(*
ID: $Id$
Author: Jeremy Dawson and Gerwin Klein, NICTA
definition 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 BinOperations imports BinGeneral BitSyntax
begin
subsection {* Logical operations *}
text "bit-wise logical operations on the int type"
instantiation int :: bit
begin
definition
int_not_def: "bitNOT = bin_rec Int.Min Int.Pls
(\<lambda>w b s. s BIT (NOT b))"
definition
int_and_def: "bitAND = bin_rec (\<lambda>x. Int.Pls) (\<lambda>y. y)
(\<lambda>w b s y. s (bin_rest y) BIT (b AND bin_last y))"
definition
int_or_def: "bitOR = bin_rec (\<lambda>x. x) (\<lambda>y. Int.Min)
(\<lambda>w b s y. s (bin_rest y) BIT (b OR bin_last y))"
definition
int_xor_def: "bitXOR = bin_rec (\<lambda>x. x) bitNOT
(\<lambda>w b s y. s (bin_rest y) BIT (b XOR bin_last y))"
instance ..
end
lemma int_not_simps [simp]:
"NOT Int.Pls = Int.Min"
"NOT Int.Min = Int.Pls"
"NOT (w BIT b) = (NOT w) BIT (NOT b)"
"NOT (Int.Bit0 w) = Int.Bit1 (NOT w)"
"NOT (Int.Bit1 w) = Int.Bit0 (NOT w)"
unfolding int_not_def by (simp_all add: bin_rec_simps)
lemma int_xor_Pls [simp]:
"Int.Pls XOR x = x"
unfolding int_xor_def by (simp add: bin_rec_PM)
lemma int_xor_Min [simp]:
"Int.Min XOR x = NOT x"
unfolding int_xor_def by (simp add: bin_rec_PM)
lemma int_xor_Bits [simp]:
"(x BIT b) XOR (y BIT c) = (x XOR y) BIT (b XOR c)"
apply (unfold int_xor_def)
apply (rule bin_rec_simps (1) [THEN fun_cong, THEN trans])
apply (rule ext, simp)
prefer 2
apply simp
apply (rule ext)
apply (simp add: int_not_simps [symmetric])
done
lemma int_xor_Bits2 [simp]:
"(Int.Bit0 x) XOR (Int.Bit0 y) = Int.Bit0 (x XOR y)"
"(Int.Bit0 x) XOR (Int.Bit1 y) = Int.Bit1 (x XOR y)"
"(Int.Bit1 x) XOR (Int.Bit0 y) = Int.Bit1 (x XOR y)"
"(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 bit.B0) = w"
"w XOR (Int.Min BIT bit.B1) = 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)
lemma int_or_Min [simp]:
"Int.Min OR x = Int.Min"
by (unfold int_or_def) (simp add: bin_rec_PM)
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 add: bin_rec_simps)
lemma int_or_Bits2 [simp]:
"(Int.Bit0 x) OR (Int.Bit0 y) = Int.Bit0 (x OR y)"
"(Int.Bit0 x) OR (Int.Bit1 y) = Int.Bit1 (x OR y)"
"(Int.Bit1 x) OR (Int.Bit0 y) = Int.Bit1 (x OR y)"
"(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 bit.B0) = w"
"w OR (Int.Min BIT bit.B1) = 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)
lemma int_and_Min [simp]:
"Int.Min AND x = x"
unfolding int_and_def by (simp add: bin_rec_PM)
lemma int_and_Bits [simp]:
"(x BIT b) AND (y BIT c) = (x AND y) BIT (b AND c)"
unfolding int_and_def by (simp add: bin_rec_simps)
lemma int_and_Bits2 [simp]:
"(Int.Bit0 x) AND (Int.Bit0 y) = Int.Bit0 (x AND y)"
"(Int.Bit0 x) AND (Int.Bit1 y) = Int.Bit0 (x AND y)"
"(Int.Bit1 x) AND (Int.Bit0 y) = Int.Bit0 (x AND y)"
"(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 bit.B0) = Int.Pls"
"w AND (Int.Min BIT bit.B1) = w"
apply (induct w rule: bin_induct)
apply simp_all[4]
apply (unfold int_and_Bits)
apply clarsimp+
done
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
(* 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"
apply (induct x rule: bin_induct)
apply simp_all[6]
apply (case_tac y rule: bin_exhaust, simp add: bit_ops_comm)+
done
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
lemma int_not_not [simp]: "NOT (NOT (x::int)) = x"
by (induct x rule: bin_induct) auto
lemmas bin_log_esimps =
int_and_extra_simps int_or_extra_simps int_xor_extra_simps
int_and_Pls int_and_Min int_or_Pls int_or_Min int_xor_Pls int_xor_Min
(* 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"
apply (induct x rule: bin_induct)
apply auto
apply (case_tac [!] y rule: bin_exhaust)
apply auto
apply (case_tac [!] bit)
apply auto
done
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
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
lemma int_and_assoc:
"(x AND y) AND (z::int) = x AND (y AND z)"
by (simp add: bbw_assocs')
lemma int_or_assoc:
"(x OR y) OR (z::int) = x OR (y OR z)"
by (simp add: bbw_assocs')
lemma int_xor_assoc:
"(x XOR y) XOR (z::int) = x XOR (y XOR z)"
by (simp add: bbw_assocs')
lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc
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
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
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
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
(*
Why were these declared simp???
