(* Title: HOL/NatBin.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
*)
header {* Binary arithmetic for the natural numbers *}
theory NatBin
imports IntDiv
begin
text {*
Arithmetic for naturals is reduced to that for the non-negative integers.
*}
instance nat :: number
nat_number_of_def [code inline]: "number_of v == nat (number_of (v\<Colon>int))" ..
abbreviation (xsymbols)
square :: "'a::power => 'a" ("(_\<twosuperior>)" [1000] 999) where
"x\<twosuperior> == x^2"
notation (latex output)
square ("(_\<twosuperior>)" [1000] 999)
notation (HTML output)
square ("(_\<twosuperior>)" [1000] 999)
subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
declare nat_0 [simp] nat_1 [simp]
lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
by (simp add: nat_number_of_def)
lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
by (simp add: nat_number_of_def)
lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
by (simp add: nat_1 nat_number_of_def)
lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
by (simp add: nat_numeral_1_eq_1)
lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
apply (unfold nat_number_of_def)
apply (rule nat_2)
done
text{*Distributive laws for type @{text nat}. The others are in theory
@{text IntArith}, but these require div and mod to be defined for type
"int". They also need some of the lemmas proved above.*}
lemma nat_div_distrib: "(0::int) <= z ==> nat (z div z') = nat z div nat z'"
apply (case_tac "0 <= z'")
apply (auto simp add: div_nonneg_neg_le0 DIVISION_BY_ZERO_DIV)
apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)
apply (auto elim!: nonneg_eq_int)
apply (rename_tac m m')
apply (subgoal_tac "0 <= int m div int m'")
prefer 2 apply (simp add: nat_numeral_0_eq_0 pos_imp_zdiv_nonneg_iff)
apply (rule of_nat_eq_iff [where 'a=int, THEN iffD1], simp)
apply (rule_tac r = "int (m mod m') " in quorem_div)
prefer 2 apply force
apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0
of_nat_add [symmetric] of_nat_mult [symmetric]
del: of_nat_add of_nat_mult)
done
(*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*)
lemma nat_mod_distrib:
"[| (0::int) <= z; 0 <= z' |] ==> nat (z mod z') = nat z mod nat z'"
apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)
apply (auto elim!: nonneg_eq_int)
apply (rename_tac m m')
apply (subgoal_tac "0 <= int m mod int m'")
prefer 2 apply (simp add: nat_less_iff nat_numeral_0_eq_0 pos_mod_sign)
apply (rule int_int_eq [THEN iffD1], simp)
apply (rule_tac q = "int (m div m') " in quorem_mod)
prefer 2 apply force
apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0
of_nat_add [symmetric] of_nat_mult [symmetric]
del: of_nat_add of_nat_mult)
done
text{*Suggested by Matthias Daum*}
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
apply (subgoal_tac "nat x div nat k < nat x")
apply (simp (asm_lr) add: nat_div_distrib [symmetric])
apply (rule Divides.div_less_dividend, simp_all)
done
subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
(*"neg" is used in rewrite rules for binary comparisons*)
lemma int_nat_number_of [simp]:
"int (number_of v) =
(if neg (number_of v :: int) then 0
else (number_of v :: int))"
by (simp del: nat_number_of
add: neg_nat nat_number_of_def not_neg_nat add_assoc)
subsubsection{*Successor *}
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
apply (rule sym)
apply (simp add: nat_eq_iff int_Suc)
done
lemma Suc_nat_number_of_add:
"Suc (number_of v + n) =
(if neg (number_of v :: int) then 1+n else number_of (Numeral.succ v) + n)"
by (simp del: nat_number_of
add: nat_number_of_def neg_nat
Suc_nat_eq_nat_zadd1 number_of_succ)
lemma Suc_nat_number_of [simp]:
"Suc (number_of v) =
(if neg (number_of v :: int) then 1 else number_of (Numeral.succ v))"
apply (cut_tac n = 0 in Suc_nat_number_of_add)
apply (simp cong del: if_weak_cong)
done
subsubsection{*Addition *}
(*"neg" is used in rewrite rules for binary comparisons*)
lemma add_nat_number_of [simp]:
"(number_of v :: nat) + number_of v' =
(if neg (number_of v :: int) then number_of v'
else if neg (number_of v' :: int) then number_of v
else number_of (v + v'))"
by (force dest!: neg_nat
simp del: nat_number_of
simp add: nat_number_of_def nat_add_distrib [symmetric])
subsubsection{*Subtraction *}
lemma diff_nat_eq_if:
"nat z - nat z' =
(if neg z' then nat z
else let d = z-z' in
if neg d then 0 else nat d)"
apply (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
done
lemma diff_nat_number_of [simp]:
"(number_of v :: nat) - number_of v' =
(if neg (number_of v' :: int) then number_of v
else let d = number_of (v + uminus v') in
if neg d then 0 else nat d)"
by (simp del: nat_number_of add: diff_nat_eq_if nat_number_of_def)
subsubsection{*Multiplication *}
lemma mult_nat_number_of [simp]:
"(number_of v :: nat) * number_of v' =
(if neg (number_of v :: int) then 0 else number_of (v * v'))"
by (force dest!: neg_nat
simp del: nat_number_of
simp add: nat_number_of_def nat_mult_distrib [symmetric])
subsubsection{*Quotient *}
lemma div_nat_number_of [simp]:
"(number_of v :: nat) div number_of v' =
(if neg (number_of v :: int) then 0
else nat (number_of v div number_of v'))"
by (force dest!: neg_nat
simp del: nat_number_of
simp add: nat_number_of_def nat_div_distrib [symmetric])
lemma one_div_nat_number_of [simp]:
"(Suc 0) div number_of v' = (nat (1 div number_of v'))"
by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
subsubsection{*Remainder *}
lemma mod_nat_number_of [simp]:
"(number_of v :: nat) mod number_of v' =
(if neg (number_of v :: int) then 0
else if neg (number_of v' :: int) then number_of v
else nat (number_of v mod number_of v'))"
by (force dest!: neg_nat
simp del: nat_number_of
simp add: nat_number_of_def nat_mod_distrib [symmetric])
lemma one_mod_nat_number_of [simp]:
"(Suc 0) mod number_of v' =
(if neg (number_of v' :: int) then Suc 0
else nat (1 mod number_of v'))"
by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
subsubsection{* Divisibility *}
lemmas dvd_eq_mod_eq_0_number_of =
dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
declare dvd_eq_mod_eq_0_number_of [simp]
ML
{*
val nat_number_of_def = thm"nat_number_of_def";
val nat_number_of = thm"nat_number_of";
val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
val numeral_2_eq_2 = thm"numeral_2_eq_2";
val nat_div_distrib = thm"nat_div_distrib";
val nat_mod_distrib = thm"nat_mod_distrib";
val int_nat_number_of = thm"int_nat_number_of";
val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
val Suc_nat_number_of = thm"Suc_nat_number_of";
val add_nat_number_of = thm"add_nat_number_of";
val diff_nat_eq_if = thm"diff_nat_eq_if";
val diff_nat_number_of = thm"diff_nat_number_of";
val mult_nat_number_of = thm"mult_nat_number_of";
val div_nat_number_of = thm"div_nat_number_of";
val mod_nat_number_of = thm"mod_nat_number_of";
*}
subsection{*Comparisons*}
subsubsection{*Equals (=) *}
lemma eq_nat_nat_iff:
"[| (0::int) <= z; 0 <= z' |] ==> (nat z = nat z') = (z=z')"
by (auto elim!: nonneg_eq_int)
(*"neg" is used in rewrite rules for binary comparisons*)
lemma eq_nat_number_of [simp]:
"((number_of v :: nat) = number_of v') =
(if neg (number_of v :: int) then (iszero (number_of v' :: int) | neg (number_of v' :: int))
else if neg (number_of v' :: int) then iszero (number_of v :: int)
else iszero (number_of (v + uminus v') :: int))"
apply (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
eq_nat_nat_iff eq_number_of_eq nat_0 iszero_def
split add: split_if cong add: imp_cong)
apply (simp only: nat_eq_iff nat_eq_iff2)
apply (simp add: not_neg_eq_ge_0 [symmetric])
done
subsubsection{*Less-than (<) *}
(*"neg" is used in rewrite rules for binary comparisons*)
lemma less_nat_number_of [simp]:
"((number_of v :: nat) < number_of v') =
(if neg (number_of v :: int) then neg (number_of (uminus v') :: int)
else neg (number_of (v + uminus v') :: int))"
by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
nat_less_eq_zless less_number_of_eq_neg zless_nat_eq_int_zless
cong add: imp_cong, simp add: Pls_def)
(*Maps #n to n for n = 0, 1, 2*)
lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
subsection{*Powers with Numeric Exponents*}
text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
We cannot prove general results about the numeral @{term "-1"}, so we have to
use @{term "- 1"} instead.*}
lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
by (simp add: numeral_2_eq_2 Power.power_Suc)
lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
by (simp add: power2_eq_square)
lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
by (simp add: power2_eq_square)
lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
apply (erule ssubst)
apply (simp add: power_Suc mult_ac)
apply (unfold nat_number_of_def)
apply (subst nat_eq_iff)
apply simp
done
text{*Squares of literal numerals will be evaluated.*}
lemmas power2_eq_square_number_of =
power2_eq_square [of "number_of w", standard]
declare power2_eq_square_number_of [simp]
lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
by (simp add: power2_eq_square)
lemma zero_less_power2[simp]:
"(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
lemma power2_less_0[simp]:
fixes a :: "'a::{ordered_idom,recpower}"
shows "~ (a\<twosuperior> < 0)"
by (force simp add: power2_eq_square mult_less_0_iff)
lemma zero_eq_power2[simp]:
"(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
by (force simp add: power2_eq_square mult_eq_0_iff)
lemma abs_power2[simp]:
"abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
by (simp add: power2_eq_square abs_mult abs_mult_self)
lemma power2_abs[simp]:
"(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
by (simp add: power2_eq_square abs_mult_self)
lemma power2_minus[simp]:
"(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
by (simp add: power2_eq_square)
lemma power2_le_imp_le:
fixes x y :: "'a::{ordered_semidom,recpower}"
shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
lemma power2_less_imp_less:
fixes x y :: "'a::{ordered_semidom,recpower}"
shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
by (rule power_less_imp_less_base)
lemma power2_eq_imp_eq:
fixes x y :: "'a::{ordered_semidom,recpower}"
shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
apply (induct "n")
apply (auto simp add: power_Suc power_add)
done
lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
by (subst mult_commute) (simp add: power_mult)
lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
by (simp add: power_even_eq)
lemma power_minus_even [simp]:
"(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
by (simp add: power_minus1_even power_minus [of a])
lemma zero_le_even_power'[simp]:
"0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
proof (induct "n")
case 0
show ?case by (simp add: zero_le_one)
next
case (Suc n)
have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
by (simp add: mult_ac power_add power2_eq_square)
thus ?case
by (simp add: prems zero_le_mult_iff)
qed
lemma odd_power_less_zero:
"(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
proof (induct "n")
case 0
then show ?case by (simp add: Power.power_Suc)
next
case (Suc n)
have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
by (simp add: mult_ac power_add power2_eq_square Power.power_Suc)
thus ?case
by (simp add: prems mult_less_0_iff mult_neg_neg)
qed
lemma odd_0_le_power_imp_0_le:
"0 \<le> a ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
apply (insert odd_power_less_zero [of a n])
apply (force simp add: linorder_not_less [symmetric])
done
text{*Simprules for comparisons where common factors can be cancelled.*}
lemmas zero_compare_simps =
add_strict_increasing add_strict_increasing2 add_increasing
zero_le_mult_iff zero_le_divide_iff
zero_less_mult_iff zero_less_divide_iff
mult_le_0_iff divide_le_0_iff
mult_less_0_iff divide_less_0_iff
zero_le_power2 power2_less_0
subsubsection{*Nat *}
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
by (simp add: numerals)
(*Expresses a natural number constant as the Suc of another one.
NOT suitable for rewriting because n recurs in the condition.*)
lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
subsubsection{*Arith *}
lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
by (simp add: numerals)
lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
by (simp add: numerals)
(* These two can be useful when m = number_of... *)
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
apply (case_tac "m")
apply (simp_all add: numerals)
done
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
apply (case_tac "m")
apply (simp_all add: numerals)
done
lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
apply (case_tac "m")
apply (simp_all add: numerals)
done
subsection{*Comparisons involving (0::nat) *}
text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
lemma eq_number_of_0 [simp]:
"(number_of v = (0::nat)) =
(if neg (number_of v :: int) then True else iszero (number_of v :: int))"
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
lemma eq_0_number_of [simp]:
"((0::nat) = number_of v) =
(if neg (number_of v :: int) then True else iszero (number_of v :: int))"
by (rule trans [OF eq_sym_conv eq_number_of_0])
lemma less_0_number_of [simp]:
"((0::nat) < number_of v) = neg (number_of (uminus v) :: int)"
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] Pls_def)
lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
subsection{*Comparisons involving @{term Suc} *}
lemma eq_number_of_Suc [simp]:
"(number_of v = Suc n) =
(let pv = number_of (Numeral.pred v) in
if neg pv then False else nat pv = n)"
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
number_of_pred nat_number_of_def
split add: split_if)
apply (rule_tac x = "number_of v" in spec)
apply (auto simp add: nat_eq_iff)
done
lemma Suc_eq_number_of [simp]:
"(Suc n = number_of v) =
(let pv = number_of (Numeral.pred v) in
if neg pv then False else nat pv = n)"
by (rule trans [OF eq_sym_conv eq_number_of_Suc])
lemma less_number_of_Suc [simp]:
"(number_of v < Suc n) =
(let pv = number_of (Numeral.pred v) in
if neg pv then True else nat pv < n)"
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
number_of_pred nat_number_of_def
split add: split_if)
apply (rule_tac x = "number_of v" in spec)
apply (auto simp add: nat_less_iff)
done
lemma less_Suc_number_of [simp]:
"(Suc n < number_of v) =
(let pv = number_of (Numeral.pred v) in
if neg pv then False else n < nat pv)"
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
number_of_pred nat_number_of_def
split add: split_if)
apply (rule_tac x = "number_of v" in spec)
apply (auto simp add: zless_nat_eq_int_zless)
done
lemma le_number_of_Suc [simp]:
"(number_of v <= Suc n) =
(let pv = number_of (Numeral.pred v) in
if neg pv then True else nat pv <= n)"
by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
lemma le_Suc_number_of [simp]:
"(Suc n <= number_of v) =
(let pv = number_of (Numeral.pred v) in
if neg pv then False else n <= nat pv)"
by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
lemma lemma1: "(m+m = n+n) = (m = (n::int))"
by auto
lemma lemma2: "m+m ~= (1::int) + (n + n)"
apply auto
apply (drule_tac f = "%x. x mod 2" in arg_cong)
apply (simp add: zmod_zadd1_eq)
done
lemma eq_number_of_BIT_BIT:
"((number_of (v BIT x) ::int) = number_of (w BIT y)) =
(x=y & (((number_of v) ::int) = number_of w))"
apply (simp only: number_of_BIT lemma1 lemma2 eq_commute
OrderedGroup.add_left_cancel add_assoc OrderedGroup.add_0_left
split add: bit.split)
apply simp
done
lemma eq_number_of_BIT_Pls:
"((number_of (v BIT x) ::int) = Numeral0) =
(x=bit.B0 & (((number_of v) ::int) = Numeral0))"
apply (simp only: simp_thms add: number_of_BIT number_of_Pls eq_commute
split add: bit.split cong: imp_cong)
apply (rule_tac x = "number_of v" in spec, safe)
apply (simp_all (no_asm_use))
apply (drule_tac f = "%x. x mod 2" in arg_cong)
apply (simp add: zmod_zadd1_eq)
done
lemma eq_number_of_BIT_Min:
"((number_of (v BIT x) ::int) = number_of Numeral.Min) =
(x=bit.B1 & (((number_of v) ::int) = number_of Numeral.Min))"
apply (simp only: simp_thms add: number_of_BIT number_of_Min eq_commute
split add: bit.split cong: imp_cong)
apply (rule_tac x = "number_of v" in spec, auto)
apply (drule_tac f = "%x. x mod 2" in arg_cong, auto)
done
lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Numeral.Min"
by auto
subsection{*Max and Min Combined with @{term Suc} *}
lemma max_number_of_Suc [simp]:
"max (Suc n) (number_of v) =
(let pv = number_of (Numeral.pred v) in
if neg pv then Suc n else Suc(max n (nat pv)))"
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
split add: split_if nat.split)
apply (rule_tac x = "number_of v" in spec)
apply auto
done
lemma max_Suc_number_of [simp]:
"max (number_of v) (Suc n) =
(let pv = number_of (Numeral.pred v) in
if neg pv then Suc n else Suc(max (nat pv) n))"
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
split add: split_if nat.split)
apply (rule_tac x = "number_of v" in spec)
apply auto
done
lemma min_number_of_Suc [simp]:
"min (Suc n) (number_of v) =
(let pv = number_of (Numeral.pred v) in
if neg pv then 0 else Suc(min n (nat pv)))"
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
split add: split_if nat.split)
apply (rule_tac x = "number_of v" in spec)
apply auto
done
lemma min_Suc_number_of [simp]:
"min (number_of v) (Suc n) =
(let pv = number_of (Numeral.pred v) in
if neg pv then 0 else Suc(min (nat pv) n))"
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
split add: split_if nat.split)
apply (rule_tac x = "number_of v" in spec)
apply auto
done
subsection{*Literal arithmetic involving powers*}
lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
apply (induct "n")
apply (simp_all (no_asm_simp) add: nat_mult_distrib)
done
lemma power_nat_number_of:
"(number_of v :: nat) ^ n =
(if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
split add: split_if cong: imp_cong)
lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
declare power_nat_number_of_number_of [simp]
text{*For arbitrary rings*}
lemma power_number_of_even:
fixes z :: "'a::{number_ring,recpower}"
shows "z ^ number_of (w BIT bit.B0) = (let w = z ^ (number_of w) in w * w)"
unfolding Let_def nat_number_of_def number_of_BIT bit.cases
apply (rule_tac x = "number_of w" in spec, clarify)
apply (case_tac " (0::int) <= x")
apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
done
lemma power_number_of_odd:
fixes z :: "'a::{number_ring,recpower}"
shows "z ^ number_of (w BIT bit.B1) = (if (0::int) <= number_of w
then (let w = z ^ (number_of w) in z * w * w) else 1)"
unfolding Let_def nat_number_of_def number_of_BIT bit.cases
apply (rule_tac x = "number_of w" in spec, auto)
apply (simp only: nat_add_distrib nat_mult_distrib)
apply simp
apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
done
lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
lemmas power_number_of_even_number_of [simp] =
power_number_of_even [of "number_of v", standard]
lemmas power_number_of_odd_number_of [simp] =
power_number_of_odd [of "number_of v", standard]
ML
{*
val numerals = thms"numerals";
val numeral_ss = simpset() addsimps numerals;
val nat_bin_arith_setup =
LinArith.map_data
(fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
{add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
inj_thms = inj_thms,
lessD = lessD, neqE = neqE,
simpset = simpset addsimps [Suc_nat_number_of, int_nat_number_of,
not_neg_number_of_Pls,
neg_number_of_Min,neg_number_of_BIT]})
*}
declaration {* K nat_bin_arith_setup *}
(* Enable arith to deal with div/mod k where k is a numeral: *)
declare split_div[of _ _ "number_of k", standard, arith_split]
declare split_mod[of _ _ "number_of k", standard, arith_split]
lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
by (simp add: number_of_Pls nat_number_of_def)
lemma nat_number_of_Min: "number_of Numeral.Min = (0::nat)"
apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
done
lemma nat_number_of_BIT_1:
"number_of (w BIT bit.B1) =
(if neg (number_of w :: int) then 0
else let n = number_of w in Suc (n + n))"
apply (simp only: nat_number_of_def Let_def split: split_if)
apply (intro conjI impI)
apply (simp add: neg_nat neg_number_of_BIT)
apply (rule int_int_eq [THEN iffD1])
apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
apply (simp only: number_of_BIT zadd_assoc split: bit.split)
apply simp
done
lemma nat_number_of_BIT_0:
"number_of (w BIT bit.B0) = (let n::nat = number_of w in n + n)"
apply (simp only: nat_number_of_def Let_def)
apply (cases "neg (number_of w :: int)")
apply (simp add: neg_nat neg_number_of_BIT)
apply (rule int_int_eq [THEN iffD1])
apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
apply (simp only: number_of_BIT zadd_assoc)
apply simp
done
lemmas nat_number =
nat_number_of_Pls nat_number_of_Min
nat_number_of_BIT_1 nat_number_of_BIT_0
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
by (simp add: Let_def)
lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
by (simp add: power_mult power_Suc);
lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
by (simp add: power_mult power_Suc);
subsection{*Literal arithmetic and @{term of_nat}*}
lemma of_nat_double:
"0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
by (simp only: mult_2 nat_add_distrib of_nat_add)
lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
by (simp only: nat_number_of_def)
lemma of_nat_number_of_lemma:
"of_nat (number_of v :: nat) =
(if 0 \<le> (number_of v :: int)
then (number_of v :: 'a :: number_ring)
else 0)"
by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
lemma of_nat_number_of_eq [simp]:
"of_nat (number_of v :: nat) =
(if neg (number_of v :: int) then 0
else (number_of v :: 'a :: number_ring))"
by (simp only: of_nat_number_of_lemma neg_def, simp)
subsection {*Lemmas for the Combination and Cancellation Simprocs*}
lemma nat_number_of_add_left:
"number_of v + (number_of v' + (k::nat)) =
(if neg (number_of v :: int) then number_of v' + k
else if neg (number_of v' :: int) then number_of v + k
else number_of (v + v') + k)"
by simp
lemma nat_number_of_mult_left:
"number_of v * (number_of v' * (k::nat)) =
(if neg (number_of v :: int) then 0
else number_of (v * v') * k)"
by simp
subsubsection{*For @{text combine_numerals}*}
lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
by (simp add: add_mult_distrib)
subsubsection{*For @{text cancel_numerals}*}
lemma nat_diff_add_eq1:
"j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
by (simp split add: nat_diff_split add: add_mult_distrib)
lemma nat_diff_add_eq2:
"i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
by (simp split add: nat_diff_split add: add_mult_distrib)
lemma nat_eq_add_iff1:
"j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)
lemma nat_eq_add_iff2:
"i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)
lemma nat_less_add_iff1:
"j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)
lemma nat_less_add_iff2:
"i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)
lemma nat_le_add_iff1:
"j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)
lemma nat_le_add_iff2:
"i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)
subsubsection{*For @{text cancel_numeral_factors} *}
lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
by auto
lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
