# Theory Rat_Pair

theory Rat_Pair
imports Complex_Main
```(*  Title:      HOL/Decision_Procs/Rat_Pair.thy
Author:     Amine Chaieb
*)

section ‹Rational numbers as pairs›

theory Rat_Pair
imports Complex_Main
begin

type_synonym Num = "int × int"

abbreviation Num0_syn :: Num  ("0⇩N")
where "0⇩N ≡ (0, 0)"

abbreviation Numi_syn :: "int ⇒ Num"  ("'((_)')⇩N")
where "(i)⇩N ≡ (i, 1)"

definition isnormNum :: "Num ⇒ bool"
where "isnormNum = (λ(a, b). if a = 0 then b = 0 else b > 0 ∧ gcd a b = 1)"

definition normNum :: "Num ⇒ Num"
where
"normNum = (λ(a,b).
(if a = 0 ∨ 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 ∨ b = 0" | "a ≠ 0" "b ≠ 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 ≠ 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' ≠ 0" "?b' ≠ 0" by - (rule notI, simp)+
from ab have stupid: "a ≠ 0 ∨ b ≠ 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' ≥ 0"
proof -
from gpos have th: "?g ≥ 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' ≥ 0"
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 ‹Arithmetic over Num›

definition Nadd :: "Num ⇒ Num ⇒ Num"  (infixl "+⇩N" 60)
where
"Nadd = (λ(a, b) (a', b').
if a = 0 ∨ b = 0 then normNum (a', b')
else if a' = 0 ∨ b' = 0 then normNum (a, b)
else normNum (a * b' + b * a', b * b'))"

definition Nmul :: "Num ⇒ Num ⇒ Num"  (infixl "*⇩N" 60)
where
"Nmul = (λ(a, b) (a', b').
let g = gcd (a * a') (b * b')
in (a * a' div g, b * b' div g))"

definition Nneg :: "Num ⇒ Num" ("~⇩N")
where "Nneg = (λ(a, b). (- a, b))"

definition Nsub :: "Num ⇒ Num ⇒ Num"  (infixl "-⇩N" 60)
where "Nsub = (λa b. a +⇩N ~⇩N b)"

definition Ninv :: "Num ⇒ Num"
where "Ninv = (λ(a, b). if a < 0 then (- b, ¦a¦) else (b, a))"

definition Ndiv :: "Num ⇒ Num ⇒ Num"  (infixl "÷⇩N" 60)
where "Ndiv = (λa b. a *⇩N Ninv b)"

lemma Nneg_normN[simp]: "isnormNum x ⟹ isnormNum (~⇩N x)"
by (simp add: isnormNum_def Nneg_def split_def)

lemma Nadd_normN[simp]: "isnormNum (x +⇩N y)"

lemma Nsub_normN[simp]: "isnormNum y ⟹ isnormNum (x -⇩N y)"

lemma Nmul_normN[simp]:
assumes xn: "isnormNum x"
and yn: "isnormNum y"
shows "isnormNum (x *⇩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 ≠ 0" "a' ≠ 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 *⇩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 ⟹ isnormNum (Ninv x)"
apply (simp add: Ninv_def isnormNum_def split_def)
apply (cases "fst x = 0")
done

lemma isnormNum_int[simp]: "isnormNum 0⇩N" "isnormNum ((1::int)⇩N)" "i ≠ 0 ⟹ isnormNum (i)⇩N"

text ‹Relations over Num›

definition Nlt0:: "Num ⇒ bool"  ("0>⇩N")
where "Nlt0 = (λ(a, b). a < 0)"

definition Nle0:: "Num ⇒ bool"  ("0≥⇩N")
where "Nle0 = (λ(a, b). a ≤ 0)"

definition Ngt0:: "Num ⇒ bool"  ("0<⇩N")
where "Ngt0 = (λ(a, b). a > 0)"

definition Nge0:: "Num ⇒ bool"  ("0≤⇩N")
where "Nge0 = (λ(a, b). a ≥ 0)"

definition Nlt :: "Num ⇒ Num ⇒ bool"  (infix "<⇩N" 55)
where "Nlt = (λa b. 0>⇩N (a -⇩N b))"

definition Nle :: "Num ⇒ Num ⇒ bool"  (infix "≤⇩N" 55)
where "Nle = (λa b. 0≥⇩N (a -⇩N b))"

definition "INum = (λ(a, b). of_int a / of_int b)"

lemma INum_int [simp]: "INum (i)⇩N = (of_int i ::'a::field)" "INum 0⇩N = (0::'a::field)"

lemma isnormNum_unique[simp]:
assumes na: "isnormNum x"
and nb: "isnormNum y"
shows "(INum x ::'a::{field_char_0,field}) = INum y ⟷ 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 ∨ b = 0 ∨ a' = 0 ∨ b' = 0" | "a ≠ 0" "b ≠ 0" "a' ≠ 0" "b' ≠ 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 ‹b ≠ 0› ‹b' ≠ 0› 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 ‹a ≠ 0› ‹a' ≠ 0› na nb
have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
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 ⟹ INum x = (0::'a::{field_char_0,field}) ⟷ x = 0⇩N"
unfolding INum_int(2)[symmetric]
by (rule isnormNum_unique) simp_all

lemma of_int_div_aux:
assumes "d ≠ 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 = "λx. 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
qed

lemma of_int_div:
fixes d :: int
assumes "d ≠ 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 ∨ b = 0" | "a ≠ 0" "b ≠ 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 ≠ 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 ⟷ normNum x = normNum y"
(is "?lhs ⟷ _")
proof -
have "normNum x = normNum y ⟷ (INum (normNum x) :: 'a) = INum (normNum y)"
by (simp del: normNum)
also have "… = ?lhs" by simp
finally show ?thesis by simp
