--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Old_Number_Theory/Int2.thy Tue Sep 01 15:39:33 2009 +0200
@@ -0,0 +1,299 @@
+(* 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