(* Title: HOL/Integ/Numeral.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
*)
header{*Arithmetic on Binary Integers*}
theory Numeral
imports IntDef Datatype
uses "../Tools/numeral_syntax.ML"
begin
text{*This formalization defines binary arithmetic in terms of the integers
rather than using a datatype. This avoids multiple representations (leading
zeroes, etc.) See @{text "ZF/Integ/twos-compl.ML"}, function @{text
int_of_binary}, for the numerical interpretation.
The representation expects that @{text "(m mod 2)"} is 0 or 1,
even if m is negative;
For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
@{text "-5 = (-3)*2 + 1"}.
*}
typedef (Bin)
bin = "UNIV::int set"
by (auto)
text{*This datatype avoids the use of type @{typ bool}, which would make
all of the rewrite rules higher-order. If the use of datatype causes
problems, this two-element type can easily be formalized using typedef.*}
datatype bit = B0 | B1
constdefs
Pls :: "bin"
"Pls == Abs_Bin 0"
Min :: "bin"
"Min == Abs_Bin (- 1)"
Bit :: "[bin,bit] => bin" (infixl "BIT" 90)
--{*That is, 2w+b*}
"w BIT b == Abs_Bin ((case b of B0 => 0 | B1 => 1) + Rep_Bin w + Rep_Bin w)"
axclass
number < type -- {* for numeric types: nat, int, real, \dots *}
consts
number_of :: "bin => 'a::number"
syntax
"_Numeral" :: "num_const => 'a" ("_")
setup NumeralSyntax.setup
abbreviation
"Numeral0 == number_of Pls"
"Numeral1 == number_of (Pls BIT B1)"
lemma Let_number_of [simp]: "Let (number_of v) f == f (number_of v)"
-- {* Unfold all @{text let}s involving constants *}
by (simp add: Let_def)
lemma Let_0 [simp]: "Let 0 f == f 0"
by (simp add: Let_def)
lemma Let_1 [simp]: "Let 1 f == f 1"
by (simp add: Let_def)
constdefs
bin_succ :: "bin=>bin"
"bin_succ w == Abs_Bin(Rep_Bin w + 1)"
bin_pred :: "bin=>bin"
"bin_pred w == Abs_Bin(Rep_Bin w - 1)"
bin_minus :: "bin=>bin"
"bin_minus w == Abs_Bin(- (Rep_Bin w))"
bin_add :: "[bin,bin]=>bin"
"bin_add v w == Abs_Bin(Rep_Bin v + Rep_Bin w)"
bin_mult :: "[bin,bin]=>bin"
"bin_mult v w == Abs_Bin(Rep_Bin v * Rep_Bin w)"
lemma Abs_Bin_inverse':
"Rep_Bin (Abs_Bin x) = x"
by (rule Abs_Bin_inverse) (auto simp add: Bin_def)
lemmas Bin_simps =
bin_succ_def bin_pred_def bin_minus_def bin_add_def bin_mult_def
Pls_def Min_def Bit_def Abs_Bin_inverse Rep_Bin_inverse Bin_def
text{*Removal of leading zeroes*}
lemma Pls_0_eq [simp]: "Pls BIT B0 = Pls"
by (simp add: Bin_simps)
lemma Min_1_eq [simp]: "Min BIT B1 = Min"
by (simp add: Bin_simps)
subsection{*The Functions @{term bin_succ}, @{term bin_pred} and @{term bin_minus}*}
lemma bin_succ_Pls [simp]: "bin_succ Pls = Pls BIT B1"
by (simp add: Bin_simps)
lemma bin_succ_Min [simp]: "bin_succ Min = Pls"
by (simp add: Bin_simps)
lemma bin_succ_1 [simp]: "bin_succ(w BIT B1) = (bin_succ w) BIT B0"
by (simp add: Bin_simps add_ac)
lemma bin_succ_0 [simp]: "bin_succ(w BIT B0) = w BIT B1"
by (simp add: Bin_simps add_ac)
lemma bin_pred_Pls [simp]: "bin_pred Pls = Min"
by (simp add: Bin_simps)
lemma bin_pred_Min [simp]: "bin_pred Min = Min BIT B0"
by (simp add: Bin_simps diff_minus)
lemma bin_pred_1 [simp]: "bin_pred(w BIT B1) = w BIT B0"
by (simp add: Bin_simps)
lemma bin_pred_0 [simp]: "bin_pred(w BIT B0) = (bin_pred w) BIT B1"
by (simp add: Bin_simps diff_minus add_ac)
lemma bin_minus_Pls [simp]: "bin_minus Pls = Pls"
by (simp add: Bin_simps)
lemma bin_minus_Min [simp]: "bin_minus Min = Pls BIT B1"
by (simp add: Bin_simps)
lemma bin_minus_1 [simp]:
"bin_minus (w BIT B1) = bin_pred (bin_minus w) BIT B1"
by (simp add: Bin_simps add_ac diff_minus)
lemma bin_minus_0 [simp]: "bin_minus(w BIT B0) = (bin_minus w) BIT B0"
