# Theory Int2

theory Int2
imports Finite2 WilsonRuss
```(*  Title:      HOL/Old_Number_Theory/Int2.thy
*)

section ‹Integers: Divisibility and Congruences›

theory Int2
imports Finite2 WilsonRuss
begin

definition MultInv :: "int => int => int"
where "MultInv p x = x ^ nat (p - 2)"

subsection ‹Useful lemmas about dvd and powers›

lemma zpower_zdvd_prop1:
"0 < n ⟹ p dvd y ⟹ p dvd ((y::int) ^ n)"
by (induct n) (auto simp add: dvd_mult2 [of p y])

lemma zdvd_bounds: "n dvd m ==> m ≤ (0::int) | n ≤ m"
proof -
assume "n dvd m"
then have "~(0 < m & m < n)"
using zdvd_not_zless [of m n] by auto
then show ?thesis by auto
qed

lemma zprime_zdvd_zmult_better: "[| zprime p;  p dvd (m * n) |] ==>
(p dvd m) | (p dvd n)"
apply (cases "0 ≤ m")
apply (insert zprime_zdvd_zmult [of "-m" p n])
apply auto
done

lemma zpower_zdvd_prop2:
"zprime p ⟹ p dvd ((y::int) ^ n) ⟹ 0 < n ⟹ p dvd y"
apply (induct n)
apply simp
apply (frule zprime_zdvd_zmult_better)
apply simp
apply (force simp del:dvd_mult)
done

lemma div_prop1:
assumes "0 < z" and "(x::int) < y * z"
shows "x div z < y"
proof -
from ‹0 < z› have modth: "x mod z ≥ 0" by simp
have "(x div z) * z ≤ (x div z) * z" by simp
then have "(x div z) * z ≤ (x div z) * z + x mod z" using modth by arith
also have "… = x"
by (auto simp add: zmod_zdiv_equality [symmetric] ac_simps)
also note ‹x < y * z›
finally show ?thesis
using assms apply arith
done
qed

lemma div_prop2:
assumes "0 < z" and "(x::int) < (y * z) + z"
shows "x div z ≤ y"
proof -
from assms have "x < (y + 1) * z" by (auto simp add: int_distrib)
then have "x div z < y + 1"
apply (rule_tac y = "y + 1" in div_prop1)
apply (auto simp add: ‹0 < z›)
done
then show ?thesis by auto
qed

lemma zdiv_leq_prop: assumes "0 < y" shows "y * (x div y) ≤ (x::int)"
proof-
from zmod_zdiv_equality have "x = y * (x div y) + x mod y" by auto
moreover have "0 ≤ x mod y" by (auto simp add: assms)
ultimately show ?thesis by arith
qed

subsection ‹Useful properties of congruences›

lemma zcong_eq_zdvd_prop: "[x = 0](mod p) = (p dvd x)"

lemma zcong_id: "[m = 0] (mod m)"

lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (mod m)"

lemma zcong_zpower: "[x = y](mod m) ==> [x^z = y^z](mod m)"
by (induct z) (auto simp add: zcong_zmult)

lemma zcong_eq_trans: "[| [a = b](mod m); b = c; [c = d](mod m) |] ==>
[a = d](mod m)"
apply (erule zcong_trans)
apply simp
done

lemma aux1: "a - b = (c::int) ==> a = c + b"
by auto

lemma zcong_zmult_prop1: "[a = b](mod m) ==> ([c = a * d](mod m) =
[c = b * d] (mod m))"
apply (auto simp add: zcong_def dvd_def)
