NOT (- 1::int) = 0"
+ "NOT (numeral w::int) = - numeral (w + Num.One)"
+ "NOT (- numeral (Num.Bit0 w)::int) = numeral (Num.BitM w)"
+ "NOT (- numeral (Num.Bit1 w)::int) = numeral (Num.Bit0 w)"
+ unfolding int_not_def by simp_all
+
+lemma int_not_not [simp]: "NOT (NOT (x::int)) = x"
+ unfolding int_not_def by simp
+
+lemma int_and_0 [simp]: "(0::int) AND x = 0"
+ by (simp add: bitAND_int.simps)
+
+lemma int_and_m1 [simp]: "(-1::int) AND x = x"
+ 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)"
+ 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"
+ unfolding int_or_def by simp
+
+lemma int_or_minus1 [simp]: "(-1::int) OR x = -1"
+ 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 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 \<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) \<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) \<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) \<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) \<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:
+ "!!x y. bin_nth (x AND y) n = (bin_nth x n & bin_nth y n)"
+ "!!x y. bin_nth (x OR y) n = (bin_nth x n | bin_nth y n)"
+ "!!x y. bin_nth (x XOR y) n = (bin_nth x n ~= bin_nth y n)"
+ "!!x. bin_nth (NOT x) n = (~ bin_nth x n)"
+ by (induct n) auto
+
+subsubsection {* Derived properties *}
+
+lemma int_xor_minus1 [simp]: "(-1::int) XOR x = NOT x"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma int_xor_extra_simps [simp]:
+ "w XOR (0::int) = w"
+ "w XOR (-1::int) = NOT w"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma int_or_extra_simps [simp]:
+ "w OR (0::int) = w"
+ "w OR (-1::int) = -1"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma int_and_extra_simps [simp]:
+ "w AND (0::int) = 0"
+ "w AND (-1::int) = w"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+(* commutativity of the above *)
+lemma bin_ops_comm:
+ shows
+ int_and_comm: "!!y::int. x AND y = y AND x" and
+ int_or_comm: "!!y::int. x OR y = y OR x" and
+ int_xor_comm: "!!y::int. x XOR y = y XOR x"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma bin_ops_same [simp]:
+ "(x::int) AND x = x"
+ "(x::int) OR x = x"
+ "(x::int) XOR x = 0"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemmas bin_log_esimps =
+ int_and_extra_simps int_or_extra_simps int_xor_extra_simps
+ int_and_0 int_and_m1 int_or_zero int_or_minus1 int_xor_zero int_xor_minus1
+
+(* basic properties of logical (bit-wise) operations *)
+
+lemma bbw_ao_absorb:
+ "!!y::int. x AND (y OR x) = x & x OR (y AND x) = x"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma bbw_ao_absorbs_other:
+ "x AND (x OR y) = x \<and> (y AND x) OR x = (x::int)"
+ "(y OR x) AND x = x \<and> x OR (x AND y) = (x::int)"
+ "(x OR y) AND x = x \<and> (x AND y) OR x = (x::int)"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemmas bbw_ao_absorbs [simp] = bbw_ao_absorb bbw_ao_absorbs_other
+
+lemma int_xor_not:
+ "!!y::int. (NOT x) XOR y = NOT (x XOR y) &
+ x XOR (NOT y) = NOT (x XOR y)"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma int_and_assoc:
+ "(x AND y) AND (z::int) = x AND (y AND z)"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma int_or_assoc:
+ "(x OR y) OR (z::int) = x OR (y OR z)"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma int_xor_assoc:
+ "(x XOR y) XOR (z::int) = x XOR (y XOR z)"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc
+
+(* BH: Why are these declared as simp rules??? *)
+lemma bbw_lcs [simp]:
+ "(y::int) AND (x AND z) = x AND (y AND z)"
+ "(y::int) OR (x OR z) = x OR (y OR z)"
+ "(y::int) XOR (x XOR z) = x XOR (y XOR z)"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma bbw_not_dist:
+ "!!y::int. NOT (x OR y) = (NOT x) AND (NOT y)"
+ "!!y::int. NOT (x AND y) = (NOT x) OR (NOT y)"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma bbw_oa_dist:
+ "!!y z::int. (x AND y) OR z =
+ (x OR z) AND (y OR z)"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma bbw_ao_dist:
+ "!!y z::int. (x OR y) AND z =
+ (x AND z) OR (y AND z)"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+(*
+Why were these declared simp???
