src/HOL/NumberTheory/Int2.thy

-(*  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