apply (rule_tac x = "ka + k * d" in exI)
apply (drule aux1)+
apply (rule_tac x = "ka - k * d" in exI)
apply (drule aux1)+
done

lemma zcong_zmult_prop2: "[a = b](mod m) ==>
([c = d * a](mod m) = [c = d * b] (mod m))"
by (auto simp add: ac_simps zcong_zmult_prop1)

lemma zcong_zmult_prop3: "[| zprime p; ~[x = 0] (mod p);
~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)"
apply (drule zprime_zdvd_zmult_better, auto)
done

lemma zcong_less_eq: "[| 0 < x; 0 < y; 0 < m; [x = y] (mod m);
x < m; y < m |] ==> x = y"
by (metis zcong_not zcong_sym less_linear)

lemma zcong_neg_1_impl_ne_1:
assumes "2 < p" and "[x = -1] (mod p)"
shows "~([x = 1] (mod p))"
proof
assume "[x = 1] (mod p)"
with assms have "[1 = -1] (mod p)"
apply (drule zcong_trans, auto)
done
then have "[1 + 1 = -1 + 1] (mod p)"
by (simp only: zcong_shift)
then have "[2 = 0] (mod p)"
by auto
then have "p dvd 2"
by (auto simp add: dvd_def zcong_def)
with ‹2 < p› show False
qed

lemma zcong_zero_equiv_div: "[a = 0] (mod m) = (m dvd a)"

lemma zcong_zprime_prod_zero: "[| zprime p; 0 < a |] ==>
[a * b = 0] (mod p) ==> [a = 0] (mod p) | [b = 0] (mod p)"
by (auto simp add: zcong_zero_equiv_div zprime_zdvd_zmult)

lemma zcong_zprime_prod_zero_contra: "[| zprime p; 0 < a |] ==>
~[a = 0](mod p) & ~[b = 0](mod p) ==> ~[a * b = 0] (mod p)"
apply auto
apply (frule_tac a = a and b = b and p = p in zcong_zprime_prod_zero)
apply auto
done

lemma zcong_not_zero: "[| 0 < x; x < m |] ==> ~[x = 0] (mod m)"
by (auto simp add: zcong_zero_equiv_div zdvd_not_zless)

lemma zcong_zero: "[| 0 ≤ x; x < m; [x = 0](mod m) |] ==> x = 0"
apply (drule order_le_imp_less_or_eq, auto)
apply (frule_tac m = m in zcong_not_zero)
apply auto
done

lemma all_relprime_prod_relprime: "[| finite A; ∀x ∈ A. zgcd x y = 1 |]
==> zgcd (setprod id A) y = 1"
by (induct set: finite) (auto simp add: zgcd_zgcd_zmult)

subsection ‹Some properties of MultInv›

lemma MultInv_prop1: "[| 2 < p; [x = y] (mod p) |] ==>
[(MultInv p x) = (MultInv p y)] (mod p)"
by (auto simp add: MultInv_def zcong_zpower)

lemma MultInv_prop2: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
[(x * (MultInv p x)) = 1] (mod p)"
assume 1: "2 < p" and 2: "zprime p" and 3: "~ p dvd x"
have "x * x ^ nat (p - 2) = x ^ (nat (p - 2) + 1)"
by auto
also from 1 have "nat (p - 2) + 1 = nat (p - 2 + 1)"
also have "p - 2 + 1 = p - 1" by arith
finally have "[x * x ^ nat (p - 2) = x ^ nat (p - 1)] (mod p)"
by (rule ssubst, auto)
also from 2 3 have "[x ^ nat (p - 1) = 1] (mod p)"
finally (zcong_trans) show "[x * x ^ nat (p - 2) = 1] (mod p)" .