declare bin_ops_comm [simp] bbw_assocs [simp]
*)
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 split: bit.split)
done
lemma le_int_or:
"!!x. bin_sign y = Int.Pls ==> x <= x OR y"
apply (induct y 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: less_eq_int_code)
done
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"
apply (induct x rule: bin_induct)
apply clarsimp
apply clarsimp
apply (case_tac bit, auto)
done
(* 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
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
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
(* 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 *}
consts
bin_sc :: "nat => bit => int => int"
primrec
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]:
"!!w. bin_nth (bin_sc n b w) n = (b = bit.B1)"
by (induct n) auto
lemma bin_sc_sc_same [simp]:
"!!w. bin_sc n c (bin_sc n b w) = bin_sc n c w"
by (induct n) auto
lemma bin_sc_sc_diff:
"!!w m. m ~= n ==>
bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)"
apply (induct n)
apply (case_tac [!] m)
apply auto
done
lemma bin_nth_sc_gen:
"!!w m. bin_nth (bin_sc n b w) m = (if m = n then b = bit.B1 else bin_nth w m)"
by (induct n) (case_tac [!] m, auto)
lemma bin_sc_nth [simp]:
"!!w. (bin_sc n (If (bin_nth w n) bit.B1 bit.B0) w) = w"
by (induct n) auto
lemma bin_sign_sc [simp]:
"!!w. bin_sign (bin_sc n b w) = bin_sign w"
by (induct n) auto
lemma bin_sc_bintr [simp]:
"!!w m. bintrunc m (bin_sc n x (bintrunc m (w))) = bintrunc m (bin_sc n x w)"
apply (induct n)
apply (case_tac [!] w rule: bin_exhaust)
apply (case_tac [!] m, auto)
done
lemma bin_clr_le:
"!!w. bin_sc n bit.B0 w <= w"
apply (induct n)
apply (case_tac [!] w rule: bin_exhaust)
apply (auto simp del: BIT_simps)
apply (unfold Bit_def)
apply (simp_all split: bit.split)
done
lemma bin_set_ge:
"!!w. bin_sc n bit.B1 w >= w"
apply (induct n)
apply (case_tac [!] w rule: bin_exhaust)
apply (auto simp del: BIT_simps)
apply (unfold Bit_def)
apply (simp_all split: bit.split)
done
lemma bintr_bin_clr_le:
"!!w m. bintrunc n (bin_sc m bit.B0 w) <= bintrunc n w"
apply (induct n)
apply simp
apply (case_tac w rule: bin_exhaust)
apply (case_tac m)
apply (auto simp del: BIT_simps)
apply (unfold Bit_def)
apply (simp_all split: bit.split)
done
lemma bintr_bin_set_ge:
"!!w m. bintrunc n (bin_sc m bit.B1 w) >= bintrunc n w"
apply (induct n)
apply simp
apply (case_tac w rule: bin_exhaust)
apply (case_tac m)
apply (auto simp del: BIT_simps)
apply (unfold Bit_def)
apply (simp_all split: bit.split)
done
lemma bin_sc_FP [simp]: "bin_sc n bit.B0 Int.Pls = Int.Pls"
by (induct n) auto
lemma bin_sc_TM [simp]: "bin_sc n bit.B1 Int.Min = Int.Min"
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, standard]
lemmas bin_sc_Suc_pred [simp] =
bin_sc_Suc_minus [of "number_of bin", simplified nobm1, standard]
subsection {* Operations on lists of booleans *}
consts
bin_to_bl :: "nat => int => bool list"
bin_to_bl_aux :: "nat => int => bool list => bool list"
bl_to_bin :: "bool list => int"
bl_to_bin_aux :: "int => bool list => int"
bl_of_nth :: "nat => (nat => bool) => bool list"
primrec
Nil : "bl_to_bin_aux w [] = w"
Cons : "bl_to_bin_aux w (b # bs) =
bl_to_bin_aux (w BIT (if b then bit.B1 else bit.B0)) bs"
primrec
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 = bit.B1) # bl)"
defs
bin_to_bl_def : "bin_to_bl n w == bin_to_bl_aux n w []"
bl_to_bin_def : "bl_to_bin bs == bl_to_bin_aux Int.Pls bs"
primrec
Suc : "bl_of_nth (Suc n) f = f n # bl_of_nth n f"
Z : "bl_of_nth 0 f = []"
consts
takefill :: "'a => nat => 'a list => 'a list"
app2 :: "('a => 'b => 'c) => 'a list => 'b list => 'c list"
-- "takefill - like take but if argument list too short,"
-- "extends result to get requested length"
primrec
Z : "takefill fill 0 xs = []"
Suc : "takefill fill (Suc n) xs = (
case xs of [] => fill # takefill fill n xs
| y # ys => y # takefill fill n ys)"
defs
app2_def : "app2 f as bs == map (split f) (zip as bs)"