+declare bin_ops_comm [simp] bbw_assocs [simp]
+*)
+
+subsubsection {* Simplification with numerals *}
+
+text {* Cases for @{text "0"} and @{text "-1"} are already covered by
+ 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 (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))"
+ 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 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"
+ "(1::int) AND - numeral (Num.Bit1 y) = 1"
+ "numeral (Num.Bit0 x) AND (1::int) = 0"
+ "numeral (Num.Bit1 x) AND (1::int) = 1"
+ "- numeral (Num.Bit0 x) AND (1::int) = 0"
+ "- numeral (Num.Bit1 x) AND (1::int) = 1"
+ 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 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)"
+ "(1::int) OR - numeral (Num.Bit1 y) = - numeral (Num.Bit1 y)"
+ "numeral (Num.Bit0 x) OR (1::int) = numeral (Num.Bit1 x)"
+ "numeral (Num.Bit1 x) OR (1::int) = numeral (Num.Bit1 x)"
+ "- numeral (Num.Bit0 x) OR (1::int) = - numeral (Num.BitM x)"
+ "- numeral (Num.Bit1 x) OR (1::int) = - numeral (Num.Bit1 x)"
+ 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 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)"
+ "(1::int) XOR - numeral (Num.Bit1 y) = - numeral (Num.Bit0 (y + Num.One))"
+ "numeral (Num.Bit0 x) XOR (1::int) = numeral (Num.Bit1 x)"
+ "numeral (Num.Bit1 x) XOR (1::int) = numeral (Num.Bit0 x)"
+ "- numeral (Num.Bit0 x) XOR (1::int) = - numeral (Num.BitM x)"
+ "- numeral (Num.Bit1 x) XOR (1::int) = - numeral (Num.Bit0 (x + Num.One))"
+ by (rule bin_rl_eqI, simp, simp)+
+
+subsubsection {* Interactions with arithmetic *}
+
+lemma plus_and_or [rule_format]:
+ "ALL y::int. (x AND y) + (x OR y) = x + y"
+ apply (induct x rule: bin_induct)
+ apply clarsimp
+ apply clarsimp
+ apply clarsimp
+ apply (case_tac y rule: bin_exhaust)
+ apply clarsimp
+ apply (unfold Bit_def)
+ apply clarsimp
+ apply (erule_tac x = "x" in allE)
+ apply simp
+ done
+
+lemma le_int_or:
+ "bin_sign (y::int) = 0 ==> x <= x OR y"
+ apply (induct y arbitrary: x rule: bin_induct)
+ apply clarsimp
+ apply clarsimp
+ apply (case_tac x rule: bin_exhaust)
+ apply (case_tac b)
+ apply (case_tac [!] bit)
+ apply (auto simp: le_Bits)
+ done
+
+lemmas int_and_le =
+ xtrans(3) [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or]
+
+(* interaction between bit-wise and arithmetic *)
+(* good example of bin_induction *)
+lemma bin_add_not: "x + NOT x = (-1::int)"
+ apply (induct x rule: bin_induct)
+ apply clarsimp
+ apply clarsimp
+ apply (case_tac bit, auto)
+ done
+
+subsubsection {* Truncating results of bit-wise operations *}
+
+lemma bin_trunc_ao:
+ "!!x y. (bintrunc n x) AND (bintrunc n y) = bintrunc n (x AND y)"
+ "!!x y. (bintrunc n x) OR (bintrunc n y) = bintrunc n (x OR y)"
+ by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr)
+
+lemma bin_trunc_xor:
+ "!!x y. bintrunc n (bintrunc n x XOR bintrunc n y) =
+ bintrunc n (x XOR y)"
+ by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr)
+
+lemma bin_trunc_not:
+ "!!x. bintrunc n (NOT (bintrunc n x)) = bintrunc n (NOT x)"
+ by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr)
+
+(* want theorems of the form of bin_trunc_xor *)
+lemma bintr_bintr_i:
+ "x = bintrunc n y ==> bintrunc n x = bintrunc n y"
+ by auto
+
+lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i]
+lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i]
+
+subsection {* Setting and clearing bits *}
+
+primrec
+ 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"
+
+(** nth bit, set/clear **)
+
+lemma bin_nth_sc [simp]:
