Merge.
(* Title: HOL/Quadratic_Reciprocity/Gauss.thy
ID: $Id$
Authors: Jeremy Avigad, David Gray, and Adam Kramer
*)
header {*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 \<Longrightarrow> p dvd y \<Longrightarrow> p dvd ((y::int) ^ n)"
by (induct n) (auto simp add: dvd_mult2 [of p y])
lemma zdvd_bounds: "n dvd m ==> m \<le> (0::int) | n \<le> 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 \<le> m")
apply (simp add: zprime_zdvd_zmult)
apply (insert zprime_zdvd_zmult [of "-m" p n])
apply auto
done
lemma zpower_zdvd_prop2:
"zprime p \<Longrightarrow> p dvd ((y::int) ^ n) \<Longrightarrow> 0 < n \<Longrightarrow> 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: "[| 0 < z; (x::int) < y * z |] ==> x div z < y"
proof -
assume "0 < z" then have modth: "x mod z \<ge> 0" by simp
have "(x div z) * z \<le> (x div z) * z" by simp
then have "(x div z) * z \<le> (x div z) * z + x mod z" using modth by arith
also have "\<dots> = x"
by (auto simp add: zmod_zdiv_equality [symmetric] zmult_ac)
also assume "x < y * z"
finally show ?thesis
by (auto simp add: prems mult_less_cancel_right, insert prems, arith)
qed
lemma div_prop2: "[| 0 < z; (x::int) < (y * z) + z |] ==> x div z \<le> y"
proof -
assume "0 < z" and "x < (y * z) + z"
then have "x < (y + 1) * z" by (auto simp add: int_distrib)
then have "x div z < y + 1"
apply -
apply (rule_tac y = "y + 1" in div_prop1)
apply (auto simp add: prems)
done
then show ?thesis by auto
qed
lemma zdiv_leq_prop: "[| 0 < y |] ==> y * (x div y) \<le> (x::int)"
proof-
assume "0 < y"
from zmod_zdiv_equality have "x = y * (x div y) + x mod y" by auto
moreover have "0 \<le> x mod y"
by (auto simp add: prems pos_mod_sign)
ultimately show ?thesis
by arith
qed
subsection {* Useful properties of congruences *}
lemma zcong_eq_zdvd_prop: "[x = 0](mod p) = (p dvd x)"
by (auto simp add: zcong_def)
lemma zcong_id: "[m = 0] (mod m)"
by (auto simp add: zcong_def)
lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (mod m)"
by (auto simp add: zcong_refl zcong_zadd)
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 (auto simp add: int_distrib)
apply (rule_tac x = "ka - k * d" in exI)
apply (drule aux1)+
apply (auto simp add: int_distrib)
done
lemma zcong_zmult_prop2: "[a = b](mod m) ==>
([c = d * a](mod m) = [c = d * b] (mod m))"
by (auto simp add: zmult_ac zcong_zmult_prop1)
lemma zcong_zmult_prop3: "[| zprime p; ~[x = 0] (mod p);
~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)"
apply (auto simp add: zcong_def)
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 zless_linear)
lemma zcong_neg_1_impl_ne_1: "[| 2 < p; [x = -1] (mod p) |] ==>
~([x = 1] (mod p))"
proof
assume "2 < p" and "[x = 1] (mod p)" and "[x = -1] (mod p)"
then have "[1 = -1] (mod p)"
apply (auto simp add: zcong_sym)
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 prems show False
by (auto simp add: zdvd_not_zless)
qed
lemma zcong_zero_equiv_div: "[a = 0] (mod m) = (m dvd a)"
by (auto simp add: zcong_def)
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 \<le> 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; \<forall>x \<in> 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)"
proof (simp add: MultInv_def zcong_eq_zdvd_prop)
assume "2 < p" and "zprime p" and "~ p dvd x"
have "x * x ^ nat (p - 2) = x ^ (nat (p - 2) + 1)"
by auto
also from prems have "nat (p - 2) + 1 = nat (p - 2 + 1)"
by (simp only: nat_add_distrib)
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 prems have "[x ^ nat (p - 1) = 1] (mod p)"
by (auto simp add: Little_Fermat)
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 zmult_ac)
lemma aux_1: "2 < p ==> ((nat p) - 2) = (nat (p - 2))"
by (simp add: nat_diff_distrib)
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)
apply (auto simp add: zcong_sym)
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)
apply (auto simp add: zmult_ac)
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 (auto simp add: zmult_ac)
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: zmult_ac)
apply (insert zcong_zmult_prop2 [of "y * MultInv p y" 1 p y x])
apply (auto simp add: zcong_sym)
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)"
by (auto simp add: zcong_scalar)
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])
apply (auto simp add: zmult_ac)
done
lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k =
(MultInv p j) * a] (mod p)"
by (auto simp add: zmult_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: zmult_ac 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