(* Title: HOL/Num.thy
Author: Florian Haftmann
Author: Brian Huffman
*)
section \<open>Binary Numerals\<close>
theory Num
imports BNF_Least_Fixpoint Transfer
begin
subsection \<open>The \<open>num\<close> type\<close>
datatype num = One | Bit0 num | Bit1 num
text \<open>Increment function for type \<^typ>\<open>num\<close>\<close>
primrec inc :: \<open>num \<Rightarrow> num\<close>
where
\<open>inc One = Bit0 One\<close>
| \<open>inc (Bit0 x) = Bit1 x\<close>
| \<open>inc (Bit1 x) = Bit0 (inc x)\<close>
text \<open>Converting between type \<^typ>\<open>num\<close> and type \<^typ>\<open>nat\<close>\<close>
primrec nat_of_num :: \<open>num \<Rightarrow> nat\<close>
where
\<open>nat_of_num One = Suc 0\<close>
| \<open>nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x\<close>
| \<open>nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)\<close>
primrec num_of_nat :: \<open>nat \<Rightarrow> num\<close>
where
\<open>num_of_nat 0 = One\<close>
| \<open>num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)\<close>
lemma nat_of_num_pos: \<open>0 < nat_of_num x\<close>
by (induct x) simp_all
lemma nat_of_num_neq_0: \<open> nat_of_num x \<noteq> 0\<close>
by (induct x) simp_all
lemma nat_of_num_inc: \<open>nat_of_num (inc x) = Suc (nat_of_num x)\<close>
by (induct x) simp_all
lemma num_of_nat_double: \<open>0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)\<close>
by (induct n) simp_all
text \<open>Type \<^typ>\<open>num\<close> is isomorphic to the strictly positive natural numbers.\<close>
lemma nat_of_num_inverse: \<open>num_of_nat (nat_of_num x) = x\<close>
by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
lemma num_of_nat_inverse: \<open>0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n\<close>
by (induct n) (simp_all add: nat_of_num_inc)
lemma num_eq_iff: \<open>x = y \<longleftrightarrow> nat_of_num x = nat_of_num y\<close>
apply safe
apply (drule arg_cong [where f=num_of_nat])
apply (simp add: nat_of_num_inverse)
done
lemma num_induct [case_names One inc]:
fixes P :: \<open>num \<Rightarrow> bool\<close>
assumes One: \<open>P One\<close>
and inc: \<open>\<And>x. P x \<Longrightarrow> P (inc x)\<close>
shows \<open>P x\<close>
proof -
obtain n where n: \<open>Suc n = nat_of_num x\<close>
by (cases \<open>nat_of_num x\<close>) (simp_all add: nat_of_num_neq_0)
have \<open>P (num_of_nat (Suc n))\<close>
proof (induct n)
case 0
from One show ?case by simp
next
case (Suc n)
then have \<open>P (inc (num_of_nat (Suc n)))\<close> by (rule inc)
then show \<open>P (num_of_nat (Suc (Suc n)))\<close> by simp
qed
with n show \<open>P x\<close>
by (simp add: nat_of_num_inverse)
qed
text \<open>
From now on, there are two possible models for \<^typ>\<open>num\<close>: as positive
naturals (rule \<open>num_induct\<close>) and as digit representation (rules
\<open>num.induct\<close>, \<open>num.cases\<close>).
\<close>
subsection \<open>Numeral operations\<close>
instantiation num :: \<open>{plus,times,linorder}\<close>
begin
definition [code del]: \<open>m + n = num_of_nat (nat_of_num m + nat_of_num n)\<close>
definition [code del]: \<open>m * n = num_of_nat (nat_of_num m * nat_of_num n)\<close>
definition [code del]: \<open>m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n\<close>
definition [code del]: \<open>m < n \<longleftrightarrow> nat_of_num m < nat_of_num n\<close>
instance
by standard (auto simp add: less_num_def less_eq_num_def num_eq_iff)
end
lemma nat_of_num_add: \<open>nat_of_num (x + y) = nat_of_num x + nat_of_num y\<close>
unfolding plus_num_def
by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
lemma nat_of_num_mult: \<open>nat_of_num (x * y) = nat_of_num x * nat_of_num y\<close>
unfolding times_num_def
by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
lemma add_num_simps [simp, code]:
\<open>One + One = Bit0 One\<close>
\<open>One + Bit0 n = Bit1 n\<close>
\<open>One + Bit1 n = Bit0 (n + One)\<close>
\<open>Bit0 m + One = Bit1 m\<close>
\<open>Bit0 m + Bit0 n = Bit0 (m + n)\<close>
\<open>Bit0 m + Bit1 n = Bit1 (m + n)\<close>
\<open>Bit1 m + One = Bit0 (m + One)\<close>
\<open>Bit1 m + Bit0 n = Bit1 (m + n)\<close>
\<open>Bit1 m + Bit1 n = Bit0 (m + n + One)\<close>
by (simp_all add: num_eq_iff nat_of_num_add)
lemma mult_num_simps [simp, code]:
\<open>m * One = m\<close>
\<open>One * n = n\<close>
\<open>Bit0 m * Bit0 n = Bit0 (Bit0 (m * n))\<close>
\<open>Bit0 m * Bit1 n = Bit0 (m * Bit1 n)\<close>
\<open>Bit1 m * Bit0 n = Bit0 (Bit1 m * n)\<close>
\<open>Bit1 m * Bit1 n = Bit1 (m + n + Bit0 (m * n))\<close>
by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult distrib_right distrib_left)
lemma eq_num_simps:
\<open>One = One \<longleftrightarrow> True\<close>
\<open>One = Bit0 n \<longleftrightarrow> False\<close>
\<open>One = Bit1 n \<longleftrightarrow> False\<close>
\<open>Bit0 m = One \<longleftrightarrow> False\<close>
\<open>Bit1 m = One \<longleftrightarrow> False\<close>
\<open>Bit0 m = Bit0 n \<longleftrightarrow> m = n\<close>
\<open>Bit0 m = Bit1 n \<longleftrightarrow> False\<close>
\<open>Bit1 m = Bit0 n \<longleftrightarrow> False\<close>
\<open>Bit1 m = Bit1 n \<longleftrightarrow> m = n\<close>
by simp_all
lemma le_num_simps [simp, code]:
\<open>One \<le> n \<longleftrightarrow> True\<close>
\<open>Bit0 m \<le> One \<longleftrightarrow> False\<close>
\<open>Bit1 m \<le> One \<longleftrightarrow> False\<close>
\<open>Bit0 m \<le> Bit0 n \<longleftrightarrow> m \<le> n\<close>
\<open>Bit0 m \<le> Bit1 n \<longleftrightarrow> m \<le> n\<close>
\<open>Bit1 m \<le> Bit1 n \<longleftrightarrow> m \<le> n\<close>
\<open>Bit1 m \<le> Bit0 n \<longleftrightarrow> m < n\<close>
using nat_of_num_pos [of n] nat_of_num_pos [of m]
by (auto simp add: less_eq_num_def less_num_def)
lemma less_num_simps [simp, code]:
\<open>m < One \<longleftrightarrow> False\<close>
\<open>One < Bit0 n \<longleftrightarrow> True\<close>
\<open>One < Bit1 n \<longleftrightarrow> True\<close>
\<open>Bit0 m < Bit0 n \<longleftrightarrow> m < n\<close>
\<open>Bit0 m < Bit1 n \<longleftrightarrow> m \<le> n\<close>
\<open>Bit1 m < Bit1 n \<longleftrightarrow> m < n\<close>
\<open>Bit1 m < Bit0 n \<longleftrightarrow> m < n\<close>
using nat_of_num_pos [of n] nat_of_num_pos [of m]
by (auto simp add: less_eq_num_def less_num_def)
lemma le_num_One_iff: \<open>x \<le> One \<longleftrightarrow> x = One\<close>
by (simp add: antisym_conv)
text \<open>Rules using \<open>One\<close> and \<open>inc\<close> as constructors.\<close>
lemma add_One: \<open>x + One = inc x\<close>
by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
lemma add_One_commute: \<open>One + n = n + One\<close>
by (induct n) simp_all
lemma add_inc: \<open>x + inc y = inc (x + y)\<close>
by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
lemma mult_inc: \<open>x * inc y = x * y + x\<close>
by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
text \<open>The \<^const>\<open>num_of_nat\<close> conversion.\<close>
lemma num_of_nat_One: \<open>n \<le> 1 \<Longrightarrow> num_of_nat n = One\<close>
by (cases n) simp_all
lemma num_of_nat_plus_distrib:
\<open>0 < m \<Longrightarrow> 0 < n \<Longrightarrow> num_of_nat (m + n) = num_of_nat m + num_of_nat n\<close>
by (induct n) (auto simp add: add_One add_One_commute add_inc)
text \<open>A double-and-decrement function.\<close>
primrec BitM :: \<open>num \<Rightarrow> num\<close>
where
\<open>BitM One = One\<close>
| \<open>BitM (Bit0 n) = Bit1 (BitM n)\<close>
| \<open>BitM (Bit1 n) = Bit1 (Bit0 n)\<close>
lemma BitM_plus_one: \<open>BitM n + One = Bit0 n\<close>
by (induct n) simp_all
lemma one_plus_BitM: \<open>One + BitM n = Bit0 n\<close>
unfolding add_One_commute BitM_plus_one ..
