(* Title: HOL/Decision_Procs/Rat_Pair.thy
Author: Amine Chaieb
*)
section \<open>Rational numbers as pairs\<close>
theory Rat_Pair
imports Complex_Main
begin
type_synonym Num = "int \<times> int"
abbreviation Num0_syn :: Num ("0\<^sub>N")
where "0\<^sub>N \<equiv> (0, 0)"
abbreviation Numi_syn :: "int \<Rightarrow> Num" ("'((_)')\<^sub>N")
where "(i)\<^sub>N \<equiv> (i, 1)"
definition isnormNum :: "Num \<Rightarrow> bool"
where "isnormNum = (\<lambda>(a, b). if a = 0 then b = 0 else b > 0 \<and> gcd a b = 1)"
definition normNum :: "Num \<Rightarrow> Num"
where
"normNum = (\<lambda>(a,b).
(if a = 0 \<or> b = 0 then (0, 0)
else
(let g = gcd a b
in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
declare gcd_dvd1[presburger] gcd_dvd2[presburger]
lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
consider "a = 0 \<or> b = 0" | "a \<noteq> 0" "b \<noteq> 0"
by blast
then show ?thesis
proof cases
case 1
then show ?thesis
by (simp add: x normNum_def isnormNum_def)
next
case ab: 2
let ?g = "gcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
let ?g' = "gcd ?a' ?b'"
from ab have "?g \<noteq> 0" by simp
with gcd_ge_0_int[of a b] have gpos: "?g > 0" by arith
have gdvd: "?g dvd a" "?g dvd b" by arith+
from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] ab
have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+
from ab have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
from div_gcd_coprime[OF stupid] have gp1: "?g' = 1" .
from ab consider "b < 0" | "b > 0" by arith
then show ?thesis
proof cases
case b: 1
have False if b': "?b' \<ge> 0"
proof -
from gpos have th: "?g \<ge> 0" by arith
from mult_nonneg_nonneg[OF th b'] dvd_mult_div_cancel[OF gdvd(2)]
show ?thesis using b by arith
qed
then have b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
from ab(1) nz' b b' gp1 show ?thesis
by (simp add: x isnormNum_def normNum_def Let_def split_def)
next
case b: 2
then have "?b' \<ge> 0"
by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
with nz' have b': "?b' > 0" by arith
from b b' ab(1) nz' gp1 show ?thesis
by (simp add: x isnormNum_def normNum_def Let_def split_def)
qed
qed
qed
text \<open>Arithmetic over Num\<close>
definition Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60)
where
"Nadd = (\<lambda>(a, b) (a', b').
if a = 0 \<or> b = 0 then normNum (a', b')
else if a' = 0 \<or> b' = 0 then normNum (a, b)
else normNum (a * b' + b * a', b * b'))"
definition Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60)
where
"Nmul = (\<lambda>(a, b) (a', b').