qed

lemma MultInv_prop2a: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
[(MultInv p x) * x = 1] (mod p)"
by (auto simp add: MultInv_prop2 ac_simps)

lemma aux_1: "2 < p ==> ((nat p) - 2) = (nat (p - 2))"

lemma aux_2: "2 < p ==> 0 < nat (p - 2)"
by auto

lemma MultInv_prop3: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
~([MultInv p x = 0](mod p))"
apply (auto simp add: MultInv_def zcong_eq_zdvd_prop aux_1)
apply (drule aux_2)
apply (drule zpower_zdvd_prop2, auto)
done

lemma aux__1: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==>
[(MultInv p (MultInv p x)) = (x * (MultInv p x) *
(MultInv p (MultInv p x)))] (mod p)"
apply (drule MultInv_prop2, auto)
apply (drule_tac k = "MultInv p (MultInv p x)" in zcong_scalar, auto)
done

lemma aux__2: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==>
[(x * (MultInv p x) * (MultInv p (MultInv p x))) = x] (mod p)"
apply (frule MultInv_prop3, auto)
apply (insert MultInv_prop2 [of p "MultInv p x"], auto)
apply (drule MultInv_prop2, auto)
apply (drule_tac k = x in zcong_scalar2, auto)
done

lemma MultInv_prop4: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
[(MultInv p (MultInv p x)) = x] (mod p)"
apply (frule aux__1, auto)
apply (drule aux__2, auto)
apply (drule zcong_trans, auto)
done

lemma MultInv_prop5: "[| 2 < p; zprime p; ~([x = 0](mod p));
~([y = 0](mod p)); [(MultInv p x) = (MultInv p y)] (mod p) |] ==>
[x = y] (mod p)"
apply (drule_tac a = "MultInv p x" and b = "MultInv p y" and
m = p and k = x in zcong_scalar)
apply (insert MultInv_prop2 [of p x], simp)
apply (auto simp only: zcong_sym [of "MultInv p x * x"])
apply (drule zcong_trans, auto)
apply (drule_tac a = "x * MultInv p y" and k = y in zcong_scalar, auto)
apply (insert MultInv_prop2a [of p y], auto simp add: ac_simps)
apply (insert zcong_zmult_prop2 [of "y * MultInv p y" 1 p y x])
done

lemma MultInv_zcong_prop1: "[| 2 < p; [j = k] (mod p) |] ==>
[a * MultInv p j = a * MultInv p k] (mod p)"
by (drule MultInv_prop1, auto simp add: zcong_scalar2)

lemma aux___1: "[j = a * MultInv p k] (mod p) ==>
[j * k = a * MultInv p k * k] (mod p)"

lemma aux___2: "[|2 < p; zprime p; ~([k = 0](mod p));
[j * k = a * MultInv p k * k] (mod p) |] ==> [j * k = a] (mod p)"
apply (insert MultInv_prop2a [of p k] zcong_zmult_prop2
[of "MultInv p k * k" 1 p "j * k" a])
done

lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k =
(MultInv p j) * a] (mod p)"
by (auto simp add: mult.assoc zcong_scalar2)

lemma aux___4: "[|2 < p; zprime p; ~([j = 0](mod p));
[(MultInv p j) * j * k = (MultInv p j) * a] (mod p) |]
==> [k = a * (MultInv p j)] (mod p)"
apply (insert MultInv_prop2a [of p j] zcong_zmult_prop1
[of "MultInv p j * j" 1 p "MultInv p j * a" k])
apply (auto simp add: ac_simps zcong_sym)
done

lemma MultInv_zcong_prop2: "[| 2 < p; zprime p; ~([k = 0](mod p));
~([j = 0](mod p)); [j = a * MultInv p k] (mod p) |] ==>
[k = a * MultInv p j] (mod p)"
apply (drule aux___1)
apply (frule aux___2, auto)
by (drule aux___3, drule aux___4, auto)

lemma MultInv_zcong_prop3: "[| 2 < p; zprime p; ~([a = 0](mod p));
~([k = 0](mod p)); ~([j = 0](mod p));
[a * MultInv p j = a * MultInv p k] (mod p) |] ==>
[j = k] (mod p)"
apply (auto simp add: zcong_eq_zdvd_prop [of a p])
apply (frule zprime_imp_zrelprime, auto)
apply (insert zcong_cancel2 [of p a "MultInv p j" "MultInv p k"], auto)
apply (drule MultInv_prop5, auto)
done

end
```