subsection {* Splitting and concatenation *}
-- "rcat and rsplit"
consts
bin_rcat :: "nat => int list => int"
bin_rsplit_aux :: "nat * int list * nat * int => int list"
bin_rsplit :: "nat => (nat * int) => int list"
bin_rsplitl_aux :: "nat * int list * nat * int => int list"
bin_rsplitl :: "nat => (nat * int) => int list"
recdef bin_rsplit_aux "measure (fst o snd o snd)"
"bin_rsplit_aux (n, bs, (m, c)) =
(if m = 0 | n = 0 then bs else
let (a, b) = bin_split n c
in bin_rsplit_aux (n, b # bs, (m - n, a)))"
recdef bin_rsplitl_aux "measure (fst o snd o snd)"
"bin_rsplitl_aux (n, bs, (m, c)) =
(if m = 0 | n = 0 then bs else
let (a, b) = bin_split (min m n) c
in bin_rsplitl_aux (n, b # bs, (m - n, a)))"
defs
bin_rcat_def : "bin_rcat n bs == foldl (%u v. bin_cat u n v) Int.Pls bs"
bin_rsplit_def : "bin_rsplit n w == bin_rsplit_aux (n, [], w)"
bin_rsplitl_def : "bin_rsplitl n w == bin_rsplitl_aux (n, [], w)"
(* potential for looping *)
declare bin_rsplit_aux.simps [simp del]
declare bin_rsplitl_aux.simps [simp del]
lemma bin_sign_cat:
"!!y. bin_sign (bin_cat x n y) = bin_sign x"
by (induct n) 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:
"!!n y. 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)
apply clarsimp
apply (case_tac n, auto)
done
lemma bin_nth_split:
"!!b c. 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)
apply clarsimp
apply (clarsimp simp: Let_def split: ls_splits)
apply (case_tac k)
apply auto
done
lemma bin_cat_assoc:
"!!z. bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)"
by (induct n) auto
lemma bin_cat_assoc_sym: "!!z m.
bin_cat x m (bin_cat y n z) = bin_cat (bin_cat x (m - n) y) (min m n) z"
apply (induct n, clarsimp)
apply (case_tac m, auto)
done
lemma bin_cat_Pls [simp]:
"!!w. bin_cat Int.Pls n w = bintrunc n w"
by (induct n) auto
lemma bintr_cat1:
"!!b. bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b"
by (induct n) 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]:
"!!b. bin_cat a n (bintrunc n b) = bin_cat a n b"
by (induct n) auto
lemma split_bintrunc:
"!!b c. bin_split n c = (a, b) ==> b = bintrunc n c"
by (induct n) (auto simp: Let_def split: ls_splits)
lemma bin_cat_split:
"!!v w. bin_split n w = (u, v) ==> w = bin_cat u n v"
by (induct n) (auto simp: Let_def split: ls_splits)
lemma bin_split_cat:
"!!w. bin_split n (bin_cat v n w) = (v, bintrunc n w)"
by (induct n) auto
lemma bin_split_Pls [simp]:
"bin_split n Int.Pls = (Int.Pls, Int.Pls)"
by (induct n) (auto simp: Let_def split: ls_splits)
lemma bin_split_Min [simp]:
"bin_split n Int.Min = (Int.Min, bintrunc n Int.Min)"
by (induct n) (auto simp: Let_def split: ls_splits)
lemma bin_split_trunc:
"!!m b c. bin_split (min m n) c = (a, b) ==>
bin_split n (bintrunc m c) = (bintrunc (m - n) a, b)"
apply (induct n, clarsimp)
apply (simp add: bin_rest_trunc Let_def split: ls_splits)
apply (case_tac m)
apply (auto simp: Let_def split: ls_splits)
done
lemma bin_split_trunc1:
"!!m b c. bin_split n c = (a, b) ==>
bin_split n (bintrunc m c) = (bintrunc (m - n) a, bintrunc m b)"
apply (induct n, clarsimp)
apply (simp add: bin_rest_trunc Let_def split: ls_splits)
apply (case_tac m)
apply (auto simp: Let_def split: ls_splits)
done
lemma bin_cat_num:
"!!b. bin_cat a n b = a * 2 ^ n + bintrunc n b"
apply (induct n, clarsimp)
apply (simp add: Bit_def cong: number_of_False_cong)
done
lemma bin_split_num:
"!!b. bin_split n b = (b div 2 ^ n, b mod 2 ^ n)"
apply (induct n, clarsimp)
apply (simp add: bin_rest_div zdiv_zmult2_eq)
apply (case_tac b rule: bin_exhaust)
apply simp
apply (simp add: Bit_def zmod_zmult_zmult1 p1mod22k
split: bit.split
cong: number_of_False_cong)
done
subsection {* Miscellaneous lemmas *}
lemma nth_2p_bin:
"!!m. bin_nth (2 ^ n) m = (m = n)"
apply (induct n)
apply clarsimp
apply safe
apply (case_tac m)
apply (auto simp: trans [OF numeral_1_eq_1 [symmetric] number_of_eq])
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
end