lemma BitM_inc_eq:
\<open>BitM (inc n) = Bit1 n\<close>
by (induction n) simp_all
lemma inc_BitM_eq:
\<open>inc (BitM n) = Bit0 n\<close>
by (simp add: BitM_plus_one[symmetric] add_One)
text \<open>Squaring and exponentiation.\<close>
primrec sqr :: \<open>num \<Rightarrow> num\<close>
where
\<open>sqr One = One\<close>
| \<open>sqr (Bit0 n) = Bit0 (Bit0 (sqr n))\<close>
| \<open>sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))\<close>
primrec pow :: \<open>num \<Rightarrow> num \<Rightarrow> num\<close>
where
\<open>pow x One = x\<close>
| \<open>pow x (Bit0 y) = sqr (pow x y)\<close>
| \<open>pow x (Bit1 y) = sqr (pow x y) * x\<close>
lemma nat_of_num_sqr: \<open>nat_of_num (sqr x) = nat_of_num x * nat_of_num x\<close>
by (induct x) (simp_all add: algebra_simps nat_of_num_add)
lemma sqr_conv_mult: \<open>sqr x = x * x\<close>
by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)
lemma num_double [simp]:
\<open>Bit0 num.One * n = Bit0 n\<close>
by (simp add: num_eq_iff nat_of_num_mult)
subsection \<open>Binary numerals\<close>
text \<open>
We embed binary representations into a generic algebraic
structure using \<open>numeral\<close>.
\<close>
class numeral = one + semigroup_add
begin
primrec numeral :: \<open>num \<Rightarrow> 'a\<close>
where
numeral_One: \<open>numeral One = 1\<close>
| numeral_Bit0: \<open>numeral (Bit0 n) = numeral n + numeral n\<close>
| numeral_Bit1: \<open>numeral (Bit1 n) = numeral n + numeral n + 1\<close>
lemma numeral_code [code]:
\<open>numeral One = 1\<close>
\<open>numeral (Bit0 n) = (let m = numeral n in m + m)\<close>
\<open>numeral (Bit1 n) = (let m = numeral n in m + m + 1)\<close>
by (simp_all add: Let_def)
lemma one_plus_numeral_commute: \<open>1 + numeral x = numeral x + 1\<close>
proof (induct x)
case One
then show ?case by simp
next
case Bit0
then show ?case by (simp add: add.assoc [symmetric]) (simp add: add.assoc)
next
case Bit1
then show ?case by (simp add: add.assoc [symmetric]) (simp add: add.assoc)
qed
lemma numeral_inc: \<open>numeral (inc x) = numeral x + 1\<close>
proof (induct x)
case One
then show ?case by simp
next
case Bit0
then show ?case by simp
next
case (Bit1 x)
have \<open>numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1\<close>
by (simp only: one_plus_numeral_commute)
with Bit1 show ?case
by (simp add: add.assoc)
qed
declare numeral.simps [simp del]
abbreviation \<open>Numeral1 \<equiv> numeral One\<close>
declare numeral_One [code_post]
end
text \<open>Numeral syntax.\<close>
syntax
"_Numeral" :: \<open>num_const \<Rightarrow> 'a\<close> (\<open>(\<open>open_block notation=\<open>literal number\<close>\<close>_)\<close>)
ML_file \<open>Tools/numeral.ML\<close>
parse_translation \<open>
let
fun numeral_tr [(c as Const (\<^syntax_const>\<open>_constrain\<close>, _)) $ t $ u] =
c $ numeral_tr [t] $ u
| numeral_tr [Const (num, _)] =
(Numeral.mk_number_syntax o #value o Lexicon.read_num) num
| numeral_tr ts = raise TERM ("numeral_tr", ts);
in [(\<^syntax_const>\<open>_Numeral\<close>, K numeral_tr)] end
\<close>
typed_print_translation \<open>
let
fun num_tr' ctxt T [n] =
let
val k = Numeral.dest_num_syntax n;
val t' =
Syntax.const \<^syntax_const>\<open>_Numeral\<close> $
Syntax.free (string_of_int k);
in
(case T of
Type (\<^type_name>\<open>fun\<close>, [_, T']) =>
if Printer.type_emphasis ctxt T' then
Syntax.const \<^syntax_const>\<open>_constrain\<close> $ t' $
Syntax_Phases.term_of_typ ctxt T'
else t'
| _ => if T = dummyT then t' else raise Match)
end;
in
[(\<^const_syntax>\<open>numeral\<close>, num_tr')]
end
\<close>
subsection \<open>Class-specific numeral rules\<close>
text \<open>\<^const>\<open>numeral\<close> is a morphism.\<close>
subsubsection \<open>Structures with addition: class \<open>numeral\<close>\<close>
context numeral
begin
lemma numeral_add: \<open>numeral (m + n) = numeral m + numeral n\<close>
by (induct n rule: num_induct)
(simp_all only: numeral_One add_One add_inc numeral_inc add.assoc)
lemma numeral_plus_numeral: \<open>numeral m + numeral n = numeral (m + n)\<close>
by (rule numeral_add [symmetric])
lemma numeral_plus_one: \<open>numeral n + 1 = numeral (n + One)\<close>
using numeral_add [of n One] by (simp add: numeral_One)
lemma one_plus_numeral: \<open>1 + numeral n = numeral (One + n)\<close>
using numeral_add [of One n] by (simp add: numeral_One)
lemma one_add_one: \<open>1 + 1 = 2\<close>
using numeral_add [of One One] by (simp add: numeral_One)
lemmas add_numeral_special =
numeral_plus_one one_plus_numeral one_add_one
end
subsubsection \<open>Structures with negation: class \<open>neg_numeral\<close>\<close>
class neg_numeral = numeral + group_add
begin
lemma uminus_numeral_One: \<open>- Numeral1 = - 1\<close>
by (simp add: numeral_One)
text \<open>Numerals form an abelian subgroup.\<close>
inductive is_num :: \<open>'a \<Rightarrow> bool\<close>
where
\<open>is_num 1\<close>
| \<open>is_num x \<Longrightarrow> is_num (- x)\<close>
| \<open>is_num x \<Longrightarrow> is_num y \<Longrightarrow> is_num (x + y)\<close>
lemma is_num_numeral: \<open>is_num (numeral k)\<close>
by (induct k) (simp_all add: numeral.simps is_num.intros)
lemma is_num_add_commute: \<open>is_num x \<Longrightarrow> is_num y \<Longrightarrow> x + y = y + x\<close>
proof(induction x rule: is_num.induct)
case 1
then show ?case
proof (induction y rule: is_num.induct)
case 1
then show ?case by simp
next
case (2 y)
then have \<open>y + (1 + - y) + y = y + (- y + 1) + y\<close>
by (simp add: add.assoc)
then have \<open>y + (1 + - y) = y + (- y + 1)\<close>
by simp
then show ?case
by (rule add_left_imp_eq[of y])
next
case (3 x y)
then have \<open>1 + (x + y) = x + 1 + y\<close>
by (simp add: add.assoc [symmetric])
then show ?case using 3
by (simp add: add.assoc)
qed
next
case (2 x)