+ "bin_nth (bin_sc n b w) n \<longleftrightarrow> b"
+ by (induct n arbitrary: w) auto
+
+lemma bin_sc_sc_same [simp]:
+ "bin_sc n c (bin_sc n b w) = bin_sc n c w"
+ by (induct n arbitrary: w) auto
+
+lemma bin_sc_sc_diff:
+ "m ~= n ==>
+ bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)"
+ apply (induct n arbitrary: w m)
+ apply (case_tac [!] m)
+ apply auto
+ done
+
+lemma bin_nth_sc_gen:
+ "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 (bin_nth w n) w) = w"
+ by (induct n arbitrary: w) auto
+
+lemma bin_sign_sc [simp]:
+ "bin_sign (bin_sc n b w) = bin_sign w"
+ by (induct n arbitrary: w) auto
+
+lemma bin_sc_bintr [simp]:
+ "bintrunc m (bin_sc n x (bintrunc m (w))) = bintrunc m (bin_sc n x w)"
+ apply (induct n arbitrary: w m)
+ apply (case_tac [!] w rule: bin_exhaust)
+ apply (case_tac [!] m, auto)
+ done
+
+lemma bin_clr_le:
+ "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 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 False w) <= bintrunc n w"
+ apply (induct n arbitrary: w m)
+ apply simp
+ apply (case_tac w rule: bin_exhaust)
+ apply (case_tac m)
+ apply (auto simp: le_Bits)
+ done
+
+lemma bintr_bin_set_ge:
+ "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)
+ apply (case_tac m)
+ apply (auto simp: le_Bits)
+ done
+
+lemma bin_sc_FP [simp]: "bin_sc n False 0 = 0"
+ by (induct n) auto
+
+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
+
+lemma bin_sc_minus:
+ "0 < n ==> bin_sc (Suc (n - 1)) b w = bin_sc n b w"
+ by auto
+
+lemmas bin_sc_Suc_minus =
+ trans [OF bin_sc_minus [symmetric] bin_sc.Suc]
+
+lemma bin_sc_numeral [simp]:
+ "bin_sc (numeral k) b w =
+ bin_sc (pred_numeral k) b (bin_rest w) BIT bin_last w"
+ by (simp add: numeral_eq_Suc)
+
+
+subsection {* Splitting and concatenation *}
+
+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
+ "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
+ "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
+ "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
+ "bin_rsplitl n w = bin_rsplitl_aux n (fst w) (snd w) []"
+
+declare bin_rsplit_aux.simps [simp del]
+declare bin_rsplitl_aux.simps [simp del]
+
+lemma bin_sign_cat:
+ "bin_sign (bin_cat x n y) = bin_sign x"
+ by (induct n arbitrary: y) auto
+
+lemma bin_cat_Suc_Bit:
+ "bin_cat w (Suc n) (v BIT b) = bin_cat w n v BIT b"
+ by auto
+
+lemma bin_nth_cat:
+ "bin_nth (bin_cat x k y) n =
+ (if n < k then bin_nth y n else bin_nth x (n - k))"
+ apply (induct k arbitrary: n y)
+ apply clarsimp
+ apply (case_tac n, auto)
+ done
+
+lemma bin_nth_split:
+ "bin_split n c = (a, b) ==>
+ (ALL k. bin_nth a k = bin_nth c (n + k)) &
+ (ALL k. bin_nth b k = (k < n & bin_nth c k))"
+ apply (induct n arbitrary: b c)
+ apply clarsimp
+ apply (clarsimp simp: Let_def split: prod.split_asm)
+ apply (case_tac k)
+ apply auto
+ done
+
+lemma bin_cat_assoc:
+ "bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)"
+ by (induct n arbitrary: z) auto
+
+lemma bin_cat_assoc_sym:
+ "bin_cat x m (bin_cat y n z) = bin_cat (bin_cat x (m - n) y) (min m n) z"
+ apply (induct n arbitrary: z m, clarsimp)
+ apply (case_tac m, auto)
+ done
+
+lemma bin_cat_zero [simp]: "bin_cat 0 n w = bintrunc n w"
+ by (induct n arbitrary: w) auto
+
+lemma bintr_cat1:
+ "bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b"
+ by (induct n arbitrary: b) auto
+
+lemma bintr_cat: "bintrunc m (bin_cat a n b) =
+ bin_cat (bintrunc (m - n) a) n (bintrunc (min m n) b)"
+ by (rule bin_eqI) (auto simp: bin_nth_cat nth_bintr)
+
+lemma bintr_cat_same [simp]:
+ "bintrunc n (bin_cat a n b) = bintrunc n b"
+ by (auto simp add : bintr_cat)
+
+lemma cat_bintr [simp]:
+ "bin_cat a n (bintrunc n b) = bin_cat a n b"
+ by (induct n arbitrary: b) auto
+
+lemma split_bintrunc:
+ "bin_split n c = (a, b) ==> b = bintrunc n c"
+ by (induct n arbitrary: b c) (auto simp: Let_def split: prod.split_asm)
+
+lemma bin_cat_split:
+ "bin_split n w = (u, v) ==> w = bin_cat u n v"
+ by (induct n arbitrary: v w) (auto simp: Let_def split: prod.split_asm)
+
+lemma bin_split_cat:
+ "bin_split n (bin_cat v n w) = (v, bintrunc n w)"
+ by (induct n arbitrary: w) auto
+
+lemma bin_split_zero [simp]: "bin_split n 0 = (0, 0)"
+ by (induct n) auto
+
+lemma bin_split_minus1 [simp]:
+ "bin_split n -1 = (-1, bintrunc n -1)"
+ by (induct n) auto
+
+lemma bin_split_trunc:
+ "bin_split (min m n) c = (a, b) ==>
+ bin_split n (bintrunc m c) = (bintrunc (m - n) a, b)"
+ apply (induct n arbitrary: m b c, clarsimp)
+ apply (simp add: bin_rest_trunc Let_def split: prod.split_asm)
+ apply (case_tac m)
+ apply (auto simp: Let_def split: prod.split_asm)
+ done
+
+lemma bin_split_trunc1:
+ "bin_split n c = (a, b) ==>
+ bin_split n (bintrunc m c) = (bintrunc (m - n) a, bintrunc m b)"
+ apply (induct n arbitrary: m b c, clarsimp)
+ apply (simp add: bin_rest_trunc Let_def split: prod.split_asm)
+ apply (case_tac m)
+ apply (auto simp: Let_def split: prod.split_asm)
+ done
+
+lemma bin_cat_num:
+ "bin_cat a n b = a * 2 ^ n + bintrunc n b"
+ apply (induct n arbitrary: b, clarsimp)
+ apply (simp add: Bit_def)
+ done
+
+lemma bin_split_num:
+ "bin_split n b = (b div 2 ^ n, b mod 2 ^ n)"
+ apply (induct n arbitrary: b, simp)
+ 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)
+ done
+
+subsection {* Miscellaneous lemmas *}
+
+lemma nth_2p_bin:
+ "bin_nth (2 ^ n) m = (m = n)"
+ apply (induct n arbitrary: m)
+ apply clarsimp
+ apply safe
+ apply (case_tac m)
+ apply (auto simp: Bit_B0_2t [symmetric])
+ done
+
+(* for use when simplifying with bin_nth_Bit *)
+
+lemma ex_eq_or:
+ "(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 True"
+ unfolding Bit_B1
+ by (induct n) simp_all
+
+lemma mod_BIT:
+ "bin BIT bit mod 2 ^ Suc n = (bin mod 2 ^ n) BIT bit"
+proof -
+ have "bin mod 2 ^ n < 2 ^ n" by simp
+ then have "bin mod 2 ^ n \<le> 2 ^ n - 1" by simp
+ then have "2 * (bin mod 2 ^ n) \<le> 2 * (2 ^ n - 1)"
+ 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 mod_mult_mult1 mod_add_left_eq [of "2 * bin"]
+ mod_pos_pos_trivial)
+qed
+
+lemma AND_mod:
+ fixes x :: int
+ shows "x AND 2 ^ n - 1 = x mod 2 ^ n"
+proof (induct x arbitrary: n rule: bin_induct)
+ case 1
+ then show ?case
+ by simp
+next
+ case 2
+ then show ?case
+ by (simp, simp add: m1mod2k)
+next
+ case (3 bin bit)
+ show ?case
+ proof (cases n)
+ case 0
+ then show ?thesis by simp
+ next
+ case (Suc m)
+ with 3 show ?thesis
+ by (simp only: power_BIT mod_BIT int_and_Bits) simp
+ qed
+qed
+
+end
+
--- a/src/HOL/Word/Bool_List_Representation.thy Mon Dec 23 16:29:43 2013 +0100
+++ b/src/HOL/Word/Bool_List_Representation.thy Mon Dec 23 18:37:51 2013 +0100
@@ -9,7 +9,7 @@
header "Bool lists and integers"
theory Bool_List_Representation
-imports Bit_Int
+imports Bits_Int
begin
definition map2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list"
--- a/src/HOL/Word/Word.thy Mon Dec 23 16:29:43 2013 +0100
+++ b/src/HOL/Word/Word.thy Mon Dec 23 18:37:51 2013 +0100
@@ -8,7 +8,7 @@
imports
Type_Length
"~~/src/HOL/Library/Boolean_Algebra"
- Bit_Bit
+ Bits_Bit
Bool_List_Representation
Misc_Typedef
Word_Miscellaneous