(* Title: HOL/Integ/Bin.thy
ID: $Id$
Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Ported from ZF to HOL by David Spelt.
*)
header{*Arithmetic on Binary Integers*}
theory Bin = IntDef + Numeral:
axclass number_ring \<subseteq> number, ring
number_of_Pls: "number_of bin.Pls = 0"
number_of_Min: "number_of bin.Min = - 1"
number_of_BIT: "number_of(w BIT x) = (if x then 1 else 0) +
(number_of w) + (number_of w)"
subsection{*Converting Numerals to Rings: @{term number_of}*}
lemmas number_of = number_of_Pls number_of_Min number_of_BIT
lemma number_of_NCons [simp]:
"number_of(NCons w b) = (number_of(w BIT b)::'a::number_ring)"
by (induct_tac "w", simp_all add: number_of)
lemma number_of_succ: "number_of(bin_succ w) = (1 + number_of w ::'a::number_ring)"
apply (induct_tac "w")
apply (simp_all add: number_of add_ac)
done
lemma number_of_pred: "number_of(bin_pred w) = (- 1 + number_of w ::'a::number_ring)"
apply (induct_tac "w")
apply (simp_all add: number_of add_assoc [symmetric])
apply (simp add: add_ac)
done
lemma number_of_minus: "number_of(bin_minus w) = (- (number_of w)::'a::number_ring)"
apply (induct_tac "w")
apply (simp_all del: bin_pred_Pls bin_pred_Min bin_pred_BIT
add: number_of number_of_succ number_of_pred add_assoc)
done
text{*This proof is complicated by the mutual recursion*}
lemma number_of_add [rule_format]:
"\<forall>w. number_of(bin_add v w) = (number_of v + number_of w::'a::number_ring)"
apply (induct_tac "v")
apply (simp add: number_of)
apply (simp add: number_of number_of_pred)
apply (rule allI)
apply (induct_tac "w")
apply (simp_all add: number_of bin_add_BIT_BIT number_of_succ number_of_pred add_ac)
apply (simp add: add_left_commute [of "1::'a::number_ring"])
done
lemma number_of_mult:
"number_of(bin_mult v w) = (number_of v * number_of w::'a::number_ring)"
apply (induct_tac "v", simp add: number_of)
apply (simp add: number_of number_of_minus)
apply (simp add: number_of number_of_add left_distrib add_ac)
done
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 False) ::'a::number_ring)"
apply (induct_tac "w")
apply (simp_all add: number_of number_of_add left_distrib add_ac)
done
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)
lemma numeral_1_eq_1 [simp]: "Numeral1 = (1::'a::number_ring)"
by (simp add: number_of)
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)
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)
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 bin.Pls)::'a::number_ring)"
by (simp add: iszero_def numeral_0_eq_0)
lemma nonzero_number_of_Min: "~ iszero ((number_of bin.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_ring))"
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], simp add: add_commute)
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_ring)"
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 (induct_tac "w", simp_all add: number_of)
lemma iszero_number_of_BIT:
"iszero (number_of (w BIT x)::'a) =
(~x & iszero (number_of w::'a::{ordered_ring,number_ring}))"
by (simp add: iszero_def compare_rls zero_reorient double_eq_0_iff
number_of Ints_odd_nonzero [OF Ints_number_of])
lemma iszero_number_of_0:
"iszero (number_of (w BIT False) :: 'a::{ordered_ring,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 True)::'a::{ordered_ring,number_ring})"
by (simp only: iszero_number_of_BIT simp_thms)
subsection{*The Less-Than Relation*}
lemma less_number_of_eq_neg:
"((number_of x::'a::{ordered_ring,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 bin.Pls ::'a::{ordered_ring,number_ring})"
by (simp add: neg_def numeral_0_eq_0)
lemma neg_number_of_Min:
"neg (number_of bin.Min ::'a::{ordered_ring,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_ring))"
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_ring))";
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_ring,number_ring})"
by (simp add: number_of neg_def double_less_0_iff
Ints_odd_less_0 [OF Ints_number_of])
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_ring,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_ring,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
(*Delete the original rewrites, with their clumsy conditional expressions*)
declare bin_succ_BIT [simp del] bin_pred_BIT [simp del]
bin_minus_BIT [simp del]
declare bin_add_BIT [simp del] bin_mult_BIT [simp del]
declare NCons_Pls [simp del] NCons_Min [simp del]
(*Hide the binary representation of integer constants*)
declare number_of_Pls [simp del] number_of_Min [simp del]
number_of_BIT [simp del]
(*Simplification of arithmetic operations on integer constants.
Note that bin_pred_Pls, etc. were added to the simpset by primrec.*)
lemmas NCons_simps = NCons_Pls_0 NCons_Pls_1 NCons_Min_0 NCons_Min_1 NCons_BIT
lemmas bin_arith_extra_simps =
number_of_add [symmetric]
number_of_minus [symmetric] numeral_m1_eq_minus_1 [symmetric]
number_of_mult [symmetric]
bin_succ_1 bin_succ_0
bin_pred_1 bin_pred_0
bin_minus_1 bin_minus_0
bin_add_Pls_right bin_add_Min_right
bin_add_BIT_0 bin_add_BIT_10 bin_add_BIT_11
diff_number_of_eq abs_number_of abs_zero abs_one
bin_mult_1 bin_mult_0 NCons_simps
(*For making a minimal simpset, one must include these default simprules
of thy. Also include simp_thms, or at least (~False)=True*)
lemmas bin_arith_simps =
bin_pred_Pls bin_pred_Min
bin_succ_Pls bin_succ_Min
bin_add_Pls bin_add_Min
bin_minus_Pls bin_minus_Min
bin_mult_Pls bin_mult_Min bin_arith_extra_simps
(*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
end