by (simp add: Bin_simps)
subsection{*Binary Addition and Multiplication:
@{term bin_add} and @{term bin_mult}*}
lemma bin_add_Pls [simp]: "bin_add Pls w = w"
by (simp add: Bin_simps)
lemma bin_add_Min [simp]: "bin_add Min w = bin_pred w"
by (simp add: Bin_simps diff_minus add_ac)
lemma bin_add_BIT_11 [simp]:
"bin_add (v BIT B1) (w BIT B1) = bin_add v (bin_succ w) BIT B0"
by (simp add: Bin_simps add_ac)
lemma bin_add_BIT_10 [simp]:
"bin_add (v BIT B1) (w BIT B0) = (bin_add v w) BIT B1"
by (simp add: Bin_simps add_ac)
lemma bin_add_BIT_0 [simp]:
"bin_add (v BIT B0) (w BIT y) = bin_add v w BIT y"
by (simp add: Bin_simps add_ac)
lemma bin_add_Pls_right [simp]: "bin_add w Pls = w"
by (simp add: Bin_simps)
lemma bin_add_Min_right [simp]: "bin_add w Min = bin_pred w"
by (simp add: Bin_simps diff_minus)
lemma bin_mult_Pls [simp]: "bin_mult Pls w = Pls"
by (simp add: Bin_simps)
lemma bin_mult_Min [simp]: "bin_mult Min w = bin_minus w"
by (simp add: Bin_simps)
lemma bin_mult_1 [simp]:
"bin_mult (v BIT B1) w = bin_add ((bin_mult v w) BIT B0) w"
by (simp add: Bin_simps add_ac left_distrib)
lemma bin_mult_0 [simp]: "bin_mult (v BIT B0) w = (bin_mult v w) BIT B0"
by (simp add: Bin_simps left_distrib)
subsection{*Converting Numerals to Rings: @{term number_of}*}
axclass number_ring \<subseteq> number, comm_ring_1
number_of_eq: "number_of w = of_int (Rep_Bin w)"
lemma number_of_succ:
"number_of(bin_succ w) = (1 + number_of w ::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)
lemma number_of_pred:
"number_of(bin_pred w) = (- 1 + number_of w ::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)
lemma number_of_minus:
"number_of(bin_minus w) = (- (number_of w)::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)
lemma number_of_add:
"number_of(bin_add v w) = (number_of v + number_of w::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)
lemma number_of_mult:
"number_of(bin_mult v w) = (number_of v * number_of w::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)
text{*The correctness of shifting. But it doesn't seem to give a measurable
speed-up.*}
lemma double_number_of_BIT:
"(1+1) * number_of w = (number_of (w BIT B0) ::'a::number_ring)"
by (simp add: number_of_eq Bin_simps left_distrib)
text{*Converting numerals 0 and 1 to their abstract versions*}
lemma numeral_0_eq_0 [simp]: "Numeral0 = (0::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)
lemma numeral_1_eq_1 [simp]: "Numeral1 = (1::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)
text{*Special-case simplification for small constants*}
text{*Unary minus for the abstract constant 1. Cannot be inserted
as a simprule until later: it is @{text number_of_Min} re-oriented!*}
lemma numeral_m1_eq_minus_1: "(-1::'a::number_ring) = - 1"
by (simp add: number_of_eq Bin_simps)
lemma mult_minus1 [simp]: "-1 * z = -(z::'a::number_ring)"
by (simp add: numeral_m1_eq_minus_1)
lemma mult_minus1_right [simp]: "z * -1 = -(z::'a::number_ring)"
by (simp add: numeral_m1_eq_minus_1)
(*Negation of a coefficient*)
lemma minus_number_of_mult [simp]:
"- (number_of w) * z = number_of(bin_minus w) * (z::'a::number_ring)"
by (simp add: number_of_minus)
text{*Subtraction*}
lemma diff_number_of_eq:
"number_of v - number_of w =
(number_of(bin_add v (bin_minus w))::'a::number_ring)"
by (simp add: diff_minus number_of_add number_of_minus)
lemma number_of_Pls: "number_of Pls = (0::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)
lemma number_of_Min: "number_of Min = (- 1::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)
lemma number_of_BIT:
"number_of(w BIT x) = (case x of B0 => 0 | B1 => (1::'a::number_ring)) +
(number_of w) + (number_of w)"
by (simp add: number_of_eq Bin_simps split: bit.split)
subsection{*Equality of Binary Numbers*}
text{*First version by Norbert Voelker*}
lemma eq_number_of_eq:
"((number_of x::'a::number_ring) = number_of y) =
iszero (number_of (bin_add x (bin_minus y)) :: 'a)"
by (simp add: iszero_def compare_rls number_of_add number_of_minus)
lemma iszero_number_of_Pls: "iszero ((number_of Pls)::'a::number_ring)"
by (simp add: iszero_def numeral_0_eq_0)
lemma nonzero_number_of_Min: "~ iszero ((number_of Min)::'a::number_ring)"
by (simp add: iszero_def numeral_m1_eq_minus_1 eq_commute)
subsection{*Comparisons, for Ordered Rings*}
lemma double_eq_0_iff: "(a + a = 0) = (a = (0::'a::ordered_idom))"
proof -
have "a + a = (1+1)*a" by (simp add: left_distrib)
with zero_less_two [where 'a = 'a]
show ?thesis by force
qed
lemma le_imp_0_less:
assumes le: "0 \<le> z" shows "(0::int) < 1 + z"
proof -
have "0 \<le> z" .
also have "... < z + 1" by (rule less_add_one)
also have "... = 1 + z" by (simp add: add_ac)
finally show "0 < 1 + z" .
qed
lemma odd_nonzero: "1 + z + z \<noteq> (0::int)";
proof (cases z rule: int_cases)
case (nonneg n)
have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
thus ?thesis using le_imp_0_less [OF le]
by (auto simp add: add_assoc)
next
case (neg n)
show ?thesis
proof
assume eq: "1 + z + z = 0"
have "0 < 1 + (int n + int n)"
by (simp add: le_imp_0_less add_increasing)
also have "... = - (1 + z + z)"
by (simp add: neg add_assoc [symmetric])
also have "... = 0" by (simp add: eq)
finally have "0<0" ..
thus False by blast
qed
qed
text{*The premise involving @{term Ints} prevents @{term "a = 1/2"}.*}
lemma Ints_odd_nonzero: "a \<in> Ints ==> 1 + a + a \<noteq> (0::'a::ordered_idom)"
proof (unfold Ints_def)
assume "a \<in> range of_int"
then obtain z where a: "a = of_int z" ..
show ?thesis
proof
assume eq: "1 + a + a = 0"
hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
with odd_nonzero show False by blast
qed
qed
lemma Ints_number_of: "(number_of w :: 'a::number_ring) \<in> Ints"
by (simp add: number_of_eq Ints_def)
lemma iszero_number_of_BIT:
"iszero (number_of (w BIT x)::'a) =
(x=B0 & iszero (number_of w::'a::{ordered_idom,number_ring}))"
by (simp add: iszero_def number_of_eq Bin_simps double_eq_0_iff
Ints_odd_nonzero Ints_def split: bit.split)
lemma iszero_number_of_0:
"iszero (number_of (w BIT B0) :: 'a::{ordered_idom,number_ring}) =
iszero (number_of w :: 'a)"
by (simp only: iszero_number_of_BIT simp_thms)
lemma iszero_number_of_1:
"~ iszero (number_of (w BIT B1)::'a::{ordered_idom,number_ring})"
by (simp add: iszero_number_of_BIT)
subsection{*The Less-Than Relation*}
lemma less_number_of_eq_neg:
"((number_of x::'a::{ordered_idom,number_ring}) < number_of y)
= neg (number_of (bin_add x (bin_minus y)) :: 'a)"
apply (subst less_iff_diff_less_0)
apply (simp add: neg_def diff_minus number_of_add number_of_minus)
done
text{*If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
@{term Numeral0} IS @{term "number_of Pls"} *}
lemma not_neg_number_of_Pls:
"~ neg (number_of Pls ::'a::{ordered_idom,number_ring})"
by (simp add: neg_def numeral_0_eq_0)
lemma neg_number_of_Min:
"neg (number_of Min ::'a::{ordered_idom,number_ring})"
by (simp add: neg_def zero_less_one numeral_m1_eq_minus_1)
lemma double_less_0_iff: "(a + a < 0) = (a < (0::'a::ordered_idom))"
proof -
have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
also have "... = (a < 0)"
by (simp add: mult_less_0_iff zero_less_two
order_less_not_sym [OF zero_less_two])
finally show ?thesis .