qed

lemma Nadd[simp]: "INum (x +⇩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 ∨ a'= 0 ∨ b =0 ∨ b' = 0" | "a ≠ 0" "a'≠ 0" "b ≠ 0" "b' ≠ 0"
by blast
then show ?thesis
proof cases
case 1
then show ?thesis
apply (cases "a = 0")
apply (cases "b = 0")
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"
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
next
case False
let ?g = "gcd (a * b' + b * a') (b * b')"
have gz: "?g ≠ 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'"]]
qed
qed
qed

lemma Nmul[simp]: "INum (x *⇩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 ∨ a' = 0 ∨ b = 0 ∨ b' = 0" | "a ≠ 0" "a' ≠ 0" "b ≠ 0" "b' ≠ 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 ≠ 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 (~⇩N x) = - (INum x :: 'a::field)"
by (simp add: Nneg_def split_def INum_def)

lemma Nsub[simp]: "INum (x -⇩N y) = INum x - (INum y:: 'a::{field_char_0,field})"

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 ÷⇩N y) = INum x / (INum y :: 'a::{field_char_0,field})"

lemma Nlt0_iff[simp]:
assumes nx: "isnormNum x"
shows "((INum x :: 'a::{field_char_0,linordered_field}) < 0) = 0>⇩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}) ≤ 0) = 0≥⇩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<⇩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}) ≥ 0 ⟷ 0≤⇩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) ⟷ x <⇩N y"
proof -
let ?z = "0::'a"
have "((INum x ::'a) < INum y) ⟷ INum (x -⇩N y) < ?z"
using nx ny by simp
also have "… ⟷ 0>⇩N (x -⇩N y)"
using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
finally show ?thesis
qed

lemma Nle_iff[simp]:
assumes nx: "isnormNum x"
and ny: "isnormNum y"
shows "((INum x :: 'a::{field_char_0,linordered_field}) ≤ INum y) ⟷ x ≤⇩N y"
proof -
have "((INum x ::'a) ≤ INum y) ⟷ INum (x -⇩N y) ≤ (0::'a)"
using nx ny by simp
also have "… ⟷ 0≥⇩N (x -⇩N y)"
using Nle0_iff[OF Nsub_normN[OF ny]] by simp
finally show ?thesis
qed

assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "x +⇩N y = y +⇩N x"
proof -
have n: "isnormNum (x +⇩N y)" "isnormNum (y +⇩N x)"
by simp_all
have "(INum (x +⇩N y)::'a) = INum (y +⇩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) +⇩N y = normNum y"
and "(a, 0) +⇩N y = normNum y"
and "x +⇩N (0, b) = normNum x"
and "x +⇩N (a, 0) = normNum x"
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⇩N" "normNum (a, 0) = 0⇩N"

assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "normNum (x +⇩N y) = x +⇩N y"
by simp

assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "normNum x +⇩N y = x +⇩N y"
proof -
have n: "isnormNum (normNum x +⇩N y)" "isnormNum (x +⇩N y)"
by simp_all
have "INum (normNum x +⇩N y) = INum x + (INum y :: 'a)"
by simp
also have "… = INum (x +⇩N y)"
by simp
finally show ?thesis
using isnormNum_unique[OF n] by simp
qed

assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "x +⇩N normNum y = x +⇩N y"
proof -
have n: "isnormNum (x +⇩N normNum y)" "isnormNum (x +⇩N y)"
by simp_all
have "INum (x +⇩N normNum y) = INum x + (INum y :: 'a)"
by simp
also have "… = INum (x +⇩N y)"
by simp
finally show ?thesis
using isnormNum_unique[OF n] by simp
qed

assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "x +⇩N y +⇩N z = x +⇩N (y +⇩N z)"
proof -
have n: "isnormNum (x +⇩N y +⇩N z)" "isnormNum (x +⇩N (y +⇩N z))"
by simp_all
have "INum (x +⇩N y +⇩N z) = (INum (x +⇩N (y +⇩N z)) :: 'a)"
by simp
with isnormNum_unique[OF n] show ?thesis
by simp
qed

lemma Nmul_commute: "isnormNum x ⟹ isnormNum y ⟹ x *⇩N y = y *⇩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 *⇩N y *⇩N z = x *⇩N (y *⇩N z)"
proof -
from nx ny nz have n: "isnormNum (x *⇩N y *⇩N z)" "isnormNum (x *⇩N (y *⇩N z))"
by simp_all
have "INum (x +⇩N y +⇩N z) = (INum (x +⇩N (y +⇩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 -⇩N y = 0⇩N ⟷ x = y"
proof -
from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0⇩N"]
have "x -⇩N y = 0⇩N ⟷ INum (x -⇩N y) = (INum 0⇩N :: 'a)"
by simp
also have "… ⟷ INum x = (INum y :: 'a)"
by simp
also have "… ⟷ x = y"
using x y by simp
finally show ?thesis .
qed

lemma Nmul0[simp]: "c *⇩N 0⇩N = 0⇩N" " 0⇩N *⇩N c = 0⇩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*⇩N y = 0⇩N ⟷ x = 0⇩N ∨ y = 0⇩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⇩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]: "~⇩N (~⇩N c) = c"