by auto
lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
by auto
lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
by auto
lemma nat_mult_dvd_cancel_disj[simp]:
"(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
by(auto)
subsubsection{*For @{text cancel_factor} *}
lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
by auto
lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
by auto
lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
by auto
lemma nat_mult_div_cancel_disj[simp]:
"(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
by (simp add: nat_mult_div_cancel1)
subsection {* legacy ML bindings *}
ML
{*
val eq_nat_nat_iff = thm"eq_nat_nat_iff";
val eq_nat_number_of = thm"eq_nat_number_of";
val less_nat_number_of = thm"less_nat_number_of";
val power2_eq_square = thm "power2_eq_square";
val zero_le_power2 = thm "zero_le_power2";
val zero_less_power2 = thm "zero_less_power2";
val zero_eq_power2 = thm "zero_eq_power2";
val abs_power2 = thm "abs_power2";
val power2_abs = thm "power2_abs";
val power2_minus = thm "power2_minus";
val power_minus1_even = thm "power_minus1_even";
val power_minus_even = thm "power_minus_even";
val odd_power_less_zero = thm "odd_power_less_zero";
val odd_0_le_power_imp_0_le = thm "odd_0_le_power_imp_0_le";
val Suc_pred' = thm"Suc_pred'";
val expand_Suc = thm"expand_Suc";
val Suc_eq_add_numeral_1 = thm"Suc_eq_add_numeral_1";
val Suc_eq_add_numeral_1_left = thm"Suc_eq_add_numeral_1_left";
val add_eq_if = thm"add_eq_if";
val mult_eq_if = thm"mult_eq_if";
val power_eq_if = thm"power_eq_if";
val eq_number_of_0 = thm"eq_number_of_0";
val eq_0_number_of = thm"eq_0_number_of";
val less_0_number_of = thm"less_0_number_of";
val neg_imp_number_of_eq_0 = thm"neg_imp_number_of_eq_0";
val eq_number_of_Suc = thm"eq_number_of_Suc";
val Suc_eq_number_of = thm"Suc_eq_number_of";
val less_number_of_Suc = thm"less_number_of_Suc";
val less_Suc_number_of = thm"less_Suc_number_of";
val le_number_of_Suc = thm"le_number_of_Suc";
val le_Suc_number_of = thm"le_Suc_number_of";
val eq_number_of_BIT_BIT = thm"eq_number_of_BIT_BIT";
val eq_number_of_BIT_Pls = thm"eq_number_of_BIT_Pls";
val eq_number_of_BIT_Min = thm"eq_number_of_BIT_Min";
val eq_number_of_Pls_Min = thm"eq_number_of_Pls_Min";
val of_nat_number_of_eq = thm"of_nat_number_of_eq";
val nat_power_eq = thm"nat_power_eq";
val power_nat_number_of = thm"power_nat_number_of";
val zpower_number_of_even = thm"zpower_number_of_even";
val zpower_number_of_odd = thm"zpower_number_of_odd";
val nat_number_of_Pls = thm"nat_number_of_Pls";
val nat_number_of_Min = thm"nat_number_of_Min";
val Let_Suc = thm"Let_Suc";
val nat_number = thms"nat_number";
val nat_number_of_add_left = thm"nat_number_of_add_left";
val nat_number_of_mult_left = thm"nat_number_of_mult_left";
val left_add_mult_distrib = thm"left_add_mult_distrib";
val nat_diff_add_eq1 = thm"nat_diff_add_eq1";
val nat_diff_add_eq2 = thm"nat_diff_add_eq2";
val nat_eq_add_iff1 = thm"nat_eq_add_iff1";
val nat_eq_add_iff2 = thm"nat_eq_add_iff2";
val nat_less_add_iff1 = thm"nat_less_add_iff1";
val nat_less_add_iff2 = thm"nat_less_add_iff2";
val nat_le_add_iff1 = thm"nat_le_add_iff1";
val nat_le_add_iff2 = thm"nat_le_add_iff2";
val nat_mult_le_cancel1 = thm"nat_mult_le_cancel1";
val nat_mult_less_cancel1 = thm"nat_mult_less_cancel1";
val nat_mult_eq_cancel1 = thm"nat_mult_eq_cancel1";
val nat_mult_div_cancel1 = thm"nat_mult_div_cancel1";
val nat_mult_le_cancel_disj = thm"nat_mult_le_cancel_disj";
val nat_mult_less_cancel_disj = thm"nat_mult_less_cancel_disj";
val nat_mult_eq_cancel_disj = thm"nat_mult_eq_cancel_disj";
val nat_mult_div_cancel_disj = thm"nat_mult_div_cancel_disj";
val power_minus_even = thm"power_minus_even";
*}
end