--- a/src/HOL/Word/Bit_Int.thy Thu Nov 17 08:07:54 2011 +0100
+++ b/src/HOL/Word/Bit_Int.thy Thu Nov 17 12:29:48 2011 +0100
@@ -87,6 +87,8 @@
end
+subsubsection {* Basic simplification rules *}
+
lemma int_not_simps [simp]:
"NOT Int.Pls = Int.Min"
"NOT Int.Min = Int.Pls"
@@ -121,20 +123,6 @@
"(Int.Bit1 x) XOR (Int.Bit1 y) = Int.Bit0 (x XOR y)"
unfolding BIT_simps [symmetric] int_xor_Bits by simp_all
-lemma int_xor_x_simps':
- "w XOR (Int.Pls BIT 0) = w"
- "w XOR (Int.Min BIT 1) = NOT w"
- apply (induct w rule: bin_induct)
- apply simp_all[4]
- apply (unfold int_xor_Bits)
- apply clarsimp+
- done
-
-lemma int_xor_extra_simps [simp]:
- "w XOR Int.Pls = w"
- "w XOR Int.Min = NOT w"
- using int_xor_x_simps' by simp_all
-
lemma int_or_Pls [simp]:
"Int.Pls OR x = x"
by (unfold int_or_def) (simp add: bin_rec_PM)
@@ -154,20 +142,6 @@
"(Int.Bit1 x) OR (Int.Bit1 y) = Int.Bit1 (x OR y)"
unfolding BIT_simps [symmetric] int_or_Bits by simp_all
-lemma int_or_x_simps':
- "w OR (Int.Pls BIT 0) = w"
- "w OR (Int.Min BIT 1) = Int.Min"
- apply (induct w rule: bin_induct)
- apply simp_all[4]
- apply (unfold int_or_Bits)
- apply clarsimp+
- done
-
-lemma int_or_extra_simps [simp]:
- "w OR Int.Pls = w"
- "w OR Int.Min = Int.Min"
- using int_or_x_simps' by simp_all
-
lemma int_and_Pls [simp]:
"Int.Pls AND x = Int.Pls"
unfolding int_and_def by (simp add: bin_rec_PM)
@@ -187,19 +161,61 @@
"(Int.Bit1 x) AND (Int.Bit1 y) = Int.Bit1 (x AND y)"
unfolding BIT_simps [symmetric] int_and_Bits by simp_all
-lemma int_and_x_simps':
- "w AND (Int.Pls BIT 0) = Int.Pls"
- "w AND (Int.Min BIT 1) = w"
- apply (induct w rule: bin_induct)
- apply simp_all[4]
- apply (unfold int_and_Bits)
- apply clarsimp+
- done
+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)"
+ 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"
+ 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"
+ 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"
+ by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp)
+
+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)
+
+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_extra_simps [simp]:
+ "w XOR Int.Pls = w"
+ "w XOR Int.Min = NOT w"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
+
+lemma int_or_extra_simps [simp]:
+ "w OR Int.Pls = w"
+ "w OR Int.Min = Int.Min"
+ by (auto simp add: bin_eq_iff bin_nth_ops)
lemma int_and_extra_simps [simp]:
"w AND Int.Pls = Int.Pls"
"w AND Int.Min = w"
- using int_and_x_simps' by simp_all
+ by (auto simp add: bin_eq_iff bin_nth_ops)
(* commutativity of the above *)
lemma bin_ops_comm:
@@ -207,19 +223,16 @@
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"
- apply (induct x rule: bin_induct)
- apply simp_all[6]
- apply (case_tac y rule: bin_exhaust, simp add: bit_ops_comm)+
- done
+ 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 = Int.Pls"
- by (induct x rule: bin_induct) auto
+ by (auto simp add: bin_eq_iff bin_nth_ops)
lemma int_not_not [simp]: "NOT (NOT (x::int)) = x"
- by (induct x rule: bin_induct) auto
+ 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
@@ -229,108 +242,64 @@
lemma bbw_ao_absorb:
"!!y::int. x AND (y OR x) = x & x OR (y AND x) = x"
- apply (induct x rule: bin_induct)
- apply auto
- apply (case_tac [!] y rule: bin_exhaust)
- apply auto
- apply (case_tac [!] bit)
- apply auto
- done
+ 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)"