let g = gcd (a * a') (b * b')
in (a * a' div g, b * b' div g))"
definition Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
where "Nneg = (\<lambda>(a, b). (- a, b))"
definition Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60)
where "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
definition Ninv :: "Num \<Rightarrow> Num"
where "Ninv = (\<lambda>(a, b). if a < 0 then (- b, \<bar>a\<bar>) else (b, a))"
definition Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60)
where "Ndiv = (\<lambda>a b. a *\<^sub>N Ninv b)"
lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
by (simp add: isnormNum_def Nneg_def split_def)
lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"
by (simp add: Nadd_def split_def)
lemma Nsub_normN[simp]: "isnormNum y \<Longrightarrow> isnormNum (x -\<^sub>N y)"
by (simp add: Nsub_def split_def)
lemma Nmul_normN[simp]:
assumes xn: "isnormNum x"
and yn: "isnormNum y"
shows "isnormNum (x *\<^sub>N y)"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
obtain a' b' where y: "y = (a', b')" by (cases y)
consider "a = 0" | "a' = 0" | "a \<noteq> 0" "a' \<noteq> 0" by blast
then show ?thesis
proof cases
case 1
then show ?thesis
using xn x y by (simp add: isnormNum_def Let_def Nmul_def split_def)
next
case 2
then show ?thesis
using yn x y by (simp add: isnormNum_def Let_def Nmul_def split_def)
next
case aa': 3
then have bp: "b > 0" "b' > 0"
using xn yn x y by (simp_all add: isnormNum_def)
from bp have "x *\<^sub>N y = normNum (a * a', b * b')"
using x y aa' bp by (simp add: Nmul_def Let_def split_def normNum_def)
then show ?thesis by simp
qed
qed
lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
apply (simp add: Ninv_def isnormNum_def split_def)
apply (cases "fst x = 0")
apply (auto simp add: gcd.commute)
done
lemma isnormNum_int[simp]: "isnormNum 0\<^sub>N" "isnormNum ((1::int)\<^sub>N)" "i \<noteq> 0 \<Longrightarrow> isnormNum (i)\<^sub>N"
by (simp_all add: isnormNum_def)
text \<open>Relations over Num\<close>
definition Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N")
where "Nlt0 = (\<lambda>(a, b). a < 0)"
definition Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")
where "Nle0 = (\<lambda>(a, b). a \<le> 0)"
definition Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N")
where "Ngt0 = (\<lambda>(a, b). a > 0)"
definition Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")
where "Nge0 = (\<lambda>(a, b). a \<ge> 0)"
definition Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55)
where "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
definition Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55)
where "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
definition "INum = (\<lambda>(a, b). of_int a / of_int b)"
lemma INum_int [simp]: "INum (i)\<^sub>N = (of_int i ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"
by (simp_all add: INum_def)
lemma isnormNum_unique[simp]:
assumes na: "isnormNum x"
and nb: "isnormNum y"
shows "(INum x ::'a::{field_char_0,field}) = INum y \<longleftrightarrow> x = y"
(is "?lhs = ?rhs")
proof
obtain a b where x: "x = (a, b)" by (cases x)
obtain a' b' where y: "y = (a', b')" by (cases y)
consider "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0" | "a \<noteq> 0" "b \<noteq> 0" "a' \<noteq> 0" "b' \<noteq> 0"
by blast
then show ?rhs if H: ?lhs
proof cases
case 1
then show ?thesis
using na nb H by (simp add: x y INum_def split_def isnormNum_def split: if_split_asm)
next
case 2
with na nb have pos: "b > 0" "b' > 0"
by (simp_all add: x y isnormNum_def)
from H \<open>b \<noteq> 0\<close> \<open>b' \<noteq> 0\<close> have eq: "a * b' = a' * b"
by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
from \<open>a \<noteq> 0\<close> \<open>a' \<noteq> 0\<close> na nb
have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
by (simp_all add: x y isnormNum_def add: gcd.commute)
from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'"
apply -
apply algebra
apply algebra