then have \<open>x + (- x + y) + x = x + (y + - x) + x\<close>
by (simp add: add.assoc)
then have \<open>x + (- x + y) = x + (y + - x)\<close>
by simp
then show ?case
by (rule add_left_imp_eq[of x])
next
case (3 x z)
moreover have \<open>x + (y + z) = (x + y) + z\<close>
by (simp add: add.assoc[symmetric])
ultimately show ?case
by (simp add: add.assoc)
qed
lemma is_num_add_left_commute: \<open>is_num x \<Longrightarrow> is_num y \<Longrightarrow> x + (y + z) = y + (x + z)\<close>
by (simp only: add.assoc [symmetric] is_num_add_commute)
lemmas is_num_normalize =
add.assoc is_num_add_commute is_num_add_left_commute
is_num.intros is_num_numeral
minus_add
definition dbl :: \<open>'a \<Rightarrow> 'a\<close>
where \<open>dbl x = x + x\<close>
definition dbl_inc :: \<open>'a \<Rightarrow> 'a\<close>
where \<open>dbl_inc x = x + x + 1\<close>
definition dbl_dec :: \<open>'a \<Rightarrow> 'a\<close>
where \<open>dbl_dec x = x + x - 1\<close>
definition sub :: \<open>num \<Rightarrow> num \<Rightarrow> 'a\<close>
where \<open>sub k l = numeral k - numeral l\<close>
lemma numeral_BitM: \<open>numeral (BitM n) = numeral (Bit0 n) - 1\<close>
by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
lemma sub_inc_One_eq:
\<open>sub (inc n) num.One = numeral n\<close>
by (simp_all add: sub_def diff_eq_eq numeral_inc numeral.numeral_One)
lemma dbl_simps [simp]:
\<open>dbl (- numeral k) = - dbl (numeral k)\<close>
\<open>dbl 0 = 0\<close>
\<open>dbl 1 = 2\<close>
\<open>dbl (- 1) = - 2\<close>
\<open>dbl (numeral k) = numeral (Bit0 k)\<close>
by (simp_all add: dbl_def numeral.simps minus_add)
lemma dbl_inc_simps [simp]:
\<open>dbl_inc (- numeral k) = - dbl_dec (numeral k)\<close>
\<open>dbl_inc 0 = 1\<close>
\<open>dbl_inc 1 = 3\<close>
\<open>dbl_inc (- 1) = - 1\<close>
\<open>dbl_inc (numeral k) = numeral (Bit1 k)\<close>
by (simp_all add: dbl_inc_def dbl_dec_def numeral.simps numeral_BitM is_num_normalize algebra_simps
del: add_uminus_conv_diff)
lemma dbl_dec_simps [simp]:
\<open>dbl_dec (- numeral k) = - dbl_inc (numeral k)\<close>
\<open>dbl_dec 0 = - 1\<close>
\<open>dbl_dec 1 = 1\<close>
\<open>dbl_dec (- 1) = - 3\<close>
\<open>dbl_dec (numeral k) = numeral (BitM k)\<close>
by (simp_all add: dbl_dec_def dbl_inc_def numeral.simps numeral_BitM is_num_normalize)
lemma sub_num_simps [simp]:
\<open>sub One One = 0\<close>
\<open>sub One (Bit0 l) = - numeral (BitM l)\<close>
\<open>sub One (Bit1 l) = - numeral (Bit0 l)\<close>
\<open>sub (Bit0 k) One = numeral (BitM k)\<close>
\<open>sub (Bit1 k) One = numeral (Bit0 k)\<close>
\<open>sub (Bit0 k) (Bit0 l) = dbl (sub k l)\<close>
\<open>sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)\<close>
\<open>sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)\<close>
\<open>sub (Bit1 k) (Bit1 l) = dbl (sub k l)\<close>
by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def numeral.simps
numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma add_neg_numeral_simps:
\<open>numeral m + - numeral n = sub m n\<close>
\<open>- numeral m + numeral n = sub n m\<close>
\<open>- numeral m + - numeral n = - (numeral m + numeral n)\<close>
by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma add_neg_numeral_special:
\<open>1 + - numeral m = sub One m\<close>
\<open>- numeral m + 1 = sub One m\<close>
\<open>numeral m + - 1 = sub m One\<close>
\<open>- 1 + numeral n = sub n One\<close>
\<open>- 1 + - numeral n = - numeral (inc n)\<close>
\<open>- numeral m + - 1 = - numeral (inc m)\<close>
\<open>1 + - 1 = 0\<close>
\<open>- 1 + 1 = 0\<close>
\<open>- 1 + - 1 = - 2\<close>
by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize right_minus numeral_inc
del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma diff_numeral_simps:
\<open>numeral m - numeral n = sub m n\<close>
\<open>numeral m - - numeral n = numeral (m + n)\<close>
\<open>- numeral m - numeral n = - numeral (m + n)\<close>
\<open>- numeral m - - numeral n = sub n m\<close>
by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma diff_numeral_special:
\<open>1 - numeral n = sub One n\<close>
\<open>numeral m - 1 = sub m One\<close>
\<open>1 - - numeral n = numeral (One + n)\<close>
\<open>- numeral m - 1 = - numeral (m + One)\<close>
\<open>- 1 - numeral n = - numeral (inc n)\<close>
\<open>numeral m - - 1 = numeral (inc m)\<close>
\<open>- 1 - - numeral n = sub n One\<close>
\<open>- numeral m - - 1 = sub One m\<close>
\<open>1 - 1 = 0\<close>
\<open>- 1 - 1 = - 2\<close>
\<open>1 - - 1 = 2\<close>
\<open>- 1 - - 1 = 0\<close>
by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize numeral_inc
del: add_uminus_conv_diff add: diff_conv_add_uminus)
end
subsubsection \<open>Structures with multiplication: class \<open>semiring_numeral\<close>\<close>
class semiring_numeral = semiring + monoid_mult
begin
subclass numeral ..
lemma numeral_mult: \<open>numeral (m * n) = numeral m * numeral n\<close>
by (induct n rule: num_induct)
(simp_all add: numeral_One mult_inc numeral_inc numeral_add distrib_left)
lemma numeral_times_numeral: \<open>numeral m * numeral n = numeral (m * n)\<close>
by (rule numeral_mult [symmetric])
lemma mult_2: \<open>2 * z = z + z\<close>
by (simp add: one_add_one [symmetric] distrib_right)
lemma mult_2_right: \<open>z * 2 = z + z\<close>
by (simp add: one_add_one [symmetric] distrib_left)
lemma left_add_twice:
\<open>a + (a + b) = 2 * a + b\<close>
by (simp add: mult_2 ac_simps)
lemma numeral_Bit0_eq_double:
\<open>numeral (Bit0 n) = 2 * numeral n\<close>
by (simp add: mult_2) (simp add: numeral_Bit0)
lemma numeral_Bit1_eq_inc_double:
\<open>numeral (Bit1 n) = 2 * numeral n + 1\<close>
by (simp add: mult_2) (simp add: numeral_Bit1)
end
subsubsection \<open>Structures with a zero: class \<open>semiring_1\<close>\<close>
context semiring_1
begin
subclass semiring_numeral ..
lemma of_nat_numeral [simp]: \<open>of_nat (numeral n) = numeral n\<close>
by (induct n) (simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
end
lemma nat_of_num_numeral [code_abbrev]: \<open>nat_of_num = numeral\<close>
proof
fix n
have \<open>numeral n = nat_of_num n\<close>
by (induct n) (simp_all add: numeral.simps)
then show \<open>nat_of_num n = numeral n\<close>
by simp
qed
lemma nat_of_num_code [code]:
\<open>nat_of_num One = 1\<close>
\<open>nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)\<close>
\<open>nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))\<close>
by (simp_all add: Let_def)
subsubsection \<open>Equality: class \<open>semiring_char_0\<close>\<close>
context semiring_char_0
begin
lemma numeral_eq_iff: \<open>numeral m = numeral n \<longleftrightarrow> m = n\<close>
by (simp only: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
of_nat_eq_iff num_eq_iff)
lemma numeral_eq_one_iff: \<open>numeral n = 1 \<longleftrightarrow> n = One\<close>
by (rule numeral_eq_iff [of n One, unfolded numeral_One])
lemma one_eq_numeral_iff: \<open>1 = numeral n \<longleftrightarrow> One = n\<close>
by (rule numeral_eq_iff [of One n, unfolded numeral_One])
lemma numeral_neq_zero: \<open>numeral n \<noteq> 0\<close>
by (simp add: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric] nat_of_num_pos)
lemma zero_neq_numeral: \<open>0 \<noteq> numeral n\<close>
unfolding eq_commute [of 0] by (rule numeral_neq_zero)
lemmas eq_numeral_simps [simp] =
numeral_eq_iff
numeral_eq_one_iff
one_eq_numeral_iff
numeral_neq_zero
zero_neq_numeral
end
subsubsection \<open>Comparisons: class \<open>linordered_nonzero_semiring\<close>\<close>
context linordered_nonzero_semiring
begin
lemma numeral_le_iff: \<open>numeral m \<le> numeral n \<longleftrightarrow> m \<le> n\<close>
proof -
have \<open>of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n\<close>
by (simp only: less_eq_num_def nat_of_num_numeral of_nat_le_iff)
then show ?thesis by simp
qed
lemma one_le_numeral: \<open>1 \<le> numeral n\<close>
using numeral_le_iff [of One n] by (simp add: numeral_One)
lemma numeral_le_one_iff: \<open>numeral n \<le> 1 \<longleftrightarrow> n \<le> One\<close>
using numeral_le_iff [of n One] by (simp add: numeral_One)
lemma numeral_less_iff: \<open>numeral m < numeral n \<longleftrightarrow> m < n\<close>
proof -
have \<open>of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n\<close>
unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
then show ?thesis by simp
qed
lemma not_numeral_less_one: \<open>\<not> numeral n < 1\<close>
using numeral_less_iff [of n One] by (simp add: numeral_One)
lemma one_less_numeral_iff: \<open>1 < numeral n \<longleftrightarrow> One < n\<close>
using numeral_less_iff [of One n] by (simp add: numeral_One)
lemma zero_le_numeral: \<open>0 \<le> numeral n\<close>
using dual_order.trans one_le_numeral zero_le_one by blast
lemma zero_less_numeral: \<open>0 < numeral n\<close>
using less_linear not_numeral_less_one order.strict_trans zero_less_one by blast
lemma not_numeral_le_zero: \<open>\<not> numeral n \<le> 0\<close>
by (simp add: not_le zero_less_numeral)
lemma not_numeral_less_zero: \<open>\<not> numeral n < 0\<close>
by (simp add: not_less zero_le_numeral)
lemma one_of_nat_le_iff [simp]: \<open>1 \<le> of_nat k \<longleftrightarrow> 1 \<le> k\<close>
using of_nat_le_iff [of 1] by simp
lemma numeral_nat_le_iff [simp]: \<open>numeral n \<le> of_nat k \<longleftrightarrow> numeral n \<le> k\<close>
using of_nat_le_iff [of \<open>numeral n\<close>] by simp
lemma of_nat_le_1_iff [simp]: \<open>of_nat k \<le> 1 \<longleftrightarrow> k \<le> 1\<close>
using of_nat_le_iff [of _ 1] by simp
lemma of_nat_le_numeral_iff [simp]: \<open>of_nat k \<le> numeral n \<longleftrightarrow> k \<le> numeral n\<close>
using of_nat_le_iff [of _ \<open>numeral n\<close>] by simp
lemma one_of_nat_less_iff [simp]: \<open>1 < of_nat k \<longleftrightarrow> 1 < k\<close>
using of_nat_less_iff [of 1] by simp
lemma numeral_nat_less_iff [simp]: \<open>numeral n < of_nat k \<longleftrightarrow> numeral n < k\<close>
using of_nat_less_iff [of \<open>numeral n\<close>] by simp
lemma of_nat_less_1_iff [simp]: \<open>of_nat k < 1 \<longleftrightarrow> k < 1\<close>
using of_nat_less_iff [of _ 1] by simp
lemma of_nat_less_numeral_iff [simp]: \<open>of_nat k < numeral n \<longleftrightarrow> k < numeral n\<close>
using of_nat_less_iff [of _ \<open>numeral n\<close>] by simp
lemma of_nat_eq_numeral_iff [simp]: \<open>of_nat k = numeral n \<longleftrightarrow> k = numeral n\<close>
using of_nat_eq_iff [of _ \<open>numeral n\<close>] by simp
lemmas le_numeral_extra =
zero_le_one not_one_le_zero
order_refl [of 0] order_refl [of 1]
lemmas less_numeral_extra =
zero_less_one not_one_less_zero
less_irrefl [of 0] less_irrefl [of 1]
lemmas le_numeral_simps [simp] =
numeral_le_iff
one_le_numeral
numeral_le_one_iff
zero_le_numeral
not_numeral_le_zero
lemmas less_numeral_simps [simp] =
numeral_less_iff
one_less_numeral_iff
not_numeral_less_one
zero_less_numeral
not_numeral_less_zero
lemma min_0_1 [simp]:
fixes min' :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>
defines \<open>min' \<equiv> min\<close>
shows
\<open>min' 0 1 = 0\<close>
\<open>min' 1 0 = 0\<close>
\<open>min' 0 (numeral x) = 0\<close>
\<open>min' (numeral x) 0 = 0\<close>
\<open>min' 1 (numeral x) = 1\<close>
\<open>min' (numeral x) 1 = 1\<close>
by (simp_all add: min'_def min_def le_num_One_iff)
lemma max_0_1 [simp]:
fixes max' :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>
defines \<open>max' \<equiv> max\<close>
shows
\<open>max' 0 1 = 1\<close>
\<open>max' 1 0 = 1\<close>
\<open>max' 0 (numeral x) = numeral x\<close>
\<open>max' (numeral x) 0 = numeral x\<close>
\<open>max' 1 (numeral x) = numeral x\<close>
\<open>max' (numeral x) 1 = numeral x\<close>
by (simp_all add: max'_def max_def le_num_One_iff)
end
text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
lemmas max_number_of [simp] =
max_def [of \<open>numeral u\<close> \<open>numeral v\<close>]
max_def [of \<open>numeral u\<close> \<open>- numeral v\<close>]
max_def [of \<open>- numeral u\<close> \<open>numeral v\<close>]
max_def [of \<open>- numeral u\<close> \<open>- numeral v\<close>] for u v
lemmas min_number_of [simp] =
min_def [of \<open>numeral u\<close> \<open>numeral v\<close>]
min_def [of \<open>numeral u\<close> \<open>- numeral v\<close>]
min_def [of \<open>- numeral u\<close> \<open>numeral v\<close>]
min_def [of \<open>- numeral u\<close> \<open>- numeral v\<close>] for u v
subsubsection \<open>Multiplication and negation: class \<open>ring_1\<close>\<close>
context ring_1
begin
subclass neg_numeral ..
lemma mult_neg_numeral_simps:
\<open>- numeral m * - numeral n = numeral (m * n)\<close>
\<open>- numeral m * numeral n = - numeral (m * n)\<close>
\<open>numeral m * - numeral n = - numeral (m * n)\<close>
by (simp_all only: mult_minus_left mult_minus_right minus_minus numeral_mult)
lemma mult_minus1 [simp]: \<open>- 1 * z = - z\<close>
by (simp add: numeral.simps)
lemma mult_minus1_right [simp]: \<open>z * - 1 = - z\<close>
by (simp add: numeral.simps)
lemma minus_sub_one_diff_one [simp]:
\<open>- sub m One - 1 = - numeral m\<close>
proof -
have \<open>sub m One + 1 = numeral m\<close>
by (simp flip: eq_diff_eq add: diff_numeral_special)
then have \<open>- (sub m One + 1) = - numeral m\<close>
by simp
then show ?thesis
by simp
qed
end
subsubsection \<open>Equality using \<open>iszero\<close> for rings with non-zero characteristic\<close>
context ring_1
begin
definition iszero :: \<open>'a \<Rightarrow> bool\<close>
where \<open>iszero z \<longleftrightarrow> z = 0\<close>
lemma iszero_0 [simp]: \<open>iszero 0\<close>
by (simp add: iszero_def)
lemma not_iszero_1 [simp]: \<open>\<not> iszero 1\<close>
by (simp add: iszero_def)
lemma not_iszero_Numeral1: \<open>\<not> iszero Numeral1\<close>
by (simp add: numeral_One)
lemma not_iszero_neg_1 [simp]: \<open>\<not> iszero (- 1)\<close>
by (simp add: iszero_def)
lemma not_iszero_neg_Numeral1: \<open>\<not> iszero (- Numeral1)\<close>
by (simp add: numeral_One)
lemma iszero_neg_numeral [simp]: \<open>iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)\<close>
unfolding iszero_def by (rule neg_equal_0_iff_equal)
lemma eq_iff_iszero_diff: \<open>x = y \<longleftrightarrow> iszero (x - y)\<close>
unfolding iszero_def by (rule eq_iff_diff_eq_0)
text \<open>
The \<open>eq_numeral_iff_iszero\<close> lemmas are not declared \<open>[simp]\<close> by default,
because for rings of characteristic zero, better simp rules are possible.
For a type like integers mod \<open>n\<close>, type-instantiated versions of these rules
should be added to the simplifier, along with a type-specific rule for
deciding propositions of the form \<open>iszero (numeral w)\<close>.
bh: Maybe it would not be so bad to just declare these as simp rules anyway?
I should test whether these rules take precedence over the \<open>ring_char_0\<close>
rules in the simplifier.
\<close>
lemma eq_numeral_iff_iszero:
\<open>numeral x = numeral y \<longleftrightarrow> iszero (sub x y)\<close>
\<open>numeral x = - numeral y \<longleftrightarrow> iszero (numeral (x + y))\<close>
\<open>- numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))\<close>
\<open>- numeral x = - numeral y \<longleftrightarrow> iszero (sub y x)\<close>
\<open>numeral x = 1 \<longleftrightarrow> iszero (sub x One)\<close>
\<open>1 = numeral y \<longleftrightarrow> iszero (sub One y)\<close>
\<open>- numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))\<close>
\<open>1 = - numeral y \<longleftrightarrow> iszero (numeral (One + y))\<close>
\<open>numeral x = 0 \<longleftrightarrow> iszero (numeral x)\<close>
\<open>0 = numeral y \<longleftrightarrow> iszero (numeral y)\<close>
\<open>- numeral x = 0 \<longleftrightarrow> iszero (numeral x)\<close>
\<open>0 = - numeral y \<longleftrightarrow> iszero (numeral y)\<close>
unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
by simp_all
end
subsubsection \<open>Equality and negation: class \<open>ring_char_0\<close>\<close>
context ring_char_0
begin
lemma not_iszero_numeral [simp]: \<open>\<not> iszero (numeral w)\<close>
by (simp add: iszero_def)
lemma neg_numeral_eq_iff: \<open>- numeral m = - numeral n \<longleftrightarrow> m = n\<close>
by simp
lemma numeral_neq_neg_numeral: \<open>numeral m \<noteq> - numeral n\<close>
by (simp add: eq_neg_iff_add_eq_0 numeral_plus_numeral)
lemma neg_numeral_neq_numeral: \<open>- numeral m \<noteq> numeral n\<close>
by (rule numeral_neq_neg_numeral [symmetric])
lemma zero_neq_neg_numeral: \<open>0 \<noteq> - numeral n\<close>
by simp
lemma neg_numeral_neq_zero: \<open>- numeral n \<noteq> 0\<close>
by simp
lemma one_neq_neg_numeral: \<open>1 \<noteq> - numeral n\<close>
using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
lemma neg_numeral_neq_one: \<open>- numeral n \<noteq> 1\<close>
using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
lemma neg_one_neq_numeral: \<open>- 1 \<noteq> numeral n\<close>
using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)
lemma numeral_neq_neg_one: \<open>numeral n \<noteq> - 1\<close>
using numeral_neq_neg_numeral [of n One] by (simp add: numeral_One)
lemma neg_one_eq_numeral_iff: \<open>- 1 = - numeral n \<longleftrightarrow> n = One\<close>
using neg_numeral_eq_iff [of One n] by (auto simp add: numeral_One)
lemma numeral_eq_neg_one_iff: \<open>- numeral n = - 1 \<longleftrightarrow> n = One\<close>
using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)
lemma neg_one_neq_zero: \<open>- 1 \<noteq> 0\<close>
by simp
lemma zero_neq_neg_one: \<open>0 \<noteq> - 1\<close>
by simp
lemma neg_one_neq_one: \<open>- 1 \<noteq> 1\<close>
using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
lemma one_neq_neg_one: \<open>1 \<noteq> - 1\<close>
using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
lemmas eq_neg_numeral_simps [simp] =
neg_numeral_eq_iff
numeral_neq_neg_numeral neg_numeral_neq_numeral
one_neq_neg_numeral neg_numeral_neq_one
zero_neq_neg_numeral neg_numeral_neq_zero
neg_one_neq_numeral numeral_neq_neg_one
neg_one_eq_numeral_iff numeral_eq_neg_one_iff
neg_one_neq_zero zero_neq_neg_one
neg_one_neq_one one_neq_neg_one
end
subsubsection \<open>Structures with negation and order: class \<open>linordered_idom\<close>\<close>
context linordered_idom
begin
subclass ring_char_0 ..
lemma neg_numeral_le_iff: \<open>- numeral m \<le> - numeral n \<longleftrightarrow> n \<le> m\<close>
by (simp only: neg_le_iff_le numeral_le_iff)
lemma neg_numeral_less_iff: \<open>- numeral m < - numeral n \<longleftrightarrow> n < m\<close>
by (simp only: neg_less_iff_less numeral_less_iff)
lemma neg_numeral_less_zero: \<open>- numeral n < 0\<close>
by (simp only: neg_less_0_iff_less zero_less_numeral)
lemma neg_numeral_le_zero: \<open>- numeral n \<le> 0\<close>
by (simp only: neg_le_0_iff_le zero_le_numeral)
lemma not_zero_less_neg_numeral: \<open>\<not> 0 < - numeral n\<close>
by (simp only: not_less neg_numeral_le_zero)
lemma not_zero_le_neg_numeral: \<open>\<not> 0 \<le> - numeral n\<close>
by (simp only: not_le neg_numeral_less_zero)
lemma neg_numeral_less_numeral: \<open>- numeral m < numeral n\<close>
using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
lemma neg_numeral_le_numeral: \<open>- numeral m \<le> numeral n\<close>
by (simp only: less_imp_le neg_numeral_less_numeral)
lemma not_numeral_less_neg_numeral: \<open>\<not> numeral m < - numeral n\<close>
by (simp only: not_less neg_numeral_le_numeral)
lemma not_numeral_le_neg_numeral: \<open>\<not> numeral m \<le> - numeral n\<close>
by (simp only: not_le neg_numeral_less_numeral)
lemma neg_numeral_less_one: \<open>- numeral m < 1\<close>
by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
lemma neg_numeral_le_one: \<open>- numeral m \<le> 1\<close>
by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
lemma not_one_less_neg_numeral: \<open>\<not> 1 < - numeral m\<close>
by (simp only: not_less neg_numeral_le_one)
lemma not_one_le_neg_numeral: \<open>\<not> 1 \<le> - numeral m\<close>
by (simp only: not_le neg_numeral_less_one)
lemma not_numeral_less_neg_one: \<open>\<not> numeral m < - 1\<close>
using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)
lemma not_numeral_le_neg_one: \<open>\<not> numeral m \<le> - 1\<close>
using not_numeral_le_neg_numeral [of m One] by (simp add: numeral_One)
lemma neg_one_less_numeral: \<open>- 1 < numeral m\<close>
using neg_numeral_less_numeral [of One m] by (simp add: numeral_One)
lemma neg_one_le_numeral: \<open>- 1 \<le> numeral m\<close>
using neg_numeral_le_numeral [of One m] by (simp add: numeral_One)
lemma neg_numeral_less_neg_one_iff: \<open>- numeral m < - 1 \<longleftrightarrow> m \<noteq> One\<close>
by (cases m) simp_all
lemma neg_numeral_le_neg_one: \<open>- numeral m \<le> - 1\<close>
by simp
lemma not_neg_one_less_neg_numeral: \<open>\<not> - 1 < - numeral m\<close>
by simp
lemma not_neg_one_le_neg_numeral_iff: \<open>\<not> - 1 \<le> - numeral m \<longleftrightarrow> m \<noteq> One\<close>
by (cases m) simp_all
lemma sub_non_negative: \<open>sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m\<close>
by (simp only: sub_def le_diff_eq) simp
lemma sub_positive: \<open>sub n m > 0 \<longleftrightarrow> n > m\<close>
by (simp only: sub_def less_diff_eq) simp
lemma sub_non_positive: \<open>sub n m \<le> 0 \<longleftrightarrow> n \<le> m\<close>
by (simp only: sub_def diff_le_eq) simp
lemma sub_negative: \<open>sub n m < 0 \<longleftrightarrow> n < m\<close>
by (simp only: sub_def diff_less_eq) simp
lemmas le_neg_numeral_simps [simp] =
neg_numeral_le_iff
neg_numeral_le_numeral not_numeral_le_neg_numeral
neg_numeral_le_zero not_zero_le_neg_numeral
neg_numeral_le_one not_one_le_neg_numeral
neg_one_le_numeral not_numeral_le_neg_one
neg_numeral_le_neg_one not_neg_one_le_neg_numeral_iff
lemma le_minus_one_simps [simp]:
\<open>- 1 \<le> 0\<close>
\<open>- 1 \<le> 1\<close>
\<open>\<not> 0 \<le> - 1\<close>
\<open>\<not> 1 \<le> - 1\<close>
by simp_all
lemmas less_neg_numeral_simps [simp] =
neg_numeral_less_iff
neg_numeral_less_numeral not_numeral_less_neg_numeral
neg_numeral_less_zero not_zero_less_neg_numeral
neg_numeral_less_one not_one_less_neg_numeral
neg_one_less_numeral not_numeral_less_neg_one
neg_numeral_less_neg_one_iff not_neg_one_less_neg_numeral
lemma less_minus_one_simps [simp]:
\<open>- 1 < 0\<close>
\<open>- 1 < 1\<close>
\<open>\<not> 0 < - 1\<close>
\<open>\<not> 1 < - 1\<close>
by (simp_all add: less_le)
lemma abs_numeral [simp]: \<open>\<bar>numeral n\<bar> = numeral n\<close>
by simp
lemma abs_neg_numeral [simp]: \<open>\<bar>- numeral n\<bar> = numeral n\<close>
by (simp only: abs_minus_cancel abs_numeral)
lemma abs_neg_one [simp]: \<open>\<bar>- 1\<bar> = 1\<close>
by simp
end
subsubsection \<open>Natural numbers\<close>
lemma numeral_num_of_nat:
\<open>numeral (num_of_nat n) = n\<close> if \<open>n > 0\<close>
using that nat_of_num_numeral num_of_nat_inverse by simp
lemma Suc_1 [simp]: \<open>Suc 1 = 2\<close>
unfolding Suc_eq_plus1 by (rule one_add_one)
lemma Suc_numeral [simp]: \<open>Suc (numeral n) = numeral (n + One)\<close>
unfolding Suc_eq_plus1 by (rule numeral_plus_one)
definition pred_numeral :: \<open>num \<Rightarrow> nat\<close>
where \<open>pred_numeral k = numeral k - 1\<close>
declare [[code drop: pred_numeral]]
lemma numeral_eq_Suc: \<open>numeral k = Suc (pred_numeral k)\<close>
by (simp add: pred_numeral_def)
lemma eval_nat_numeral:
\<open>numeral One = Suc 0\<close>
\<open>numeral (Bit0 n) = Suc (numeral (BitM n))\<close>
\<open>numeral (Bit1 n) = Suc (numeral (Bit0 n))\<close>
by (simp_all add: numeral.simps BitM_plus_one)
lemma pred_numeral_simps [simp]:
\<open>pred_numeral One = 0\<close>
\<open>pred_numeral (Bit0 k) = numeral (BitM k)\<close>
\<open>pred_numeral (Bit1 k) = numeral (Bit0 k)\<close>
by (simp_all only: pred_numeral_def eval_nat_numeral diff_Suc_Suc diff_0)
lemma pred_numeral_inc [simp]:
\<open>pred_numeral (inc k) = numeral k\<close>
by (simp only: pred_numeral_def numeral_inc diff_add_inverse2)
lemma numeral_2_eq_2: \<open>2 = Suc (Suc 0)\<close>
by (simp add: eval_nat_numeral)
lemma numeral_3_eq_3: \<open>3 = Suc (Suc (Suc 0))\<close>
by (simp add: eval_nat_numeral)
lemma numeral_1_eq_Suc_0: \<open>Numeral1 = Suc 0\<close>
by (simp only: numeral_One One_nat_def)
lemma Suc_nat_number_of_add: \<open>Suc (numeral v + n) = numeral (v + One) + n\<close>
by simp
lemma numerals: \<open>Numeral1 = (1::nat)\<close> \<open>2 = Suc (Suc 0)\<close>
by (rule numeral_One) (rule numeral_2_eq_2)
lemmas numeral_nat = eval_nat_numeral BitM.simps One_nat_def
text \<open>Comparisons involving \<^term>\<open>Suc\<close>.\<close>
lemma eq_numeral_Suc [simp]: \<open>numeral k = Suc n \<longleftrightarrow> pred_numeral k = n\<close>
by (simp add: numeral_eq_Suc)
lemma Suc_eq_numeral [simp]: \<open>Suc n = numeral k \<longleftrightarrow> n = pred_numeral k\<close>
by (simp add: numeral_eq_Suc)
lemma less_numeral_Suc [simp]: \<open>numeral k < Suc n \<longleftrightarrow> pred_numeral k < n\<close>
by (simp add: numeral_eq_Suc)
lemma less_Suc_numeral [simp]: \<open>Suc n < numeral k \<longleftrightarrow> n < pred_numeral k\<close>
by (simp add: numeral_eq_Suc)
lemma le_numeral_Suc [simp]: \<open>numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n\<close>
by (simp add: numeral_eq_Suc)
lemma le_Suc_numeral [simp]: \<open>Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k\<close>
by (simp add: numeral_eq_Suc)
lemma diff_Suc_numeral [simp]: \<open>Suc n - numeral k = n - pred_numeral k\<close>
by (simp add: numeral_eq_Suc)
lemma diff_numeral_Suc [simp]: \<open>numeral k - Suc n = pred_numeral k - n\<close>
by (simp add: numeral_eq_Suc)
lemma max_Suc_numeral [simp]: \<open>max (Suc n) (numeral k) = Suc (max n (pred_numeral k))\<close>
by (simp add: numeral_eq_Suc)
lemma max_numeral_Suc [simp]: \<open>max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)\<close>
by (simp add: numeral_eq_Suc)
lemma min_Suc_numeral [simp]: \<open>min (Suc n) (numeral k) = Suc (min n (pred_numeral k))\<close>
by (simp add: numeral_eq_Suc)
lemma min_numeral_Suc [simp]: \<open>min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)\<close>
by (simp add: numeral_eq_Suc)
text \<open>For \<^term>\<open>case_nat\<close> and \<^term>\<open>rec_nat\<close>.\<close>
lemma case_nat_numeral [simp]: \<open>case_nat a f (numeral v) = (let pv = pred_numeral v in f pv)\<close>
by (simp add: numeral_eq_Suc)
lemma case_nat_add_eq_if [simp]:
\<open>case_nat a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))\<close>
by (simp add: numeral_eq_Suc)
lemma rec_nat_numeral [simp]:
\<open>rec_nat a f (numeral v) = (let pv = pred_numeral v in f pv (rec_nat a f pv))\<close>
by (simp add: numeral_eq_Suc Let_def)
lemma rec_nat_add_eq_if [simp]:
\<open>rec_nat a f (numeral v + n) = (let pv = pred_numeral v in f (pv + n) (rec_nat a f (pv + n)))\<close>
by (simp add: numeral_eq_Suc Let_def)
text \<open>Case analysis on \<^term>\<open>n < 2\<close>.\<close>
lemma less_2_cases: \<open>n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0\<close>
by (auto simp add: numeral_2_eq_2)
lemma less_2_cases_iff: \<open>n < 2 \<longleftrightarrow> n = 0 \<or> n = Suc 0\<close>
by (auto simp add: numeral_2_eq_2)
text \<open>Removal of Small Numerals: 0, 1 and (in additive positions) 2.\<close>
text \<open>bh: Are these rules really a good idea? LCP: well, it already happens for 0 and 1!\<close>
lemma add_2_eq_Suc [simp]: \<open>2 + n = Suc (Suc n)\<close>
by simp
lemma add_2_eq_Suc' [simp]: \<open>n + 2 = Suc (Suc n)\<close>
by simp
text \<open>Can be used to eliminate long strings of Sucs, but not by default.\<close>
lemma Suc3_eq_add_3: \<open>Suc (Suc (Suc n)) = 3 + n\<close>
by simp
lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
context semiring_numeral
begin
lemma numeral_add_unfold_funpow:
\<open>numeral k + a = ((+) 1 ^^ numeral k) a\<close>
proof (rule sym, induction k arbitrary: a)
case One
then show ?case
by (simp add: Num.numeral_One numeral_One)
next
case (Bit0 k)
then show ?case
by (simp add: Num.numeral_Bit0 numeral_Bit0 ac_simps funpow_add)
next
case (Bit1 k)
then show ?case
by (simp add: Num.numeral_Bit1 numeral_Bit1 ac_simps funpow_add)
qed
end
context semiring_1
begin
lemma numeral_unfold_funpow:
\<open>numeral k = ((+) 1 ^^ numeral k) 0\<close>
using numeral_add_unfold_funpow [of k 0] by simp
end
context
includes lifting_syntax
begin
lemma transfer_rule_numeral:
\<open>((=) ===> R) numeral numeral\<close>
if [transfer_rule]: \<open>R 0 0\<close> \<open>R 1 1\<close>
\<open>(R ===> R ===> R) (+) (+)\<close>
for R :: \<open>'a::{semiring_numeral,monoid_add} \<Rightarrow> 'b::{semiring_numeral,monoid_add} \<Rightarrow> bool\<close>
proof -
have \<open>((=) ===> R) (\<lambda>k. ((+) 1 ^^ numeral k) 0) (\<lambda>k. ((+) 1 ^^ numeral k) 0)\<close>
by transfer_prover
moreover have \<open>numeral = (\<lambda>k. ((+) (1::'a) ^^ numeral k) 0)\<close>
using numeral_add_unfold_funpow [where ?'a = 'a, of _ 0]
by (simp add: fun_eq_iff)
moreover have \<open>numeral = (\<lambda>k. ((+) (1::'b) ^^ numeral k) 0)\<close>
using numeral_add_unfold_funpow [where ?'a = 'b, of _ 0]
by (simp add: fun_eq_iff)
ultimately show ?thesis
by simp
qed
end
subsection \<open>Particular lemmas concerning \<^term>\<open>2\<close>\<close>
context linordered_field
begin
subclass field_char_0 ..
lemma half_gt_zero_iff: \<open>0 < a / 2 \<longleftrightarrow> 0 < a\<close>
by (auto simp add: field_simps)
lemma half_gt_zero [simp]: \<open>0 < a \<Longrightarrow> 0 < a / 2\<close>
by (simp add: half_gt_zero_iff)
end
subsection \<open>Numeral equations as default simplification rules\<close>
declare (in numeral) numeral_One [simp]
declare (in numeral) numeral_plus_numeral [simp]
declare (in numeral) add_numeral_special [simp]
declare (in neg_numeral) add_neg_numeral_simps [simp]
declare (in neg_numeral) add_neg_numeral_special [simp]
declare (in neg_numeral) diff_numeral_simps [simp]
declare (in neg_numeral) diff_numeral_special [simp]
declare (in semiring_numeral) numeral_times_numeral [simp]
declare (in ring_1) mult_neg_numeral_simps [simp]
subsubsection \<open>Special Simplification for Constants\<close>
text \<open>These distributive laws move literals inside sums and differences.\<close>
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ \<open>numeral v\<close>] for v
lemmas distrib_left_numeral [simp] = distrib_left [of \<open>numeral v\<close>] for v
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ \<open>numeral v\<close>] for v
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of \<open>numeral v\<close>] for v
text \<open>These are actually for fields, like real\<close>
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of \<open>numeral w\<close>] for w
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of \<open>numeral w\<close>] for w
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of \<open>numeral w\<close>] for w
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of \<open>numeral w\<close>] for w
text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>. It looks
strange, but then other simprocs simplify the quotient.\<close>
lemmas inverse_eq_divide_numeral [simp] =
inverse_eq_divide [of \<open>numeral w\<close>] for w
lemmas inverse_eq_divide_neg_numeral [simp] =
inverse_eq_divide [of \<open>- numeral w\<close>] for w
text \<open>These laws simplify inequalities, moving unary minus from a term
into the literal.\<close>
lemmas equation_minus_iff_numeral [no_atp] =
equation_minus_iff [of \<open>numeral v\<close>] for v
lemmas minus_equation_iff_numeral [no_atp] =
minus_equation_iff [of _ \<open>numeral v\<close>] for v
lemmas le_minus_iff_numeral [no_atp] =
le_minus_iff [of \<open>numeral v\<close>] for v
lemmas minus_le_iff_numeral [no_atp] =
minus_le_iff [of _ \<open>numeral v\<close>] for v
lemmas less_minus_iff_numeral [no_atp] =
less_minus_iff [of \<open>numeral v\<close>] for v
lemmas minus_less_iff_numeral [no_atp] =
minus_less_iff [of _ \<open>numeral v\<close>] for v
(* FIXME maybe simproc *)
text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of \<open>numeral v\<close>] for v
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ \<open>numeral v\<close>] for v
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of \<open>numeral v\<close>] for v
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ \<open>numeral v\<close>] for v
text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
named_theorems divide_const_simps \<open>simplification rules to simplify comparisons involving constant divisors\<close>
lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
pos_le_divide_eq [of \<open>numeral w\<close>, OF zero_less_numeral]
neg_le_divide_eq [of \<open>- numeral w\<close>, OF neg_numeral_less_zero] for w
lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
pos_divide_le_eq [of \<open>numeral w\<close>, OF zero_less_numeral]
neg_divide_le_eq [of \<open>- numeral w\<close>, OF neg_numeral_less_zero] for w
lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
pos_less_divide_eq [of \<open>numeral w\<close>, OF zero_less_numeral]
neg_less_divide_eq [of \<open>- numeral w\<close>, OF neg_numeral_less_zero] for w
lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
pos_divide_less_eq [of \<open>numeral w\<close>, OF zero_less_numeral]
neg_divide_less_eq [of \<open>- numeral w\<close>, OF neg_numeral_less_zero] for w
lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
eq_divide_eq [of _ _ \<open>numeral w\<close>]
eq_divide_eq [of _ _ \<open>- numeral w\<close>] for w
lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
divide_eq_eq [of _ \<open>numeral w\<close>]
divide_eq_eq [of _ \<open>- numeral w\<close>] for w
subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
text \<open>Simplify quotients that are compared with a literal constant.\<close>
lemmas le_divide_eq_numeral [divide_const_simps] =
le_divide_eq [of \<open>numeral w\<close>]
le_divide_eq [of \<open>- numeral w\<close>] for w
lemmas divide_le_eq_numeral [divide_const_simps] =
divide_le_eq [of _ _ \<open>numeral w\<close>]
divide_le_eq [of _ _ \<open>- numeral w\<close>] for w
lemmas less_divide_eq_numeral [divide_const_simps] =
less_divide_eq [of \<open>numeral w\<close>]
less_divide_eq [of \<open>- numeral w\<close>] for w
lemmas divide_less_eq_numeral [divide_const_simps] =
divide_less_eq [of _ _ \<open>numeral w\<close>]
divide_less_eq [of _ _ \<open>- numeral w\<close>] for w
lemmas eq_divide_eq_numeral [divide_const_simps] =
eq_divide_eq [of \<open>numeral w\<close>]
eq_divide_eq [of \<open>- numeral w\<close>] for w
lemmas divide_eq_eq_numeral [divide_const_simps] =
divide_eq_eq [of _ _ \<open>numeral w\<close>]
divide_eq_eq [of _ _ \<open>- numeral w\<close>] for w
text \<open>Not good as automatic simprules because they cause case splits.\<close>
lemmas [divide_const_simps] =
le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
subsection \<open>Setting up simprocs\<close>
lemma mult_numeral_1: \<open>Numeral1 * a = a\<close>
for a :: \<open>'a::semiring_numeral\<close>
by simp
lemma mult_numeral_1_right: \<open>a * Numeral1 = a\<close>
for a :: \<open>'a::semiring_numeral\<close>
by simp
lemma divide_numeral_1: \<open>a / Numeral1 = a\<close>
for a :: \<open>'a::field\<close>
by simp
lemma inverse_numeral_1: \<open>inverse Numeral1 = (Numeral1::'a::division_ring)\<close>
by simp
text \<open>
Theorem lists for the cancellation simprocs. The use of a binary
numeral for 1 reduces the number of special cases.
\<close>
lemma mult_1s_semiring_numeral:
\<open>Numeral1 * a = a\<close>
\<open>a * Numeral1 = a\<close>
for a :: \<open>'a::semiring_numeral\<close>
by simp_all
lemma mult_1s_ring_1:
\<open>- Numeral1 * b = - b\<close>
\<open>b * - Numeral1 = - b\<close>
for b :: \<open>'a::ring_1\<close>
by simp_all
lemmas mult_1s = mult_1s_semiring_numeral mult_1s_ring_1
setup \<open>
Reorient_Proc.add
(fn Const (\<^const_name>\<open>numeral\<close>, _) $ _ => true
| Const (\<^const_name>\<open>uminus\<close>, _) $ (Const (\<^const_name>\<open>numeral\<close>, _) $ _) => true
| _ => false)
\<close>
simproc_setup reorient_numeral (\<open>numeral w = x\<close> | \<open>- numeral w = y\<close>) =
\<open>K Reorient_Proc.proc\<close>
subsubsection \<open>Simplification of arithmetic operations on integer constants\<close>
lemmas arith_special = (* already declared simp above *)
add_numeral_special add_neg_numeral_special
diff_numeral_special
lemmas arith_extra_simps = (* rules already in simpset *)
numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
minus_zero
diff_numeral_simps diff_0 diff_0_right
numeral_times_numeral mult_neg_numeral_simps
mult_zero_left mult_zero_right
abs_numeral abs_neg_numeral
text \<open>
For making a minimal simpset, one must include these default simprules.
Also include \<open>simp_thms\<close>.
\<close>
lemmas arith_simps =
add_num_simps mult_num_simps sub_num_simps
BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
abs_zero abs_one arith_extra_simps
lemmas more_arith_simps =
neg_le_iff_le
minus_zero left_minus right_minus
mult_1_left mult_1_right
mult_minus_left mult_minus_right
minus_add_distrib minus_minus mult.assoc
lemmas of_nat_simps =
of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
text \<open>Simplification of relational operations.\<close>
lemmas eq_numeral_extra =
zero_neq_one one_neq_zero
lemmas rel_simps =
le_num_simps less_num_simps eq_num_simps
le_numeral_simps le_neg_numeral_simps le_minus_one_simps le_numeral_extra
less_numeral_simps less_neg_numeral_simps less_minus_one_simps less_numeral_extra
eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
lemma Let_numeral [simp]: \<open>Let (numeral v) f = f (numeral v)\<close>
\<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
unfolding Let_def ..
lemma Let_neg_numeral [simp]: \<open>Let (- numeral v) f = f (- numeral v)\<close>
\<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
unfolding Let_def ..
declaration \<open>
let
fun number_of ctxt T n =
if not (Sign.of_sort (Proof_Context.theory_of ctxt) (T, \<^sort>\<open>numeral\<close>))
then raise CTERM ("number_of", [])
else Numeral.mk_cnumber (Thm.ctyp_of ctxt T) n;
in
K (
Lin_Arith.set_number_of number_of
#> Lin_Arith.add_simps
@{thms arith_simps more_arith_simps rel_simps pred_numeral_simps
arith_special numeral_One of_nat_simps uminus_numeral_One
Suc_numeral Let_numeral Let_neg_numeral Let_0 Let_1
le_Suc_numeral le_numeral_Suc less_Suc_numeral less_numeral_Suc
Suc_eq_numeral eq_numeral_Suc mult_Suc mult_Suc_right of_nat_numeral})
end
\<close>
subsubsection \<open>Simplification of arithmetic when nested to the right\<close>
lemma add_numeral_left [simp]: \<open>numeral v + (numeral w + z) = (numeral(v + w) + z)\<close>
by (simp_all add: add.assoc [symmetric])
lemma add_neg_numeral_left [simp]:
\<open>numeral v + (- numeral w + y) = (sub v w + y)\<close>
\<open>- numeral v + (numeral w + y) = (sub w v + y)\<close>
\<open>- numeral v + (- numeral w + y) = (- numeral(v + w) + y)\<close>
by (simp_all add: add.assoc [symmetric])
lemma mult_numeral_left_semiring_numeral:
\<open>numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)\<close>
by (simp add: mult.assoc [symmetric])
lemma mult_numeral_left_ring_1:
\<open>- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'a::ring_1)\<close>
\<open>numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'a::ring_1)\<close>
\<open>- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'a::ring_1)\<close>
by (simp_all add: mult.assoc [symmetric])
lemmas mult_numeral_left [simp] =
mult_numeral_left_semiring_numeral
mult_numeral_left_ring_1
subsection \<open>Code module namespace\<close>
code_identifier
code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
subsection \<open>Printing of evaluated natural numbers as numerals\<close>
lemma [code_post]:
\<open>Suc 0 = 1\<close>
\<open>Suc 1 = 2\<close>
\<open>Suc (numeral n) = numeral (inc n)\<close>
by (simp_all add: numeral_inc)
lemmas [code_post] = inc.simps
subsection \<open>More on auxiliary conversion\<close>
context semiring_1
begin
lemma num_of_nat_numeral_eq [simp]:
\<open>num_of_nat (numeral q) = q\<close>
by (simp flip: nat_of_num_numeral add: nat_of_num_inverse)
lemma numeral_num_of_nat_unfold:
\<open>numeral (num_of_nat n) = (if n = 0 then 1 else of_nat n)\<close>
apply (simp only: of_nat_numeral [symmetric, of \<open>num_of_nat n\<close>] flip: nat_of_num_numeral)
apply (auto simp add: num_of_nat_inverse)
done
end
hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
end