- apply (auto simp: bbw_ao_absorb int_or_comm)
- apply (subst int_or_comm)
- apply (simp add: bbw_ao_absorb)
- apply (subst int_and_comm)
- apply (subst int_or_comm)
- apply (simp add: bbw_ao_absorb)
- apply (subst int_and_comm)
- apply (simp add: bbw_ao_absorb)
- done
+ 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)"
- apply (induct x rule: bin_induct)
- apply auto
- apply (case_tac y rule: bin_exhaust, auto,
- case_tac b, auto)+
- done
-
-lemma bbw_assocs':
- "!!y z::int. (x AND y) AND z = x AND (y AND z) &
- (x OR y) OR z = x OR (y OR z) &
- (x XOR y) XOR z = x XOR (y XOR z)"
- apply (induct x rule: bin_induct)
- apply (auto simp: int_xor_not)
- apply (case_tac [!] y rule: bin_exhaust)
- apply (case_tac [!] z rule: bin_exhaust)
- apply (case_tac [!] bit)
- apply (case_tac [!] b)
- apply (auto simp del: BIT_simps)
- done
+ 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 (simp add: bbw_assocs')
+ 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 (simp add: bbw_assocs')
+ 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 (simp add: bbw_assocs')
+ 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)"
- apply (auto simp: bbw_assocs [symmetric])
- apply (auto simp: bin_ops_comm)
- done
+ 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)"
- apply (induct x rule: bin_induct)
- apply auto
- apply (case_tac [!] y rule: bin_exhaust)
- apply (case_tac [!] bit, auto simp del: BIT_simps)
- done
+ 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)"
- apply (induct x rule: bin_induct)
- apply auto
- apply (case_tac y rule: bin_exhaust)
- apply (case_tac z rule: bin_exhaust)
- apply (case_tac ba, auto simp del: BIT_simps)
- done
+ 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)"
- apply (induct x rule: bin_induct)
- apply auto
- apply (case_tac y rule: bin_exhaust)
- apply (case_tac z rule: bin_exhaust)
- apply (case_tac ba, auto simp del: BIT_simps)
- done
+ by (auto simp add: bin_eq_iff bin_nth_ops)
(*
Why were these declared simp???
declare bin_ops_comm [simp] bbw_assocs [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)
@@ -359,20 +328,6 @@
lemmas int_and_le =
xtr3 [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or]
-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)"
- apply (induct n)
- apply safe
- apply (case_tac [!] x rule: bin_exhaust)
- apply (simp_all del: BIT_simps)
- apply (case_tac [!] y rule: bin_exhaust)
- apply (simp_all del: BIT_simps)
- apply (auto dest: not_B1_is_B0 intro: B1_ass_B0)
- done
-
(* interaction between bit-wise and arithmetic *)
(* good example of bin_induction *)
lemma bin_add_not: "x + NOT x = Int.Min"
@@ -382,34 +337,21 @@
apply (case_tac bit, auto)
done
-(* truncating results of bit-wise operations *)
+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)"
- apply (induct n)
- apply auto
- apply (case_tac [!] x rule: bin_exhaust)
- apply (case_tac [!] y rule: bin_exhaust)
- apply auto
- done
+ 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)"
- apply (induct n)
- apply auto
- apply (case_tac [!] x rule: bin_exhaust)
- apply (case_tac [!] y rule: bin_exhaust)
- apply auto
- done
+ 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)"
- apply (induct n)
- apply auto
- apply (case_tac [!] x rule: bin_exhaust)
- apply auto
- done
+ 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: