(* Title: HOL/Library/Abstract_Rat.thy
Author: Amine Chaieb
*)
header {* Abstract rational numbers *}
theory Abstract_Rat
imports Complex_Main
begin
types 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_int[presburger]
declare gcd_dvd2_int[presburger]
lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
proof -
have " \<exists> a b. x = (a,b)" by auto
then obtain a b where x[simp]: "x = (a,b)" by blast
{assume "a=0 \<or> b = 0" hence ?thesis by (simp add: normNum_def isnormNum_def)}
moreover
{assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
let ?g = "gcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
let ?g' = "gcd ?a' ?b'"
from anz bnz 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 zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)]
anz bnz
have nz':"?a' \<noteq> 0" "?b' \<noteq> 0"
by - (rule notI, simp)+
from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" .
from bnz have "b < 0 \<or> b > 0" by arith
moreover
{assume b: "b > 0"
from b 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' anz bnz nz' gp1 have ?thesis
by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
moreover {assume b: "b < 0"
{assume b': "?b' \<ge> 0"
from gpos have th: "?g \<ge> 0" by arith
from mult_nonneg_nonneg[OF th b'] zdvd_mult_div_cancel[OF gdvd(2)]
have False using b by arith }
hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
from anz bnz nz' b b' gp1 have ?thesis
by (simp add: isnormNum_def normNum_def Let_def split_def)}
ultimately have ?thesis by blast
}
ultimately show ?thesis by blast
qed
text {* Arithmetic over Num *}
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 \<equiv> (\<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 \<equiv> \<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 \<equiv> \<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]: "\<lbrakk> isnormNum y\<rbrakk> \<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-
have "\<exists>a b. x = (a,b)" and "\<exists> a' b'. y = (a',b')" by auto
then obtain a b a' b' where ab: "x = (a,b)" and ab': "y = (a',b')" by blast
{assume "a = 0"
hence ?thesis using xn ab ab'
by (simp add: isnormNum_def Let_def Nmul_def split_def)}
moreover
{assume "a' = 0"
hence ?thesis using yn ab ab'
by (simp add: isnormNum_def Let_def Nmul_def split_def)}
moreover
{assume a: "a \<noteq>0" and a': "a'\<noteq>0"
hence bp: "b > 0" "b' > 0" using xn yn ab ab' by (simp_all add: isnormNum_def)
from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a*a', b*b')"
using ab ab' a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)
hence ?thesis by simp}
ultimately show ?thesis by blast
qed
lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
by (simp add: Ninv_def isnormNum_def split_def)
(cases "fst x = 0", auto simp add: gcd_commute_int)
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 {* Relations over Num *}
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_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
proof
have "\<exists> a b a' b'. x = (a,b) \<and> y = (a',b')" by auto
then obtain a b a' b' where xy[simp]: "x = (a,b)" "y=(a',b')" by blast
assume H: ?lhs
{assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
hence ?rhs using na nb H
by (simp add: INum_def split_def isnormNum_def split: split_if_asm)}
moreover
{ assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0"
from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: isnormNum_def)
from H bz b'z have eq:"a * b' = a'*b"
by (simp add: INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
by (simp_all add: isnormNum_def add: gcd_commute_int)
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_int[OF gcd1(2) raw_dvd(2)]
coprime_dvd_mult_int[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 have ?rhs by simp}
ultimately show ?rhs by blast
next
assume ?rhs thus ?lhs by simp
qed
lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)"
unfolding INum_int(2)[symmetric]
by (rule isnormNum_unique, simp_all)
lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) =
of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
proof -
assume "d ~= 0"
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 `d ~= 0`)
qed
lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
(of_int(n div d)::'a::field_char_0) = of_int n / of_int d"
apply (frule of_int_div_aux [of d n, where ?'a = 'a])
apply simp
apply (simp add: dvd_eq_mod_eq_0)
done
lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"
proof-
have "\<exists> a b. x = (a,b)" by auto
then obtain a b where x[simp]: "x = (a,b)" by blast
{assume "a=0 \<or> b = 0" hence ?thesis
by (simp add: INum_def normNum_def split_def Let_def)}
moreover
{assume a: "a\<noteq>0" and b: "b\<noteq>0"
let ?g = "gcd a b"
from a b have g: "?g \<noteq> 0"by simp
from of_int_div[OF g, where ?'a = 'a]
have ?thesis by (auto simp add: INum_def normNum_def split_def Let_def)}
ultimately show ?thesis by blast
qed
lemma INum_normNum_iff: "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y" (is "?lhs = ?rhs")
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_inverse_zero})"
proof-
let ?z = "0:: 'a"
have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
then obtain a b a' b' where x[simp]: "x = (a,b)"
and y[simp]: "y = (a',b')" by blast
{assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" hence ?thesis
apply (cases "a=0",simp_all add: Nadd_def)
apply (cases "b= 0",simp_all add: INum_def)
apply (cases "a'= 0",simp_all)
apply (cases "b'= 0",simp_all)
done }
moreover
{assume aa':"a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0"
{assume z: "a * b' + b * a' = 0"
hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp
hence "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)
hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa' by simp
from z aa' bb' have ?thesis
by (simp add: th Nadd_def normNum_def INum_def split_def)}
moreover {assume z: "a * b' + b * a' \<noteq> 0"
let ?g = "gcd (a * b' + b * a') (b*b')"
have gz: "?g \<noteq> 0" using z by simp
have ?thesis using aa' bb' z gz
of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]] of_int_div[where ?'a = 'a,
OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]]
by (simp add: Nadd_def INum_def normNum_def Let_def add_divide_distrib)}
ultimately have ?thesis using aa' bb'
by (simp add: Nadd_def INum_def normNum_def Let_def) }
ultimately show ?thesis by blast
qed
lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero}) "
proof-
let ?z = "0::'a"
have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
then obtain a b a' b' where x: "x = (a,b)" and y: "y = (a',b')" by blast
{assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" hence ?thesis
apply (cases "a=0",simp_all add: x y Nmul_def INum_def Let_def)
apply (cases "b=0",simp_all)
apply (cases "a'=0",simp_all)
done }
moreover
{assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
let ?g="gcd (a*a') (b*b')"
have gz: "?g \<noteq> 0" using z by simp
from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]]
of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]]
have ?thesis by (simp add: Nmul_def x y Let_def INum_def)}
ultimately show ?thesis by blast
qed
lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"
by (simp add: Nneg_def split_def INum_def)
lemma Nsub[simp]: shows "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"
by (simp add: Nsub_def split_def)
lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (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_inverse_zero})" by (simp add: Ndiv_def)
lemma Nlt0_iff[simp]: assumes nx: "isnormNum x"
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x "
proof-
have " \<exists> a b. x = (a,b)" by simp
then obtain a b where x[simp]:"x = (a,b)" by blast
{assume "a = 0" hence ?thesis by (simp add: Nlt0_def INum_def) }
moreover
{assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
have ?thesis by (simp add: Nlt0_def INum_def)}
ultimately show ?thesis by blast
qed
lemma Nle0_iff[simp]:assumes nx: "isnormNum x"
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x"
proof-
have " \<exists> a b. x = (a,b)" by simp
then obtain a b where x[simp]:"x = (a,b)" by blast
{assume "a = 0" hence ?thesis by (simp add: Nle0_def INum_def) }
moreover
{assume a: "a\<noteq>0" hence b: "(of_int b :: 'a) > 0" using nx by (simp add: isnormNum_def)
from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
have ?thesis by (simp add: Nle0_def INum_def)}
ultimately show ?thesis by blast
qed
lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
proof-
have " \<exists> a b. x = (a,b)" by simp
then obtain a b where x[simp]:"x = (a,b)" by blast
{assume "a = 0" hence ?thesis by (simp add: Ngt0_def INum_def) }
moreover
{assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
have ?thesis by (simp add: Ngt0_def INum_def)}
ultimately show ?thesis by blast
qed
lemma Nge0_iff[simp]:assumes nx: "isnormNum x"
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
proof-
have " \<exists> a b. x = (a,b)" by simp
then obtain a b where x[simp]:"x = (a,b)" by blast
{assume "a = 0" hence ?thesis by (simp add: Nge0_def INum_def) }
moreover
{assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
have ?thesis by (simp add: Nge0_def INum_def)}
ultimately show ?thesis by blast
qed
lemma Nlt_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)"
proof-
let ?z = "0::'a"
have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" using nx ny by simp
also have "\<dots> = (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_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)"
proof-
have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" using nx ny by simp
also have "\<dots> = (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_inverse_zero})"
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_inverse_zero})"
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, simp add: Nadd_def split_def)
apply (subst Nadd_commute, simp add: Nadd_def split_def)
done
lemma normNum_nilpotent_aux[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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_inverse_zero})"
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_inverse_zero})"
shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
lemma Nadd_normNum1[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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_inverse_zero})"
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_inverse_zero})"
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_int mult_commute)
lemma Nmul_assoc:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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_inverse_zero})"
assumes x: "isnormNum x" and y:"isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"
proof-
{ fix h :: 'a
from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
also have "\<dots> = (INum x = (INum y :: 'a))" by simp
also have "\<dots> = (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_inverse_zero})"
assumes nx:"isnormNum x" and ny: "isnormNum y"
shows "(x*\<^sub>N y = 0\<^sub>N) = (x = 0\<^sub>N \<or> y = 0\<^sub>N)"
proof-
{ fix h :: 'a
have " \<exists> a b a' b'. x = (a,b) \<and> y= (a',b')" by auto
then obtain a b a' b' where xy[simp]: "x = (a,b)" "y = (a',b')" by blast
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])
by (simp add: INum_def split_def isnormNum_def split: split_if_asm)
}
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