qed
lemma odd_less_0: "(1 + z + z < 0) = (z < (0::int))";
proof (cases z rule: int_cases)
case (nonneg n)
thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
le_imp_0_less [THEN order_less_imp_le])
next
case (neg n)
thus ?thesis by (simp del: int_Suc
add: int_Suc0_eq_1 [symmetric] zadd_int compare_rls)
qed
text{*The premise involving @{term Ints} prevents @{term "a = 1/2"}.*}
lemma Ints_odd_less_0:
"a \<in> Ints ==> (1 + a + a < 0) = (a < (0::'a::ordered_idom))";
proof (unfold Ints_def)
assume "a \<in> range of_int"
then obtain z where a: "a = of_int z" ..
hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
by (simp add: a)
also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
also have "... = (a < 0)" by (simp add: a)
finally show ?thesis .
qed
lemma neg_number_of_BIT:
"neg (number_of (w BIT x)::'a) =
neg (number_of w :: 'a::{ordered_idom,number_ring})"
by (simp add: neg_def number_of_eq Bin_simps double_less_0_iff
Ints_odd_less_0 Ints_def split: bit.split)
text{*Less-Than or Equals*}
text{*Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals*}
lemmas le_number_of_eq_not_less =
linorder_not_less [of "number_of w" "number_of v", symmetric,
standard]
lemma le_number_of_eq:
"((number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y)
= (~ (neg (number_of (bin_add y (bin_minus x)) :: 'a)))"
by (simp add: le_number_of_eq_not_less less_number_of_eq_neg)
text{*Absolute value (@{term abs})*}
lemma abs_number_of:
"abs(number_of x::'a::{ordered_idom,number_ring}) =
(if number_of x < (0::'a) then -number_of x else number_of x)"
by (simp add: abs_if)
text{*Re-orientation of the equation nnn=x*}
lemma number_of_reorient: "(number_of w = x) = (x = number_of w)"
by auto
subsection{*Simplification of arithmetic operations on integer constants.*}
lemmas bin_arith_extra_simps =
number_of_add [symmetric]
number_of_minus [symmetric] numeral_m1_eq_minus_1 [symmetric]
number_of_mult [symmetric]
diff_number_of_eq abs_number_of
text{*For making a minimal simpset, one must include these default simprules.
Also include @{text simp_thms} *}
lemmas bin_arith_simps =
Numeral.bit.distinct
Pls_0_eq Min_1_eq
bin_pred_Pls bin_pred_Min bin_pred_1 bin_pred_0
bin_succ_Pls bin_succ_Min bin_succ_1 bin_succ_0
bin_add_Pls bin_add_Min bin_add_BIT_0 bin_add_BIT_10 bin_add_BIT_11
bin_minus_Pls bin_minus_Min bin_minus_1 bin_minus_0
bin_mult_Pls bin_mult_Min bin_mult_1 bin_mult_0
bin_add_Pls_right bin_add_Min_right
abs_zero abs_one bin_arith_extra_simps
text{*Simplification of relational operations*}
lemmas bin_rel_simps =
eq_number_of_eq iszero_number_of_Pls nonzero_number_of_Min
iszero_number_of_0 iszero_number_of_1
less_number_of_eq_neg
not_neg_number_of_Pls not_neg_0 not_neg_1 not_iszero_1
neg_number_of_Min neg_number_of_BIT
le_number_of_eq
declare bin_arith_extra_simps [simp]
declare bin_rel_simps [simp]
subsection{*Simplification of arithmetic when nested to the right*}
lemma add_number_of_left [simp]:
"number_of v + (number_of w + z) =
(number_of(bin_add v w) + z::'a::number_ring)"
by (simp add: add_assoc [symmetric])
lemma mult_number_of_left [simp]:
"number_of v * (number_of w * z) =
(number_of(bin_mult v w) * z::'a::number_ring)"
by (simp add: mult_assoc [symmetric])
lemma add_number_of_diff1:
"number_of v + (number_of w - c) =
number_of(bin_add v w) - (c::'a::number_ring)"
by (simp add: diff_minus add_number_of_left)
lemma add_number_of_diff2 [simp]: "number_of v + (c - number_of w) =
number_of (bin_add v (bin_minus w)) + (c::'a::number_ring)"
apply (subst diff_number_of_eq [symmetric])
apply (simp only: compare_rls)
done
hide (open) const Pls Min B0 B1
end