apply simp
apply algebra
done
from zdvd_antisym_abs[OF coprime_dvd_mult[OF gcd1(2) raw_dvd(2)]
coprime_dvd_mult[OF gcd1(4) raw_dvd(4)]]
have eq1: "b = b'" using pos by arith
with eq have "a = a'" using pos by simp
with eq1 show ?thesis by (simp add: x y)
qed
show ?lhs if ?rhs
using that by simp
qed
lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> INum x = (0::'a::{field_char_0,field}) \<longleftrightarrow> x = 0\<^sub>N"
unfolding INum_int(2)[symmetric]
by (rule isnormNum_unique) simp_all
lemma of_int_div_aux:
assumes "d \<noteq> 0"
shows "(of_int x ::'a::field_char_0) / of_int d =
of_int (x div d) + (of_int (x mod d)) / of_int d"
proof -
let ?t = "of_int (x div d) * (of_int d ::'a) + of_int (x mod d)"
let ?f = "\<lambda>x. x / of_int d"
have "x = (x div d) * d + x mod d"
by auto
then have eq: "of_int x = ?t"
by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
then have "of_int x / of_int d = ?t / of_int d"
using cong[OF refl[of ?f] eq] by simp
then show ?thesis
by (simp add: add_divide_distrib algebra_simps \<open>d \<noteq> 0\<close>)
qed
lemma of_int_div:
fixes d :: int
assumes "d \<noteq> 0" "d dvd n"
shows "(of_int (n div d) ::'a::field_char_0) = of_int n / of_int d"
using assms of_int_div_aux [of d n, where ?'a = 'a] by simp
lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0,field})"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
consider "a = 0 \<or> b = 0" | "a \<noteq> 0" "b \<noteq> 0" by blast
then show ?thesis
proof cases
case 1
then show ?thesis
by (simp add: x INum_def normNum_def split_def Let_def)
next
case ab: 2
let ?g = "gcd a b"
from ab have g: "?g \<noteq> 0"by simp
from of_int_div[OF g, where ?'a = 'a] show ?thesis
by (auto simp add: x INum_def normNum_def split_def Let_def)
qed
qed
lemma INum_normNum_iff: "(INum x ::'a::{field_char_0,field}) = INum y \<longleftrightarrow> normNum x = normNum y"
(is "?lhs \<longleftrightarrow> _")
proof -
have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
by (simp del: normNum)
also have "\<dots> = ?lhs" by simp
finally show ?thesis by simp
qed
lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0,field})"
proof -
let ?z = "0::'a"
obtain a b where x: "x = (a, b)" by (cases x)
obtain a' b' where y: "y = (a', b')" by (cases y)
consider "a = 0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" | "a \<noteq> 0" "a'\<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
by blast
then show ?thesis
proof cases
case 1
then show ?thesis
apply (cases "a = 0")
apply (simp_all add: x y Nadd_def)
apply (cases "b = 0")
apply (simp_all add: INum_def)
apply (cases "a'= 0")
apply simp_all
apply (cases "b'= 0")
apply simp_all
done
next
case neq: 2
show ?thesis
proof (cases "a * b' + b * a' = 0")
case True
then have "of_int (a * b' + b * a') / (of_int b * of_int b') = ?z"
by simp
then have "of_int b' * of_int a / (of_int b * of_int b') +
of_int b * of_int a' / (of_int b * of_int b') = ?z"
by (simp add: add_divide_distrib)
then have th: "of_int a / of_int b + of_int a' / of_int b' = ?z"
using neq by simp
from True neq show ?thesis
by (simp add: x y th Nadd_def normNum_def INum_def split_def)
next
case False
let ?g = "gcd (a * b' + b * a') (b * b')"
have gz: "?g \<noteq> 0"
using False by simp
show ?thesis
using neq False gz
of_int_div [where ?'a = 'a, OF gz gcd_dvd1 [of "a * b' + b * a'" "b * b'"]]
of_int_div [where ?'a = 'a, OF gz gcd_dvd2 [of "a * b' + b * a'" "b * b'"]]
by (simp add: x y Nadd_def INum_def normNum_def Let_def) (simp add: field_simps)
qed
qed
qed
lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a::{field_char_0,field})"
proof -
let ?z = "0::'a"
obtain a b where x: "x = (a, b)" by (cases x)
obtain a' b' where y: "y = (a', b')" by (cases y)
consider "a = 0 \<or> a' = 0 \<or> b = 0 \<or> b' = 0" | "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
by blast
then show ?thesis
proof cases
case 1
then show ?thesis
by (auto simp add: x y Nmul_def INum_def)
next
case neq: 2
let ?g = "gcd (a * a') (b * b')"
have gz: "?g \<noteq> 0"
using neq by simp
from neq of_int_div [where ?'a = 'a, OF gz gcd_dvd1 [of "a * a'" "b * b'"]]
of_int_div [where ?'a = 'a , OF gz gcd_dvd2 [of "a * a'" "b * b'"]]
show ?thesis
by (simp add: Nmul_def x y Let_def INum_def)
qed
qed
lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x :: 'a::field)"
by (simp add: Nneg_def split_def INum_def)
lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a::{field_char_0,field})"
by (simp add: Nsub_def split_def)
lemma Ninv[simp]: "INum (Ninv x) = (1 :: 'a::field) / INum x"
by (simp add: Ninv_def INum_def split_def)
lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y :: 'a::{field_char_0,field})"
by (simp add: Ndiv_def)
lemma Nlt0_iff[simp]:
assumes nx: "isnormNum x"
shows "((INum x :: 'a::{field_char_0,linordered_field}) < 0) = 0>\<^sub>N x"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
show ?thesis
proof (cases "a = 0")
case True
then show ?thesis
by (simp add: x Nlt0_def INum_def)
next
case False
then have b: "(of_int b::'a) > 0"
using nx by (simp add: x isnormNum_def)
from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
show ?thesis
by (simp add: x Nlt0_def INum_def)
qed
qed
lemma Nle0_iff[simp]:
assumes nx: "isnormNum x"
shows "((INum x :: 'a::{field_char_0,linordered_field}) \<le> 0) = 0\<ge>\<^sub>N x"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
show ?thesis
proof (cases "a = 0")
case True
then show ?thesis
by (simp add: x Nle0_def INum_def)
next
case False
then have b: "(of_int b :: 'a) > 0"
using nx by (simp add: x isnormNum_def)
from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
show ?thesis
by (simp add: x Nle0_def INum_def)
qed
qed
lemma Ngt0_iff[simp]:
assumes nx: "isnormNum x"
shows "((INum x :: 'a::{field_char_0,linordered_field}) > 0) = 0<\<^sub>N x"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
show ?thesis
proof (cases "a = 0")
case True
then show ?thesis
by (simp add: x Ngt0_def INum_def)
next
case False
then have b: "(of_int b::'a) > 0"
using nx by (simp add: x isnormNum_def)
from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
show ?thesis
by (simp add: x Ngt0_def INum_def)
qed
qed
lemma Nge0_iff[simp]:
assumes nx: "isnormNum x"
shows "(INum x :: 'a::{field_char_0,linordered_field}) \<ge> 0 \<longleftrightarrow> 0\<le>\<^sub>N x"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
show ?thesis
proof (cases "a = 0")
case True
then show ?thesis
by (simp add: x Nge0_def INum_def)
next
case False
then have b: "(of_int b::'a) > 0"
using nx by (simp add: x isnormNum_def)
from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
show ?thesis
by (simp add: x Nge0_def INum_def)
qed
qed
lemma Nlt_iff[simp]:
assumes nx: "isnormNum x"
and ny: "isnormNum y"
shows "((INum x :: 'a::{field_char_0,linordered_field}) < INum y) \<longleftrightarrow> x <\<^sub>N y"
proof -
let ?z = "0::'a"
have "((INum x ::'a) < INum y) \<longleftrightarrow> INum (x -\<^sub>N y) < ?z"
using nx ny by simp
also have "\<dots> \<longleftrightarrow> 0>\<^sub>N (x -\<^sub>N y)"
using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
finally show ?thesis
by (simp add: Nlt_def)
qed
lemma Nle_iff[simp]:
assumes nx: "isnormNum x"
and ny: "isnormNum y"
shows "((INum x :: 'a::{field_char_0,linordered_field}) \<le> INum y) \<longleftrightarrow> x \<le>\<^sub>N y"
proof -
have "((INum x ::'a) \<le> INum y) \<longleftrightarrow> INum (x -\<^sub>N y) \<le> (0::'a)"
using nx ny by simp
also have "\<dots> \<longleftrightarrow> 0\<ge>\<^sub>N (x -\<^sub>N y)"
using Nle0_iff[OF Nsub_normN[OF ny]] by simp
finally show ?thesis
by (simp add: Nle_def)
qed
lemma Nadd_commute:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "x +\<^sub>N y = y +\<^sub>N x"
proof -
have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)"
by simp_all
have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)"
by simp
with isnormNum_unique[OF n] show ?thesis
by simp
qed
lemma [simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "(0, b) +\<^sub>N y = normNum y"
and "(a, 0) +\<^sub>N y = normNum y"
and "x +\<^sub>N (0, b) = normNum x"
and "x +\<^sub>N (a, 0) = normNum x"
apply (simp add: Nadd_def split_def)
apply (simp add: Nadd_def split_def)
apply (subst Nadd_commute)
apply (simp add: Nadd_def split_def)
apply (subst Nadd_commute)
apply (simp add: Nadd_def split_def)
done
lemma normNum_nilpotent_aux[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
assumes nx: "isnormNum x"
shows "normNum x = x"
proof -
let ?a = "normNum x"
have n: "isnormNum ?a" by simp
have th: "INum ?a = (INum x ::'a)" by simp
with isnormNum_unique[OF n nx] show ?thesis by simp
qed
lemma normNum_nilpotent[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "normNum (normNum x) = normNum x"
by simp
lemma normNum0[simp]: "normNum (0, b) = 0\<^sub>N" "normNum (a, 0) = 0\<^sub>N"
by (simp_all add: normNum_def)
lemma normNum_Nadd:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "normNum (x +\<^sub>N y) = x +\<^sub>N y"
by simp
lemma Nadd_normNum1[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "normNum x +\<^sub>N y = x +\<^sub>N y"
proof -
have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)"
by simp_all
have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)"
by simp
also have "\<dots> = INum (x +\<^sub>N y)"
by simp
finally show ?thesis
using isnormNum_unique[OF n] by simp
qed
lemma Nadd_normNum2[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "x +\<^sub>N normNum y = x +\<^sub>N y"
proof -
have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)"
by simp_all
have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)"
by simp
also have "\<dots> = INum (x +\<^sub>N y)"
by simp
finally show ?thesis
using isnormNum_unique[OF n] by simp
qed
lemma Nadd_assoc:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
proof -
have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))"
by simp_all
have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)"
by simp
with isnormNum_unique[OF n] show ?thesis
by simp
qed
lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
by (simp add: Nmul_def split_def Let_def gcd.commute mult.commute)
lemma Nmul_assoc:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
assumes nx: "isnormNum x"
and ny: "isnormNum y"
and nz: "isnormNum z"
shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
proof -
from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
by simp_all
have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)"
by simp
with isnormNum_unique[OF n] show ?thesis
by simp
qed
lemma Nsub0:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
assumes x: "isnormNum x"
and y: "isnormNum y"
shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
proof -
from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
have "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)"
by simp
also have "\<dots> \<longleftrightarrow> INum x = (INum y :: 'a)"
by simp
also have "\<dots> \<longleftrightarrow> x = y"
using x y by simp
finally show ?thesis .
qed
lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
by (simp_all add: Nmul_def Let_def split_def)
lemma Nmul_eq0[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
assumes nx: "isnormNum x"
and ny: "isnormNum y"
shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
obtain a' b' where y: "y = (a', b')" by (cases y)
have n0: "isnormNum 0\<^sub>N" by simp
show ?thesis using nx ny
apply (simp only: isnormNum_unique[where ?'a = 'a, OF Nmul_normN[OF nx ny] n0, symmetric]
Nmul[where ?'a = 'a])
apply (simp add: x y INum_def split_def isnormNum_def split: if_split_asm)
done
qed
lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
by (simp add: Nneg_def split_def)
lemma Nmul1[simp]: "isnormNum c \<Longrightarrow> (1)\<^sub>N *\<^sub>N c = c" "isnormNum c \<Longrightarrow> c *\<^sub>N (1)\<^sub>N = c"
apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
apply (cases "fst c = 0", simp_all, cases c, simp_all)+
done
end